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BIOMATHlMATICS TRAINING PROGKAM
A GENETIC ANALYSIS OF SERUM CHOLESTEROL;·.AND BLOOD PRESSUE LEVELS IN A LARGE PEDIGREE;.
,'../~'.Y;.'
by
Kelvin Kwoklen Lee
Department of BiostatisticsUniversity of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1174
JUNE 1978
ABSTRACT'
KELVIN KWOKLEN LEE. A Genetic Analysis of Sennn Cholesterol andBlood Pressure Levels in a Large Pedigree. (Under thedirection of Robert C. Elston.)
Serum cholesterol and blood pressure data from a five-generation
pedigree from Bay City, Michigan with 235 members are analyzed. The
method of pedigree analysis does not require that the pedigree be
divided up into nuclear families, nor does it rely on arbitrary cutoff
points to dichotomize or trichotomize quantitative data. The method
involves calculating the likelihood of observing the phenotypes in
the pedigree based on a genetic hypothesis. Max~ likelihood esti
mates of parameters are obtained, and hypothesis testing is based on
the likelihood ratio criterion.
This study considers three specific underlying genetic models:
(1) major gene model - a single identifiable gene that can account for
a significant portion of the phenotypic variance; (2) polygenic model
the phenotype is controlled by a large number of equal and additive
gene effects; (3) a mixed model - allows for segregation of a major
gene together with polygenic and environmental background.
Hitherto, analyses of pedigree data using the mixed model have
not been attempted for lack of an efficient algorithm to calculate
the likelihood. As a first approach, an easily calculable conditional
likelihood function is maximized.
The results of the analyses indicate an autosomal dominant gene
for hypercholesterolemia segregating in this pedigree. The data are
consistent with an autosomal recessive gene segregating for systolic
hypertension at least in the main branc!l of the pedigree. Little
evidence for a major gene segregating for diastolic hypertension is
detected in this pedigree.
ACN'JOWLEDGMENTS
I wish to express my deep appreciation to my advisor, Dr. R.C.Elston, who suggested the topic of this dissertation and who provided
invaluable guidance, constant encouragement, anti patient tmderstanding.
Appreciation is also expressed to the other members of my advisory
committee, Drs.R.C. Elandt-Johnson, R.R. Kuebler, M.J. Symons, and
H.A. Tyroler; they made many valuable and helpful suggestions.
I am grateful to Dr. Kurt Hirschhorn for generously providing the
data which are analyzed in this dissertation.
The programming aid of Ellen Kaplan is gratefully acknowledged.
Without her programming support, this work may never have been com-
pleted. Thanks go to Geoffrey Day of the Radiation Effects Research
Fotmdation in Hiroshima, Japan, for his talented drawing of many of
the figures. Thanks to Susan Stapleton who typed this dissertation.
This investigation was supporteJ by NIH Training Grant No. T-Ol
GM00038 from the National Institute of General Medical Sciences. This
support is gratefully appreciated, as is the computer time provided by
the Biostatistics Department.
Finally, I thank my parents for their encouragement, tmderstanding,
and support throughout the years.
ACKNOWLEDGME1'ITS
LIST OF TABLES
LIST OF FIGURES
Chapter
TABLE OF CONfENTS
. . . . . . . . .P,age
ii
viii
xi
I . INTRODUCfION AND LITERA11JRE REVIEW. .
1.1 Introduction .
1.2 Essential Hypertension.
1.2.1 Definitions ..
1.2.2 The Pickering School
1.2.3 The Platt School
1.2.3.1 The Pre-1960 Hypothesis.
1.2.3.2 The Post-1960 Hypothesis
1
1
1
1
3
7
8
10
1.2.4 The Contributions of Other Investigators.. 12
1.3 Familial Hypercholesterolemia .
1.3.1 Classification of Familial Hyper1ipidemias . . . . . . .
1.3.2 Type II Hyper1ipoproteinemia..
1.3.3 The Genetics of Familial Hypercho1estero1e~a. .
1.4 Synopsis of the Problem...
II DESCRIPTION OF THE KINDRED..
2.1 Source of the Kindred..
2.2 Analysis of the Data as if From a Sample ofIndependent Individuals ....
2.2.1 Sex Differences ....
2.2.2 Relationships with Age.
21
21
26
28
33
35
35
36
47
47
III DESCRlPTION OF THE PEDIGREE
Chapter
3.3.2.2 Serum Cholesterol
2.2.3 Skewness and Kurtosis ..
viPage
57
60
66
69
69
73
76
78
80
80
80
90
90
94
94
99
105
105
106
111
118
118
124
128
134
Inter-trait Correlation .
3.3.2.3 Systolic Blood Pressure andSerum Cholesterol .
3.3.1.3 Sperry Cholesterol ..
3.3.1.4 Zak Cholesterol
2.2.4
4.4.3 Systolic Blood Pressure ..
4.4.4 Diastolic Blood Pressure
3.3.2 Bivariate Log-Normal Distributions.
3.3.2.1 Blood Pressure ...
2.3 Random Mating.....
4.1 Introduction . . . . · · . .4.2 The Major Gene Model
4.3 Method of Analysis . · · . .4.4 Results of Univariate Analyses .
4.4.1 Sperry Cholesterol . . · ·4.4.2 Zak Cholesterol . .
3.3 Fitting a Mixture of Normal Distributions
3.3.1 Univariate Log-Normal Distributions.
3.3.1.1 Systolic Blood Pressure.
3.3.1.2 Diastolic Blood Pressure
3.2 Age Distribution in the Two Pedigrees
3.1 The Pedigree Structure.
TIIE MAJOR GENE HYParnESIS .IV
Chapter
4.5 Results of Bivariate Analyses .
viiPage
134
4.5.1 Sperry Cholesterol and Zak Cholesterol.
4.5.2 Systolic and Diastolic Blood Pressure
v4.6 Conclusions .
TIlE POLYGENIC HYPOTI-IESIS
5.1 The Polygenic ~IDdel..
. . . . . . . . .
137
137
143
146
146
5.1.1 Sperry and Zak Cholesterol. 151
5.1.2 Systolic and Diastolic Blood Pressure. . . 153
5.1.3 Other Traits
5.2 Conclusions..
VI THE MIXED r-DDEL • • • •
6.1 Method of Analysis
. . . .. . . . .
. .
ISS
161
163
166
6.1.1 Genotypic Classification of Individuals
6.2 Sperry and Zak Cholesterol
167
169
170
177
178
. . .. . . .• •
6.3 Systolic Blood Pressure. • • .
6.4 Conclusions••••
VII Sm.MARY A'ID CONCLUSIONS •
APPENDICES
1. Secondary Hypertension: Hypertension Occurring as aManifestation of a Known Disease. . . . . . 186
2. List of Variables Observed in 1947.
3. List of Variables Observed in 1958 . . . . . . . .187
188
4. Sex, Age, Height, Weight, Systolic Blood Pressure,Diastolic Blood Pressure, Sperry and Zak Cholesterol, Beta and Prebetalipoprotein Values forMembers of the Pedigree Observed in 1958. . . 189
BIBLIOGRAPHY • . • • • • • • • • • . • . • • . • . . 193
LIST OF TABLES
Table Page
1.1 Diagnosis of Hyper1ipoproteinemia. . . 25
2.1 Age Distribution by Sex. . . . . . 37
2.2 Summary Statistics for Original and Natural Logarithmic-Transformed Variables. . . . . . . . . . . . . 42
2.3 Mean and Standard Error of Logarithmic-TransformedVariables By Sex . . . . . . . . . . . . . . . 48
2.4 Mean and Standard Error of Logarithmic-TransformedVariables by Age and Sex . . . . . . . . . . . 49
2.5 LL~ear and Quadratic Regression Coefficients of Agefor Logarithmic-Transfonned Variables by Sex . . . 56
2.6 Skewness and Kurtosis for Original and Age-adjustedLogarithmic-Transformed Variables. . . . . . 59
2.7 Correlation and Partial Correlation CoefficientsBetween Logarithmic-Transfonned VariablesInvolving Systolic and Diastolic Blood Pressureand Sperry and Zak Cho1estero1s . . . . . . . :62
2.8 Correlation and Partial Correlation CoefficientsBetween Logarithmic-Transformed Variables. . . 64
2.9 Inter-spouse Correlations of Age-Adjusted Logarithmic-Transformed Variables. . . . . . . 67
3.1 Age Distribution by Sex and Pedigree
3.2 Mean Age by Sex and Pedigree .....
3.3 Availability of Data for Six Traits by Sex andPedigree . . . . . . . . . . . . . . . . .
3.4 Maximum Likelihood Estimates of the Parameters for aMixture of Univariate Log-Normal Distributions
74
75
77
a. Trait: Systolic Blood Pressure - t~es. . 81b. Trait: Systolic Blood Pressure - Females. 81c. Trait: Diastolic Blood Pressure - Males . . 84d. Trait: Diastolic Blood Pressure - Females 84e. Trait: Sperry Cholesterol - Males . 87f. Trait: Sperry Cholesterol - Females . 87g. Trait: Zak Cholesterol - t~les 91
h. Trait: Zak Cholesterol - Females. . 91
ix
Table Page3.5 Maximum Likelihood Estimates of the Parameters for a
Hixture of Bivariate Log Nonnal Distributions
a. Trait: Systolic and Diastolic Blood Pressure -Males . . · · · · · · · · · · · · · · · · · . 9S
b. Trait: Systolic anu Diastolic Blood Pressure -Females . · · · · · · · · · · · · · · · 96
c. Trait: Sperry and Zak Cholesterol - Hales · 97
d. Trait: Sperry and Zak Cholesterol - Females 98
3.6 Maximum Likelihood Estimates of the Parameters forthe Two Local ~Iaxima for
a. Systolic Blood Pressure and Sperry Cholesterol -~1a.les· • . . . . . . . • • • . • . • . . • 101
b. Systolic Blood Pressure and Zak Cholesterol -Males . . . . . . . . . . . . . . . . . . 101
c. Systolic Blood Pressure and Sperry Cholesterol -Females . . . . . . . . . . . . . . . . . 102
d. Systolic Blood Pressure and Zak Cholesterol -Fet:nales . . . . . . . . . . . . . . . 102
4.1 The Genetic Transition Hatrix for a One-Locus, Two -Allele System. . . . . • • . . . . . . . . . 109
4.2 Parameters of the ~~del and Their Interpretation. .. 112
4.3 Maximum Likelihood Estimates From Univariate PedigreeAnalysis of Sperry Cholesterol Data
a.b.c.
Right Pedigree .Left Pedigree.
Both Pedigrees • · . . . .
· . . . · . .· . . . . · . .
119120
121
4.4 Maximum Likelihood Estimates From Univariate PedigreeAnalysis of Zak Cholesterol Data
a. Right Pedigree · · · · · • · 125
b. Left Pedigree. · · • · · · · 126
c. Both Pedigrees · · · · · · • · · · · · · · · · · · 127
4.5 Maximum Likelihood Estimates From Univariate PedigreeAnalysis of Systolic Blood Pressure Data
a. Right Pedigree • · · · · · · · · · · · · 129
b. Left Pedigree · · · · · · · 130
c. Both Pedigrees · · · · · · .. . · · · 131
x
Table Page4.6 Maximum Likelihood Estimates From Univariate Pedigree
Analysis of Diastolic Blood Pressure Data
a. Right Pedigree . .b. Left Pedigree. . . .
4.7 Maximum Likelihood Estimates of Bivariate PedigreeAnalysis of Sperry Cholesterol and Zak CholesterolLevels .
135136
a.b.c.
Right Pedigree . .Left Pedigree
Both Pedigrees .
138139140
4.8 Maximum Likelihood Estimates of Bivariate PedigreeAnalysis of Systolic and Diastolic Blood PressureLevels
a. Right Pedigree . . .b. Left Pedigree
141142
5.1 Maximum Likelihood Estimates of the Parameters for thePolygenic Model for Sperry and Zak Cholesterol byPedigree . . . . . . . . . . . . . . . . . . . .. 152
5.2 ~~imum Likelihood Estimates of the Parameters for thePolygenic Model for Systolic and Diastolic BloodPressure by Pedigree . . . . . . . . . . . . . .. 154
5.3 Maximum Likelihood Estimates of the Parameters for thePolygenic Model for Other Traits by Pedigree . .. 156
6.1 Maximum Likelihood Estimates of the Parameters for theMixed Model for Sperry and Zak Choles terol byPedigree . . . . . . . . . . . . . . . . . . . .. 171
6.2 Variance Component Estimates, Proportion of the TotalVariance, and Total Heritability Estimates for theVarious Traits by Pedigree . . . . . . . . . . .. 172
6.3 Maximum Likelihood Estimates of the Parameters for theMixed Model for Systolic Blood Pressure byPedigree 175
38
LIST OF FIGURES
Figure Page
1.1 The Five Types of Lipid and Lipoprotein Patterns forPatients with Familial Hyperlipoproteinemia. 23
2.1 Cumulative Plot of Age by Sex
2.2 Cumulative Plot by Sex
a. Systolic Blood Pressure and Ln(SBP) 43b. Sperry Cholesterol and Ln(Sperry) . . . . . . • . 44c. Weight and Ln(Weight) . . . . . . • . . 45d. Height and Ln(Height) .... . 46
3.1 a. Right Pedigree.
b. Left Pedigree
3.2 Empirical and Theoretical Cumulative Plots AfterFitting a Mixture of Log-Nonnal Distributions
70
71
a. Trait: Systolic Blood Pressure - ~4ales. · · 82
b. Trait: Systolic Blood Pressure - Females · · · · 83
c. Trait: Diastolic Blood Pressure - ~~les · · · · 85
d. Trait: Diastolic Blood Pressure - Females · · 86
e. Trait: Sperry Cholesterol - Males · · · · · 88
f. Trait: Sperry Cholesterol - Females 89
g. Trait: Zak Cholesterol. . · · · . . 92
3.3 Plots of the Estimated Distribution Means for SystolicBlood Pressure and Serum Cholesterol by Sex Correspondingto the Two Local Maxima of the Likelihood. . . . 104
6.1 Component and Total Theoretical Density Functions
a. Sperry Cholesterol . . . . · · · · · · ·b. Zak Cholesterol . . . . · ·c. Systolic Blood Pressure · · · . . · · · ·
173
173176
GlAPTER I
INTRODUCTIO:-J Ai'JD LITERATl..jRE REVIEW
1.1 Introduction
ThrougIl tile years, hign senun cholesterol levels and high blood
pressure have been recognized as two of the many risk factors for
coronary heart disease. Indeed, in 1970, the Report of the Inter
Society Commisson for heart Disease Resources (1970) named three risk
factors, hypercholesterolemia and hypertension along with smoking, as
being major risk factors for premature atherosclerotic disease,
especially coronary heart disease. Due to the serious dimensions of
morbidity and mortality attributable to both essential hypertension
and familial hypercholesterolemia, tilere have been many investigations
concerning these two conditions. There is no question that heredity
plays a role in both conditions. The debate concerns how big that
role is, and what is the genetic mechanism. In this chapter, a review
of the role heredity plays in both essential hypertension and familial
hypercilolesterolemia will be given.
1.2 Essential Hypertension
1.2.1 Definitions
In order to study tile heredity of essential hypertension properly,
one must differentiate between it and secondary hypertension. Essential
hypertension has always been defined by exclusion since no pathognomonic
biocllemical or metabolic abnormality has yet been identified. If hyper
tension is preceded by a specific cause or a specific lesion, then it
2
is tenned secondary hypertension (~'~ndlowitz 1961) (See Appendix 1 for
a list of causes). Essential hypertension is actually whatever remains
after exclusion, consequently, essential hypertension Iileans hypertension
without evident cause and is usually characterizeo by elevated arterial
pressure. It has also been called primary hypertension because the
hypertension precedes any cardiovascular changes.
ifuat is the cut-off point dividing the hypertensives and non
hypertensives? A scan of the literature will reveal almost as many
division lines as there are investigators. The dividing lines range
from about 120/80 to about 180/110 (Pickering 1961). In 1959, tIle
Conference on Methodology in Epidemiological Studies in Cardio-
vascular Diseases met in Princeton, i~ew Jersey (Pollack and Krueger
1960). The Conference, recognizing that clinical usage demanded
arbitrary but tmiform criteria of nonnal and of abnormal arterial blood
pressure, suggested the follO\iing criteria: Any person with systolic
pressure at or above 160 nun Hg or (inclusive or) diastolic pressure
at or above 95 nun Hg definitely is hypertensive. Those with systolic
pressure below 140 nun Hg and diastolic pressure below 90 mn Hg are
considered to be norrnotensives. The residual blood pressure levels
represent the questionables and are left up to the individual investi
gator. Although these criteria have not been tmiversally accepted,
many studies, including the Framingham Study (Kannel and Gordon 1970) ,
the Evans COLDlty Study (Cassel lY71) , and the U.S. National Health Survey
0~ational Center for Healtll Statistics 1966), have adopted them so
that some comparisons of results are possible.
Concerning the genetics of essential hypertension, there are pri
marily two Schools of Thought. One School, whose main proponent is Sir
3
Robert Platt, says that essential hypertension is a specific disease
entity and that the population can be separated into subgroups, those
with essential hypertens ion and those who are nonnotens ives . This
School has further hypothesized that the disease is detennined by a
gene with incomplete dominance. The other School, whose main proponent
is Sir George Pickering, maintains that essential hypertension is not
a specific disease entity, that a person inherits not a disease
essential hypertension, but rather a large number of genes which deter
mines a particular level of blood pressure, and those individuals
categorized as haVing essential hypertension are simply the ones whose
blood pressures fallon the upper end of a continuous unimodal frequency
distribution. This School says that essential hypertension is deter
mined by multifactorial inheritance, a combination of genetics and
environment .
This controversy, lively and bitter at times, has spanned more
than two decades. For the remainder of section 1. 2, the argt.mlents
advanced in support of and in opposition to the two Schools will be
presented. The important contributions made by'other investigators
will be cited.
1. 2.2. The Pickering School
Sir George Pickering ana his supporters (Pickering 1968; Cruz-Coke
1960; Hamilton et al 1954c)compare essential hypertension to human
stature; both are classical examples of polygenic inheritance. In
their view, any cut-off line between the hypertensive and non-hyper
tensive segments of the population can only be arbitrary (Hamilton et
al1954a).
Since it is known that mean blood pressure levels increase with
4
age for the two sexes, it would be misleading to compare two populations
of different ages and sexes. Hamilton, Pickering, Fraser-Roberts, and
Sowry (1954b) corrected for age mlo sex differences by computing age
and sex-adjusted scores. The effects of age are dealt with by adjusting
all readings by detennining how much each person's blood pressure level
is above or below the appropriate mean for his age and sex, and then
multiplying the deviation by a factor to make it equivalent to the
deviation at some standard age. Using these scores, they build evidence
in support of the multifactorial inheritance theory for essential
hypertension:
1) The frequalCY distribution of the adjusted blood pressures is a
continuous unimodel curve (Hamilton et al 1954c; Murphy et al 1966).
The distribution is not quite Gaussian; it is positively skewed.
Hamilton et al (1954b) studied the distribution of the adjusted scores
for diastolic pressures in three different populations - one group
represents the population-at-large; the second group consists of those
who are first-degree relatives (i.e. sibs, parents, and cllildren) of
propositi liith normal blood pressures (whom Hamil ton et al define as
those with diastolic pressures not exceeding 85 nun Hg) ; the final group
consists of first-degree relatives of propositi with essential hyper
tension (diastolic pressures of 100 rom Hg or more). The distributions
of the adjusted blood pressure scores for the first two groups are
almost identical - unimodal and positively skewed. The curve for the
last group, the relatives of the hypertensives, is still tDlimodal, but
the distribution is shifted to the right. Pickering and his supporters
argue that the consistent wlirnodal distributions indicate that people
with essential hypertension are those whose blood pressures fall in the
5
upper end of a "bell-shaped" distribution. Furthennore, any line of
demarcation to divide the population into two subgroups, the nonno
tensives and the hypertensives, can only be arbitrary; essential
hypertension represents a quantitative, not quaZitative, deviation from
the nonn. In this respect, essential hypertension would be like
stature or intelligence, a multifactorial trait.
2) Pickering and his group looked at the relationship between the
blood pressures of propositi and their first-degree relatives (Hamilton
1954c; -Mial1 et al 1967; Pickering 1967, 19(8). They noticed a
similarity which they tried to quantify by calculating coefficients of
resemblance; which are the regressions of.age-and sex-adjusted scores
for first-degree relatives on the· age-and sex-adjusted scores for pro
positi. The computed coefficients of resemblance between the adjusted
scores of all relatives and all propositi were 0.224 for systolic
pressures and 0.178 for diastolic pressures. 1~en the coefficients
were computed for relatives of hypertensive propositi, they were fairly
constant at about 0.2; this means that if the pressure of a subj ect
deviated from the nonn by 10 mm Hg, then the pressure of his relatives
differed from the nonn on the average by 2 mm Hg. The Pickering School
concludes from this that environmental factors playa major role in
detennining arterial pressure since the coefficient of resemblance is
relatively small - 0.2 for arterial pressure as against O.S for stature
(Oldham et al 1960). This constant coefficient of resemblance says
that the lower the pressure of the propositi, the lower the pressure
of their first-degree relatives of all kind; the higher the propositi
pressure, the higher the pressure for the relatives. This evidence
indicates that the inheritance of arterial pressure is quantitative,
6
or polygenic, and that the inheri tance is of the same kind whether the
arterial pressure is less than the nonm or in the essential }~ertension
range.
Acheson and Fowler (1967) have disputed this evidence. They say
that it is misleading to compare arterial pressure with height since
excess height is not associateci with excess mortality; furthermore,
height is normally distributed in the population, while arterial
pressure has a distribution which is skewed to the right. Acheson and
Fowler also criticized computing coefficients of resemblances using age
and sex-aujusted scores based on single blood pressure measurements
which tend to be highly variable. Miall and Oldham (1963) have recal
culated the coefficients basing them on two measurements. The results
are 0.399 for systolic and 0.302 for diastolic blood pressure, higher
than the 0.2 for both systolic and diastolic pressures when only one
measurement was taken .. They claim that even these may be underestimates
of the true familial resemblance because the use of age- and sex
adjusted scores does not correct for selective mortality; young people
with the higher pressures of genetic origin are more apt to have
lost older relatives with hypertension than older relatives with normo
tension-; Therefore, Acheson and Fowler surmised that genetics may
really play a larger role than the Pickering School have granted.
3) Hamilton, Pickering, Fraser-Roberts, and Sowry (1954c) regressed
systolic and diastolic scores on age for three groups of males and
females: a population sample from a skin disease clinic, relatives
of normal propositi, and relatives of hypertensive propositi. Except
for systolic blood pressures in males, the rates of increase of blood
pressures with age are almost the same for all three samples. The
7
Pickering Scllool advanced these results to argue tllat, ignoring male
systolic pressures, it is not the rate of rise with age that is
important in hypertension inheritance; there is a propensity for
higher pressures at aLL ages. This conclusion is to be contrasted to
that of the Platt School which says that individuals with essential
hypertension are characterized by a sudden rise of arterial pressure
during middle age.
4) The last piece of evidence appeals to logic and intuition. It is
well known that arterial pressure depends on many physiological factors
including cardiac output, radius of vessels, viscosity of the blood,
secretions of the adrenal gland, the electrolyte content of the blood,
the state of the baro-receptors, etc. (Pickering 1968). This dependence
on so many ftictors nas led the Pickering School to the opinion that it
is tmlikely that the inheritance of arterial pressure can be character
ized by one gene. In the opinion of the Pickering School, it is
tmlikely that anyone will be able to find a specific biochemical lesion
for essential hypertension.
1.2.3 The Platt School
Sir Robert Platt and his supporters consider essential hypertension
to be a distinct disease entity. Their arglDllents consist of :two kinds,
their own and those in rebuttal to evidence advanced by the Pickering
School.
Whereas the Pickering School sees the frequency distribution of
blood pressure in the population to be a tmimodal one where the top
10-20%, by middle age, have attained a blood pressure so high as to
carry hazards to survival, the Platt School proposes that there are
really two or more populations instead of only one; there are people
8
who genetically are more prone to develop hypertension in middle age,
and others who are not. The Platt School agrees with the Pickering
School that there is no natural dividing line between normal and
abnormal blood pressures. However, since there does not exist a more
specific test for essential hypertension, one is forced to base con
clusions principally on studies of blood pressures.
The Platt School agrees with clinicians that the age of risk for
essential hypertension is 45-60. Platt (1967) suggests that those with
essential hypertension have blood pressures that have risen steeply
during the middle years, and those who are normotensives demonstrate
no significant rise of blood pressure. Therefore, the Platt School
argues strongly that it is important to study only sibs instead of all
first-degree relatives, since the children of hypertensives are unlikely
to have reached ages 45- 60, and parents of hypertens ives will have
already experienced a selective mortality.
1.2.3.1. The Pre-1960 Hypothesis
Until about 1960, the Platt School (Platt 1959, 1961) hypothesized
that essential hypertension was inherited as a result of a major
dominant gene. Hence, siblings of hypertensive propositi should
segregate into two groups, those who inherited and those who did not
inherit the gene from their parents; it then should follow that a plot
of the blood pressures of siblings of hypertensive propositi should
reveal a bimodal distribution. The Pickering School would argue for
a unimodal distribution. Platt (1959) reanalyzed the data collected
by Hamilton and his co-workers (1954a) and by Sobye (1948) by looking
at just the siblings of hypertensive propositi. The resulting curves
do not appear to be unimodal; it is difficult to distinguish between
9
tlIeir being bimodal or trimodal. Indeed, the curves display troughs at
150 Jml Hg systolic and 90 nun Hg diastolic which conveniently happens
to be the dividing line between nonnal and high blood pressures cited
by many clinicians.
The results of a study by Morrison and Horris (1959, 1960) of 302
London bus drivers and conductors support Platt's findings. ~10rrison
mId Morris studied the blood pressure distributions of clIildren of
hypertens ive and non-hypertens i ve parents. According to the single gene
hypotllesis, children of hypertensive parents should segregate into
roughly two groups, and the distribution of their blood pressures should
be bimodal. On the other hand,children of nonnotensive parents should
also be normotensive and should have a unimodal blood pressure distri
bution. The results of the Morrison and Morris study support the one
gene hypothesis. The distribution of the blood pressures of tlle drivers
and conductors with hypertens ive parents showed bimodality, whereas tlle
distribution for drivers and conductors of non-hypertensive parents was
approximately normal. These findings have been disputed because of
tlle unusual criterion(age at death)that Morrison and ~brris used to
classify the parents as being hypertensive or non-hypertensive.
Lowe and McKeown (1962) conducted a study similar to that, of
Morrison and Morris of 5239 men working in an electrical engineering
firm. They found no bimodality' in the distribution of the blood
pressures of the middle-aged men who had one or both parents dead.
Ostfe1d and Paul (1963) examined 1989 men of ages 40-55 working for
Western Electric Company in Chicago. Using tlle same method of sub
dividing the parents, Ostfe1d and Paul also were unable to reproduce
tlle results obtained by ~-brrison and ~brris.
10
1.2.3.2. Tne Post-1960 Hypothesis
In about 1963, Platt (1903) perfonned his own study of 350 sibs of
178 hypertensive propositi. Examining the frequency distributions of
the blood pressures by age, he fOlll1d that the distribution is not
Gaussian and does not become Gaussian after a logarithmic transformation.
Platt observed that, with increasing age, there developed a bulge in
the distribution curves in the middle ranges of blood pressures, cen
teririg at a systolic pressure of about 160 mm Hg. By ages SO-59, most
of the sibs were in this middle range. There also developed, with
increasing age, a bulge at the high end of the distribution. In light
of these irregular trimodal distributions, Platt had to modify his
dominant inheritance hypothes is of pre-1960. He proposed the hypo
thesis that essential hypertension is inherited as a gene of incomplete
domin&lce. Those at the high end of the distribution are severe hyper
tensives, inheriting the gene for hypertension in the homozygous form;
persons in the middle range represent those with moderate llypertension,
inheriting the gene for hypertension in the heterozygous form;
finally, those in the lower end are the nonnotensives, inheriting two
nonnal genes.
The results of longitudinal studies of different populations of
sibs by Cruz-Coke (1959) and Perera (1960) support Platt's theory.
They showed that the sibs can be divided into two groups - those whose
arterial pressures rose little with age and those whose arterial pressures
rose steeply with age. In fact, a plot of the logarithm of the
systolic pressure resulted in a curve with three modes.
Platt found further support in this controversy from the Evans
County, Georgia Study conducted in 1960-1962 by McDonough, Garrison,
11
and Hames (1964). This study examined the frequency distributions of
systolic and diastolic pressures in 621 whites and 379 blacks of ages
55-74. The three investigators, perfoming a curve-fitting exercise,
attempted to find the minimum lllD'I1lJer of subgroups compatible wi th four
conditions: (1) Summing the curves for the subgroups must result in the
parent distribution. (2) The distribution for each subgroup should be
nearly normal. (3) The subgroup in the lower tail should have
pressures not mudl different from those seen at younger ages (repre
senting the normotensives whose blood pressures exhibit little rise
with age) . (4) Comparing equivalent subgroups, there should be no
white-black differences in mean blood pressure. The three investigators
found that two subgroups did not satisfy all of the conditions whereas
thr.ee subgroups did. The subgroup at the lower tail represents the
normotensives, possessing two normal genes assuming Platt's hypothesis
of incomplete dominance; those in the middle subgroup represent those
inheriting the gene for hypertension in heterozygous form, and those
in the upper tail represents those inheriting the gene for hyper
tension in the homozygous form. The estimated frequencies, obtained
from curve-fitting, of the three subgroups are surprisingly close to
what would be expected under Hardy-Weinberg equilibrium.
It must be noted that the curves for the three subgroups display
nDJch overlap. If Platt's hypothesis of three genotypes is correct,
then blood pressure is a poor discriminator; most blood pressure levels
could be the expression of more than one genotype; misclassification
cou~d result. More sensitive and specific methods for separating the
genotypes are needed. Furthermore, Inerely dividing a distribution into
subgroups can offer no confirmation of their physical existence
(~1urphy 1964).
12
1.2.4. The Contributions of Other Investigators
Hall (1966), in his PhD dissertation in 1966, compared the direct
and indirect methods of measuring arterial pressures. The direct
method of measuring blood pressure involves cormecting the artery
directly to a manometer with a hollow tubing. The indirect method,
the sphygmomanometer, is better known and involves a soft rubber cuff,
a colUlIU1 of mercury, and a stethoscope. Hall adjusted for differences
in ages, in tricep skinfold, in sub-scapular skinfold, and in mid-ann
circumference by including these variables along with direct blood
pressure reading as independent variables in a regression model with
the difference between direct and indirect readings as the dependent
variable. He thus could obtain an equation involving the difference
between direct and indirect readings as a function of direct blood
pressure readings, after adjusting for the ~ther independent variables.
A plot of the equation showed that the difference between direct and
indirect values increased with increasing direct measurements. In
other words, with increasing arterial pressures, the indirect readings
were increasingly underestimating the direct readings. Since almost
all frequency distributions of blood pressures have been based on
indirect measurements, Hall's result:suggests that it may be necessary
to modify the shapes of these frequency distributions. The overall
effect will be to extend the right-hand tails of the frequency curves
and, as a result, perhaps to sharpen the divisions separating possible
subgroups. Unfortunately, for a large population, it is difficult
and impractical to make direct measurements on every person in the
study.
Hall used the data from the Evans County Study to estimate
13
Hall assumed that the heterogeneous population consistedZ
(fll' 0 ),
analytically the same parameters that McDonough et al (1964) estimated
graphically.
of three normally distributed subpopulations with parametersZ Z
(flZ' 0), and (fl3' 0) in the proportions aI' aZ' and a3, respec-
tively (I a.= 1). Hall used the method of maximum likelihood to. 11
estimate the six parameters, (fll' flZ' fl3' aI' aZ' 0). It should be
noted that the standard errors for the estimates of the proportions
are quite large, particularly for the black population. In co~aring
Hall's estimates with those of McDonough et al (1964) obtained by free-
hand curve-fitting, one can observe that the proportions are similar for
blacks, but not for whites. In looking at the estimated means, Hall
fOlD1.d that the subpopulation means for blacks and whites are very
similar. However, there are greater proportions of blacks in the sub
populations with the higher arterial pressures than of whites. As a
result, the overall mean blood pressures for blacks are higher than
those for whites, with the higher frequency of a hypertens ion gene
possibly accolD1.ting for the observed differences.
For many years, the controversy between the Platt and Pickering
Schools has reached a standstillj neither School has been willing to
concede much to the other. As a result of this lack of progress, some
researchers have decided to try other approaches to try to resolve the
differences.
Some reasoned that if essential hypertension is determined by a
single major genetic factor, then by the "one-gene, one..enzyme hypo
thesis", a unitary defect in a biochemical mechanism is worth seeking
(McKusick 1960a, 1960b). If fOlD1.d, it coulc;i be the basis for the
treatment of essential hypertension. In 1959, Mendlowitz et al (1959)
14
boldly singled out a deficiency of the enzyme, O-methyl transferase,
which is important in the degradation of norepinephrine. But, by 1964,
they had dismissed this hypothesis in favor of a gene that modifies
catecholamine metabolism (Mendlowitz et al 1964, 1970). However, other
enzymes like renin and angiotensinase, and hormones like aldosterone
each have their own proponents (Pickering 1968). COllsequently, there
is disagreement regarding which enzyme or hormone is important in
determining essential hypertension.
The Platt and Pickering Schools both have maintained that the final
proof regarding whether essential hypertension is a distinct disease
entity or not may have to await a prospective study lasting 20 years
or more on a large lDlselected population to see whether, with increasing
age, the population will segregate into two subgroups, those whose
pressures rose steeply and those wllose pressures remain little changed.
There have been several longitudinal studies of the kind suggested.
Miall and Lovell (1967) reported the results of a longitudinal
study in South Wales. They fOlDld that age, per se, plays no direct
part in determining the rate of change of blood pressure, and that
changes in blood pressure are more closely related to the attained level
of blood pressure than to age. In other words, the higher the
individual's blood pressure, the greater will be the rate of increase
in his blood pressure with time. These results are in accord with the
Platt School.
Another longitudinal study was conducted by Harlan, Osborne, and
Graybiel (1962); they observed a relatively homogeneous group of white
males over anl8-year period. The 1056 healthy white Navy pilots were
first examined in 1940. None of them had a blood pressure level
15
over 132/86 at tllat time. They were re-examined in 1951-1952 and 1957
1958. After closely scrutinizing the frequency distributions of the
blood pressure levels, the three investigators concluded that there
was no evidence of a natural bimodality to suggest any evidence of
qualitatively different populations; furthermore, they suggested that
their study confirmed the fact that hypertension is a quwltitative
difference in blood pressure determined by a nroltiplicity of factors,
both genetic and environmental.
Unfortunately, this study and several like it suffer from two basic
flaws. One is that no attempt is usually made to exclude secondary
hypertensives. However, this is not as serious as the second flaw -
the study populations are usually highly selected. Sampling biases
exclude certain subgroups from the sample. The population in Harlan,
Osborne, and Graybiel's study excluded individuals who had high blood
pressures to begin with. Males with high blood pressures were dis
qualified and could not become i~avy pilots. As a consequence of this
deficiency, any presumed hypertensive subgroup may be so small as to
be obscured in the tail of the larger subgroup. The same phenomenon
may be observed in using insurance policyholders as the study popula
tion. Those with high blood pressures may be selected out since they
can be refused insurance or may have to pay higher premiwns for
insurance.
Feinlieb et al (1969) reported the results of a longitudinal study
of the relationship between blood pressure and age; the study was
based on data from the Framingham Study. Among the questions they
wanted to answer are two that are pertinent to the present discussion.
First, how does blood pressure change on a longitudinal basis; second,
16
to what extent <10 changes in blood pressure over time depend upon an
initial blood pressure? The Framingham Study is well suited to answer
these questions since the people in the study represent a cohort of
over 5,000 persons who have been examined biennially for almost 20
years. The report of Feinlieb et al covers the first seven examinations,
and it should be noted that secondary hypertensives have not been
eliminated.
The cross-sectional patterns of blood pressure in this study
agree with those of other population studies; with increasing age,
systolic blood pressures tend to rise; the same is true of diastolic
blood pressures, at least in women, and also in men up to about age 60.
To examine these trends in greater detail, Feinlieb et al divide the
study population into age cohorts. They find that, for both men and
women, the longitudinal trends of systolic blood pressures with age
are similar to the cross-sectional trends. The same is true of female
diastolic blood pressure patterns. However, for diastolic blood
pressures in males, although there is a basic trend of a rise of
diastolic blood pressure with age for all the cohorts, there is also a
tendency for the younger cohorts to have higher diastolic blood
pressures than the older cohorts; no explanation could be found for
this phenomenon.
To answer the question of the extent to which changes in blood
pressure over time depends on an initial blood pressure, Feinlieb
and his co-investigators divide the population into systolic blood
pressure cohorts according to the systolic blood pressure levels at a
particular instant in time; these cohorts are then examined for longi
tudinal trends. They find that, regardless of whether the initial
17
systolic blood pressure was 100-109 or 160-169, the longitudinal trends
in systolic blood. pressure are parallel. These longitudinal trends
remain even after the systolic blood pressure cohorts are subdivided
according to age. From this, Feinlieb et al conclude that the change
in an individual's blood pressure level later in life does not depend
on his blood pressure level earlier in life; there is a propensity for
an increase in blood pressure with age regardless of any earlier levels
of biood pressure. This result is markedly different from that of ~1iall
and Lovell (1967); there is no tendency for the population to separate
into two groups, those whose blood pressures rose steeply and those
whose blood pressures remained little cllanged.
Studies that have been used to determine the relative magnitude
of genetic and non-genetic effects on a trait include twin studies.
Investigators have been attracted to twins possibly because the analysis
seems simple. The basis for twin studies is that monozygotic (MZ) twins
are identical in their genetic constitution so that any differences
between them can be ascribable to non-genetic influences; dizygotic (DZ)
twins are related to each other in the same way as ordinary full
siblings. The twin study method assumes that the zygosity of the
pairs of twins has been determined correctly.
Using the observed among- and within-pair variances for the two
types of twins, twin study investigators have obtained estimates of
the heritability, the proportion of the total variation in a trait
accounted for by heritable effects. With regard to blood pressure,
the results of twin studies have not been consistent.
Osborne, DeGeorge, and Mathers (1963), in their study of the
blood pressures in S3 pairs of twins, found no significant difference
18
between the intrapair variances for MZ twins and DZ twins. Downie,
Boyle, et al (1969) also found no differences in the intrapair variance
in their series of 109 pairs of twins. Both groups conclude that
variability in blood pressure levels is predominantly under environ
mental influences.
Mtllhany, Shaffer,and rfines (1975) found that, in tl~ir series of
200 pairs of twins, genetic factors play an important role in determining
blood pressure levels. Their estimates of heritability for systolic
blood pressure are 0.73 and 0.56 for females and males, respectively,
and for diastolic blood pressure, 0.61 and 0.41, respectively. Borhani,
Feinlieb et al (1976) gathered 514 white male twin pairs from the
records of the Veterans Administration. Based on a method to estimate
the heritability which eliminates possible biases that may result
because the total variance in MZ twins was smaller than in DZ twins,
their estimates of ileritabi1ity are 0.8 for systolic blood pressure and
0.6 for diastolic blood pressure, both indicative of a major contribu
tion of genetic factors.
A recent paper by Elston and Bok1age (1978) is particularly relevant
to this brief examination of twin studies. They studied the fundamental
assumptions underlying the twin method and find that, of the many
assumptions, some have been discredited, some have not been tested,
and same are untestab1e. Consequently, they conclude that they have
serious reservations about estimates of heritability based onZy on
twin studies and question wilether, in most cases, the results of
genetic twin studies are applicable to the general population.
Finally, mention should be made of a study that is rich in both
ambition and potential: The Detroit Project Studies of Blood Pressure
19
using the family set method. The goal of the project is to test concur
rently medical, environmental, sociopsychological, and genetic hypo
theses for blood pressure variation (Harburg, Erfurt et al 1977 ) •
The study design consists of selecting four census areas in
Detroit to represent extremes of stress areas for blacks and whites;
the areas are designated black high stress, black low stress, white
high stress, and white low stress. rligh and low stress areas are areas
which differ markedly with respect to socio-economic variables (e.g.
income, education, occupation) and instability variables (e.g. crime,
marital instability, residential instabiIity). Wi thin each of the
four areas, family sets are collected. A family set consists of five
persons: an index case, his or her sibling, his or her first cousin,
his or her spouse, and an unrelated individual from the same area, of
the same sex and of a similar age who is a potential index case. Three
persons of the set are genetically related (index case, sib, and first
cousin) while the other two share an environmental connection with
the index case. The spouse serves as a "proximal environmental"
control while the 'unrelated person controls for environmental factors
which are within the same socio-environmental area as the index case.
There have been several reports of results from the Detroit
Project studies. One considers the relationship between socio
ecological stress areas and blood pressure (Harburg, Erfurt et al 1973).
Tne investigators found that black males living in a high stress area
have the highest blood pressure levels of all eight race-sex-res,idence
groups. Their blood pressure levels are significantly higher than
black males living in a low stress area. There is no significant
differential in blood pressure between this latter group and other
20
white groups. These results suggest an envirorunental, or more specifi
cally a socio-psychological influence on blood pressure.
More recently, there is a report examining family aggregation of
hypertension where systolic hypertension is defined as ~ 160 mm Hg
and diastolic hypertension as ~ 90 rom Hg (ScllUll, Harburg et al 1977).
The investigators are able to find only a weak tendency for diastolic
hypertension and less for systolic hypertension to aggregate in family
sets. Since this finding is in disagreement with other studies which
show familial aggregation of hypertension (Thomas and Cohen 1955;
Gearing, Clark, et al 1962; Ostfeld and Paul 1963), the authors suggest
that prior studies may have confounded envirorunental and genetic
correlations.
The family set has been used to estimate the heritability of blood
pressure by which is meant the proportion of the total variation in
blood pressure that can be accounted for by heritable effects
(Chakraborty, Schull, et al 1977). Consider a family set consisting
of the index case, sib, first cousin, and the unrelated control. The
covariance between members of the family set with respect to the trait
can be expressed as a function of an additive genetic variance, a
dominance variance, and an envirorunental variance (Falconer 1960).
From estimates of these variance components, the estimate of herit
ability can be computed. For the Detroit Project data, although the
estimates of lleritability are quite erratic, they show a tendency to be
relatively low. The investigators conclude from this that nongenetic
variables contribute more to observed blood pressure variation than do
genetic differences between individuals. Advocates of a large heri
table component in blood pressure variability certainly cannot find
much in these results to agree with.
21
1.3 Familial Hypercholesterolemia
Before the discovery and availability of sophisticated biochemical
procedures, familial and non-familial hyper1ipidemias, diseases which
are characterized by an increase of one or more plasma lipids and of
which familial hypercholesterolemia is but a subgroup, were first
discovered through its secondary manifestations, lipid deposits in
tendons and subcutaneous tissue called xanthomatosis. However, recent
advances in the laboratory have allowed the familial hyper1ipidenias
to be subdivided into subgroups using plasma levels of cholesterol and
triglycerides and plasma lipoprotein patterns.
1.3.1. Classification of Familial Hyperlipidemias
The most cOIllIOOn lipid in plasma is usually phospholipid; it is
believed that its flUlction is to bind other lipids to plasma proteins.
The next JOOst conunon lipid in plasma is cholesterol, with about
three-fourths of the total cholesterol usually esterified with long
chain fatty acids. The third most canmon plasma lipid is triglyceride.
There are two major sources of triglyceride; one is exogenous or from
the diet; the other is endogenous which origina"ces mainly from the
liver. After these three classes of plasma 1ipids, there are several
other lipids, but of smaller concentration. Among these are the free
fatty acids, carotenoids, vitamin A, and glycolipids (Frederickson and
Lees 1972).
The endogenous lipids and those from the diet must be transported
through the blood vessels. However, since the maj or plasma lipids,
phospholipid, cholesterol, and triglyceride, are not soluble in serum,
they do not circulate free in the senun. Rather, they circulate bOlUld
to proteins by forming stable lipid-protein complexes called lipo-
22
proteins. Thus, lipoproteins are the units of lipid transport (Levy
1971) .
Advances in laboratory procedures since 1950 have olanged the
classification of familial and nonfamilial hyperlipidemias into one
for hyperlipoproteinemias. The lipoprotein patterns can be distin
guished using an ultracentrifuge or paper electrophoresis.
Using ultracentrifugation, the lipoproteins can be separated into
four groups (Stone and Levy 1972; Frederickson and Lees 1972):
1) High density lipoproteins (HOL) - commonly called alpha
lipoproteins, density> 1.063 grn/ml.
2) Low density lipoproteins (LDL) - common referred to as beta
lipoproteins, density between 1.006 and 1.063 grn/ml.
3) Very low density lipoproteins (VLDL) - commonly known as prebeta
lipoproteins, density between 0.95 and 1.006 grn/ml.
4) Chylomicrons - density < 0.95 gm/ml.
As a laboratory procedure, ultracentrifugation tends to be
expensive and difficult to use. Paper electrophoresis, while rela
tively rapid, simple, and inexpensive, does not have the resolution
power of an ultracentrifuge (Frederickson and Lees 1972). As the dif
ferent lipoproteins migrate towards the anode, four separate bands can
be distinguished. The non-migrating band consists of the chylomicrons.
Then, with increasing distance from the origin, come the bands of the
beta-lipoproteins, prebeta-lipoproteins, and the alpha-lipoproteins
(Stone and Levy 1972; Frederickson and Lees, 1972). See Figure 1.1.
Of the four lipoproteins, only chylomicrons, beta-lipoproteins,
and prebeta-lipoproteins are important as far as classifying the
familial hyperlipoproteinemias are concerned. These three lipo
proteins are interrelated in that they consist of the same lipids, but
23
Normal II III IV v
------ ------ ------
/3 /3 ~ , ...: .... ;_~.' 'r' : , ~:.', •
,,,./3 ,,../3
. .';'. i'·~..·:~. i :' •.:.~,... :..:;..:..:.. ~::; ,:"I,."
~
+ t t t I + IC TG C TG C TG C TG C TG
Usual changeIn
plasma lipids
Figure 1.1 The Five Types of Lipid and Lipoprotein Patternsfor Patients with Familial Hyper1ipoproteinemia
C = Cholesterol TG = Triglyceride [FromFrederickson and Lees (1966) p. 435]
24
in differu1g proportions (Frederickson and Lees 1972). The chy1omicrons
are the vehicles for the exogenous lipids, especially the trig1ycerides,
in the blood. The prebeta-1ipoproteins transport principally endo
genous triglyceride. The beta-lipoproteins transport about 75% of the
cholesterol in the serum. As a result of the different compositions,
increased beta-lipoproteins are associated with increases in cholesterol
and phospholipid while increased chy1omicrons and prebeta-lipoproteins,
are associated with increases in triglyceride.
Five familial hyperlipidemias can be distinguished using the
lipoprotein patte~. However, it should be noted that this transla
tion of hyperlipidemia into hyperlipoproteinemia does not imply that
these diseases are determined by mutations at loci regulating the
structure or metabolism of liPOProteins. Investigators feel that,
although the present system of classifying familial hyperlipidemias is
convenient, it will be replaced by a system based on etiology as more
information appears. The value of including lipoprotein patterns in
the classification procedure lies in a small but definite increase in
specificity above that possible in using only plasma lipid concentra
tion measurements (Frederickson and Lees 1972).
The lipoprotein patterns can be discerned by applying a combina
tion of three procedures:
(1) Examination· of the standing plasma after the plasma has been
kept at 4°C for 18-24 hours. A creamy layer at the top indicates the
presence of chylomicrons. A turbid infranate is indicative of increased
prebeta-lipoproteins; a clear infranate can mean increased beta
lipoproteins as these small molecules are completely soluble and do not
refract light (Stone and Levy 1972).
25
(2) Measurement of the plasma cholesterol and triglyceride con
centrations. For each of the five hyperlipoproteinemias, there is a
different cholesterol to triglyceride ratio. Figure 1.1 illustrates
this schematically, while Table 1.1 displays some typical ratios.
Type Definitive lipoprotein Appearanc~ of: Standing Cholesterol-to-pattern Plasma at 4°C triglyceride
Ratio
I Chylomicron present; Creamy supernatant jnormal or decreased clear infranatant 1:9beta- and prebeta-
II Increased beta-jnormal No creamj clear oror increased prebeta-j slightly turbidno chylomicron 4:1
III Abnormal beta-and Creamy supernatant mayprebeta-j abnormal be present; turbid orchylomicron cloudy infranant 1:1
IV Increased prebeta-; No cream;normal beta-; no turbidchylomicron 9:10
V Increased prebeta-j Creamy supernatant jnormal beta-; turbid infranatant~lylomicrons present 1:5
Table 1.1 Diagnosis of Hyperlipoproteinemia[from Stone and Levy(1972) p. 346]
(3) Paper electrophoresis. Plasma samples are obtained from
individuals after they have been on a l6-hour fast. Figure 1.1 shows
the electrop}IDretic patterns associated with each of the five
hyperlipoproteinemias.
26
Table 1.1 summarizes the way in which the five familial h~)er
lipidemias can be distinguished using the lipoprotein patterns:
Type I Hyperchylomicronemia
Type II Hyperbetalipoproteinemia
Type III Combined Hyperbetalipoproteinemia and Hyperprebetalipo
proteinemia
Type IV Hyperprebetalipoproteinemia
Type V Combined Hyperchylomicronemia and Hyperprebetalipoproteinemia
1.3.2 Type II Hyperlipoproteinemia
Type II Hyperlipoproteinemia has been called, for reasons stated
below, familial hypercholesterolemia, hyperbetalipoproteinemia,
familial xanthoma, and familial hypercholesterolemic xanthomatosis.
lID individual with hyperbetalipoproteinemia has higher concentra
tions of beta-lipoproteins, and since beta-lipoproteins transport
principally plasma cholesterol, he has higher levels of plasma choles
terol as well. The triglyceride level is little affected. Type II
is the most common type of familial hyperlipoproteinemia known
(Frederickson and Lees 1972). The clinical manifestations of this
disease include deposition of lipid in the skin and tendons
(xanthomatosis), corner of the eyelids (xanthelasma), eyes (corneal
arcus), and vascular endothelium (atheromatosis) (Frederickson and
Lees 1972; Harlan, Graham and Estes 1966), although not every mani
festatio~ is present in every case.
This disease was probably first reported by Rayer [1836] when
he observed one of its clinical manifestations, cutaneous and
tendinous xanthomas in 1836. Soon thereafter, clinicians noted that
there was considerable aggregation of cases of xanthomas in families,
27
and, as a result, familial xanthoma became well-established as an
entity. In 1873, a connection between xanthomas and blood lipids was
suggested by Quinquad (Chauffard and LaRoche 1910). In 1913, after
observing similar lesion in the arteries of cholesterol - fed rabbits,
a relationship between xanthomas and hypercholesterolemia was hypo
thesized. A study by Burns (1920) showed that cutaneous xanthomas were
always associated with hypercholesterolemia; consequently, familial
hypercholesterolemic xanthomatosis became the designation for the
disease. Svendsen (1940) declared that the primary expression of the
disease was hypercholesterolemia and that physicians should consider
increased cholesterol levels instead of cutaneous lesions as being
the characteristic sign of the disorder. In the early 1950's, McGinley,
Jones, and Gofman (1952) showed that individuals with xanthomas and
xanthelasma also have increases in beta-lipoproteins. Their studies
were perhaps the first to point out the connection between lipoprotein
patterns and hyper1ipidemias which has resulted in the present scheme
of classifying hyper1ipidemias.
Besides the manifestations mentioned above, one other is coronary
heart disease. However, there is disagreement as to the exact rela
tionship between hypercholesterolemia and coronary heart disease.
Although Harlan et al {l966) observed deaths fram heart disease in
the second and third decades associated with extensive xanthomatosis,
they also found that familial hypercholesterolemia was compatible with
survival into the sixth, seventh, and eighth decades. In 1967, Jensen
and his colleagues (1967) found a significantly higher death rate from
coronary heart disease in family members with hypercholesterolemia
than in family members with normal levels of cholesterol. Piper and
28
Orrild (1956) and Slack and Nevin (1968) fOlUld morbidity rates [rom
coronary artery disease significantly higher in hypercholesterolemics
than in their normocho1estero1emic relatives.
The clinical manifestations of Type II hyperlipoproteinemia
usually appear at an early age. Wilkinson and his coworkers (1948)
and Epstein and his colleagues (1959), in separate studies on the same
population, found that hypercholesterolemia usually was evident before
age 10, in many cases by age of one year.
A specific biochemical defect for Type II hyperlipoproteinemia
has been reported. Goldstein and Brown (1974, 1975) report that
familial hypercholesterolemia, in vitro, is due to a mutation involving
a regulatory protein. They identify on the cell surface of cultures of
normal human fibroblasts a regulatory molecule, the low density lipo
protein (LDL) receptor. In normal cells, the binding of LDL to the
receptor reduces cholesterol synthesis by suppressing 3-hydroxy - 3
methylglutaryl CoA reductase which is a rate-controlling enzyme. The
binding also enhances the rate of degradation of the lipoprotein. The
homozygotes for familial hypercholesterolemia lack the LDL receptor,
while in heterozygotes there is a reduction in the number of LDL
receptors.
1.3.3. The Genetics of Familial Hypercllolesterolemia
There is no longer mum controversy over the concept that a single
autosomal gene mutant for Type I I hyperlipoproteinemia gives the bearer
almost 100% certainty that he will have hyperbetalipoproteinemia.
There is almost total agreement that the disorder is due to a dominant
gene.
29
There have been numerous genetic studies of patients with familial
hypercholesterolemia. Investigators, in studying the genetics of
hypercholesterolemia, have used generally the same basic approach.
First they determine the cholesterol level in each member of a family
or of several families. Then, using a predetermined (either statis
tically computed or, more often, adopting one cited by clinicians) cut
off point, eadl member is categori zed as being hypercholesterolemic or
having normal cholesterol values by observing whether his cholesterol
value is above or below the cut-off point, respectively. If the family
spans more than two generations, it is broken down into two-generational
families. The parents are then categorized by mating types, i.e. both
hypercholesterolemic, both normal, or one normal and one affected.
For each mating type, the investigators test for Mendelian segregation
ratios among the offspring. The difficulty in interpreting the results
of the studies has been due to a lack of uniformity in defining hyper
cholesterolemia; part of the problem has been the many different
laboratory procedures used in determining cholesterol levels. 1ft
addition, there has been a deficiency of matings of certain phenotypes.
Wilkinson and his colleagues (1948) were one of the first to
conduct an extensive genetic study of hypercholesterolemia. They
investigated the condition in a family of over 200 members in 1948.
Based on their observation that about a half of the offspring resulting
from a mating between a hypercholesterolemic parent and a parent with
nonna1 cholesterol values became hyperdlo1esterolemic, they proposed
that the condition was determined by a dominant gene. In fact, after
observing the occurrences of xanthomatosis in the family, they con
cluded that the gene produced a moderate increase of serum cholesterol
30
wilen present in a single dose (heterozygous) and a large increase in
serum cholesterol along with severe xanthomatosis and a higher suscep
tib ility to coronary heart disease when present in a double dose
(homozygous). The contention that xanthomatosis is fOl.md only in
homozygous individuals was reaffirmed by Hirschhorn and \~ilkinson
(1959) in a separate study in 1958. Aldersberg et al (1949), Herndon
(1954), and Godal et al (1956) have agreed with this interpretation
based on results obtained fram studying other families.
During the 1950's and, early 1960's, investigators began to question
this interpretation. Among the dissenters were Alvord (1949), Stecher
and Hersh (1949), Leonard (19S6), Piper and Orrild (1956), Wheeler
(1957), Harris - Jones et al (1957), and Guravidl (1962). They began
to find patients with xanthomatosis (supposedly horoozygous abnormal)
who either had an offspring or a parent with normal cholesterol levels.
They also found two xrolthomatous parents producing some offspring who
had normal dlolesterol levels. In 1966, Harlan et al (1966), found
that out of 42 children of xanthomatous parents, 19 had normal
dlolesterol levels. As a result of these discrepancies, the theory
was advanced that the inheritance of hyperdlolesterolemia can be
explained on the basis of a simple dominant gene and that xanthomatosis
and high cholesterol readings are different expressions of the same
gene.
In 1964, Khaclladurian (1964) noticed that, in his study of 10
Arab sibships, both parents of young children afflicted with high
cholesterol readings and extensive xanthomatosis suffered from
abnonna1 cno1esterol concentrations. He reasoned that if he were to
assume that such children were homozygous affected, then both of their
31
parents and all of their offspring would have to be hypercholesterolemic.
He then attempted to locate similar individuals in other studies. Six
homozygous affected individuals were located in Epstein et aI's study
(1959) and one each from Meilman et al (1964), Piper and Orrild (1956),
and Adlersberg et al (1949). In eacll case, both parents were h)~er
cholesterolemic. He was unable to find any homozygous affected
producing any offspring, perhaps because of death from coronary neart
disease in the first or second decade. From these findings, Khachadurian
was able to define the phenotype of a hOlOOzygously affected individual:
he has markeu hypercllolesterolemia and extensive xanthomatosis usually
developing before age 15. Xanthomascan also develop in the heterozygous
individual, but the lesions develop later in life, are smaller and
fewer in number. The levels of beta-lipoprotein in the heterozygous
individuals are about twice those in the normal individuals, and the
homozygotes have levels that are two to three times higher than those
in the heterozygotes. Although the heterozygote often dies prematurely
of vascular diseases, the homozygote rarely survives to adulthood. In
other words, the homozygous abnormal genotype is often associated with
the more severe expressions of the disease.
In 1972, Jensen and Blankenhorn (1972) challenged the conclusion
that a single dominant gene is the IOOde of inheritance for familial
hypercholesterolemia. Their evidence consisted principally of pointing
out instances where the results of previous studies did not meet the
strict criteria of Mendelian inheritance, and they concluded that a
more probable mode of inheritance for familial hypercholesterolemia is
polygenic inheritance. They claimed that positively skewed Gaussian
distributions are more easily explained by polygenic inheritance
weighted with a few hypercholesterolemic genes than as a composite
32
curve of two distinct populations. They fOLU1d that the cholesterol
levels of hypercholesterolemic children are closer to that of the
midparent (the mean of the cholesterol levels for both parents) than
to that of the hypercholesterolemic parent. Furthermore, the polygenic
theory could explain the observed phenomenon of skipped generations
more reasonably than the single dominant gene theory. Jensen and
Blankenhorn found children with high cholesterol levels which exceeded
the sum of the parental levels which they cite as evidence of heterosis.
Finally, they cite evidence of observed outbreeding in several studies.
In part as a response to the polygenic theory of Jensen and
B1ackenhorn (1972), Schrott, Goldstein, et a1 (1972) studied the
inheritance of familial hypercholesterolemia in a large kindred
spanning four generations with 92 descendants. They observed that the
distribution of serum cholesterol in a family where hypercholesterolemia
is present is bimodal. Using a cutoff point to separate normals from
affecteds, analysis of various mating types produced segregation ratios
which are consistent with monogenic inheritance. Third, the serum
cholesterol level distribution in third degree relatives of hyper
cho1estero1emics was still bimodal. From these three pieces of
evidence, Schrott, Goldstein, et al concluded that familial hyper
dlo1estero1emia is inherited as an autosomal dominant gene.
Elston, Namboodiri, et a1 (1975) used pedigree analysis to study
the genetic transmission of hypercholesterolemia in a 195 member
kindred. In this study of a pedigree, they did not break the five
generational pedigree into two-generational families,and serum
cholesterol was analyzed as a quantitative trait, avoiding the necessity
of using cutoff points to dichotomize or trichotomize the data; these
33
two considerations should result in a more powerful analysis. They
found that a mixture of bvo lognormal distributions fits the cllolesterol
data better than a single lognormal distribution. From their pedigree
analysis, they concludedthat there is a dominant gene segregating for
hypercllolesterolemia in their kindred.
During the past several years, there have been reports of evidence
for linkage between a hypercholesterolemia locus and the C3 locus
(Ott, Schrott, et al 1974; Elston, !~amboodiri et al 1976; Berg and
Heiberg 1977). This reported linkage, the results of pedigree analysis,
along with the discovery of the biochemical mechanism,constitute rather
conclusive evidence that a dominant gene is the mode of inheritance of
hypercholesterolemia.
1.4 Synopsis of the Problem
With respect to the genetics of familial hypercholesterolemia, there
is conclusive evidence that it is determined by a single autosomal
dominant gene. One of the purposes of this study is to attempt to
corroborate this by reanalyzing, using more modem methods, the serum
cholesterol data collected by Wilkinson and his co-workers (1948) in
1947 and Epstein and his colleagues (1959) in 1958 from a mU1tigenera
tiona1 family living in or near Bay City, Michigan.
With regard to the genetics of essential hypertension, in spite
of a vo1tmri.nous literature, the role played by inherited factors remains
unresolved. The question of whether a person with essential hyper
tension has inherited a distinct disease or merely a predisposition made
manifest by environmental factors has not been answered. We will analyze
the blood pressure data collected from the same multigenerational Bay
City pedigree using the same methods as for the serum cholesterol data.
34
Chapter II will be a descriptive study of the data collected from
this pedigree where sex differences, transformations, inter-trait
correlations will be examined. In Chapter III, the pedigree structure
will be described. At the same time, attempts will be made to fit a
mixture of more than one distribution to the blood pressure data and
to the serum cholesterol data. In addition to determining whether
there is bimodality or trimodality to the distributions of blood pres
sure 'and serum cholesterol, the estimates of the means, variances, and
admixture proportions of the component distributions will be used in
the genetic analysis.
Chapter IV, V and VI will be the analysis of the data assuming
various underlying genetic models. Chapter IV will consider the major
gene model; a major gene is a single identifiable gene which can
accoun~ for a significant amount of the phenotypic variation. Chapter
V will consider the polygenic rodel where the phenotype is assumed to
be determined by a large number of equal and additive gene effects.
In Chapter VI, the underlying genetic model will be a mixed model by
which is meant a model that allows for segregation of a major gene
together with a polygenic and environmental background. In a sense,
this model combines features of the models of ChaptersIV and V.
CHAPTER II
DESCRIPTION OF TtIE KINDRED
2.1 Source of the Kindred
Members of the kindred which is the subject of the present study
lived in or near Bay City, Michigan~ Bay City is located in central
Michigan on the shores of Saginaw Bay which connects it wi th Lake
Huron. The serum cholesterol data for this pedigree have been analyzed
twice before, once in 1947 by Wilkinson and his colleagues (1948) and
the other time in 1958 by Epstein and his co-workers (1959).
As of 1958, the entire pedigree, whicll spans five generations,
consisted of 383 persons. Scattered among this pedigree are some
rather large sibships. For example, the largest sibship consists of
eighteen members; other examples are two sibships of size 13, one of
size 11, and one of size 10. SUcll large sibships should prove advan
tageous in doing a genetic study.
However, due to death, migration, or recalcitrance, out of the
383 persons in the pedigree, only 284 were examined in 1947 or 1958,
or both. Of the ninety-nine who were not examined, twenty-four were
said to have died before 1947; the remainder either moved from
the area or refused to participate in the study.
The 284 members who were examined can be subclassified into three
groups: 49 who were examined in 1947 only, 123 who were examined in
1958 only, and 112 who were examined in both years. Regarding the
36
49 who were examined only in 1947, thirteen had died between 1947 and
1958, most of the remainder had migrated out of Bay City, while a few
refused to co-operate in the subsequent 1958 study.
Appendix 2 lists the variables which comprise the data for the
161 persons examined in 1947. Two laboratory methods were used to
determine serum cholesterol. Most of the serum cholesterol deter
minations were analyzed using the Bloor method (Todd and Sanford 1943)
while about a dozen were determined by the Schoenheimer-Sperry method
(Sperry 1945). Since serum cholesterol was the variable of interest
in 1947, there were not much data available on other variables. Where
as almost all of the 161 individuals who were examined in 1947 had
cholesterol (160) and cholesterol ester (156) measurements made, less
than half of them (69) had blood pressure data taken. 111erefore, due
to the differing procedures of determining serum cholesterol and to
the lack of other data, the analysis will be confined to the 235
persons examined in 1958.
2.2 Analysis of the Data as if From a Sample of Independent Individuals
Appendix 3 gives a list of the variables which comprise the data
collected for the 235 individuals examined in 1958. The records for
each person include a medical history, results of a physical examina
tion, and laboratory results. Data for each person were extracted from
these records and keypunched onto computer cards. An attempt was made
to identify secondary hypertensives by searching through the records
for any mention of conditions listed in Appendix 1; none was found.
Of course, not every person had data available on every variable.
Although these 235 individuals are part of a five generation kindred,
for the rest of this chapter, the underlying pedigree structure will
37
be ignoreu, and the 1958 data will be examined as if the 235 persons
constitute a sample of inllependent individuals.
Table 2.1 Age Distribution by Sex
Hales Females Total
Number of Number of Number of~ Persons % Persons % Persons %
< 10 35 31.5 35 28.2 70. 29.8
10 - 19 18 16.2 32 25.8 50 21.3
20 - 29 8 7.2 10 8.1 18 7.6
30 - 39 24 21.6 28 22.6 52 22.1
40 - 49 15 13.5 9 7.2 24 10.2
50 - 59 5 4.5 3 2.4 8 3.4
60+ 6 5.4 7 5.6 13 5.5
Total 111 100.0 124 100.0 235 100.0
The 1958 population consists of 111 males and 124 females, and
their age distribution is shown in Table 2.1. The age distributions by
sex are similar; the males are slightly older than the females, with,
the mean age of the males being 24.5 years and that of the females
23.2 years; this difference is not statistically significant. Figure
2.1 shows the cumulative plot of age for the entire population as well
as for each sex. The cumulative plot gives, for each age along the
abscissa, the proportion of the population with an age less than or
equal to it. Half of the population are younger than 20, and only 10%
are older than 50. As both Table 2.1 and Figure 2.1 show, there is
an Unusual scarcity of individuals in the 20-29 age group. One can
38
75
1....... #'
1.0
"""• #'."MAL' /., ....MALE
.,.0.5 ._r
0.5
Z0I-::>al
a:I-~ 00
0 25 50 75 0 25 50 75w> 1.
."..",.;AGE
l-e{.J::> TOTAL~::>u
AGE
Figure 2.1 Cumulative Plot of Age by Sex
39
only sunnise the reason or reasons for this. Possible explanations
include military service, away at college, or couples marry and move
away. HoWever, these reasons seem less plausible when one discovers
that there is a dip in the age distribution for the 1947 population
at age group 10-19, and the dip in 1958, ten years later, is just a
continuation of it. Perhaps what is being observed is the result of
birth rate suppression during the Depression and World War.
A medical history was obtained from all subjects, and a physical
examination was perfonned during which height, weight, and blood
pressure were measured, and a non-fasting blood sample drawn. Each
blood sample was divided in half. One half was shipped to a New York
laboratory, where the serum cholesterol level w~ detennined using
the Schoenheimer-Sperry method (Sperry, 1945). For the purposes of
the present study, these values will be called Sperry cholesterol
values. The other half was shipped to the Lipid Metabolism Laboratory
in the Department of Medicine at the Medical College of South Carolina
in Charleston, where the serum cholesterol was measured using the Zak
method (Zak et al 1952); these will be called Zak cholesterol values.
Furthermore, in Charleston, the serum was fractionated in an ultra
centrifuge giving a high density alpha lipoprotein (alphaLP) component
(density> 1.063 gm/ml) ,a low density beta LP component (density
between 1.006 and 1.063 gm/ml) ,and a very lCM density prebeta LP
component (density < 1.006 grn/ml). It should be noted that the latter
component also includes the chylomicrons (density < 0.95 gm/ml), but,
for convenience, it will be called the prebeta LP component. The
quoted upper limits of nonna1 for the above measurements are: Sperry
cholesterol 210 mgt, Zak cholesterol 240 mgt, alpha LP 90 mgt, beta LP
40
140 mg%, and prebeta LP 30 mg% (Wilkinson, Hand, and Fliegelman 1948;
Epstein, Block, Hand and Francis 1959).
Let xl ,x2"" ,xn be a sample of n observations. In addition
to the mean and the standard error of the mean, two summary statistics
which measure departures from normality can be computed (Snedecor and
Cochran 1967).
The first of these statistics is the coefficient of skewness,
denoted by v'Ol or gl' with a positive value indicating that the
distribution is skewed to the right (toward the higher values) and a
negative value for distributions skewed to the left. Let
mp =
nI
i=l(x.-X)p/n
1
thbe the p sample moment about the mean. Then the skewness can be
computed from
[2.1]
!f the x I s are a sample from a normal population, gl is approximately
normal with mean zero and variance 6/n. For sample sizes less than
200, significance levels of &1 are tabulated (for example, the
Biometrika Tables, Pearson and Hartley 1954).
The other statistic is the kurtosis, denoted by g2 or bZ-3,
which can be computed from
[2.2]
The ratio m4/m~ has value 3 for the nomal distribution. Therefore
positive values of &2 indicate that the distribution is more peaked
41
than the nonnal distribution and negative values result from distri
butions that have a flatter top than the nonnal. For samples from the
normal distribution, g2 is asymptotically normal with mean a and
variance 24/n. Since the distribution of g2 approaches normality
very slady, significance levels can be found in tables, like the
Biometrika Tables.
The summary statistics for the variables are shown in Table 2.2.
All the variables except for weight and diastolic blood pressure had
distributions, on the original scale of measurement, with significant
skewness. Since the later genetic analyses will assume normality for
the variables, it is important to transform them in a way to signifi
cantly reduce, if not to eliminate, the skewness. A commonly used
transformation for this is the logarithmic one. Table 2.2 shows that
by making the logarithmic transformation, the skewness for some vari
ables (diastolic blood pressure, phospholipid, uric acid, alpha LP and
prebeta LP) disappeared, while for others (systolic blood pressure,
Sperry and Zak cholesterol, cholesterol ester, and beta LP), although
skewness did not disappear, it was at least reduced. Skewness and
kurtosis will again be considered in section 2.2.3 ; since many of the
traits have significant age effects, it is more valid to look at the
distribution of the traits after they have been adjusted for the effects
of age. Figure 2.2 displays the cumulative plots for several traits
(systolic blood pressure, Sperry cholesterol, height, and weight) both
on the original measurement scale and after the logarithmic transfor
mation.
42
Table 2.2 Summary Statistics for Original and NaturalLogarithmic-Transformed Variables
Number of StandardVariable Scale Persons i\1ean Error Skewness Kurtosis
**Age Orig. 235 23.8 1.2 0.64 -0.48** **Blood Pressure Orig 188 121.3 1.4 0.93 1. 53
(Systolic) Log 4.79 0.01 0.39** 0.27*Blood Pressure Orig. 188 78.1 0.9 0.38 0.41
(Diastolic) Log 4.34 0.01 -0.14 0.08** **Cholesterol Orig. 200 204.5 5.1 1. 64 2.79
(Sperry) Log 5.27 0.02 0.78** 0.46** **Cholesterol Orig. 155 240.4 6.0 1.39 1.45
(Zak) Log 5.44 0.02 0.76** 0.13**Weight Orig. 222 114.0 3.7 -0.02 -1.14
Log 4.58 0.04 -0.70** -0.71****Height Orig. 210 59.1 0.7 -0.78 -0.30
Log 4.06 0.01 -1.11** 0.48**Hemoglobin Orig. 196 13.3 0.1 -0.62 -0.39
Log 2.57 0.01 -1. 02** 1.00**** **Cholesterol Orig. 199 151.1 3.8 1.67 2.98
Ester Log 4.97 0.02 0.75** 0.55**
Phospholipid Orig. 196 300.8 3.0 0.50 0.19Log 5.70 0.01 0.13 -0.24
Uric Acid **Orig. 189 3.4 0.1 0.87 0.36Log 1.18 0.03 0.12 -0.58
** *Alpha LP Orig. 152 74.2 0.8 0.73 0.86Log 4.30 0.01 0.30 0.30
** **Beta LP Orig. 152 137.1 5.4 1. 53 1. 73
Log 4.82 0.03 0.62** -0.01** **
Prebeta LP Orig. 153 21.4 1.1 1. 78 5.17Log 2.89 0.05 -0.35 0.78*
Significance levels are indicated by
0.01 < P < 0.05
** P < 0.01
43
.* ..-MALe .1
1'1I·~
.. r
5.25
5.25
5.25
5.05
15.05
5.0154.86
4.85
4.85
LN (SBP)
4.1515
4.65
4.615
/(.,..........
TOTAL
,JI
..... 1'*
... ;.. ..
FEMALE .1
{tI'
.~
,Jo4.45
1.0
0.5
o4.45
1.0
0.5
0.5
190
o190 4.45
1.0... .
165 190
1615
140
140
115
1115
•• .. -MALE .1
.IrJ
,. r
90 1115 140 165
SYSTOLIC BLOOD PReSSURe (mmHg)
o90
0.5
TOTAL
1.01.-------.,...-----,.---...,...-""4""'..............,....
90
1.0
zo~
:lal
a:IUI
is 05w .>I-«~
:l~:lU
Figure 2.2 Cumulativ~ Plot by Sex
a. Systolic Blood Pressure and Ln(SBP)
44
1.0,.------r-----""'"!"'---_........;to;
0.5
0.5
1.0,.------r------.----:_::-eo--t;. .-
~
0.5
0.5
0 095 225 355 485 4.6 5.1 5.6 6.2
1.0 ... 1.0lII'-~•• •• ,0;" i
.;*'z MALE f MALE
S!I-:>
I1ZI
a:I-Ul
00.5wO. S
>I-
j"et..J:>~:>0
~;+0
225 330 435 4.75 5.20 5.70 6.10.. ..,. .. • 1.0+.. ~ •.#' .-
FEMALE ,* FEMALE ,10/,10
5.1 5.6
LN (SPERRY)
6.2
Figure 2.2 b. Sperry Cholesterol and Ln (Sperry)
4S
TOTAL
1.0r-------.-----....,.----:~
TOTAL
.5 .5
(/ ;~
,I'
,/
" 0/.
20 95 170 245 3.10 3.90 4.70 5.50
WEIGHT IN POUNDS LN (WEIGHT)
1.0r-"----..,....----~--~--....
·5 .5
//170 245 3.90 4.70 5.50
I/O'1.0
MALE MALEZ0~
::JCD
a::~enis .5 ...l .5w> .+ ,. .;v~ .,;';c( /*..J::J
y) ~+++I~:JU
#'•025 95 165 235 3.20 3.95 4.70 5.45
1.0
/1.0
FEMALE FEMALE
Figure 2.2 c. Weight and Ln (Weight)
46
TOTAL
.5
1.0r----..,...-----T"--~........
TOTAL
.5
1.0 ~----r------'----_",,-+
,./ ,./; .;
°30.0 45.0 60.0 75.0 03.4 3.7 4.0 4.31.0 1.0
Z MALE MALE
2I-:::lal
II:I-!!!0w .5 .5>l-e(
~+ .",...J:::l )1 )1:t:::l(J
,; ~.;
•• •• ~
030.0 45.0 60.0 75.0 °3.4 3.7 4.0 4.3
1.0 1.0 ,FEMALE FEMALE
o~~---~--- ~----=,2.0 44.5 58.0 88.0
MelGMT IN INeMES
.5
",. .+•
,+.'•• t
/
.5
~ ++•3.75 4.00
LN (MeIOMT)
4.25
Figure 2.2 d. Height and Ln(Height)
47
2.2.1 Sex Differences
As was mentioned before, there is no significant difference in
the age distributions or the mean age between sexes. Table 2.3 shows
the means and standard errors of the variables by sex.
With regards to the four main variables of interest, there are
no significant sex differences in the means for systolic or diastolic
blood pressure levels nor for Sperry or Zak cholesterol levels. In
addition, no significant sex differences are found in the means for
cholesterol ester, phospholipid, or beta lipoprotein.
However, the males were significantly taller and heavier, had
higher hemoglobin, uric acid. and prebetalipoprotein levels, and lower
alpha lipoprotein levels than the females.
2.2.2 Relationships with Age
Table 2.4 shows the relat ionships of the variables wi th age and
sex. Due to the small number of cases, especially in the higher age
groups, it will be difficult to make valid comparisons of the trends
by sex, but certain trends are evident.
For each sex, systolic blood pressure increases with age. In
general, the data are consistent with the results of other studies in
that the males have higher systolic blood pressures than the females
until about age 50, when the curves cross and after that the females
have higher blood pressures. For the females, the relationship with
age is fairly linear, while in males, the increase is not as steep
after age 30 as before.
For both males and females, there is an increase of diastolic
blood pressure with age; the increase in females is linear but the
diastolic pressure in males seems to peak at age group 50-59 and is
48
Table 2.3 Mean and Standard Error of Logari thmic-Transformed Variables by Sex
Number of StandardVariable Sex Persons Mean Error Test
Age M III 24.55 1. 78(Original) F 124 23.19 1. 58 NS
Blood Pressure M 90 4.79 0.02(Systolic) F 98 4.78 0.02 NS
Blood Pressure M 90 4.35 0.02(Diastolic) F 98 4.34 0.02 NS
Cholesterol M 93 5.29 0.03(Sperry) F 107 5~26 0.03 NS
Cholesterol M 70 5.48 0.03(Zak) F 85 5.41 0.03 NS
Weight M 103 4.67 0.06F 119 4.51 0.05 *
Height ~1 99 4.10 0.02F 111 4.03 0.02 *
Hemoglobin M 90 2.62 0.02F 106 2.53 0.02 **
O1.olesterol M 92 4.98 0.03Ester F 107 4.95 0.03 NS
Phospho- M 92 5.72 0.01lipid F 104 5.68 0.01 NS
Uric Acid M 86 1. 26 0.04F 103 1.10 0.03 **
Alpha LP ~1 69 4.28 0.02F 83 4.32 0.01 *
Beta LP M 69 4.89 0.05F 83 4.77 0.04 NS
Prebeta LP M 69 3.05 0.07F 84 2.76 0.07 **
Significance levels are indicated by:
NS - Not significant * 0.01 < P < 0.05 ** P < 0.01
49
Table 2.4 Mean and Standard Error of Logarithmic-TransformedVariables by Age and Sex
Systolic BPSexes Pooled Males Females
Age No. Mean SE No. Mean SE No. Mean SE
< 10 26 4.60 .02 14 4.62 .02 12 4.57 .02
10 - 19 47 4.70 .02 18 4.66 .02 29 4.72 .02
20 - 29 18 4.83 .02 8 4.83 .03 10 4.82 .02
30 - 39 52 4.83 .02 24 4.87 .03 28 4.80 .02
40 - 49 24 4.85 .02 15 4.86 .03 9 4.85 .02
50 - 59 8 4.97 .06 5 4.93 .05 3 5.04 .13
60+ 13 5.01 .04 6 4.94 .05 7 5.07 .05
Diastolic BP
Age
< 10 26 4.14 .02 14 4.15 .03 12 4.14 .02
10 - 19 47 4.24 .02 18 4.23 .03 29 4.25 .02
20 - 29 18 4.39 .01 8 4.39 .03 10 4.39 .01
30 - 39 52 4.42 .02 24 4.44 .02 28 4.40 .02
40 - 49 24 4.44 .02 15 4.44 .03 9 4.44 .03
50 - 59 8 4.52 .04 5 4.51 .05 3 4.53 .09
60+ 13 4.49 .04 6 4.43 .07 7 4.54 .05
50
SperrySexes Pooled Males Females
Age ~o. Mean SE No. Mean SE . No. t-lean _SE
< 10 38 5.17 .04 18 5.16 .04 20 5.18 .06
10 - 19 47 5.10 .04 17 5.05 .08 30 5.13 .05
20 - 29 18 5.32 .07 8 5.44 .14 10 5.22 .05
30 - 39 52 5.33 .04 24 5.36 .04 28 5.31 .06
40 - 49 24 5.41 .06 15 5.43 .08 9 5.39 .10
50 - 59 8 5.57 .14 5 5.57 .19 3 5.58 .23
60+ 13 5.40 .08 6 5.25 .06 7 5.52 .11
Zak
Age
< 10 23 5.34 .04 9 5.36 .06 14 5.32 .06
10 - 19 39 5.27 .04 13 5.27 .08 26 5.27 .04
20 - 29 13 5.47 .07 6 5.59 .13 7 5.37 .07
30 - 39 47 5.50 .03 23 5.54 .04 24 5.47 .06
40 - 49 15 5.62 .08 10 5.61 .11 5 5.64 .14
50 - 59 7 5.72 .13 4 5.74 .19 3 5.70 .20
60 + 11 5.54 .07 5 5.41 .05 6 5.66 .11
Weight
Age
< 10 65 3.79 .04 30 3. 79 .07 35 3.79 .05
10 - 19 46 4.57 .04 17 4.61 .06 29 4.55 .05
20 - 29 17 4.98 .04 7 5.12 .04 10 4.88 .04
30 - 39 50 5.07 .03 24 5.21 .02 26 4.95 .04
40 - 49 23 5.05 .03 14 5.10 .04 9 4.96 .04
50 - 59 8 5.10 .06 5 5.19 .04 3 4.94 .05
60+ 13 5.08 .04 6 5.16 .05 7 5.01 .04
51
HeightSexes Pooled Males Females
Age No. Mean SE No. Mean SE No. Mean SE
< 10 57 3.80 .02 26 3.81 .03 31 3.80 .02
10 - 19 46 4.09 .01 17 4.10 .02 29 4.08 .01
20 - 29 17 4.19 .01 7 4.24 .02 10 4.15 .01
30 - 39 49 4.19 .01 24 4.23 .01 25 4.14 .01
40 - 49 21 4.21 .01 14 4.24 .01 7 4.15 .01
50 - 59 7 4.21 .02 5 4.22 .02 2 4.18 .01
60+ 13 4.16 .01 6 4.21 .01 7 4.13 .01
Phospholipid
Age
< 10 34 5.67 .02 17 5.68 .02 17 5.66 .04
10 - 19 47 5.66 .02 17 5.71 .03 30 5.64 .02
20 - 29 18 5.71 .03 8 5.76 .04 10 5.67 .04
30 - 39 52 5.70 .01 24 5.69 .02 28 S.71 .02
40 - 49 24 5.71 .04 15 5.77 .04 9 5.62 .05
50 - 59 8 5.86 .05 5 5.83 .08 3 5.90 .03
60+ 13 5.72 .05 6 5.67 .07 7 5.76 .07
Uric Acid
~
< 10 36 1. 00 .05 16 1.07 .08 20 0.95 .07
10 - 19 46 1.12 .05 17 1.13 .09 29 1.11 .06
20 - 29 18 1.26 .08 8 1.37 .11 10 1.18 .12
30 - 39 52 1. 21 .05 24 1.34 .07 28 1.10 .06
40 - 49 20 1.38 .08 12 1.43 .10 8 1. 31 .12
50 - S9 6 1. 20 .12 4 1. 32 .13 2 0.97 .13
60+ 11 1.30 .11 5 1. 41 .10 6 1. 22 .18
52
HemoglobinSexes Pooled ~1ales Females
Age i~O. Mean SE No . Mean SE No. Mean SE
< 10 39 2.53 . 02 19 2.52 .04 20 2.54 .03
10 - 19 47 2.52 .03 17 2.62 .04 30 2.47 .04
20 - 29 18 2.69 .02 8 2.70 .02 10 2.68 .03
30 - 39 49 2.60 .02 22 2.70 .01 27 2.52 .03
40 - 49 24 2.59 .03 15 2.59 .04 9 2.60 .05
50 - 59 7 2.57 .06 4 2.59 .10 3 2.54 .09
60+ 12 2.58 .04 5 2.67 .04 7 2.51 .OS
Alpha LP
Age
< 10 23 4.29 .03 9 4.31 .05 14 4.28 .04
10 - 19 39 4.31 .02 13 4.28 .05 26 4.32 .02
20 - 29 13 4.32 .03 6 4.34 .04 7 4.32 .04
30 - 39 44 4.28 .02 22 4.25 .02 22 4.32 .03
40 - 49 15 4.34 .03 10 4.33 .03 5 4.36 .06
50 - 59 7 4.26 .02 4 4.24 .02 3 4.29 .04
60+ 11 4.27 .04 5 4.16 .05 6 4.36 .04
Prebeta LP
Age
< 10 23 2.68 .11 9 2.77 .14 14 2.62 .16
10 - 19 39 2.58 .09 13 2.61 .14 26 2.57 .12
20 - 29 13 2.98 .13 6 3.01 .11 7 2.95 .24
30 - 39 45 2.95 .09 22 3.22 .11 23 2.70 .12
40 - 49 15 3.13 .17 10 3.24 .22 5 2.92 .28
50 - 59 7 3.41 .12 4 3.40 .16 3 3.43 .22
60+ 11 3.38 .15 5 3.38 .33 6 3.37 .12
53
lower after that. The curves are almost identical tD'ltil about age 50,
and for higher ages the females have higher diastolic pressure.
However, there are relatively small numbers of individuals in ~le
older ages.
For each sex and for both Sperry and Zak cholesterol, there are
increases with age. Comparing the sexes, for the first two decades of
life the cholesterol levels are very similar in the two sexes; for ages
20 to about 50, the male cholesterol levels are higher, after which
the curves cross, and the females have the higher cholesterol levels.
The relations between cholesterol ester and beta lipoprotein with age
are the same as for cholesterol, which underscores the fact that all
these variables are correlated with each other.
For the first twenty or thirty years of life, there is a steep
increase in both height and weight for both males and females. After
age 30, the heights plateau for both males and females; the rate of
increase in weight decreases for females, and in males, weight
continues to increase tD'ltil about age 40 and then plateaus or perhaps
decreases. After age 20, the males are taller and heavier than the
females.
There is no consistent trend of hemoglobin with age for females.
In males, hemoglobin levels increase tD'ltil age 30, plateau tD'ltil age
40, and then decrease slowly. There is a general tendency for the
hemoglobin levels in males to be higher than those in females.
There is no consistent age effect for phospholipids in males, and
there is a weak linear increase in females. There is also a weak
linear increase of uric acid levels with age in females; however for
males, uric acid levels increase until age 50, after which they seem
S4
to decrease a little. At all age levels, the males have higher uric
acid levels than the females.
Although there is a general tendency for females to have higher
alpha lipoprotein levels, neither males nor females show any trend of
alpha lipoprotein with age. On the other hand, prebeta lipoprotein
levels increase with age in both males and females, with the males
having the higher levels at all ages. The level in males shows a
general increase for almost the entire age span, whereas the level
in females remains relatively constant until about age 20 before
increasing with age afterwards.
In order to quantify the relationships between these variables and
age, and also to have a method available subsequently for adjusting
for age, a regression analysis was done. Let YI'Y2""'Yn and
xl ,x2, ... ,xn denote tile value of the trait and the age, respectively,
for n individuals. The regression equation of Y on X can be
written
[2.3]
where
Y. is the logarithm of the variable for .th individual,11
So is a constant,
X. is the age of the .th individual,11
e. is random variation, assumed distributed NCO, (12),1
131 is the linear regression coefficient of age, and
62 is the quadratic regression coefficient of age.
A least square procedure which minimizes Ley i - 80is used to estimate the regression coefficients 80 , 81
ss
" ,,2 2- 81Xl - 62Xi )
and 82 and
their standard errors. First HO: 61 = 0, 62 = 0 is tested. If
either 61 or 62 is not significantly greater than zero, then that
term is removed fram the regression equation, and then the regression
estimates and the standard errors are computed anew and tested. Table
2.5 gives the linear and quadratic regression coefficients for each
trait by sex. If a particular coefficient is missing, this is to
indicate that that particular coefficient is not significantly different
from zero.
For most traits, there are no sex differences in the degree of
the age relationship. However, there are a few differences;. for
systolic and for diastolic blood pressure, the linear and quadratic
age effects are significant in males while only the linear effect is
significant in females. There is no age effect for hemoglobin levels
in females, but both linear and quadratic effects are significant
in males. The most striking difference is for prebeta LP; for the
males, only the linear effect is significant, and for the females,
only the quadratic effect is significant. For alpha LP, there is no
age effect in either sex.
The total sum of squares about the mean, Ss.T' can be partitioned
into two portions:
sSr [2.4]
where Y is the mean of the Y's
The second term on the right handregression for the .th1 person.
and y.1
is the value of the fitted
56
Table 2.5 Linear and Quadratic Regression Coefficients ofAge for Logarithmic-Transformed Variables by Sex
Variable Sex Linear Quadratic Constant R2(%)
'IeSystolic BP M .01239 -.0001 4.538 48.47
F .00703 4.585 57.66
Diastolic BP M .01720 -.00017 4.036 50.14F .00666 4.153 49.77
Sperry Cholest M .00602 5.113 11. 85F .00651 5.087 13.47
Zale Cholest M .00537 5.319 10.38F .00718 5.216 20.98
Neight ~1 .08117 -.00090 3.486 88.48F .06810 -.00073 3.558 81.64
Height M .026S3 -.00030 3.706 81.82F .02220 -.00026 3.737 72.24
illHemoglobin M .00794 -.00010 2.502 9.66
F .NO AGE EFFECT 2.530
Cholesterol r.1 .00550 4.822 9.91Ester F .00645 4.786 12.79
Phospholipid M i~O AGE EFFECT 5.720F .00186* 5.629 5.03
Uric Acid M .00731 1. 059 12.74F .00413* 0.997 4.05
Alpha LP M ~O AGE EFFECT 4.28F ~O AGE EFFECT 4.32
Beta LP M .00712* 4.668 7.88F .00950 4.526 16.46
Prebeta LP M .01514 2.588 19.98F .00018 2.577 12.41
Note: All linear and quadratic regression coefficient estimates
are significant at the 1% level, with the exception ofthose marked with 'Ie which are significant at the 5%level.
57
side of equation 2.4 is the sum of squares of the deviations of the
points Y on the fitted line from their mean. This quantity is denoted
by SSR' the sum of squares attributable to regression. A statistic
R2 can be computed using the formula
2R = SSR/SST [2.5]
R2 indicates how much of the total variation in Y can be accounted
for by the fitted regression. In the case of traits and ages, if R2
is large, this indicates that much of the variation of the trait can
be explained by age.
Age accounts for 80-90% and 70-80% of the total variation for
weight and height, respectively. For systolic and diastolic blood
pressure, R2 is 48-58%; for Sperry or Zak cholesterol, cllo1estero1
ester, and beta lipoprotein, age explains 8-20% of the total variation.
Except for prebeta LP (R2 is 12-20%), the other remaining traits
have low R2.
2.2.3 Skewness and Kurtosis
At the beginning of section 2.2, there was a brief discussion of
the necessity for finding a transformation which made the distribution
of the trait more like the normal distribution. It was noted that the
logarithmic transformation was one commonly use for this purpose.
Furthermore, two statistics, skewness (gl) and kurtosis (g2) , could be
used to measure departures from normality (equation 2.1 and 2.2) .
In section 2.2.2, it was shown that many of the traits have
significant age effects. The question arises: what, if the traits
are adjusted for age, would the distributions look like? Would they
58
be more like the normal distribution? Consequently, the traits were'" A "2
age adjusted by computing the residuals, Yi - 60 - 61Xi - 82XiA A A
where 60, 61, and 62 are the regression coefficients tabulated
in Table 2.5.
The skewness and kurtosis of the distribution of the resulting
residuals are exhibited in Table 2.6. For every trait except
hemoglobin, the skewness either disappears or is at least reduced.
After taking logari thIns and age adj usting, only hemoglobin and prebeta
lipoprotein still have significant kurtosis. Thus, except for
hemoglobin, doing a logarithmic transformation causes the distribution
to become more like the normal distribution. For the exception
hemoglobin, the distribution of the original values looks unusual:
Hemoglobin Numbe r 0f cases %
< 9 2 1.0
9 - 9.9 10 5.1
10 - 10.9 10 5.1
11 - 11.9 23 11.7
12 - 12.9 4 2.0
13 - 13.9 41 20.9
14 - 14.9 44 22.4
15 - 15.9 16 8.2
16 - 16.9 46 23.5
Total 196 100.0
After trying several transformations, it was f01.IDd that the square
transformation reduced the skewness froTIl -0.62 to -0.35, the
latter value being significant at the 5% level.
59
Table 2.6 Skewness and Kurtosis for Original and Age-Adjusted Logarithmic-Transformed Variables
. Number ofTrait Scale Individuals Skewness Kurtosis
** **Blood Pressure Original 188 0.93 1. 53(Systolic) Log(age-adj.) 0.59** 0.56
*Blood Pressure Original 188 0.38 0.41(Diastolic) Log(age-adj.) 0.07 0.26
** **Cholesterol Original 200 1.64 2.79(Sperry) Log(age-adj.) 0.84** 0.57
** **Cholesterol Original 155 1. 39 1.45(Zak) Log(age-adj.) 0.82** 0.21
**Weight Original 222 -0.02 -1.14Log(age-adj.) 0.23 -0.26
**Height Original 210 -0.78 -0.30Log(age-adj.) -0.19 0.51
**Hemoglobin Original 196 -0.62 -0.39Log(age-adj.) -1. 01** 1.18**
** **Cholesterol Original 199 1.67 2.98Ester Log(age-adj.) 0.83** 0.63
**Phospholipid Original 196 0.50 0.19Log(age-adj.) 0.11 -0.35
**Uric Original 189 0.87 0.36Acid Log(age-adj .) 0.11 -0.49
** *Alpha LP Original 152 0.73 0.86Log(age-adj.) 0.35 0.40
** **Beta LP Original 152 1. 53 1. 73Log(age-adj.) 0.69** 0.03
** **Prebeta LP Original 153 1. 78 5.17Log(age-adj.) -0.47* 1.20**
Test results are lndicated by
Not significant* 0.01 < P < 0.05** P < 0.01
60
Z.Z.4 Inter-trait Correlations
Assuming that two traits, Xl and XZ' have a bivariate nonnal
distribution, a measure of the linear relationship between them is
the correlation coefficient
[Z.6J
with the sample est~late r l2 calculated from
[Z.7]
A test of HO: p(XI , XZ) = 0 vs. ~II: p(XI , XZ) ~ 0 can be done by
computing r lZ and referring to standard tables. An alternative pro
vided by Fisher is to transfonn r lZ to a quantity Z
and Z is distributed almost N(~ in i~~
[2.8]
-!-). n is the sample size.n-3 '
Suppose there are three variables Xl' XZ' and X3, and it was
required to compute the correlation between Xl and Xz in a cross
section of individuals all having the same value of variable X3• A
partial correlation coefficient P12. 3 measures the part of the correla
tion between Xl and Xz that is not simply due to their relationships with
X3• The sample estimate can be computed fromr 12 - r 13 r 23r = [2.9]
12.3 L 2 2Y'(1-r13 ) (1-r23 )
This can be referred to standard tables. An alternative way to compute
61
r12 . 3 is to regress Xl on X3 and X2 on X3, and then compute
the correlation coefficient on the residuals (Xl - Xl) and (XZ - X2).
The two methods lead to identical results.
If there are four variables Xl' X2, X3, and X4, PlZ.34
measures the correlation between Xr and Xz while holding X3 and
X4 at fixed levels. The sample estimate can be calculated
r 1Z . 34 =2 2
1(1 - r14 .3 )(1 - r Z4 .3 )
[2.10]
This can be referred to standard tables (Snedecor and Cochran 1967).
Tables 2.7 and 2.8 display the correlation coefficients for 78
pairs of traits, and many significant correlations are noted. However
a significant correlation may not mean that two variables are directly
related to one another. As section 2.2.2 showed, many of the variables
are related to age so that perhaps the significant correlation between
traits is due to each trait being correlated with age. Partial
correlation coefficients are computed, adjusting for age and ageZ.
Sperry and Zak cholesterol values are highly correlated (0.94)
which is not W1expected since the two laboratory procedures are
different measurements of the same thing. The two are significantly
correlated with both systolic blood pressure (0.30, 0.31 respectively)
and diastolic blood pressure (0.22, 0.23), but not to either weight
or height; any correlation between cholesterol and weight or height
disappears after adjusting for age. Sperry and Zak cholesterol are
highly correlated with cholesterol ester (0.99, 0.92), phospholipid
(0.31,0.31), beta-lipoprotein (0.92,0.96), and prebeta lipoprotein
(0.28,0.32). The high correlation with beta lipoprotein is also to
62
Table 2.7 Correlation and Partial Correlation CoefficientsBetween Logaritlunic-Transformed VariablesInvolving Systolic and Diastolic Blood PressuresmId Sperry and Zak Cholesterol
Partial CorrelationVariables H Correlation Partialling Age, Age2
Systolic BP andDiastolic BP 188 .80** .63**Sperry Cholesterol 181 .45** .30**Zak Cholesterol 145 .49** .31**Weight 181 .65** .24**Height 177 .59** .18**Hemoglobin 177 .14* .04Cholesterol Ester 181 .44** .30**Phospholipid 180 .16* .04uric Acid 172 .29** .12Alpha LP 142 -.01 .02Beta LP 142 .45** .29**Prebeta LP 143 .42** .21**
Diastolic BP andSperry Cholesterol 181 .40** .22**Zak Cholesterol 145 .44** .23**Weight 181 .66** .17*Height 177 .59** .07Hemoglobin 177 .19** .10Cholesterol Ester 181 .39** .23**Phospholipid 180 .20** .10Uric Acid 172 .27** .08Alpha LP 142 .01 .04Beta LP 142 .38** .18**Prebeta LP 143 .38** .18**
Sperry Cholesterol andZak Cholesterol 15S .94** .94**Neight 192 .31** .01Height 187 .23** -.10Hemoglobin 193 .15* .10Cholesterol Ester 199 .99** .99**Phospholipid 196 .35** .31**Uric Acid 189 .11 .00Alpha LP 152 .04 .06Beta LP 152 .93** .92**Prebeta LP 153 .38** .28**
Table 2. 7 Con 't
63
Partial Correlation
Variables N Correlation Partialling Age, Age2
Zak Cholesterol andWeight 149 .35** -.02Height 146 .25** -.17*Hemoglobin 150 .14* .08Cholesterol Ester 155 .92** .91**Phospholipid 155 .35** .31**Uric Acid ISO .13 .00Alpha LP 152 .04 .06Beta LP 152 .97** .96**Prebeta.LP 153 .42** .32**
Significance levels indicated by
** P < 0.01
* 0.01 < P < 0.05
64
Table 2.8 Correlation and Partial Correlation CoefficientsBetween Logarithmic-Transformed Variables
\'11' hi HE~D Q-:lOE PHOS UA ALP BETA PREB
WT .95** .24** .31** .14* .35** -.05 .31** .31**
HT .82** .23** .23** .10 .35** -.01 .24** .23**
HEM) .17* .12 .14* .14* .03 -.01 .15* .12
CHOE .04 -.07 .09 .33** .12 .08 .91** .35**
PHOS -.02 -.07 .12 .29** - .13* -.01 .34** .15*
UA .19** .20** -.03 .02 -.20** -.11 .13 .14*
ALP -.n -.01 -.01 .09 .00 - .10 -.02 - .19**
BETA -.02 - .11 .09 .90** .30** .02 -.01 .29**
PREB .01 -.08 .07 .25** .08 .04 -.19**.18*
Number of pairs in each cell range from 143 to 209 .
Common Correlation Coefficients are Above the Diagonal.
Partial Correlation Coefficients (Partialling Age, Age 2) are
Be low the Diagonal .
\IT =Weight
HT = Height
HEM) = Hemoglobin
CHOE = Cholesterol Ester
PHOS = Phospholipid
UA = Uric Acid
ALP = Alpha LP
BETA = Beta LP
PREB = Prebeta LP
Si~lificance levels are indicated by
** P < 0.01
* 0.01 < P < 0.05
65
be expected since beta lipoproteins transport about 75% of the
cholesterol in the serum.
Systolic and diastolic blood pressure are correlated (0.63).
Systolic blood pressure is significantly correlated with weight (0.24)
and lleight (0.18), but diastolic pressure is not. In addition to being
associated with Sperry and Zak cholesterol as mentioned above, both
systolic and diastolic pressures are correlated with cholesterol ester
(0.30 and 0.23 , respectively), beta lipoprotein (0.29, 0.18), and
prebeta lipoprotein (0.21,0.18). There is no significant correlation
with phospholipid, uric acid, hemoglobin, or alpha lipoprotein
especially after age adjusting.
Indeed, except for a weak negative correlation between alpha and
prebeta liproprotein (-0.19) and a weak positive correlation between
weight and hemoglobin (0.17), neither alpha lipoprotein nor hemoglobin
are significantly related to any of the other traits. Uric acid is
significantly correlated only with weigllt (0.19), height (0.20), and
phospholipid (- 0.20) .
Except for the correlations already mentioned, neither height nor
weight is significantly related to any other trait; in particular,
neither is correlated with any of the "cholesterol-related" traits, such
as Sperry and Zak cholesterol, cholesterol ester, phospholipid, and
beta LP.
Cholesterol :esters and beta lipoprotein are highly correlated
(0.90). i'Jeither is significantly correlated with weight, height,
hemoglobin, uric acid, and alpha LP; both are significantly correlated
with systolic and diastolic pressures, Sperry and Zak cholesterol,
phospholipid, and prebeta LP.
66
With most of the pairs of traits, there are no significant
differences in the partial correlation coefficients by sex. There are
15 pairs of traits which have heterogeneous correlations by sex. For
almost all of these pairs, the difference is not substantial; the
correlations for the two sexes have the same sign but one is significant,
and the other one is not. There is a big difference only for the
correlation between hemoglobin and alpha LP; for males, the correlation
is significant and negative (-0.21), and for females, it is significant
and positive (0.22).
This last part of the descriptive study, in addition to providing
an initial estimate of the correlations for bivariate pedigree analyses,
will be helpful in interpreting the results. For example, if a
monogenic model were to fit the cholesterol data, then the fact that
cholesterol, cllolesterol ester, and beta LP pairwise have high corre
lations may suggest the existence of pleiotropic gene effects.
2.3 Random ~~ting
In this kindred, there are 65 pairs of spouses. For the sub
sequent genetic analysis, it would be both interesting and important
to determine with respect to which traits there is random mating and
which traits there is assortative mating. Assortative mating means
that mated pairs are more alike (positive assortative mating) or more dis
similar (negative assortative mating) for some phenotypic trait than
would be expected if they were picked at random from the population.
A measure of assortative mating is the inter-spouse correlation, which
is listed for the traits in Table 2.9.
There is a high inter-spouse correlation with respect to age (0.971);
people generally marry another person of a similar age. Because of this,
Table 2.9 Inter-spouse Correlations of Age-adjustedLogarithmic-Transformed Variables
Variable Ntunber of Pairs Correlation
Age (Original Scale) 49 0.97**
Weight 46 0.3J'
Height 44 0.58**
Systolic BP 49 0.44**
Diastolic BP 49 0.22
Sperry Cholesterol 49 0.22
Zak Ololesterol 38 0.41**
Hemoglobin 44 -0.12
Ololesterol Ester 49 0.21
Phospholipid 49 0.38**
Uric Acid 41 0.55**
Alpha LP 36 0.24
Beta LP 36 0.46**
Prebeta LP 37 O.ll
67
uS
it is essential to adjust the traits for age before computing the inter
spouse correlations; otherwise spurious results may be obtained due to
the effect of age on the traits. The method of age adjustr:lent is to
compute Y. - Y. where Y1' is the fitted regress ion of the tra it Y
1 12on age and age ; these residuals are used in calculating the correlations.
There is strong assortative mating for height (0.58) and somewhat
weaker for weight (0.33). There is an unexpected high correlation
between spouses for uric acid. However, Table 2.7 has shown that uric
acid and height are correlated, and assortative mating for height has
been noted. Likewise, assortative mating exists for systolic blood
pressure, Zak cholesterol, phospholipid, and beta LP, but not for the
other variables, specifically diastolic blood pressure, Sperry
cholesterol, hemoglobin, cholesterol ester, alpha LP, and prebeta LP.
The effect of assortative mating will be discussed at the end of
Chapter IV.
QiAPTER III
reSCRIPTION OF lliE PEDIGREE
The kindred described in Chapter 2 forms a pedigree which spans
five generations. The pedigree is displayed in Figure 3.1. For the
purposes of the subsequent genetic analyses, the pedigree has been
split into two simple pedigrees each originating with a pair of parents.
The two co~onent pedigrees have been labeled the "Right Pedigree"
(Figure 3.1a) and the "Left Pedigree" (Figure 3.1b).
3.1 The Pedigree Structure
The co~onent pedigrees as displayed in Figure 3.1 include the
284 persons examined in 1947 and/or 1958, along with fifteen persons
who, although never examined, are included to complete the
structure. For example, if the parents of examined children are
omitted because they were never observed, the pedigree structure would
look bizarre and inco~lete. Persons who were never observed are
indicated in Figure 3.1 by completely darkened symbols, and those who
were examined only in 1947 are indicated by a semi-darkened symbols.
Those who are unmarked, the 235 individuals examined in 1958, will be
the subjects of the genetic analyses. Measures on several traits for
these 235 individuals are listed in Appendix 4.
A reference to a specific member of the pedigree is accomplished
using a combination of a letter, a Roman numeral, and an Arabic ntunber.
The letter, either "R" or "L", specifies whether it is the Right or
I
"III
IV
V
I ,
", ••• ,' •• "auH .... IIJ1 • • II .. " ..... ', •••• .II. ..
KEY•• NEVER OBSERVED
II () OBSERVED IN '947 ONL Y
..,1 ••• .," ........ " .... """14
III
IV
V •••,U ••••• ~ •• n •• MnnnunNnN~•• aaMM." •• ......... n •••• ...... ...
e
Figure 3.1a Right Pedigree
e e
'-Jo
).
~0
o "~ ~ct: ..
:I
~ :t'" -ClQ Qo ~~ ct:::. ~
). ~ ~
~ ee.e
::
(1)(1)!-<eo.....~(1)
0..
+oJ4-l
~
:::I ..0
~
l"")
(1)!-<::3eo.....~
71
72
Left pedigree; the Roman numeral specifies the particular generation,
and the Arabic number locates the specific person in that generation.
Thus, R IV 35 denotes the 35th person of the fourth generation in the
Right Pedigree.
The two pedigrees are each centered around a large sibship. The
Left Pedigree is centered around a sibship of ten which comprise the
third generation. The Right Pedigree is centered around a sibship of
twelve which dominates the third generation. The parents of eadl of
these sibships, L II 1-2 and R II 3-4, respectively, came to this
country from Alsace-Lorraine, first settling in Illinois before moving
to Michigan in about 1923. Of the 235 individuals examined in 1958,
only R II 3 was born outside the United States. Although the families
are closely knit, there is no consanguinity in this kindred (Epstein,
Block, et al 1959). There is one pair of dizygotic twins (L V 7 and 8)
which, for the purposes of the analyses, will be treated as ordinary
sibs. As was mentioned in Chapter 2, included in this kindred are
several large sibships; there is one with fifteen sibs (R IV 1-22),
one with twelve sibs (R III 3-23), two with ten sibs each (R IV 73-88
and L III 1-18), and one with nine sibs (R IV 55-71).
The two pedigrees can be joined at two places; there are four
persons who are COIIDnon to both pedigrees. R II I 3, 4 are the same as
L III 19, 18, and R III IS, 16 are the same as L III 8, 7. It should
be noted that the children and grandchildren of R II I 3, 4 and R I II
IS, 16 could also be included in the Left Pedigree. However, the
strategy adopted is to maximize the Right Pedigree and use it as the
main focus for the search for a maj or gene. Then the search will be
repeated on the Left Pedigree to dleck for consistency of results.
73
Finally, both pedigrees will be analyzed, treating the component
pedigrees as independent pedigrees; this should be a reasonable
approximation since the two pedigrees have only [our persons in common.
In preparation to doing genetic analysis on the pedigree, the
data set must be in a form such that the pedigree structure can be
reconstructed quickly. First, each individual in the pedigree is
given a sequence number. Then, for each individual, a record on disk
is created containing the following:
1. Sequence number of the individual.
2. Sequence number of the spouse, if any.
3. Sequence number of the next sib, if any.
4. Sequence number of the first child, if any.
5. The number of children.
6. Sequence number of the individual's father.
7. Sequence number of the individual's mother.
8. Sex
9. Age
10. Measures on traits.
3.2 Age Distribution in the Two Pedigrees
Table 3.1 shows the age distribution in the two pedigrees by sex,
and the mean ages are tabulated in Table 3.2. The tables point out that
those in the Left Pedigree are older than those in Right Pedigree. Al though
the males in the Lp.ft Pedigree are, on the average, 3.6 years older than
those in the Right Pedigree, the big difference is in the females;
those in the Left Pedigree are, on the average, almost fifteen years
older. There are no females ,in the Left Pedigree under 10 years of
age, while females in this age group constitute 35% of the females in
Table 3.1 Age Distribution by Sex and Pedigree
Right Pedigree
Males Females Total
Age No. '% No. % No. %
< 10 28 33.7 35 35.0 63 34.4
10 - 19 13 15.7 25 25.0 38 20.8
20 - 29 4 4.8 6 6.0 10 5.5
30 - 39 21 25.3 22 22.0 43 23.5
40 - 49 9 10.8 5 5.0 14 7.6
50 - 59 3 3.6 2 2.0 5 2.7
60+ 5 6.0 5 5.0 10 5.5
Total 83 100.0 100 100.0 183 100.0
Left Pedigree
Age
< 10 7 23.3 0 0.0 7 12.5
10 - 19 ,.. 16.7 7 26.9 12 21.4;)
20 - 29 3 10.0 3 11.5 6 10.7
30 - 39 4 13.3 7 26.9 11 19.6
40 - 49 6 20.0 4 15.4 10 17.8
SO - S9 3 10.0 1 3.8 4 7.1
60+ 2 6.7 4 15.4 6 10.7
Total 30 100.0 26 100.0 56 100.0
74
Table 3.2 Mean Age by Sex and Pedigree
TestRight Left Pedigree
Sex Pedigree Pedigree Differences
No. 83 30
Males Mean 24.3 27.9 NS
SE 2.1 3.7
75
Females
Pooled
TestSexDifferences
No.
Mean
SE
No.
Mean
SE
100
20.8
1.7
183
22.3
1.3
NS
26
35.6
3.4
56
31.5
2.6
NS
**
**
Significance levels are indicated by
NS Not significant
** p < 0.01
76
the Right Pedigree. These results LUlderscore the importance of
adj usting for age.
Not everyone had measurements made for all the traits. Table 3.3
shows the availability of data for the four main traits along with
height and weight.
3.3 Fitting a Mixture of Normal Distributions
Prior to the pedigree analysis, and to obtain initial estimates
for it, analyses were done to determine if a mixture of two or three
normal distributions would fit the data for each of the four main
traits significantly better than one distribution. In addition, for
certain pairs of traits, mixtures of bivariate normal distributions
were fitted.
If a mixture of two or three distributions were to fit the data
significantly better than one distribution, it would be attractive to
think of each distribution as corresponding to the distribution of
phenotypes for a specific genotype. In fact, during the pedigree
analysis, we assume that the conditional probability density flIDction2(pdf) of observing phenotype x, given genotype u, is N(].1u' 0 ) ,
where the initial estimates of].1u and 02 are obtained from the curve
fitting in this section.I
Analogous to sections 2.2.3 and 2.2.4, let Y. be the measure1
of the trait (in the original scale) for the i th individual. Then,
the age-adjusted natural logarithmic transformed trait value for the
i th individual in the pedigree is
I
Y. = .en Y.1 1
i = I, ... ,n
77
Table 3.3 Availability of Data for Six Traits by Sexand Pedigree
Pedigree
Left Right BothM F Total M F Total M F Total
Observedin 1958 30 26 - S6 83 100 183 ·111 124 235
BloodPressure
SperryCholesterol
ZakCholesterol
Weight
Height
23 25
24 26
18 22
26 23
25 23
48
50
40
49
48
69 75 144
71 83 154
54 65 119
79 98 177
76 90 166
90 98 188
93 107 200
70 85 ISS
103 119 222
99 III 210
78
where X. is the age of the i th individual and1
81, 82 are the appropriate estimated regression
coefficients from Table 2.5.
The curve-fitting will be performed on the Y..1
3.3.1 Univariate Log-Normal Distributions
Ignoring the genetic relationships, Y., (i = 1,2, ... ,n), can be1
assumed to be independent and identically distributed random variables
with p.d.f. g(y). The p.d.f. of a mixture of k distributions from
a conmon family of distributions f(y;!V can be written as
g(y) k= '. 1 a.f(y; e.)l.J= J .....J[3.1]
The admixture proportions a., j = 1,2, ... , k-lJ
j = 1,2, .•. ,k will be estimated using thee. ,.....J
maximum likelihood procedure. There is no difficulty in allowing
kwhere '. 1 a. = 1.l.J= J
and the parameters
g(y) to be a mixture of normal distributions so that
12 -1 ( 2 2)fey; ~j)= (2no ) exp -(y - ~j) /20 .
A cormnon variance 02 will be used rather than a different variance
for each component distribution; this is because assuming different
variances leads to singularities on the likelihood surface, e.g. the
estimation procedure may collapse one of the components to a single
observation, resulting in the variance for that component being zero.
For a mixture of k normals, the likelihood is written:
nL = IT
i=l
79
[3.2]
The likelihood is written as a fWlction of 2k fWlctionally inde
pendent parameters: k means, (k-l) independent admixture
proportions, and a conunon variance. These 2k parameters will be
simultaneously estimated using a maximum likelihood subroutine package
~IK devised by Kaplan and Elston (1972). We are interested in
testing HO: data consist of k component distributions versus HI:
k' components (k' > k). The test will be based on the likelihood
ratio criterion. Wolfe (1971) investigated the distribution of
-2 in A where A = y~, and where ~ and Lk , are the likeli
hoods Wlder HO and HI' respectively. After discovering that the
distribution of -2 in A did not sufficiently match the usual X2
approximation, his Monte Carlo investigations suggest that - 2 C
in(Lk/~') is approximately distributed as a x2 with 2m(k' - k)
degrees of freedom, where
C = [n - I - (~k')/2]/n,
m = number of variables,
k' = number of components Wlder HI' and
n = sample size.
It should be noted that, for any reasonably large n, C will be near
1 in value; in the present study, C ranges from 0.936 to 0.975. The
effect of letting C = I will be an anti-conservative test, i.e. the
actual significance level is larger than the nominal one.
For k components, the parameters are (~l""'~k' 0, al , ••• ,
ak-l) which will be estimated using ML procedures.
80
3.3.1.1 Systolic Blood Pressure
The maximum likelihood estimates (MLE) of the parameters for a
mixture of k univariate log-normal distributions for systolic blood
pressure are given in Table 3.4a for males and Table 3.4b for females.
Figure 3.Za and 3.Zb show the empirical and theoretical cumulative
plots for males and for females, respectively.
To test HO: k = 1 vs HI: k' = 2 or HO: k = Z vs HI: k'= 3,
-ZC in Ll/LZ and -ZC in LZ/L3, respectively, should be compared
to a X2 with 2 d.f.
For males, neither a mixture of two nor a mixture of three dis-
tributions fits the systolic blood pressure data better than one
distribution while for females, a mixture of two distributions fits the
data significantly better (0.01 < P < 0.05) than only one distribu
tion. Almost 19% of the females are in the higher distribution.
3.3.1.2 Diastolic Blood Pressure
The MLE of the parameters are given in Table 3.4c for males and
Table 3Ad for females. Figures 3. 2c and 3. 2d show the empirical and
the theoretical cumulative plots of diastolic blood pressure for males
and for females, respectively. For neither the males nor the females
is the hypothesis of one log-normal distribution fitting the diastolic
blood pressure data rejected; a mixture of two distributions does not
fit significantly better than one alone.
3.3.1.3 Sperry Ololesterol
The MLE of the parameters are given in Table 3.4e and Table 3.4f
while the cumulative plots of Sperry cholesterol levels for males and
females are shown in Figures 3.2e and 3.Zf, respectively.
81
Table 3.4 Maximum Likelihood Estimates of the Parametersfor a Mixture of Univariate Log Normal Distributions
a. Trait: Systolic Blood Pressure - Males
Number of Distributions
Parameters One Two Three
4.537
4.698
4.982
0.06
0.485
0.491
0.024(2)
4.70(4)
0.10
4.618
4.957
0.10
0.975
0.OZ5(1)
5..04(3)
0.08
4.627
0.11
1.00
cr
(Xl
(Xz
(X3
xZ (Z d. f.)
Signif. level
(i) S.E. = 0.03
(2) S.E. = O.OZ
(3) Tests that a mixture of two distributions fits better than one.
(4) Tests that a mixture of three distributions fits better than two.
b. Trait: Systolic Blood Pressure - Females
Parameters
1J1 4.593 4.560 4.533
1JZ 4.740' 4.665
113 4.81Z
cr 0.10 0.07 0.06
(Xl 1.000 0.81Z 0.618
(Xz 0.188 0.314
(X3 0.068(1)Z 8.37 2.64X (Z d.f.)
Signif. level 0.02 >0.10
(1) S.E. = 0.05
82
1.0r-----.....-----...-;:~==--_"T
0.5
O-=:;; L.- --'
5.05 4.35 4.58 4.81 5.05LN (SYSTOLIC BPI
4.814.58
4.5B 4.81 5.05
LN (SYSTOLIC BPI
THREEDISTRIBUTIONS
ONEDISTRIBUTION
1.0 r-----.....-----,....:;::;II-OZ-~
0.5
0.5
zo~
::lCD
a:~en(5w>~
«.J::l~::l(J
Figure 3.2 Empirical and Theoretical Cumulative PlotsAfter Fitting a Mixture of Log NonnalDistributions
a. Trait: Systolic Blood Pressure - Males
1.0r-----...-----,.-~::::=o...__,
83
, .Or-----...,.-----,...----:::::P-'"'t
4.60 4.715 4.90
LN (SYSTOLIC BPI
zo~
:JCD
II:~
!!!cw>~c(.J:J~:Jo
0.5
ONEDISTRIBUTION
4.60 4.75 4.90
0.5
4.60 4.75
LN (SYSTOLIC BPI
4.90
Figure 3.2b Trait: Systolic Blood Pressure- Females
Table 3.4
e. Trait: Diastolic Blood Pressure - ~~les
Number of Distributions
84
Parameters
o
One
4.107
0.12
1.000
0.72
Two
4.100
4.361
0.12
0.974
0.026(1)
0.84
Three
3.977
4.137
4.377
0.09
0.247(2)
0.715
0.038(3)
(1) S.E. = 0.06
(2) S.E. = 0.16
(3) S.E. = 0.04
d. Trait: Diastolic Blood Pressure - Females
Parameters
)..11 4.161 3.890 3.889
)..IZ 4.165 4.160
)..13 4.347
0 0.11 0.11 0.10
CL1 1.000 0.014(1) 0.017(1)
CL Z 0.986 0.953
CL3 0.030(Z)2 0.62 0.32X (Z d.L)
(1) S.E. = 0.03
(2) S.E. = 0.11
85
1.0r-----...-----""T"-:"""'"""":::;::;,..-...
TWODISTRIBUTIONS
1.0 ,..------r------.----:::::;r--t
0.5
O~~__--:,,:- --:,,= ~
4.150 J.715 4.0 4.25 4.50
LN (DIASTOLIC BP)
4.215
ONEDISTRIBUTION
0.5
0.5
0_.::;;; .... &.- ....
J.75 4.0 4.25 4.150
LN (DIASTOLIC BPI
z2~;)lD
a::~
'"a 3.75 4.0w 1.0>~
~ THREEi DISTRI BUTIONS
;)U
Figure 3.2c Trait: Diastolic Blood Pressure - Males
86
0.5
0.5
TWODISTRIBUTIONS
O__-=~""';'---, ....J. :-:'
3.80 4.03 4.26 4.150LN (DIASTOLIC BP)
1.,.r-------r----,..---~-..,
0.15
THREEDISTRIBUTIONS
O~...~~_--:~ --:~:--__--:~
3.BO 4.03 4.28 4.150LN (DIASTOLIC BPI
ONEDISTRIBUTION
1.0..-----.......-----,..--:-:P_~
zoI::>III
a:I-'a 0 ..._:!!::::__..,...,."... --:~:__---~
a 3.80 4.28 4.150
~ 1.0 .-----~-----.----~-~
IotoJ::>~::>(J
Figure 3.2d Trait: Diastolic Blood Pressure - Females
87
Table 3.4
e. Trait: Sperry Cholesterol - Males
Number of DistributionsParameters One Two Three
lJ1 5.124 5.041 4.861
lJ2 5.673 5.149
lJ3 5.697
a 0.29 0.19 0.13
a ' 1.000 0.868 0.3271
a2 0.132 0.546
Cx3 0.127
x2(2 d.f.) 16.46 2.78
Signif. level < .001 >.10
f. Trait: Sperry Cholesterol - Females
Parameters
lJ1 5.091 4.988 4.693
lJ2 5.581 4.999
lJ3 5.584
a 0.28 0.17 0.16
a1 1.000 0.827 0.031 (1)
a2 0.173 0.795
a3 0.1732 23.62 0.84X (2 d.f.)
Signif. level <.001 >.10
(1) S.E. = 0.07
88
6.004.95 5.30 5.65
LN (SPERRY CHOLESTEROL)
1.0~----,-----r---""""--=""
0.5
6.00
0.5
0.5
4.95 5.30 5.65 6.00LN (SPERRY CHOLESTEROL)
zQI::JlD
a:I~C
w r-----.,.-----.,...----r---::::=_~> 1.0le(..J::J~::J(J
Figure 3.2e Trait: Sperry Cholesterol - Males
89
O .....:::::.....:.....__~ -'- --J
4.5 5.0 5.5 6.0LN (SPERRY CHOLESTEROL)
1.0.-------,------r----:;=_-+
0.5
6.05.5
0.5
5.0 5.5 8.0
LN (SPERRY CHOLESTEROL)
1.0
zo~
::JCD
a:~
~oW r-----...,...-----,.....-...,.,-o:z:oo-.....> 1.0~c(..J::J~:JU
Figure 3.2£ Trait: Sperry Cholesterol - Females
90
The results are consistent in the two sexes. For each sex, a
mLxture of two distributions fits the Sperry cholesterol data signifi-
cantly better than one distribution. There is almost the same
proportion of each sex in the higher distribution, 13% of males and
17% of females.
3.3.1.4 Zak Cholesterol
The maximum likelihood estimates of the parameters are given in
Table 3.4g for males and Table 3.4h for females, and the empirical and
the theoretical cumulative plots of Zak cholesterol for males and for
females are shown in Figure 3. 2g . As in the case wi th Sperry
cholesterol, the results with Zak cholesterol are consistent in the
two sexes. For each sex, a mixture of two log-normals fits the Zak
cholesterol data significantly better than one distribution, but a
mixture of three distributions does not fit significantly better than
two. In addi tion, there is a.lmos t the same proportion of individuals
of each sex in the higher distribution, 19% of males and 13% of females.
It is reassuring to know that two laboratory procedures whidl are
supposedly measuring the same serum cholesterol levels can produce
such consistent results.
3.3.2 Bivariate Log-Normal Distributions
In examining pairs of traits, analyses were done to determine if
a mixture of two or three bivariate log-normal distributions would fit
the data better than one distribution. Analogous to section 3.3.1,
the Y. are now random vectors of two random variables corresponding
to th:'two traits. Xi" (~~]. The :i' (i = 1•.•.•nl. can be
assumed to be independent and identically distributed random vectors
with p.d.f. gel)' The p.d.f. of a mixture of k distributions
91
Table 3.4
g. Trait: Zak Cholesterol - Males
Number of DistributionsParameters One Two Three
lJ l 5.335 5.229 5.099
lJ2 ·5.774 5.357
lJ3 5.807
a 0.26 0.16 0.10
CL1 1.000 0.806 0.390
CL2 0.194 0.436
CL3 0.1742 .
16.66 5.84X (2 d.L)
Si~nif. level <.001 .05
h. Trait: Zak Cholesterol - Females
Parameters
lJ1 5.219 5.147 5.045
lJZ 5.699 5.239
lJ3 5.718
a 0.24 0.15 0.12
CL1 1.000 0.870 0.407(1)
CLZ 0.130 0.471(1)
CL30.123
2 20.40 1.34X (2 d.L)Signif. level <.001 >.10
(1) S.E. = 0.22
92
MAL.E
THAEEOISTAIBUTIONS
1.0..-------r------.......----'::l"'·
0.5
O-'::::~ """' ~ --I
5.60 5.e5 4.eO 5.25 5.60 5.i5L.N (ZAK CHOL.ESTEAOL.)
1.0
Z TWO0 OISTAIBUTIONS~
:lIII
a::~III
00.5w
>~c(..J:l~;)U
04.90 5.25
FEMAL.E
1.o------.......-----r---~=_ 1.0r------.,.------r-----:"'::::--+
z2~
:lCD
a::~
~ow 0.5>~c(..J:l~:lU
IU5
0.5
0-=;;.-. ....... -"'-- -'
5.55 !l.eo 4.75 !l.1!l 5.55 5.90
L.N (ZAK CHOL.ESTEAOL.l
Figure 3.2g Trait: Zak Cholesterol
93
from a common family of distributions f(l;~) can be written as
g (v) = \~ 1 a· fey; 8)."" L. J= J ...-
[3.3]
There is no difficulty in allowing gel) to be a mixture of bivariate
normal distributions so that
for j = 1,2, and 3,
[3.4]
(~l' Jwhere ,l,I. = JJ ~2j
and
The correlation between Y1 and Y2 is p, and 0'1 and 0'2 are the
standard deviations for Y1 and Y2' respectively. A common variance
covariance matrix L will be assumed for the component distributions.
For a sample of n independent random vectors X with p.d.f.
given in equat ion 3. 3, the likelihood funct ion is wri tten
n kL = n L· 1 a. f (l·; ij., L ) .
i-I J= J 1 J[3.5]
It can be written as a function of 3k + 2 parameters: 2k
means, k-l admixture proportions, the common variance for each variate
along with acoounon correlation between variates. The 3k + 2 para
meters will be estimated simultaneously using MAXL1K (Kaplan and
Elston 1972). Wolfe's test (Wolfe 1971), as described in section
94
3.3.1, can be used to test 110: data consist of k component distri
butions versus HI: k' components (k' > k). Again wi th A = Lk/Lk"
-2C in A is approximately distributed as a x2 with 2m(k'-k)
degrees of freedom.
3.3.2.1 Blood Pressure
After age-adjusting the natural logarithmic-transformed systolic
and diastolic blood pressure, an attempt is made to fit a mixture of
bivariate log-normals to the data. Tables 3. Sa and. 3. Sb show the maxirrn..un
likelihood estimates of the parameters for males and females,
respectively.
For males, a mixture of two bivariate distributions does not fit
the systolic and diastolic blood pressure data significantly better
than one distribution. However, for females a mixture of two distri-
butions fits the data significantly better than one distribution, but
a mixture of three distributions does not fit significantly better than
two. There are about 20% of the females in the higher distribution,
and comparing these means, common standard deviations, and proportions
with those in Tables 3.4b and 3.4d, the local maximum for the bivariate
distributions corresponds more to the systolic blood pressure than to
the diastolic blood pressure univariate values. Therefore, if there
is a genetic polymorphism for blood pressure, evidently it expresses
itself clearly only for systolic blood pressure in females.
3.3.2.2 Serum Cholesterol
After age-adjusting the natural logarithmic transformed Sperry
and Zak cholesterol data, a mixture of bivariate log-normals is fitted
to the data. Tables 3.Sc and 3.Sd give the maximum likelihood estimates
of the parameters for males and females, respectively.
e e e
Table 3.5 ~fuocimum Likelihood Estimates of the Parametersfor a Mixture of Bivariate Log Normal Distributions
a. Trait: Systolic and Diastolic Blood Pressure - Males
Number of Distributions
P
2X (4 d.f.)
1.000
.564
0.989
0.011 (1)
Three
Srstolic Diastolic
4.567 4.107
4.652 4.107
5.036 4.082
0.10 O.lZ
.343
.646
.011 (1).
.673
0.16
Diastolic
4.107
4.081
0.12
.626
Two
0.11
Systolic
4.622.
5.034
3.58
.12
Diastolic
4.108
0.11
One
Systolic
4.625
a
a l
a Za 3
l!1
l!2
l!3
Parameters
(1) S.E. = .01
\.0trl
Tahle 3.5
b. Trait: Systolic and Diastolic Blood Pressures - Females
Number of Distributions
Diastolic
4.162
Parameters
l:!l
l:!2
l:!3
(J
al
a 2
a3
p
One
Systolic
4.594
0.10
1.000
0.657
0.11
Two
Systolic
4.566
4.701
0.08
0.796
0.204
0.871
Diastolic
4.169
4.130
0.11
Three
Systolic Diastolic
4.567 3.962
4.567 4.171
4.762 4.210
.08 0.10
0.072
0.792
0.136
0.839
2X (4 d.L)
Signif. level
e
26.46
<.001
7.30
>.10
e e
\DQ\
e e
Table 3.5c. Trait: Sperry and Zak Cholesterol - Males
Number of Distributions
e
Parameters
1!1
1!2
1!3
a
a 1
a 2
a 3
p
One
Sperry
5.169
0.30
1.000
0.939
Zak
5.340
0.27
Two
Sperry
5.053
5.639
0.18
0.814
0.186
0.830
Zak
5.232
5.784
0.16
Three
Sperry
4.944
5.169
5.669
0.15
0.416
0.411
0.173
0.727
Zak
5.108
5.365
5.809
0.10
2X (4 d.f.)
Signif. level
13.51
.009
7.42
>.10
\D-....J
Table 3.5d. Trait: Sperry and Zak Cholesterol - Females
Number of Distributions
Parameters
~l
~Z
~3
a
One
Sperry
5.087
0.Z9
Zak
5.223
0.24
Two
Sperry
4.997
5.638
0.19
Zak
5.147
5.700
0.15
Three
Sperry
4.897(1)
5.066(2)
5.651
0.17
Zak
5.035(2)
5.221(2)
5.714
0.12
Ul
Uzu3
p
ZX (4 J.f.)
Signif. level(1) S.E.:: 0.1
(2) S.E.:: O.Z
1.000
0.936
19.56
<.001
.870
.130
0.839
1.58
>.10
.340
.536
.124
.805
\.Q
00
e e e
99
For both the males and the females, a mixture of two bivariate
distributions fits the data better than one, but a mixture of three
distributions does not fi t better than a mixture of two. About 19%
and 13% of the males and females, respectively, belong to the'higher
distribution. Further.more, regardless of whatever initial estimates
are tried for the males and for the females (e.g. first try initial
estimates where there are 2 different means for Sperry cholesterol and
the same means for Zak cholesterol and then where there are 2 different
means for Zak cholesterol and the same means for Sperry cholesterol),
the search of the likelihood surface always converges to the same
maximum likelihood estimates; i.e. there is only one maximum. In
addition, comparing these estimates with those obtained from fitting
mixtures of univariate distributions (Tables 3.4e -3.4h) reveals that
the two sets of means, common variance, and proportions are quite
similar. The estimate of p, the cammon correlation between Sperry
and Zak cholesterol, is similar to the 0.94 in Table 2.7. An inter
pretation of this is that if there is a major gene for Sperry
cholesterol and a major gene for Zak cholesterol, it is the same gene
for both measurements. This case is to be contrasted with the
situation described next.
3.3.2.3 Systolic Blood Pressure and Serum Cholesterol
The univariate analyses have shown that a mixture of two distri
butions fits the data significantly better than one distribution for
Sperry cholesterol, Zak cholesterol, and systolic blood pressure; and
in the case of the latter trait, only for females. There was no
evidence of a genetic polymorphism for diastolic blood pressure in
either sex nor for systolic blood pressure in males.
100
1be bivariate analysis of blood pressure showed that a mixture of
two distributions fits better than one for females only; moreover, the
local maximum in the bivariate case corresponds to the maximum for
systolic blood pressure in the univariate case.
In order to study blood pressure and serum cholesterol jointly,
a mixture of two bivariate log normal distributions is fitted to the
age-adjusted log-transformed systolic blood pressure and either Sperry
cholesterol or Zak cholesterol; diastolic blood pressure is not used
since all attempts in this study to detect a polymorphism have been
unsuccessful.
The purpose of studying systolic blood pressure and serum
cholesterol jointly is to determine whether there is only one maximum,
or if there are two local maxima. The procedure is to start with two
sets of initial estimates. In the first set, the initial estimates of
the means for sys tolic blood pressure are set equal, and the means for
serum cholesterol are different; in the second set, the situation is
reversed. Then, for each set of initial estimates, a search of the
likelihood surface is made for a maximum. In the case of Sperry and
Zak cholesterol in section 3.3.2.2, the estimation procedure found
only one maximum regardless of which set of initial estimates was used.
For males and for females, two local maxima were found for systolic
blood pressure and serum cholesterol. The estimates of the parameters
for the two maxima are shown, for males, in Table 3.6a for Sperry
cholesterol and in Table 3.6b for Zak cholesterol. Tables S.6c and 3.6d
show the estimates for females for systolic blood pressure and Sperry
cholesterol and Zak cholesterol, respectively. In each case, maximum I
corresponds to the maximum for cholesterol while maximum 2 corresponds
~,
101
Table 3.6 Maximum Likelihood Estimates of the Parameters
a. Trait: The Two local Maxima for Systolic Blood Pressure andSperry Cholesterol - Males
Local Maxima
1Parameters Systolic BP
2Sperry Systolic ap Sperry
5.105
5.105
0.29
4.617
4.941
0.10
5.016
5.652
0.19
4.616
4.700
0.11a
ld1
ldz
*L.1
p
0.859
0.141
0.222
96.32
is the likelihood corresponding to the
0.968
0.032
0.378
. th .1 maxllTILDTl.
b. Trait: The Two Local Maxima for Systolic Blood Pressure andZak Cholesterol - ~fuUes
Local Maxima21
Parameters Systolic BP Zal<
ldl 4.628 5.223
B2 4.610 5.774
a 0.11 0.15
<ll 0.792
<lZ 0.208
p 0.295
L/L2 397.50
Systolic BE4.623
4.935
0.10
0.957
0.043
0.368
5.340
5.296
0.27
102
c. Trait: The Two Local Maxima for Systolic Blood Pressure andSperry Cholesterol - Females
Local I\ta.xima
Parameters
~l
1!2
a
1Systolic BP
4.594
4.605
0.10
0.838
0.162
0.368
6616.36
Sperry
4.980
5.579
0.17
2Systolic BP
4.561
4.743
0.07
0.811
0.189
0.131
Sperry
5.044
5.217
0.36
u.. Trait: The Two Local Maxima for Systolic Blood Pressure and ZakCholesterol - Females
Local Maxima
Parameters1
Systolic BP Zal<2
Systolic BP
0.15a
4.594
4.642
0.10
5.134
5.693
4.557
4.732
0.071
5.179
5.324
0.24
0.853
0.147
0.215
2218.55
0.746
0.254
0.112
103
to one for systolic blood pressure. Judging from the likelihoods, the
local maxUnuffi corresponding to cholesterol is much higher. than the
one corresponding to systolic blood pressure. Comparing these
estimates with those obtained from fitting mixtures of univariate
distributions reveals that the two sets of means, cornmon variance, and
proportions are similar. In most cases, the cornmon correlation is
close to the correlation of 0.30 between systolic blood pressure and
Sperry and Zak cholesterol levels (Table 2.7). Estimates of the distri
bution means at these local maxima are plotted in Figure 3.3.
This analysis suggests that, if there is a maJor gene for systolic
blood pressure and a major gene for sennn cholesterol, there are two
separate genes, since fitting a mixture of bivariate normal distribu
tions reveals two local maxima.
104
A. SYSTOLIC BLOOD PRESSURE VS SPERRY CHOLESTEROL
MALE FEMALE
5.5 r------,,...-----,.---.......---.., 5.5
6.0
•5.5
•
5.0
•
•
• • 4_.4.5 l-.__---' --'- --'- ....I 4.5,L-.---'-------'---.........----J
5.0 5.5 6.0 5.0
LN (SPERRY CHOLESTEROL)
01XIU.J
g 5.0III>!!!z.J
B. SYSTOLIC BLOOD PRESSURE VS ZAK CHOLESTEROL
MALE
!US r------,----....----r---..,FEMALE
5.5,....-----,.---.....,....---..,.------,
5.0
CL1XI0.J0
5.0~III •>-!!!z.J
• • • •••
•6.05.5
4.5 l-.__---! --'- -L. ....I 4.5L-.---''-------'---.........---~
5.0 5.5 6.0 5.0
LN (ZAK CHOLESTEROL)
KEY: • MEANS FOR LOCAL MAXIMUM CORRESPONDING TO CHOLESTEROL
• MEANS FOR LOCAL MAXIMUM CORRESPONDING TO SYSTOLIC BLOOD PRESSURE
Figure 3.3 Plots of the Estimated Distribution Means forSystolic Blood Pressure and Serum CholesterolSex Corresponding to the Two Local Maxima ofthe Likelihood.
OiAPTER IV
TI1E ~fAJOR GENE HYP01HESIS
4.1 Introduction
The traditional method of analyzing pedigree data has been to
break the pedigree into many two-generational families and then to
look for Mendelian segregation ratios. After the investigator uses
some criterion to classify each individual to a specific genotype,
he will detennine, depending on the mating types of the parents,
whether the genotype distribution in the offspring is consistent wi th
a particular genetic hypothesis. This method of analysis wastes infor
mation in that various relationships (e.g. grandparent-grandchild,
uncle-niece) are ignored. In addition, the families are not indepen
dent, the same individual may be a child in one family and a parent
in another.
Sometimes two-generational data are not sufficient; misleading
conclusions can be drawn. In a classical study by Sewall Wright (1934)
of the inheritance of polydactyly in guinea pigs, he was able to
demonstrate that, for two generations after crossing two inbred strains,
the data mimic simple Mendelian inheritance. However, after further
breeding, it was concluded that p~lygenic inheritance was more appro
priate. MOre recently, Li1ienfeld (1959), also looking at two
generational data, was able to shmv that the inheritance of a trait,
like attending medical school, which is presumed to be determined
principally by socio-cultural factors, is consistent with transmission
106
by an autosomal recessive gene.
Analyzing the whole pedigree intact will yield more genetic infor
mation. Elston and Stewart (1971) have developed a general approach
to the genetic analysis of pedigree data, and their metllod will be
applied to the Bay City pedigree. In this chapter, the approach will
be briefly described, and then the method will be used to seardl for
a major gene for systolic blood pressure, diastolic blood pressure,
and sennn cholesterol separately as well as for the two blood pressures
jointly and for the two measures of serum choles terol jointly. The
next chapter, Chapter V, will be an examination of the polygenic model,
and Chapter VI will consider the mixed model,segregation of a major
gene toge't!~er with polygenic and environmental background.
4.2 The Major Gene Model
By major gene is meant a single gene that can account for a
significant portion of the phenotypic variation. In a one-autosomal
locus, two-allele system, let the two alleles be represented by A
and B, and further, for convenience, let the genotypes be indexed
M = I, AB = 2, and BB = 3.
Consider a pedigree with n individuals and a measure of a
quantitative trait x., (i = 1, ... ,n)1
on each individual. The
mathematical model for x. under a major gene model, given genotype t,1
can be wri tten as
[4.1]
where met) Ct = 1,2,3) is the major gene effect IDld e is the
random environmental effect which is assumed distributed NCO, 02).
107
The major gene effect m is distributed
Genotype
M AB BBIndex l 1 2 3
Effect mCl) 1J1 1J 2 1J3
Frequency I/Jl 1/J1 1/J2 1/J3 Le I/Jl = 1
The mean and variance of mare
ECm) = r3I/Jl mCl) = m
l=l[4.2]
3 2V(m) = r I/Jl m(l)
l=l
-2 2- m = om [4.3]
The quantity 0; is called the variance due to the major gene, and its
estimate will be used to co~ute an estimate of heritability. For
convenience, mCl
) will be written me since there will be no ambiguity
as to what the subscript is referring.
It is assumed in deriving the likelihood that, given the parental
genotypes, the genotypes of the offspring are independent of one another.
Further assume that, conditional on their own genotypes, the
phenotypes of the offspring are independent of one another.
Thus, the likelihood L of observing a sibship of size n with
measures xl' ... ,xn given parental major gene effects ms and mt is
= II f(x·lm , mt )i 1 S
= II r3 f(x·lm) f(m 1m mt )i u=l 1 u u S
=
108
where mu is the major gene effect of the offspring.
Spouses are assumed to be marrying into the pedigree, i. e. they
have no parents who are members of the pedigree. Under the assumption
of random mating, i.e. an individual's phenotype is independent of
his spouse's phenotype and genotype, it is not difficult to consider
spouses in the model. Given an individual's parents have major gene
effect ms and mt , the likelihood of observing him wi th measure
x and his spouse with measure y is
= f(xlms mt ) fey)
3 3I f(x I~) f(m 1m mt ) I f(Ylmy) f(m )~l u s ~l ~
where mv is the major gene effect of the spouse. Therefore, the
likelihood L of observing a sibship and their spouses, given ms ' mt
is
L = IT I3f(x·lm) f(mulms mt ) I 3
f(Y·lm) femv)i ~1 1 u ~l 1 v
[4.4]
For persons with no spouse, the second summation is set equal to one.
This likelihood can be expressed as a function of three quantities
eaCt~ of which will be rewritten to confonn with the notation in the
Elston and Stewart (1971) paper:
1. The genetic mechanism, f(m 1m mt ),u s is denoted simply by
Pstu ' the probability that, conditional on the parents'
genotype being s and t, an individual has genotype u.
For the one-locus, two-allele model these probabilities
109
can be arranged into a 3x3 genetic transition matrix in
which each element is a vector (p$t 1 Pst 2 Pst 3)·
Furthermore, we can express each of the Pst u in the
matrix as a function of three transmission probabilities:
i) T1
= Pr(AA ~ A), probability a parent of genotype
AA will transmit an A allele to his offspring
ii) T2 = Pr(AB ~ A), probability a parent of genotype AB
will transmit an A allele to his offspring, and
iii) T3
= Pr(BB ~ A), probability a parent of genotype BB
will transmit an A allele to his offspring.
The relationship between the pIS and the TIS is:
p = (l-T )(l-T )st 3. s t
s, t = 1,2,3 [4.5]
For the simple Mendelian hypothesis, and T =3
0, and the genetic transition matrix is as shown in Table 4.1.
Table 4.1 The Genetic Transition Matrix for a One-Locus,Two-Allele System
t
s
1 =M
2 = AB
3 = BB
1- M
(1 0 0)
(1 1. 0)2 2
(0 1 0)
2 = AB(1 !. 0)2 2
(1. ! 1)424
(0 t t)
3 = BB(0 1 0)
(0 1. 1.)2 2
(0 0 1)
The phenotype-genotype relationship,2.
is denoted by &u(xi ) or gv(Yi)'
conditional p.d.f., given genotype
110
f(xi!mu) or f(Yi1m),
Thus, gJx) is the
u, of observing x. For
a specific trai t under the one maj or locus bvo -allele model,
there will be three p.J.f. g (x), one for each postulatedugenotype, and the distribution of x given genotype u is
taken to be ~(~u' 02). There is no difficulty in allowing
~(x) to be age and/or sex dependent, or alternatively, x
can be age and/or sex adjusted.
3. The genotypic distribution among persons "external" to the
pedigree, f (m). By "external" is meant that these
individuals have no parents who are themselves members of the
pedigree. Hence, persons "external" to the pedigree include
spouses of individuals in the pedigree and the two original
parents. In most cases, their genotypes will not be known,
and we let fCIDvJ, denoted by Wv ' be the proportion of
individuals "external" to the pedigree with genotype v
(v = 1,2,3).
Under the assumption of random mating and the basic assumption
that, given the genotypes of both parents, the genotypes and phenotypes
of the offspring are independently distributed, Elston and Stewart
(1971) have derived the likelihood of observing a particular set of
pedigree data as a function of the three quantities described above.
The derived likelihood involves a series of summations and products.
There is a Fortran program GENPED (Kaplan and Elston 1975) which will
construct this likelihood.
J
111
4.3 Method of Analysis
For systolic and diastolic blood pressure, as well as Sperry
cholesterol and Zak cholesterol, the analyses done in Chapter II suggest
that a natural logarithmic transformation is appropriate. Therefore,
the pedigree analysis will be done on the natural logarithmic trans-
formed variable, appropriately age-adjusted. If y. is the measure1
of the trait for the i th indiVidual in the pedigree, and x. is his1
age, the variable of interest is the measure of the trait adjusted
to age 30:" "2 2y! = .en y. - 131ex. -30) - 82ex. - 30 )
1 1 1 1
where y' is the log-transformed value adjusted to age 30,
131 is the linear age correction coefficient
82 is the quadratic age correction coefficient.
[4.6]
The quadratic coefficient 82 is used only if necessary, and necessity
is determined by whether, in a regression of .en y against x and x2,
the regression coefficient 132 is statistically significant. Table
2.5 indicates that 82 is needed only for systolic blood pressure and
diastolic blood pressure in males.
The parameters of the model and their interpretation are listed
in Table 4.2. To account for sex differences, there will be one set
of ~'s, 02, and 8' s for males and another set for females. In
this way, the model considers the heterogeneity between sexes, but at
the expense of adding five or six more parameters.
The likelihood of observing a particular set of pedigree data can
be expressed as a function of the parameters listed in Table 4.2. Denote
112
Table 4.2 Parameters of the 010del and Their Interpretation
1- T1: Tile probability that an individual of type AA will transmit A.
2. T2: The probability that an individual of type AB will transmit A.
3. T~: 'Gle probability that an individual of type BB will transmit A..)
4. 1j!1: The relative frequency of type M in "parental" population.
S. 1j!,,: The relative frequency of type AB in "parental" population..:..
6. 1j!3: The relative frequency of type BB in "parental" population.
7. ~l : The mean of the natural log of the trait for individuals of
type AA.
8. ).J2: The mean of the natural log of the trait for individuals of
type AB.
9. ~3: The mean of the natural log of the trait for individuals of
type BB.
10. 0
11. PI:
12. 62:
Tne standard deviation of the natural log of the trait.
The linear age correction coefficient.
The quadratic age correction coefficient (if necessary) .
113
this likelihood as
where ~ is a vector of the parameters. Suppose that there are k
parameters, and write 8 = (8 , 8) where r + s = k. We are- -r-s
interested in testing the hypothesis (r ~ 1) that
HO: ~l" = ~rO agains t HI: ~r ~ ~rO· [4.7]
The test will be done using the Likelihood Ratio (LR) method proposed
by Neyman and Pearson (1928). The method requires the maximum 1ikeli
hood{ML) estimators of (~r' ~s)' giving the uncondi tiona! maximum
of the likelihood function
A '"
L (P 18 , 8 ).-r -s
We will call this the maximum ~f the likelihood under the generoal
unroestT'iated model, i.e. no restrictions have been placed on the
parameters. The LR method also requires finding the ~ln.. estimators of
e when HO holds t giving the condi tional maximum of the LF-s
In most cases, ~s ~ ~s· The test of HO is based on the likelihood
ratio
A =
'"A
L (P I~rO' ~s)
L(PI~r' ~s)[4.8]
114
Wilks (1938) showed that asymptotically, when 1I0
holds, the distribu
tion of -2 to A tends to a x2-distribution with r degrees of
freedom. The term -2 to A, where A is given by equation 4.8, can'" '" '"be rewritten 2 (in L(ple , e ) - to L(ple 0' e))j this is twice the.... r ....s ....r ....s
difference in the two log likelihoods.
There is an alternative to the likelihood ratio test. Under
regularity conditions, the vector of ~~ estimators e is asymptoti-....r
cally rnultinonmally distributed with variance-covariance matrix V
(Kendall and Stuart 1973). Then
is asymptotically distributed as a x2 with r degrees of freedom,
where a consistent estimate of V can be obtained by numerical double
differentiation of the likelihood at its maximum (Kaplan and Elston
1972). It can be shown that the likelihood ratio test and this test
based on the ML estimators of the parameters are asymptotically
equivalent (Kendall and Stuart 1973). Since the distribution of Qr
may approach normality slowly, and since, if r is large, obtaining
a consistent estimate of V by numerical double differentiation can
be quite time-consuming on the computer, hypotheses will be tested
using the likelihood ratio test.
Subject to the constraints 0 s T., ~. s 1 (i = 1,2,3),l. l.
and all the variances being non-negative, the maximum like-l 1jJ. = 1,. l.l.lihood of the pedigree data, maximizing over all the unknown parameters,
can be obtained both under the general unrestricted model and WIder
HO' using the maximum likelihood subroutine package MAXLIK devised by
Kaplan and Elston (1972) which essentially conducts a search of the
115
likelihood surface. MAXLIK allows for various constraints and restric-
tions .
We are interested in testing the following hypothesis:
1. The presence of Harody-Weinberg equi Zibriwn proportions, HO: W2 =
21WlW3' Twice the difference between the likelihoods under the
general unrestricted model and under HO will be compared with a
x2 with 1 d.f.
2. The presence of
l3 = O. Thus,
1simple Mendelian inheritanae, HO: T1 = 1, l2 = 2'
-2 in A will be compared with a x2 with 3 d.f.
3. The presence of a purely environmental hypothesis~ HO: II = l2 =
l3 = T. This is equivalent to saying that the probability of
transmitting the A allele is not dependent on the genotypes of
the parents. Rejection of this hypothesis indicates that there is
transmission from one generation to the next. The resulting
statistic -2 in A will be compared with a x2 with 2 d.f.
4. The presence of a dominant (HO: ~l = ~2) or a reaessive
(HO: ~2 = ~3) hypothesis. In each case, -2 in A will be compared
with a x2 with 1 d.f.
Initial estimates are needed to use the subroutine package ~~IK.
It is not unreasonable to use as initial estimates the final estimates
obtained in fitting the mixture of log nonnal distributions to the
pedigree data (Chapter III). There is no failsafe guarantee that
MAXLIK \vill find the absolute maximum; the program may produce a local
maximum. As a check, it is prudent to enter MAXLIK with various
initial estimates, noting whether it converges to the same maximum
each time.
.' .,'.' '0" . ,.. "
116
There is no problem to extend the Elston and Stewart method to the
bivariate case. Consider a pedigree with n individuals and measures
of two quantitative traits ~ = (xl' x2) on each individual. Eacll
measure in ~ can be adjusted for age using equation 4.6. Analogous
to equation 4.1, the mathematical model for X., ( i =1 , ... ,n)-1
under a
major gene model,given genotype u, can be written as
[4.9]
wllere ID(u) (u = 1,2,3) is the major gene effect and £ is the random
environmental effect which is assumed distributed N(Q, L ), where L
is of the fonn:
L =POIOZ
The numbers 1 and Z refer to the two trait~ and P is the intertrait
correlation.
The major gene effect m is distributed
Genotypes
AA AB BB
Index .e. 1 Z 3
Effect met) ~l ~2 1!3
Frequency WI Wz W3 L Wt = 1.e
The genetic mecruwlism, as expressed by Pst u' is identical to
the univariate case. To express the pllenotype-genotype relationships
under the one major locus, two-allele model, the distribution of ~
given genotype u is taken to be N(Mu ' L)' The genotype distribution
BIOMATHEMATICS TRAINING PROGRAM 117
among persons "external" to the pedigree is again IjJ (v = 1,2,3).v
Although the number of parameters is increased, the method of testing
hypotheses using the likelihood ratio test is the same as for the
univariate case.
It is possible to estimate what portion of the total phenotypic
variance is attributable to the major gene. Assume that the common
variance represents the cOlllllon environmental variance oZ. Thee
variance for the major gene effect 02 is given by equation 4.3.m
Thus, the proportion attributable to the major gene is 02/(02 + 0eZ)m mand can be estimated using the ML estimates in place of the parameters.
Elston, Namboodiri, and Kaplan (1978) suggest that, based on
siJm.Jlation experiments (Go, Elston, and Kaplan 1977), genetic segrega
tion at a major locus can be reasonably inferred if the following
criteria are satisfied during pedigree analysis:
1. The hypothesis of Mendelian inheritance HO: Ll = 1, L2 =i, L3 = 0
cannot be rejected, and the estimated probabilities are close to
these values.
2. The environmental hypothesis, HO: Ll = TZ = T3 = T, is rejected.
3. The likelihood of the pedigree under the dominant hypothesis HO:
ul = Uz is very different from that under the recessive hypo
thesis HO: Uz = u3·
4. The data, ignoring any pedigree structure, fit a mixture of two log
normal distributions significantly better than a single log normal
distribution.
These criteria will be kept in mind in analyzing each of the
traits.
118
For each of the various models (e.g. Mendelian inheritance,
envirorunental hypothesis), the M... estimates of the unlalown parameter
vector .2 of L(P!.2) will be obtained for the Left Pedi1.~ree, the
Right Pedigree, and for Both Pedigrees combined. A likelihood ratio
test can be constructed to test HO: ~L = ~R vs. HI: .2L ~ QR'
where the subscript denotes whether the parameters are for the Left or
the Right Pedigree. The likelihood ratio method requires the uncondi
tional maximum of the likelihood function. Now, since the measures of the
trait in the Left Pedigree and in the Right Pedigree are essentially
independent (there are only four persons common to both pedigrees),
this maximum is approximated by the product of the unconditional
maximum for each pedigree, L(PLPRI.2L.2R) = L(PLI.2L) * L(PR1.2~ .
This is compared with the conditional maximum when HO holds,
L(PLPRI.2L = .2R). The likelihood ratio is
L(PLPRI.2L = .2R)
L(PLI.2L) L(PRI.2R)
The distribution of -2 tn A is asymptotically a x2 - distribution
with k degrees of freedom where, in this case, k is the number of
parameters in ~ •
4.4 Results of Univariate Analyses
4.4.1 Sperry Cholesterol
The maximum likelihood estimates are tabulated in Table 4.3, first
for the Right Pedigree (Table 4.3a) , then for the smaller Left Pedigree
(Table 4.3b) , and finally, for both pedigrees combined assuming that
they are independent (Table 4. 3c) .
119
Table 4.3. Maximum Likelihood Estimates From UnivariatePedigree Analysis of Sperry Cholesterol
a. Right PedigreeHardy-We inberg
Unrestricted Equilibrium ~~nde1ian EnvironmentalTl .631 .555 1.000 .110
T2 .350 .411 .500 .110
T- .0001# .0001# .000 .110.)
WI .051 .001 .000" .026
llJ2.0001# .053 .050 .0001#
llJ .. .949 .946 .950 .974.)
Males
*~l and ~2 5.842 5.837 5.827 5.840
*~ .. 5.216 5.216 5.216 5.219.;)
(J .194 .194 .196 .202
81 .005 .005 .005 .005
Females
*~1 and ~2 5.805 5.807 5.802 5.798
*~3 5.203 5.203 5.203 5.203
(J .163 .162 .163 .161
81 .007 .007 .007 .008
lnL 153.212 152.759 150.136 133.745
2 .906 6.152 38.934X
Signif. level > .10 >.10 <.001
1#Converged to a bound
'IeMeans are adjusted to age 30
120
b. Left PeciigreeHardy-Weinberg
Unrestricted EquilibriLDTl Mendelian Environmental
1 11. 000# 1.000# 1.000 .116
1 2 .423 .469 .500 .116
1 3.008 .005 .000 .116
WI .103 .003 .107 .037
W2 .000# .110 .000# .026
W3 .897 .887 .893 .937
~la1es
*lJ1 and lJ2 5.822 5.820 5.821 5.824
*lJ 35.164 5.164 5.164 5.164
(J .138 .138 .138 .138
61 .007 .007 .007 .007
Females
*lJ1 and lJ 2 5.696 5.698 5.701 5.611
*lJ- 5.121 5.121 5.122 5.099.;)
(J .158 .159 .158 .154
61.005 .005 .005 .007
tnL 52.860 52.218 52.699 49.717
2 1.284 .322 6.286X
Signif. level >.10 >.10 .043
#Converged to a bound
*Means are adjusted to age 30
121
c. Both Pedigrees
Hardy-WeinbergUnrestricted Equilibrium Mendelian Envi roruncnta1
.632 .SS4 1.000 - .10;--Tl
T2 .368 .428 .500 .107
T3.000# .000# .000 .107
1jJ1 .068 .001 .00011 .013
1jJ2 .000# .070 .068 .012
1jJ3 .932 .929 .932 .976
Males.-
lJ1 and lJ 5.830 5.827 5.821 5.841.- 2
lJ35.203 5.203 5.203 5.207
(J .184 .184 .185 .187
81.006 .006 .006 .005
Females.-
lJ1 and lJ2 5.779 5.782 5.777 5.779.-
lJ35.178 5.179 5.179 5.182
(J .168 .168 .169 .172
81 .006 .006 .006 .007
£.nL 199.905 199.249 196.693 177.543
2 1.312 6.424 44.724X
Signif. level >.10 .093 <.001
2~eterog X14 12.334
'Converged to a bound
"Means are adjusted to 30
11.456 12.284 11. 838
122
For each set of data, the first column contains the maximum like li-
hood estimates under the general unrestricted model obtained with the
constraints that the l' S and the lJJ' s could each vary between 0 and
1, and the sum of the ~'s is 1. The second column contains esti-
mates when the likelihood is maximized wi th the above constraints and
the restriction that the ~'s are at Hardy-Weinberg equilibrium. The
third column consists of the estimates when the maximization is done
1with 1 1, 1 2, and 1 3 fixed at 1, 2' and 0, respectively; this
represents the Mendelian hypothesis. Finally, the last column corres
ponds to the maximization being done with all the 1'S set equal
(the environmental hypothesis).
For all three data sets, the test for departure of the ~'s from
Hardy-l~einberg equilibrium is not significant. In the Right Pedigree
and for Both Pedigrees, the environmental hypothesis is emphatically
rejected. In the Left Pedigree, a test of HO: 1 1 = 1 2 = T3 results
in a chi square of 6.286. Ordinarily, for this case, -2 .en A is
distributed asymptotically as a chi square with 2 degrees of freedom.
However, in maximizing the likelihood for the general unrestricted
model, two of the parameters (Tl and ~2) may not be at local maxima
since they have converged to a bound. There is little theory concerning
the distribution of -2 in A in this situation. There have been sug
gestions that the asymptotic distribution is a x2 but with the number
of degrees of freedom being less than two. We will decide whether or
not to reject a hypothesis by relying on a conservative approach.
Whatever the actual cumulative distribution of -2 in A is when certain
parameters have converged to a boundary, it is botmded by the
cumulative distribution of a chi square with 2 degrees of freedom.
123
Using this to approximate the actual distribution of -2 fu A, the
actual significance level Sllould be lower than the nominal one; thus
this is a conservative test. For the test of HO: Tl = TZ : T3 with
a resulting chi square of 6.Z86, the nominal significance level is
0.04, and the actual one is lower. Consequently, this suggests that
there is transmission from one generation to the next.
In the Right Pedigree, the likelihood of the pedigree under the
dominant hypothesis is 13.4 times larger than that under the recessive
hypothesis. In the Left Pedigree, this value is 4.8, and for the two
pedigrees combined, the ratio of the likelihoods is 39.5. These
comparisons show that a dominant hypothesis is preferable for Sperry
cholesterol over a recessive one.
The Mendelian hypothesis cannot be rejected for any of these three
data sets. Again we have to rely on a conservative approach since two
parameters converge to bounds.
In Chapter III, it was shown that a mixture of two log nomal
distributions fits the Sperry cholesterol data significantly better
than one distribution.
Referring to the four criteria listed at the end of section 4.3,
each has been satisfied, and therefore we can conclude from the
analyses that there is a major gene segregating for hypercholesterolemia
in this pedigree. In examining the ~'s, we can see that over 90%
of those "external" to the pedigree have the homozygous recessive
genotype BB, about 7% have genotype AB, and less than 1% have the
homozygous dominant genotype AA. This latter small value is certainly
consistent with the idea that persons homozygous for the hyper
cholesterolemia gene may be subject to selection through premature
mortality.
124
The percent of the Sperry cholesterol variance accounted [or by the
major gene is 42% for males and 44~o for females. The non- significant
heterogeneity ali square values indicate that no heterogeneity was
detected between the estimates for the Left Pedigree and those for the
Right Pedigree.
4.4.2 Zak Cholesterol
The maximum likelihood estimates are tabulated in Table 4.4. The
results of the analyses with Zak cholesterol closely ~irror those with
Sperry cholesterol; t~is is not surprising as the two measures of
cholesterol are highly correlated (correlation of 0.94 in Table 2.7).
In the three data sets, the test for departure of the ~'s from
Hardy-Weinberg equilibrium is not significant. The environmental hypo
thesis is again rejected in the Right Pedigree and in Both Pedigrees,
indicating the existence of vertical transmission. In the Left Pedigree,
the chi square for the environmental hypothesis is 5.104. Since two of
the parameters in the unrestricted case have converged to a bound, the
nominal significance level is about 0.08. On this basis, we cannot
reject the environmental hypothesis. Elston, Namboodiri, and Kaplan
(1978) point out that it is possible for a major gene to be segregating
and find that the environmental hypothesis fits the data. This could
happen if all the parental mating types are the same, in which case the
children's distribution of genotypes would be the same. In this case,
it would be impossible to distinguish between the effect of a major
gene and an environmental effect. For the Left Pedigree in which 90%
of the population are of genotype BB and where the number of individuals
is small, it is not unlikely for all the parental mating types to be
the same.
125
Table 4.4 ~~irnum Likelihood Estimates From Univariate PedigreeAnalysis of Zak Cholesterol
a. Right PedigreeHardy-Weinberg
Unrestricted Equilibrium Mendelian Environmental
(I .635 .488 1.000 .105
(2 .277 .391 .500 .105
(3 .000# .000# .000 .105
4;1 .089 .002 .000# .000#
4;2 .000# .089 .071 .000#
4;- .911 .909 .929 1. 000#.:>
Males
'*\.11 and \.1 2 5.924 5.922 5.930 5.949
'*\.135.388 5.390 5.393 5.417
a .157 .157 .158 .184
81.005 .005 .005 .006
Females
'*\.11 and f.lZ 5.918 5.916 5.883 5.942
*\.1 3 5.379 5.379 5.377 5.383
a .139 .140 .145 .137
81.007 .007 .007 .007
fuL 136.150 135.100 132.934 122.169
2 2.100 6.432 27.962X
Signif. level >.10 .092 <.001
itConverged to a bound
#I~leans are adjusted to age 30
126
b. Left Pedigree
Hardy-HeinbergUnrestricted Equilibrium Mendelian Envi rorunent al
Tl1.000# 1.000 1.000 .096
T2 .374 .418 .500 .096
T3 .007 .004 .000 .096
WI .124 .006 .128 .051
1Ji 2.000# .139 .000# .048
W3 .876 .855 .872 .902
Males
*~1 and l.J 5.922 3.921 5.891 5.923
2I\:
~3 5.327 5.326 5.322 5.327
a .123 .123 .125 .123
61.007 .007 .008 .007
Females
*~1 and l.J2 5.772 5.771 5.772 5.757
*l.J 35.265 5.265 5.265 5.263
a .126 .126 .126 .128
61.006 .006 .006 .006
fuL 46.712 46.032 46.073 44.160
2 1.360 1. 278 5.104X
Signif. level >.10 >.10 .078
#Converged to a bound
*Means are adjusted to age 30
127
c. Both Pcdigrces
Hardy-WcinbergUnrestricted J~~Lui libritun ~~ndclian Environmcntal
T1 .659 .521 1.000 .105
T2 .297 .410 .500 .105
T3 .000* .000* .000 .105
1/Ji .103 .003 .000* .000#
1/JZ .000* .104 .096 .033
1/J3 .897 .893 .904 .967
l-1a1es
*1J1 and 1J2 5.928 5.926 5.928 5.940
*1J3 5.375 5.375 5.375 5.381
a .152 .151 .152 .155
61 .005 .005 .005 .005
Females
*1J1 and 1J2 5.854 5.857 5.831 5.914
*1J3 5.343 5.344 5.342 5.353
a .148 .149 .152 .149
61 .006 .006 .006 .006
inL 175.906 174.469 172.444 159.953
2 2.874 6.924 31.906X14Signif. level .090 .074 <.001
Heterog x2 13.912 13.326 13.126 12.75214
IIConverged to a bound
*Means are adjusted to age 30
128
In the Right Pedigree as ,~cll as in Left Pedigree, the likelihood
of each pedigree under the dominant hypothesis is more than twice as
large as that under the recessive hypothesis. For Both Pedigrees
together, the ratio of the likelihoods is 11.0. This indicates a
preference for the dominant hypothesis over the recessive hypothesis.
A chi square of 1.278 for the Left Pedigree indicates an adequate
fit of hypercholesterolemia to the major gene hypothesis. The tests
of the Mendelian hypothesis result in a chi square of 6.432 for the
Right Pedigree and a chi square of 6.924 for Both Pedigrees. Ordinarily,
these are chi squares with three degrees of freedom. However, in each
case, two of the parameters under the unrestricted model (T 3 and 1Ji2
)
converged to a bound as did one parameter (lJil) under the Mendelian
model. The nominal significance levels are 0.09 for the Right Pedigree
and 0.07 for both pedigrees combined. On this basis, the genetic
hypothesis cannot be rejected. In Chapter III, it was shown that a
mixture of two log normal distributions fits the Zak cholesterol data
significantly better than one distribution (Table 3.4 g and h) .
As with Sperry cholesterol, the four criteria for inferring that
a major gene for Zak cllolesterol is segregating in this pedigree have
been fUlfilled. The percent of the variance for Zak cholesterol
accounted for by the major gene is 54% for males and 47% for females.
Again, there is no evidence to reject the hypothesis that the estimates
for the Right Pedigree and for the Left Pedigree are homogeneous.
4.4.3 Systolic Blood Pressure
The maximlUll likelihood estimates are tabulated in Table 4.5. In
the Left Pedigree, the likelihood of the pedigree under the recessive
hypothesis (HO: u2 = u3) is twice as large as that under the dominant
129
Table 4.5 ~~imum Likelihood Estimates From UnivariatePedigree Analysis of Systolic Blood Pressure
a. Right Pedigree
Unrestricted Mendelian Envirorunental
T1 .168 1.000 .470
T2 .619 .500 .470
T3.000# .000 .470
WI .225 .234 .194
W2 .775 .766 .697
W3 .000# .000# .109
Males
]JI\: 4.881 4.833 4.7651
*]J2 and ]J3 4.759 4.763 4.786
a .102 .107 .111
81 .006 .006 .006
FemalesI\:
]J1 4.944 4.918 4.936
I\: 4.749 4.738 4.746]J2 and ]J3
.067 .063 .066a
81.007 .007 .007
lIlL 257.948 255.523 255.446
2 4.850 5.004X
Signif. level >.10 .082
it a bound'Converged to
*Means are adjusted to age 30
130
b. Left Pedigree
Unrestricted ~lende1ian Envirorunenta1
1 11.000# 1.000 .717
1 2 .659 .500 .717
1 3 .000# .000 .717
~1 .087 .108 .102
~2 .899 .892 .800
~3 .013 .000# .098
Males
*~1 4.988 4.992 4.966
*~2 and ~ .. 4.773 4.775 4.770
.)
a .076 .076 .080
61 .006 .006 .006
Females
*~l 4.913 4.936 4.907
'Ie
~2 and ~3 4.760 4.791 4.744
a .057 .068 .052
61 .008 .007 .009
.fuL 84.916 84.467 84.492
2 .898 .848X
Signif. level >.10 >.10
#Converged to a bound
*Means are adj usted to age 30
c. Both Pedigrees
Unrestricted Mendelian Envirorunental
T11. 000# 1.000 .484
TZ .631 .500 .484
T3 .000" .000 .484
1jJ1 .154 .175 .178
IjJZ .834 .8Z5 .800
1jJ3 .01Z .000" .0Zl
Males1:
]..11 4.803 4.843 4.909
*]..12 and ]..13 4.787 4.773 4.761
0 .115 .111 .099
61 .006 .006 .006
Females1:
]..11 4.910 4.919 4.933
*]..IZ and ]..13 4.741 4.748 4.758
0 .063 .064 .070
61 .008 .008 .007
131
in L 337.875
2X -.-..,.
Signif. levelZHeterog X 9.97814
/#Converged to a bound
1:Means are adjusted to age 30
336.901
1.948
>.106.178
334.539
6.672
.03610.798
132
hypothesis. For the larger Right Pedigree, the ratio of the likelihoods
in favor of the recessive hypothesis is almost 17; for Both Pedigrees
combined, the ratio is 46.2. The evidence shows a clear preference
for the recessive mode of inheritance over the dominant mode. Further-
more, there is no evidence of any departures from Hardy-Weinberg
equilibrium.
For the Right Pedigree, the chi square for testing the environ
mental hypothesis, HO: Ll = L2 = L3' is 5.004. Since two of the
parameters, when the unrestricted likelihood is maximized, converged to
a bound, the nominal significance level is 0.082 with the actual
significance level being somewhat lower. Strictly speaking, the
environmental hypothesis cannot be rejected. Although the Mendelian
hypothesis cannot be rejected (chi square is 4.85), two points must be
noted. Under the unrestricted model, the parameter estimate Ll is
0.168 when theoretically it should be 1.00. Standard errors for the
estimates can be computed numerically by double differentiation of
the likelihood surface at its maximum using ~~XLIK (Kaplan and ElstonA
1972). The computed standard error for Ll is 0.262 which indicates
that Ll is significantly different from 1.00. Secondly, the
difference in the two means for the males is not statistically
significant, suggesting that one distribution will fit the data for
males.
For the Left Pedigree, neither the environmental hypothesis nor
the Mendelian hypothesis is rejected. Since the likelihoods of the
pedigree data under the unrestricted model, under the Mendelian model,
and under the environmental model are very similar, this suggests that
the likelihood surface is very flat; consequently, there is little
133
evidence for a major gene segregating for systolic hypertensio~in the
Left Pedigree.
For Both Pedigrees combined, the environmental hypothesis is
rejected (0.01 < P < 0.05) but the Mendelian hypothesis is not rejected.
There is bimodality in the data for females, but the means of the dis
tributions for males are not significantly different; this is certainly
consistent with the results obtained N'hen mixtures of distributions
were fitted ignoring the pedigree structure (Tables 3.4a and 3.4b).
This is also reflected in the fact that the percentage of the variation
for systolic blood pressure accounted for by a major gene is only 5.4%
for males but 50.6% for females. The test that the estimates of the
Left Pedigree and those of the Right Pedigree are homogeneous is not
rejected.
Although there does not appear to be a ~jor gene for systolic
hypertension segregating in the Left Pedigree the interpretations of
the results for the Right Pedigree cannot be unequivocal. Recalling
the four criteria for inferring genetic segregation at a major locus,
all of them, to some extent, have been satisfied by the Right Pedigree.
For the Right Pedigree, the genetic hypothesis cannot be rejected
although the estimate of 1'1 is significantly different from one.
The environmental hypothesis is not rejected, however the nominal
significance level is 0.082 but the actual one is smaller. There is
clear preference for the recessive hypothesis. A mixture of two
distributions fits the systolic blood pressure data better than a single
distribution for females but not for males. This evidence suggests
that perhaps there may be a major gene for systolic blood pressure
segregating in females in the Right Pedigree. The analysis done on
both pedigrees combined does not disagree with this conclusion.
134
4.4.4 Diastolic Blood Pressure
The maximum likelihood estimates of the parameters are tabulated
in Table 4.6. In both the Left Pedigree and the Right Pedigree, the
dominant and the recess i ve modes of inheri tance are about equally
likely. However, when the two component pedigrees are combined, the
likelihood under the dominant hypothesis is almost 350 times larger
than the likelihood under the recessive hypothesis. This is a
curious and probably a spurious result.
In the Right Pedigree as well as in the Left Pedigree, both the
major gene ID1d the environmental models fit the data; neither hypo
thesis can be rejected. Such a flat likelihood surface indicates
that there is little evidence that there is a major gene segregating
for diastolic blood pressure in this pedigree. Furthermore, recall
that, ignoring the pedigree structure, in neither the males nor the
females does a mixture of two distributions fit the diastolic blood
pressure data significantly better than one distribution (Tables 3.4c
and 3.4d) .
4.5 Results of Bivariate Analyses
The reason for looking at more than one trait at a time is that,
in some instances, doing so may lead to a better separation of groups.
This is not very likely in the case of Sperry cholesterol and Zak
cholesterol as the two traits are so highly correlated (correlation of
0.94). But, in the case of systolic and diastolic blood pressures
(correlation of 0.63), analyzing the traits jointly may produce a
clearer genetic ~1alysis than examining each trait separately. Further
more, blere may be a major gene controlling the correlated portion of
the two traits .
135
Table 4.6 Maximum Likelihood Estimates From UnivariatePedigree Analysis of Diastolic Blood Pressure
a. Right Pedigree
Unrestricted r.1ende1ian Environmental
L11.000# 1.000 .667
L2 .578 .500 .667
T~ .000# .000 .667.)
\til .580 .493 .318
\tI2 .009 .507 .050
\tI3 .411 .000# .632
Males
*].11 4.452 4.446 4.447
* 4.339].12 and ].I~ 4.317 4.302.)
(J .101 .098 .107
61 .015 .015 .016
62 -.0002 -.0001 -.0002
Females
*].11 4.352 4.356 4.356
*f.l2 and f.l~ 4.305 4.298 4.320.)
(J .10S .104 .106
61.007 .007 .007
fuL 243.473 242.323 241. 774
2 2.300 3.398X
Signif. level >.10 >.10
#Converged to a bound
*Means are adjusted to age 30
136
b. Left Pedigree
Unrestricted Mendelian Environmental
T1 .651 1.000 .368
TZ .434 .500 .368
T3.000# .000 .368
WI .072 .070 .067
4J2 .927 .829 .833
W3 .001 .100 .100
Males
*~1 4.745 4.593 4.734
*~2 and ~3 4.401 4.392 4.400
a .091 .097 .091
81.004 .004 .004
Females
*~1 4.634 4.634 4.634
*~2 and ~3 4.400 4.400 4.401
a .047 .047 .047
81 .003 .003 .003
fuL2
X
88.411 87.898
1.026
87.972
.878
Signif.1eve1
#Converged to a bound
*Means are adjusted to age 30
>.10 >.10
137
There is one addition to the list of parameters in Table 4.2:
p, the conunon correlation between the two traits. Of course, the
means (lJ' s) will now be vectors with two elements, one for each
trait.
4.5.1 Sperry Cholesterol and Zak Cholesterol
The maximum likelihood estimates of the bivariate pedigree
analysis are tabulated in Table 4.7. The dominant mode of inheritance
for both traits which was assl.D11ed in doing the univariate analysis is
retained for the bivariate analysis. For both traits, the means of
the component distributions and the conmm variances in Table 4.7
are almost identical to those obtained by fitting mixtures of bivariate
log normal distributons ignoring the peuigree structure. For the Left
Pedigree ~ld the Right Pedigree, and for both pedigrees combined, there
is no significant departure from Mendelian segregation. The frequency
distribution of the three genotypes obtained in bivariate analyses is
siITlilar to those obtained for both Sperry cholesterol and Zak
cholesterol in univariate analyses. An estimated 8~o of those "external"
to this pedigree have the gene for hypercholesterolemia. These
results only reinforce the conclusion that there is a major gene
segregating for high cholesterol regardless of whether measurement
is made by the Sperry method or by the Zak method.
4.5.2 Systolic and Diastolic Blood Pressures
The maximl.D11 likelihood estimates of the parameters are tabulated
in Table 4.8. When the pedigree relationships are ignored, a mixture
of two bivariate distributions fits the blood pressure data signifr
cantly better than one distribution for females only, and not for
males (Tables 3. 5a and 3. 5b). Even for the females, the estimated means
138
Table 4.7 ~mximum Likelihood Estimates of Bivariate PedigreeAnalysis of Sperry Cholesterol and Zak CholesterolLevels
MendelianSperry Choles Zak Choles
1.000
.500
.000
.000"
.058
.942
a. Right Pedigree
UnrestrictedSperry Choles zak Choles
1"1 .698
1"? .316..1 3
.000#
~1 .058
~2.000#
~3 .942
Males;Ie
].J.1 and ].J.2 5.332 5.939;Ie
].J.- 5.238 5.396.)
(J .183 .160
p .804
81 .005 .005
Females;Ie
].J.l and ].J.2 5.832 5.893;Ie
].J.- 5.196 5.377.)
(J .166 .145
p .843
61 .007 .007
fuL 352.6082
X3
itConverged to a bound
;Ie
Means are adj us ted to age 30
5.832
5.239
.182
.005
5.829
5.197
.166
.007
.804
.845
350.095
5.026
P >.10
5.940
5.396
.159
.005
5.890
5.378
.145
.007
139
b. Left Pedigree
Unrestricted MendelianSperl)" tholes zal< Choles Sperry Cho1es Zak Cho1es
T1 1.000# 1. 000
T2 .374 .500
T3 .007 .000
~1 .125 .128
IV2 .000# .000#
IV- .875 .872.)
t-ta1es
'*J.l1 and J.l 2 5.730 5.892 5.735 5.893
'*J.l3 5.158 5.319 5.156 5.316
CJ .150 .124 .148 .122
p .910 .903
81 .010 .008 .010 .008
Females
"J.ll and J.l2 5.701 5.772 5.701 5.772
*lJ3 5.114 5.265 5.114 5.264
CJ .170 .126 .169 .126
p .730 .730
81 .005 .006 .005 .006
fuL 125.965 125.561
2 .808X_.:> p >.10
#converged to a bound
'*Means are adjusted to age 30
140
c. Both Pedigrees eUnrestricted Mendelian
Sperry Choles Zak Choles Sperry Choles Zak CholesII .678 1.000
l2 .319 .500
l3 .000# .000
1Ji1.082 .000#
1Ji2.000# .080
1Ji3.918 .920
Males'Ie
f..!1 and f..!2 5.823 5.936 5.824 5.935'Ie
f..! ... 5.221 5.377 5.221 5.377.)
a .179 .153 .178 .153
p .823 .824
81 .006 .005 .006 .005
Females
'"f..!1 and f..!2 5.793 5.861 5.786 5.855'Ie
f..!3 5.171 5.346 5.172 5.346
a .174 .150 .175 .151
p .828 .831
81 .006 .006 .006 .006
fuL 466.124 463.044
2 6.160x....) >.10P
2 24.898 25.224Heterog X24
itConverged to a bound
'IeMeans are adjusted to age 30
141
Table 4.8 Maximum Likelihood Estimates of Bivariate PedigreeAnalysis of Systolic and Diastolic Blood PressureLevels
a. Right Pedigree
Unrestricted MendelianSystolic BP Diastolic BP Systolic BP Diastolic BP
Tl .430 1.000
T2 .489 .SOO
T- .248 .000~
WI .190 .163
W2 .810 .785
W3 .000 .052
Males
"~1 4.851 4.246 4.852 4.343
"~2 and ~3 4.769 4.381 4.763 4.365
(J .108 .111 .106 .120
p .735 .591
61 .006 .013 .006 .012
62 -.0001 -.0001
Females
*~1 4.899 4.314 4.920 4.344
"~2 and ~3 4.753 4.336 4.747 4.326
(J .084 .107 .080 .109
p .860 .839
61 .008 .007 .007 .008
tnL 537.102 530.847
2 12.510X3- ---
p = .006
"Means are adjusted to age 30
142
b. Left Pedigree
Unrestricted MendelianSystolic BP Diastolic BP Systolic BP Diastolic BP
1 11.000# 1.000
1 2 .459 .500
1 3.000 .000
tJi1.091 .085
tJi 2 .909 .915
tJi 3.000 .000
Males
'"loll 4.994 4.547 4.993 4.547
*lol2 and lol3 4.776 4.378 4.776 4.377
a .077 .094 .077 .093
p .433 .428
61 .006 .004 .006 .004
Females'it
loll 4.981 4.634 4.981 4.633:'I
lJ2 and lJ3 4.804 4.400 4.804 4.401
a .068 .047 .068 .047
p .407 .407
61 .006 .003 .006 .003
fuL 189.275 189.185
2 0.180X3P >.10
#Converged to a bound
*Means are adj usted to age 30
143
for diastolic blood pressure in 'Table 3.5b are very similar, suggesting
that the bivariate fit is dominated by systolic blood pressure.
For the Right Petiigree (Table 4. Sa), the Mendelian hypothes is is
rejected as the chi square statistic is 12.510. For the males, there
is really only one bivariate distribution for blood pressure; for
females, the means for diastolic blood pressure are similar. This
reflects the results from bivariate curve fitting analysis.
For the Left Pedigree, the major gene hypothesis is not rejected,
but neither is the environmental hypothesis. With a likelihood
surface being so flat, there is no evidence for the existence of a
major gene for systolic anti diastolic blood pressure jointly.
4.4 Conclusions
The univariate pedigree analysis indicates that there is a major
gene segregating in tIris large pedigree for hypercholesterolemia.
This is confinued by the bivariate analysis. The univariate analysis
uncovers little evidence for a major gene segregating for diastolic
blood pressure, but there is a suggestion that perhaps there may be
a major gene for systolic blood pressure segregating among females
in the Right Pedigree.
It should be noted that when Elston and Stewart (1971) derived
the likelihood for doing pedigree analysis, random mating was assumed.
As Table 2.9 showed, there is evidence of assortative mating for these
traits as indicated by the inter-spouse correlations: 0.44 for systolic
blood pressure, 0.41 for Zak Cholesterol, and 0.22 for diastolic blood
pressure and for Sperry cholesterol levels; the first two inter-spouse
correlations are statistically significant.
The question is what is the effect of assortative mating on the
144
analysis. ~~cLean, Morton, and Lffi~ (1975) performed a series of
simulation experiments on nuclear family data to study the power of
segregation analysis of quantitative traits and the robustness when
various assumptions of the model are violated.
To test tIle effect of assortative mating, they generated data
with total genetic correlation between mates. The samples were
generated with no major locus, and their results show that, even in
the presence of an inter-spouse correlation of one, the likelihood
ratio for testing for a major gene never approached significance.
However, it should be noted that they were considering the likelihood
of the phenotypes of a sibship ~onditional on the parental phenotypes
so whether or not assortative mating can simulate a major locus in the
unconditional situation is not yet resolved.
Li (1975) points out that, while the observed correlation between
husband and wife with respect to a trait is the phenotypic correlation,
what is of interest is the genetic correlation between spouses. Under
the assumption that gene effects are additive both intra-locus mld
inter-locus (i.e. no donunance nor epistasis) and that environmental
and genetic effects are uncorrelated, Li states that the relationship
between the phenotypic correlation r and the genetic correlationpp
m can be expressed by
m = h2 r pp
where h2 is the heritability.
Therefore, although the inter-spouse correlations (phenotypic
correlations) for systolic blood pressure is 0.44 and for Zak cholesterol
is 0.41, the genetic correlations, based on estimates of heritability
145
from Chapter VI,are 0.2 and 0.28, respectively. Although the genetic
correlations are smaller than phenotypic correlations, their effect
on pedigree analysis must await fur:her simulation experiments.
Finally, as Table 2.6 shows, for systolic blood pressure and
Sperry and Zak cholesterol, even after taking the log-transfonmation
of the measures of these traits ~ld age-adjusting, the resulting
distributions still are significantly skewed. Is it possible that, in
the presence of skewness, there is a serious danger of detecting
spurious maj or loci? Go, Elston, and Kaplan (1978) use r·1onte Carlo
methods to simulate data with skewness in order to test the robustness
of pedigree segregation analysis. They conclude that "skewness per se
will not lead to the detection of a spurious locus." Their studies
show that only in the presence of polygenic inheritance as well as
sibling environmental correlation is there a possibility that skewness
in the data may falsely detect a major locus.
QIAPTER V
TIlE POLYGENI C I1YFarllESI S
In this chapter, the polygenic model will be considered. Under
this model, we assume that a genotypic value is made up of an "infinite"
number of equal and additive gene effects. Conceptually, we can think
of the phenotype as being determined by the genes at an infini te
number of unlinked loci. At each locus, assume that there are only two
alleles, one which has no effect and the other which increases the
measure of the quantitative trait under study. Further, assume that,
at each locus,the magnitude of the gene effect is equal, and that the
total gene effect at all the loci is simply the sum of the gene effects
at each lOCUS, i.e. there is no interlocus interaction.
Elston and Stewart (1972) have derived the likelihood of a set of
pedigree data under this polygenic model. Their procedure will be
briefly described in the next section, and will be used to determine
the significance of additive genetic effects on serum cholesterol
levels, systolic and diastolic blood pressures under this polygenic
model. In addition, several other variables whose measurements were
obtained on this pedigree will be similarly studied.
5.1 The Polygenic f'.lodel
Consider a pedigree with n individuals and a measure of a
quantitative trait x. ,1
(i = l, ... ,n) on each individual. The
mathematical model for x. under polygenic inheritance can be written1
147
asx.. = ~ + g. + e.
1 1 1[5.1]
where ~ is the overall mean, g.1
is the effect due to a large number
of additive genetic factors, ~ld e. is an environmental effect; the1
two random effects are assumed to act independently. The polygenic
the polygenic effect can be partitioned into two
the additive genetic variance.
ise
Since the
(J2 is calledg
is distributed
are to be estimated.
and the environmental effect
Thus the phenotype x.1
to additive genetic factors,
..,o~),
and 02e
N(O,
20g
is dueg
2 2Ocr + 0e)'
eo
For offspring,
polygenic effect
effect is distributed
distributed N(O, 0;);
independent components,
g. = b + y.1 1
[5.2]
where b is the midparental effect and Yi is the individual deviation
from the midparental effect. Both b and y. are normally distri1
buted with mean zero, and under random mating, b and y. each1
contribute half the variance of g.:1
Thus, if the parental genotypic effects are gM and gF' under
panmixia, the p.d.f. of the genotypic effect within each sibship is
1 2N(2 (gr-.tgF)' °g/2) .
The notation ¢(u, ( 2) will be used to denote the ordinate at u
of the distribution N(O, ( 2):
1
¢(u, ( 2) = (2n0 2) -7 exp [_u2/(202)].
148
The ordinate at u of the distribution N(jJ, 02) is thus denoted by
1? 2 -...,. 2 2
¢(u-jJ, 0-) = (2no) W exp[-(u-jJ) /(20 )]
~ote that ¢(u-lJ, 02) = ¢(jJ-u, 0
2). In light of this, the ordinate at?
u of the distribution ~~(J.I, OW)
2of the distribution ~(u-jJ, 0 ),
is equivalent to the ordinate at zero
and both can be written ¢(u-jJ, 02).
In Chapter IV, it was stated that the likelihood of the set of
pedigree data under the major gene model could be expressed as a
function of three quantities: The genetic mechanism (expressed as a
genetic transition matrix), the phenotype-genotype relationship in the
form of ~(x), the conditional p.d.f. of observing phenotype x
given the uth genotype, and the p.d.f. of the genotypes among those
"external" to the pedigree, i.e. the original parents and individuals
marrying into the pedigree. For the likelihood under the polygenic
model, there are three corresponding quantities.
1. If the parental genotypic effects are gH and gp and the
genotypic effect for their offspring is
the latter being the p.d.f. of the genotypic
the probabilities Pstu' the p.d.f.
02;2) which is the ordinate at g.g 1
gM+gF 20l (2 ,;;gl 2) ,
of the distribution
2.
effect among the offspring given g~1 and gp'
The phenotype-genotypic value relationship is expressed by f(x. Ig·),1 1
the conditional p.d.f. of Xi given genotypic effect gi' The
distribution of x. conditional on g. is taken to be N(jJ + g.,1 1 1
and thus for an individual with measure x. ,1
f(x·lg·) =1 1
149
3. For individuals "external" to the pedigree in that they have no
parents in the pedigree,
is NCO, O~), and thus
the p.d.f. of their genotypic effect h
Zfeh) = ¢(h, 0g)'
It is assumed in deriving the likelihood that, given the parental
genotypic values, the genotypic values and the phenotypes of the off
spring are independent of one another. Thus, the likelihood L of
observing a sibship of size n with measures
n= IT fCxi!gr-t' gF)
i=l
X., xz' ... , x given1 n
=(
IT J f Cx. Ig .) f (g. IgAP gp). 1 1 1 1'.1 g.
1
where f means that everything following it is to be integrated withgi
respect to cr. from -00 to + 00.°1
With the assumption of random mating, i.e. an individual's pheno-
type is independent of the phenotype and genotypic value of his
spouse, it is not difficult to consider spouses in the model. Given
his parents' genotypic effects are g~1 and gp' the likelihood of
observu1g an individual with measure x and his spouse with measure
Y is
= fex Ig~l' gp) fCy)
150
Eence, the likelihood of a sibship and their spouses, given parental
genotypic effects g~l and gF' is
L = ~ f f (x. Ig.) f (g. I g~l gF) f f (y. Ih.) f (h.) .. 1 1 1 1. h. 1 1 11= gi 1
For persons with no spouse, the second integral is set equal to one.
Under the random mating assumption and the implicit assumption
that there is no separable common sibling environmental effect nor
common parent-offspring environmental effect, Elston and Stewart (1972)
derived the likelihood of observing a set of pedigree data under the
polygenic model as a series of products and integrals and as a function
of the three quantities f(g.lgM, gF)' f(x·lg·),1 • 1 1
and f(h.),1
as
defined above. The integrations are performed over all possible
genotypic values, and hence the limits of integration are from -00 to
+ 00. There is a Fortrml program (Green 1972) which constructs the
likelihood for simple pedigrees, i.e. pedigrees that originate from
a single pair of individuals.
2 2There are three parameters to the model: 0g' 0e' and ~. Subject
to the constraints that the variances are non-negative, the maximum
likelihood of the pedigree data can be obtained, maximizing over all
the unknown parameters.
1\e are also interested in estimating the polygenic heritability,
defined as the proportion of the total variance accounted for by the
222additive genetic variance, 0g/Cog + 0e); this estimate is obtained
by replacing the parameters in this expression by their maximum likeli
hood estimates. The likelihood ratio test will be used to test HO:
against H :a2o > O.g
151
As in Chapter IV, the logarithmic-transformed values of the traits
(properly age- and sex-adjusted) are used.
formed value of a trait for the
Let y. be the log-trans1
i th person in the pedigree and let
First, the values on both males and females
using sex-dependent regression coefficients. If
are
y. - bl
(w. - 30)1 m 1
adjusted value is
value for a female is y. 1
(blm
, blf) and (b 2m , b2f)
individual is a male, the
- 302); the corresponding
30) - b7f(w~ - 302) where_ 1
2-b 2m (Wi
blf(wi -
w. be his or her age.1
are adjusted to age 30
the i th
the least squares estimates of the regression coefficients for age and
age2, respectively. Then the values on females are further adjusted
to have the same mean and variance as males at age 30, using the
formula:
where Yf is the final adjusted value for females,
Yf,30 and Ym,30 are the female and male means,
respectively, at age 30,
Sf and sm are the standard deviations for females
and males, respectively, and
Yf,30 is the individual female value adjusted to
age 30.
5.1.1 Sperry and Zak Cholesterol
The ~~ estimates of the parameters for the polygenic model are
tabulated in Table 5.1. For Sperry cholesterol levels, the hypothesis
02 = a is rejected (P < 0.01). The estimate of polygenicg
152
Table 5..! Maximum Likelihood Estimates of the Parametersfor the Polygenic ~1odel for Sperry and ZakCholesterol by Pedigree
PedigreeTrait Left Both Right Ileterog 2
X-.)
Sperry Cholesterol
02 .074* .071 .068g2 .009* .028 .032°e
J..I 5.252* 5.290 5.304
Heritability (%) 89.1 72.1 67.8 3.44
.fuL 45.121 170.671 126.923
2 to test HO: 2 0 30.64 20.90Xl ° =g
Zak Cholesterol
2 .056 .050 .045°g
02 .020 .035 .040e
J..I 5.422 5.478 5.497
Heritability (%) 74.0 58.8 53.4
.fuL 37.257 135.832 100.156 3.16
2 HO
: 02 = 0 4.60 13.48 7.56Xl to test g
Signif. level .032 <.001 .006
*Obtained tmder constraint 2 + 2 20g °e =sT
.e153
heritability is 67.8% for the Right Pedigree, 89.1~ for the Left
Pedigree, and 72.1% for Both Pedigrees combined.
NWllerical problems were encowltered in obtaining the ML estimates
of the parameters in the Left Pedigree. The estimate for the environ-
mental variance 2oe converged to zero, which largely contributed to
obtaining overflmv-errors on the computer. In order to circumvent
these difficulties, the maximization was redone with the constraint
respectively, and the estimate for the
Under the assumption that the measures in the
that the estimates of the additive genetic variance and the
mental variance should sum to the sample total variance si,
2 2 U d h' . h .0e = sT' n er t IS constraInt, t e estlmates
2 25.252 for 0 , 0, and ~,g e
heritability is 89.1%.
are
environ-
. 2I.e. 0 +g
0.74, 0.009, and
Left Pedigree and in the Right Pedigree are independent, using the
method as outlined in Chapter IV, the heterogeneity x2 was computed
to be 3.44; thus, there is no indication that the estimates of the
parameters for the two pedigrees are not homogeneous.
The results for Zak cholesterol are almost identical to those
for Sperry cholesterol except that the estimate of heritability for
Sperry cholesterol is about 14% higher than the one for Zak cholesterol.
The hypothesis of no additive genetic variance is rejected (5% level
for the Left Pedigree and 1% level for the Right Pedigree and for Both
Pedigree combined). Again, there is no indication that the estimates
for the separate pedigrees are heterogenous.
5.1.2 Systolic and Diastolic Blood Pressure
The ML estimates of the parameters are shown in Table 5.2. For
systolic blood pressure, it is estimated that additive genetic effects
account for only 17-21% of the phenotypic variation, while for diastolic
2Heterog x
::J
Table 5.2
Trait
154
Maximum Likelihood Estimates of the Parametersfor the Polygenic ~b<1el for Systolic andDiastolic Blood Pressure by Pedigree
P d'. e 19reeLeft Both Right
Systolic BP2
0g
0;1..l
Heritability (%)
xi to test HO:
.002 .003 .003
.011 .010 .010
4.852 4.822 4.811
16.8 21.2 20.8
78.061 324.992 251.528 9.19(p=.027)
0.26 2.84 2.14
Diastolic BP2 .000# .004 .0030g...,
0" .012 .012 .012e
1..l 4.451 4.421 4.386
Heritability (%) 0.0 27.1 21.8
lIlL 80.548 308.311 234.941 14.36(p=. 002)
2 . 2 0 0.00 3.62 1.96Xl to test HO' o =g
DEstimate converged to a boundary value, zero.
155
blood pressure, the estimate is 22-27%. For both systolic and diastolic
blood pressures, the hypothesis HO: a~ = 0 cannot be rejected. This
suggests that, for these two traits, any familial correlation is not
significantly due to additive gene action. Furthermore, the hetero
geneity x2 of 14.36 for diastolic blood pressure indicates that the
estimates for the separate pedigree are not homogeneous; in particular,
the estimated mean is higher in the Left Pedigree.
5.1.3 Other Traits
Since the polygenic model can be expressed as a function of only
three parameters and since the Fortran program to obtain maximum like
lihood estimates of these three parameters is fairly efficient in terms
of computer time needed, it is interesting to investigate the herit-
ability of the other traits for '''hich data are available. The ML
estimates of the parameters are tabulated in Table 5.3.
Polygenic heritability is nearly non-existent in this pedigree for
height and for weight since the estimate for additive genetic variance
has either converged to zero or is very nearly zero. There is no
heterogeneity in the estimates between the component pedigrees for
either height or weight.
These results for height and weight are unexpected because other
studies (a recent example is Rao, Maclean, Morton and Yee 1975) have
found a significant additive gene effect for height and weight. In
fact, Rao et al (1975), in their segregation analysis of nuclear
families, fitted a mixed.model (segregation of a major gene together
with a polygenic background) to height and to cube root of weight and
concluded that there is no significant major locus for height or for
weight and that the polygenic heritability is significant. Their
156
Table 5.3 MaxirllUJil Likelihood Estimates of the Parametersfor the Polygenic Model for Other Traits byPedigree
Pedi£rees,
H .. 2Trait Left Beth Right e ...erog X7....
Height2 0.000# 0.0004 0.000#0a
'"2 0.006 0.0068 0.007ae
\.I 4.244 4.229 4.223
Heritability (%) 0.0 5.3 0.0
.en L 95.493 428.308 332.925 0.22
2 ..,.. 0 0.00 0.52 0.00Xl to test HO: a =g
Weight..,
er- 0.000# 0.0001 0.001g
02 0.047 0.048 0.048e
J.l 5.109 5.116 5.121
Heritability (%) 0.0 0.0 1.2
.fuL 50.362 239.667 189.683 0.76
2 2 = 0 0.00 0.00 0.12Xl to test HO: ag
Uric Acid2 .058 .094 .0940g
a~ .026 .029 .032
J.l 1.416 1.295 1.245
Heritability (%) 69.6 76.2 74.4
tnL 40.414 142.247 106.755 9.84..,
02 =(P = O. 02)
xi to test HO: a 9.52 37.14 29.72g
Signif. level .002 <.001 <.001
IEsttffiate converged to a boundary value, zero.
157
PedigreeTrait Left Both Right 2Heterog X3
Alpha-Lipoprotein,cr .. .020 .008 .006g
cr2 .001 .012 .013e
j.l 4.295 4.277 4.266
Heritability (%) 93.5 42.0 31. 7
.en L 64.392 236.569 176.133 7.91(P '"' .048)
2 2 0 4.82 3.46 1. 54Xl to test HO: crg =
Signif. level .028 .063 >.10
Beta-Lipoprotein2 .136 .138 .137ag2 .039 .071 .081cre
j.l 4.800 4.886 4.913
Heritability (%) 77 .6 66.0 62.6
tnL 21.822 66.399 45.795 2.44(P > .10)
2 .,Xl to test HO: cr" '"' 0 5.05 14.75 8.95
g
Signif. 1eve1 .025 <.001 .001
Cholesterol Esters2 .100 .073 .073crg
cr2 .003 .028 .031ej.l 4.940 4.983 4.994
Heritability (%) 97.6 72.5 70.1
tnL 45.255 167.972 123.926 2.42(P > .10)
2 . 2 0 8.40 29.90 22.50Xl to test HO' crg '"'
Signif. level .004 <.001 <.001
158
PedigreeTrait Left Both Right 2Heterog X...
.)
Prebeta-Lipoprotein2 .062 .123 .149crg
cr2 .084 .174 .197e
J.1 2.993 3.048 3.067
Heritabili ty (%) 42.3 41.4 43.0
.tIlL 21.444 29.176 12.791 10.12(P '" . 018)
2 2 0 1.92 4.72 2.90Xl to test HO: crg =
Signif. level >.10 .030 .089
Phospholipid2 .005 .005 .0060:cr0
cr2 .014 .014 .013e
J.1 5.701 5.720 5.724
Heritability (%) 26.8 28.1 31.2
.tIlL 72.930 305.222 234.172 3.76(P • .053)
2 2 0 1.02 3.90 3.96Xl to test HO: cr =g
Signif. level >.10 .047 .045
159
estimates of the heritabilities are 0.396 + 0.026 for height and 0.363
~ 0.027 for cube root of weight.
However, it must be emphasized that the results obtained in the
present investigation are specific to this pedigree; they should not be
generalized to apply to other pedigrees or other families. In this2pedigree, as shown in Table 2.5, age and age account for about 75%
and 85% of the total variation for height and weight, respectively.
Hence, once age is accolDlted for, the residual variation for height
and for weight is very small, and in light of this small residual
variance, it may not be surprising that efforts to further partition
the variance have been frustrated.
For uric acid, the estimated heritability is 70-76%. The x2
value of 9.84 indicates that there is significant heterogeneity (0.01 <
P < 0.05) in the estimates between the Left Pedigree and the Right
Pedigree, and a comparison shows that the estimate of the overall mean2 2is larger for the Left Pedigree while the estimates of 0g and 0e
are larger for the Right Pedigree.
The analysis of alpha-lipoprotein in the Right Pedigree shows that
the additive genetic variance is not significantly different from zero.
The chi square of 7.91 indicates significant heterogeneity (5% level)
in the estimates between the two sides of the pedigree. In fact, o~
is larger and 0; is smaller in the Left Pedigree. However, in the
Left Pedigree, the estimation of the variance components is based on
40 persons with available values while there are almost three times
that many in the Right Pedigree. Thus, the preponderant weight of the
evidence favors the conclusion that there is no significant additive
160
genetic variance in the inheritance of alpha-lipoprotein.
There is a significant additive genetic variance in the inheritance
of beta-lipoproteins, with the estimated heritability of about 66%.
These results for beta-lipoproteins mirror those obtained for Sperry
and Zak cholesterol, as the correlation between beta-lipoproteins and
either of the two measurements of cholesterol is more than 0.90
(Table 2.7).
Table 2.7 also shows that cholesterol ester levels and Sperry
cholesterol are almost completely correlated; there is a correlation
between cholesterol ester and either Zak cholesterol or beta-lipoprotein
of about 0.90. The estimated polygenic heritability for cholesterol
ester is 97.6% in the smaller Left Pedigree and 70.1% in the Right
Pedigree. Although there is heterogeneity in the heritability estimates,
there is no doubt that the additive genetic variance for cholesterol
ester is significantly larger than zero.
For prebeta-1ipoprotein, the chi square of 10.12 indicates some
heterogeneity between the estimates of the Left Pedigree and the Right
Pedigree. Indeed, the variance component estimates for the Right Pedigree
are more than twice as large_as those for the Left Pedigree. However,
there is little heterogeneity in the proportion of the total variance
accounted for by additive gene action - more than 40%, but the test of
2HO: 0g = a does not attain statistical significance until both
pedigrees are combined.
There is no evidence of heterogeneity in the ~~ estimates for the
phospholipid data. The estimated heritability is about 30%. The
hypothesis that the additive genetic variance is zero is rejected in
the Right Pedigree (5% level) but not in the Left Pedigree due possibly
161
to the small number of individuals with phospholipid data. Tllercfore,
the weight of the evidence suggests that there is an additive genetic
component in the inheritance of phospholipid.
5.2 Conclusions
The analyses in this chapter indicate that, under this polygenic
model, additive genetic effects are significant in this pedigree for
Sperry cholesterol, Zak cholesterol, uric acid, beta-lipoprotein, and
cholesterol ester. Since the model does not specifically allow for
environmental correlations between relatives, this must be interpreted
with caution. In this analysis, additive genetic effects are largely
confounded with environmental correlations. Under these same reserva
tions, for prebeta-lipoprotein and phospholipid levels, the analyses
suggest that there is a significant additive genetic component, but the
results are not as clear-cut as for the variables enumerated above. For
height, weight, alpha-lipoprotein, systolic blood pressure, and diastolic
blood pressure, the estimated additive genetic variance is either very
small or has converged to the bound of zero. These results are
specific to this pedigree.
The analyses in Chapter IV indicated that, while there is little
evidence for a major gene for diastolic blood pressure, there is a
suggestion that perhaps a major gene is segregating for systolic blood
pressure, at least in the Right Pedigree. Also, the evidence of
Chapter IV is rather persuasive that there is a major gene segregating
in this pedigree for hypercholesterolemia. In this chapter, it was
shown that there is a significant additive genetic variance for
cholesterol, and for systolic blood pressure, the estimated herit
ability is about 20%. The question is: Can the genetic variance be
162
apportioned into two components, the major gene component and the poly
genic component?
In the next chapter, we will try to answer this question by
attempting to analyze the cholesterol data as well as the blood pressure
data using a Inixed model. In the preSCJ1Ce of a polygenic background,
are the data consistent \~ith the presence of a segregating major gene?
If there is a major gene segregating, is there a significant additive
genetic variance in the residual variance?
Q-iAPTER VI
THE MIXED H)DEL
The mixed model will be considered ill this chapter. This is
a model that allows for segregation of a major gene together with a
polygenic and environmental background. In a sense, this model combines
features of the major gene model from Chapter IV with features of the
polygenic model from Chapter V.
Consider an autosomal locus with two alleles, A and B, and for
convenience, let the genotypes be numbered M = 1, AB = 2, and
BB = 3. Assume that each of these genotypes makes a specific contri
bution to the measure of the trait x. of the i th individual in the1
pedigree. Labeling this major genotypic effect m, the mathematical
model for the phenotype Xi under the mixed model can be written as
[6.1]
\vhere g. is the polygenic effect and e. the environmental effect1 1
considered in Chapter V. In this model, m has a discrete distri-
bution (see section 4.2), and g. and e. have continuous distributions.1 1
The three variables are assumed to be mutually independent. Each m
takes on the value of one of the three means ~l' ~2' or ~3'
corresponding to genotypes M, An, and BB, respectively, around \1hich
there is random polygenic and environmental variation. The polygenic
effect g is distributed N(O, a~), and e is distributed NCO, a~).
x. ,1
conditional on a particular ffi,
164
is distributed ~(m, o~ +
For offspring, g. can be partitioned into two independent1
components, as in Chapter V,
As before, b and y.1
g. = b + y ..1 1
are each distributed NCO, 02/2)g
[6.2]
Therefore,
as individual's genotypic value consists of two parts: a major geno-
typic effect mu(u = 1,2,3) and a polygenic effect gi'
It is assumed in deriving the likelihood that, given the parental
genotypic values, the p.d.f. of the genotypic effect and the phenotypes
of the offspring are independent. Let ms and mt (s,t = 1,2,3)
refer to the parental major genotypic effects. Thus, the likelihood
of observing a sibship of size n with measures Xl" ",xn given
ms ' mt , and the mid-parental effect b is
= IT f (x. Ib, m , mt
) .i 1 S
Let mu (u=1,2,3) refer to the major genotypic effect of a child. Now,
the probability that the ith child of a sibship has measure x.1
3given ms ' illt , and b is 2 1 f(x. 1m , b) fCm 1m , mt ). The secondu= 1 U U 5
qumltity of this eA~ression ca~ be rewritten as the more familiar
P t of C~apter IV.s u Thus,
LCxlm , mt , b)- s
ihe polygenic effect of the ith offspring is g ..1
165
L(xlm , mt , b) = 11 I p t r f(x·lm, g.) f(e·lb) [6.3]- s ius u )g. 1 U 1 1
1
where Jgi is used to mean that everything following it is to be inte
grated with respect to gi from - ~ to +~. lVith minor modification,
these p.d.f. 's are similar to those in Chapter V.
The first p.d.f.
value relationship.
f(x.lm , g.) expresses the phenotype-genotypic1 u 1
The distribution of xi conditional on the
pOlygenic -effect &i and the major genotypic effect ffiu2be N em + g., a) and thus f (x. Ig., m ) = ¢(x. - m -u 1 ell u· 1 u
distribution of g. given b is ~I(b, ii/2), and thus1 g
is taken to
2gi' ae)· The
f(gi 1b) =
2Hg. - b, a /2) .1 g
Under the assumption of random mating, i.e. an individual's phcno-
type is independent of the phenotype and genotypic value of his
spouse, the likelihood of observing a spouse with measure y( major
genotypic effect mv and polygenic effect h) is
L(y) = 13 ~v f f(Ylh, mv) f(h) [6.4]v=1 h
where f(Ylh, mv) = ¢(y - mv - h, a;), f(h) = ¢(h, a~), and ~v
is the proportion with major genotypic value mv (v = 1,2,3). For
individuals in ~he pedigree with no spouse, the likelihood in equation
6.4 is set equal to one.
Furthermore,implicit1y assuming no separable cornmon sibling
environmental effect nor common parent-offspring environmental effect,
Elston and Stewart (1972) indicated how the likelihood of observing a
set of pedigree data could be expressed under the mixed model as a
function of Pst u' f(xlg, ffiu)' f(glb), ~v' f(Ylh, mv)' and f(h)
166
where s, t, u, x, g, and b refer to persons in the pedigree, and
v, y, and h refer to persons "external" to the pedigree. The
resulting likelihood involves prouucts, summations, and integrations
and contains five parameters: 3 means, \.ll' \.lZ' and \.l3' and Z variance
1"\"2 and 1"\"2components , v vg e
6.1 ~~thod of Analysis
Consider a pedigree with n individuals numbered from 1 to n.
Let xi:: phenotype of i th individual, and let ;S = (xl', ... ,xn) be
the vector of phenotypes. t!ith 2 alleles at an autosomal locus, there
are three possible major genotypes for each individual. Let m be an
1 h ·th I . h· f h . thn x vector w ose 1 e ement IS t e maJor genoty'pe 0 tel
individual. The vector m will be called the genotypic configuration
for the pedigree. Since each person can be assigned any of 3 genotypes,
there are 3n genotypic configurations possible. The likelihood of
observing the pedigree, ,under the mixed model, can be written
L :: L f(~ Im) P (m)m
[6.5]
where the (multiple) summation is over all genotypic configurations.
~lany of the 3n genotypic configurations will not be compatible \vith
the pedigree structure, so for them, P(m) :: O. Even so, with any
moderate size pedigree, the number of terms soon exceeds the capacity
of modern day computers. Therefore, until the likelihood can be
computed, perhaps using numerical integration, methods to approximate
the likelihood ''lill be sougllt.
Ott (1978) suggests t~~ing a random sample of the genotypic con-
figurations. Suppose ng vectors were sampled. Since these vectors
were sampled at random, they are independent, and each has the same
probabili ty of occurring
is approximated by
167
ling' Thus, the likelihood in equation 6.5\' k k thL f ~ 1m) where m is the k sampledk ,..,
configuration, and the summation is over all the sampled vectors. l1ith
this random sampling scheme, some of the mk ~ay have ?(mk) close tokzero, and for others, P(E) may be relatively large.
An apparent improvement to random sampling may be to stratify the
3n vectors of m into those m for which P(m) is relatively large
and those for which P(m) is relatively small. Although this
s tratifieci sampling may be <l reasonable al ternative to random sampling,
there is no simple algorithm yet known to select, say the l most
probable genotypic configurations. Therefore, as a first look at the
mixeci model, we have decided to approximate the likelihood of observing
a set of pedigree data by computing the likelihood conditional on the
"most probable genotypic configuration".
6.1.1 Genotypic Classification of Individuals
Under a specific genetic hypothesis HO' for example Mendelian
inheritance, the likelihood of observing a set of pedigree data can be
computed and is denoted by L(PIHO' 2), where e,.., are the other para-
meters of the model. Specifically for the w~jor gene model, the
likelihood can be written L (P IT1=1, T 2= i, T 3""0, Q) where Q includes
the means, variances, regression coefficients, etc. Suppose the .th1
member in the pedigree has phenotype x ..1
Then the likelihood can be
expressed as the sum of 3 conditional likelihoods, each being the
likelihood of the pedigree where the i th individual has genotype
5 = I, 2,3 (.-\A, AB, BB).
[6.6]
168
The posterior probability that the i th individual should have genotype
t, given what is known about his relatives, can thus be estir;Klted as
"q (t IP, ~) = [6.7]
In this way,individual.
for the.th1
is the ~ estimates of the lUlknOwn parameters Q. Thus if
i th individual, then
where ~
q(uIP,~) is the max q(tIP, 2)t=1,2,3
u is the most probable genotype for the
the most probable genotype can be obtained for all members of the
pedigree, and let m* be the nxl vector of the most probable eeno
types. If m* is not the most probable genotypic configuration, it
should nevertheless be reasonabJy close to it. The vector m* should-be checked for consistency of genotypes within the pedigree, i.e. we
must not have P(m*) = o. The vector of phenotypes x and the vector-m* are used in the mixed model analysis.
The phenotypes used in the following analysis are the age and sex-
adjusted values described in Section 5.1. The Fortran program used in
the polygenic analysis has been modified to allow for the existence of
three means corresponding to the three major genotypic effects. Thus,
conditional on the estimated genotypic configuration, the likelihood
of the pedigree under the mixed model can be computed and maximtnn
likelihood estimates of the parameters obtained. The likelihood ratio
test will be used to test biO hypotheses: HO: ~l = ~2 = ~3 vs.
HI : ~l" ~2 or ~2" ~3' and HO: <1~ = 0 vs. HI : <1~ > O.
Rejection of the first hypothesis indicates that, conditional on the
estimated genotypic configuratiol1, more than one distribution needs to
be fitted to the phenotypes; evidence of this alone is not sufficient
169
to prove the existence of a major gene, but it is certainly consistent
with it. Rejection of the second hypothesis inJicates that, after
allo\~ing for the presence of a major gene effect, there is a signifi
cant additive genetic component to the residual variance.
6.2 Sperry and Zak Cholesterol
under the ~lendelian hypothesis C'r1 = 1, T 2 = ~, T 3 = 0) and
Hardy-Weinberg equilibritD11, the likelihood of the pedigree was used
to assign the most probable genotypes to all the individuals. The
resulting genotypic configuration for both measures of cholesterol may
well be the most probable one as, with very fe\v exceptions, the prob
ability of the most probable genotype for each individual, q(uIP, 8)from equation 6.7, is at least 0.90.
Using this configuration, the maximum likelihood estimates of the
parameters under the mixed model for Sperry and Zak cholesterol are
shm~ in Table 6.1. The estimates of total heritability are tabulated
in Table 6.2.
For Sperry cholesterol, the hypothesis HO: ~l = ~2 = ~3 is
decisively rejected indicating that the data are consistent with the
existence of a major gene segregating for hypercholesterolemia. In
the presence of tIns major locus, the hypothesis HO: a~ = 0 is also
rejected indicating a significant additive genetic component to the
residual variance. About 70% of the total variability in Sperry
cholesterol is accounted for by heritable effects, if environmental
correlations between relatives do not affect the trait. It should be
pointed out that for Sperry cholesterol in the Left Pedigree, after
the variability due to the major gene and the additive genes has been
accounted for, there is very little variability due to random environ-
170
mental effects. This is also true of Zak cholesterol in the Left
Pedigree.
For Zak cholesterol, the hypothesis HO: ~l = ~2 = ~3 is also
decisively rejected. There is signific~~t heterogeneity between the
estimates for the Left Pedigree and those for the Right Pedigree. In
addition to the much smaller estimate of 02 in the Left Pedigree,e
the estimated means are a little larger in the Right Pedigree. For
the Right Pedigree, in the presence of the major gene, there is an
additive gene component but it does not attain statistical significance.
As with Sperry cholesterol, about 70% of the total variability in
Zak cholesterol values can be accounted for by heritable effects.
Figure 6.1 shows a plot of the density functions for Sperry
cholesterol (Figure 6.la) and for Zak cholesterol (Figure 6.lb)
illustrating the mixture of two normal distributions. These plots are
of the cholesterol values for individuals in the larger Right Pedigree;
tIle log-transformed values have been adjusted to age 30. The best
cutoff point is tIle point on the abscissa at which the two component
p.d.f. 's intersect. As one C~l see, the area of overlap is small; the
two components are fairly clearly separated. About 5% of the Sperry
cholesterol values and about 9% of the Zak cholesterol values in this
pedigree are in the higher distribution.
6.3 Systolic Blood Pressure
The assigning of the most probable genotypes to all individuals
in the pedigree in the case of systolic blood pressure was not as
unequivocal as was the case for Sperry and Zak cholesterol, since the
difference in the probability between the most probable genotype and
the next probable genotype is small for many individuals. This suggests
171
Table 6.1 ~~imum Likelihood Estimates of the Parametersfor the r,Iixed Model for Sperry and ZakCholesterol by Pedigree
Pedigree ..,Trait Left Both Right l-ieterog X4
Sperry Cholesterol2 .030 .020 .0180g
02 .002 .014 .017e
fll and fl*Z 5.832 5.833 5.834
fl* 5.188 5.209 5.2153fuL 72 .231 268.574 199.682 6.68
(P > .10)2 .. tX -stat1st1c to tes :
i) HO: ].J.l=fl2=].J.3 53.53 195.81 145.52
•• ) t.: 2 9.17 18.70 10.5911. 1iO: 0=0g
Zak Cholesterol
02 .017 .018 .013g
02 .004 .012 .017e
fl1 and fl~ 5.945 5.963 5.976
fl* 5.349 5.385 5.4003
fuL 63.450 215.522 159.052 13.96(P = .007)
2 .. t tX -stat1st1c 0 tes :
i) EO: fll=fl2=fl3 52.39 159.38 117.79
ii) nO: o~ = 0 9.08 10.92 2.60
*Means are adjusted to age 30
172
Table 6.2 Variance Component Estimates, Proportion of theTotal Variance, and Total Heritability Estimatesfor the Various Traits by Pedigree
2 2 2 TotalTrait Pedigree °mg 0g °e Heritability %
Sperry Cliol. Left .042 .030 .002 97.556.5% 41.0% 2.6%
Right .020 .018 .017 68.335.9% 32.4% 31.8%
Both .026 .020 .014 76.542.7% 33.8% 23.5%
Zak Cho1. Left .044 .017 .004 94.668.3% 26.3% 5.4%
Right .027 .013 .017 71.147.7% 23.4% 28.9%
Both .034 .018 .012 80.552.4% 28.1% 19.4%
Systolic BP Left .007 .000 .005 57.457.4% 0.0% 42.6%
Right .005 .000 .006 45.045.0% 0.0% 55.0%
Both .006 .000 .007 44.344.3% 0.0% 55.7%
2 variance attributable to a major gene°rog2 additive genetic variance0g
2 environmental variance°e
Figure 6.1 Cor.~onent and Total Theoretical Density FWlctions
173
a. Sperry Cholesterol
Estimated • I
• I,
t t• ,• • I
2 i percentile• I , I
128 237,
I,II I
97 i percentileI I266 I 493
Mean 184 342
Best Cutoff Point 295
b. Zak Cholesterol
Estimated II I I,I I I
2 ~ percentileI• I I158
I I97 ~ percentile 311 I 553IHean 2 1 394
Best Cutoff Point 333
174
that there may be many genotypic configurations for systolic blood
pressure clustering close to the most probable configuration.
Table 6.3 tabulates the maximum likelihood estimates of the para
meters for the mixed model for systolic blood pressure. The total
heritability estimates are listed in Table 6.2. The hypothesis 110:
~l = ~2 = ~3 is rejected indicating that the data are consistent
witll a major locus for systolic blood pressure. Furthermore, all the
genetic variance in this pedigree, which constitutes about 45% of the
total trait variability, is accounted for by this major locus;
the estimate for cr~ converged to a bound, zero. However, it is
conceivable that if the sibling correlation is much larger than the
parent-offspring correlation, then this situation could mimic genetic
dominance. The large heterogeneity x2 is due to the fact that
the estimates for the means are larger in the Left Pedigree; however
note that the displacement (~l - ~3) is about the same in the Left
Pedigree and the Right Pedigree.
Figure 6.lc shows a plot of the density functions for systolic
blood pressure values in the Right Pedigree. Note that the area of
overlap for systolic blood pressure is larger than for serum choles
terol. Presuming a major locus for essential hypertension, figure 6.lc
suggests that blood pressure, as measured indirectly by the portable
sphygmomanometer, is not a good discriminator. Recall that Hall (1966)
suggest that measuring the blood pressure directly may have the effect
of more clearly separating subgroups. Also note that the estimated best
cutoff point for the Right Pedigree is 135 rom Hg , very similar to
the value of 140 rom Hg used by many clinicians and investigators.
175
Table 6. 3 ~1aximum Likelihood Estimates of the Parametersfor the !'fuced ~1ode1 for Systolic Blood Pressureby Pedigree
Systolic BP2
0g
02e
~i
~2 and ~3
lnL
2 . to tx-stat1s 1C 0
Pedigree
Left Both Right
.000# .000# .000#
.005 .007 .006
5.012 4.951 4.950
4.803 4.780 4.777
99.253 378.414 290.809
Heterog x24
23.30(P < .001)
42.38 106.84 78.56
#Converged to a boundary value, zero
*Means are adjusted to age 30
Figure 6.1
c. Systolic Blood Pre5sure
176
,Estimated
,I, fI
2 -i percentile 101I II
I! I97 1percentile
I1391 165
II
Mean 119 141
Best Cutoff Point 135
177
No attempt 'vas made to fit the diastolic blood pressure data to
the mixed model. In Chapter IV, there was no evidence of a major gene
segregating for diastolic blood pressure, and the polygenic analysis
in Chapter V failed to detect any significant additive genetic effect.
In addition, there was manifest heterogeneity between the Left and
Right Pedigree so that results from further analysis, if not meaning
less, would be difficult to interpret.
6.4 Conclusions
For both Sperry and Zak cholesterol, the genetic variance can be
apportioned into two components, a maj or gene component and, with
the possible exception of Zak cholesterol in the Right Pedigree, an
additive genetic component. However, it should be kept in mind that
the model does not include any effects to specifically allow for
environmental correlations between relatives so that the additive
genetic effects in this analysis may be inflated in the presence of
any envirop~ental correlations. The systolic blood pressure data are
consistent with the existence of a major gene.
rnAPTER VI I
S1J1-MARY AND CONCLUSIONS
Data from a five-generational pedigree with 235 individuals from
Bay City, Hichigan were analyzed without having to rely on arbitrary
cut-off points to dichotomize or trichotomize the quantitative data.
MacLean, l\forton, and Lew' (1975) have shown that, based on simulation
studies of nuclear families, there is an appreciable loss of infonna
tion when quantitative data are converted to a dichotomy or trichotomy.
The models used in the analyses are all multifactorial in the
sense that both genetic and environmental influences are considered to
be involved in detennining the phenotype. That is, regardless of
whether the genetic mechanism is a segregating major gene or segre
gation of genes at many loci or a mLxture of a segregating major gene
together with a polygenic background, environmental effects are also
included in the model.
Preliminary to doing any pedigree analysis, initial analyses
,vere done ignoring the familial structure of the data. Mixtures of two
or more univariate normal distributions were fitted to the serum
cholesterol and blood pressure data by maxirrn.Im likelihood methods in
order to test, using the likelihood ratio criterion, for significant
departure from single nonnal distributions. Significant departures
were found for serum cholesterol and for systolic blood pressure in
females, but not for systolic blood pressure in males nor for diastolic
blood pressure.
179
This dissertation considers three specific underlying genetic
models in performing pedigree analyses on quantitative traits. The
first model is the major gene moJel by which is meant a single gene
that can account for a significant portion of the phenotypic variance.
As developed in Chapter IV, for each individual in the pedigree, the
age-adjusted quantitative trait was assumed to come from one of three
lognormal distributions with a common variance, with the three pheno
typic distributions corresponding to the three genotypes of a two
allele locus. The likelihood of observing the phenotypes in the
pedigree was expressed as a function, among other parameters, of
three transmission probabilities (1'S), and goodness of fit of the
Mendelian hypothesis to the data was tested by comparing the maximum
likelihood obtained when the 1'S were allowed to vary between zero
and one with the maximum likelihood obtained when the 1'S were fixed
at their Menuelian values (11 = 1, 1 2:= ~, 1 3 = 0).
In Chapter V, the polygenic model was considered. Under this
r:lodel, the phenotype was determined by a large number of equal and
additive gene effects. There are three parameters to the model: an
overall r.l.ean j.J, an additive genetic variance 02 , and an environ-g
mental variance 02• The heritability, which is the portion of thee .
total variability of the trait in the pedigree that is due to heritable
effects, can be estimated assuming that environmental effects common
to relatives do not affect the trait. The likelihood ratio test was
used to test for a significant additive genetic variance.
Finally, in Chapter Vl, the mixed model was considered; this
model allrn~s for segregation of a major gene together with a polygenic
and environmental background. The importance of this model is that it
180
simultaneously considers the two extreme genetic hypotheses - major gene
on the one hanJ and polygenic inheritance on the other. Using this
model, we can analyze the data to determine if most of the genetic
variation is due to one locus or if many loci are involved.
iiitherto, analyses of pedigree data using the mixed model have
not been attempted for lack of an efficient algorithm to calculate the
likelihood of a pedigree under the mixed model. The problem is, with
n individuals in the pedigree and with 2 alleles at an autosomal locus,
there are three possible major genotypes for each individual and 3n
genotypic configurations possible for the pedigree. IIi th any moderate
size pedigree, the number of terms soon exceeds the limits of modern
day computers. Therefore, as a first look at the mixed model, we
decided to approximate the likelihood function by computing the likeli
llood conditional on one genotypic configuration, namely the most
probable configuration.
The most probable genotypic configuration for the pedigree is
approximated by obtaining tile most probable genotype for each
individual in the pedigree. The most probable genotype is the one for
which the posterior probability that an individual should have that
genotype is maximized. The result is an n x 1 vector of the most
probable genotypes for all individuals in the pedigree. If this vector
is not the most probable genotypic configuration for this pedigree,
it should nevertheless be reasonably close to it. Thus, conditional on
this estimated genotypic configuration, the likelihood of the pedigree
under the mixed model was computed and maximum likelihood estimates
of the parameters, three means and two variance components, obtained.
The likelihood ratio test was used to test whether, conditional on the
181
estimated genotypic configuration, more than one distribution needed
to be fitted to the phenotypes; a significant departure from a single
distribution is consistent with the existence of a major gene. A test
of the hypothesis that the additive genetic variance is zero was done;
rejection of the hypothesis indicates that, after allowing for the
presence of a major gene, there remains a significant additive genetic
component to the residual variance.
The analyses indicated an autosomal dominant gene for hyper
cholesterolemia segregating in this pedigree, regardless of whether
the serum cholesterol levels are measured by the Sperry method ~r the
Zak method. It is estimated that about 70% of the total variability
in serum cholesterol in this pedigree can be accounted for by heritable
effects. The existence of the dominant gene for hypercholesterolemia
in this pedigree agrees with the results of studies by Elston,
Namboodiri et a1 (1975) and Schrott, Goldstein et a1 (1972). This
result has been confirmed by the discovery of the biochemical
mechanism (Goldstein and Brown 1974, 1975) and the reported linkage
between a hypercholesterolemia locus and the C3 locus (Ott, Schrott
et al 1974; Elston, Namboodiri et al 1976; Berg and Heiberg 1977).
The same methods which detected the dominant gene for hyper
cholesterolemia in this pedigree were used to analyze the blood
pressure data. Apriori, since this pedigree was ascertained because
of hypercholesterolemia, the chances of also finding a major locus
segregating for higIl blood pressure is relatively low. Despite this,
the data are consistent with an autosomal recessive gene segregating
for high systolic blood pressure, at least in the larger Right Pedigree.
It is estimated that under the mixed model, about 45% of the total
182
variability in systolic blood pressure in the Right Pedigree can be
accounted for by heritable effects. The mixed analysis indicates
that, in the presence of the major gene, there is no additive genetic
crnlvonent to the residual variance; the residual variance is all
accounted for by environmental effects. The major gene analysis
estimates that about 23% of those marrying into the Right Pedigree
have this gene in the homozygous form. In an analysis which ignored
the pedigree structure, two distributions fit the systolic blood
pressure data significantly better than one in females, but not in
males.
Analogous analyses show that in the Left Pedigree neither the
Mendelian hypothesis or the environmental hypothesis could be rejected,
suggesting that the likelillood surface is flat and that there is little
evidence for a major gene for systolic blood pressure segregating in
the Left Pedigree.Little evidence for a major gene segregating for diastolic blood
pressure is detected in this large pedigree. In addition to the mani
fest heterogeneity between the Left and Right Pedigrees, the environ
mental hypotllesis and the Mendelian hypothesis have similar likeli
hoods, again suggesting a flat likelihood surface.
Various studies have reported an association between weight (or
obesity) and blood pressure. Consequently, the possibility that the
major gene detected for high systolic blood pressure in the Right
Pedigree may be due to an association of systolic blood pressure with
weight must be considered. In our analyses, weight is adjusted for
when adjustments are made for age. In this pedigree, age and age2
account for over 80% of the variability in weight. Although tlle
183
ordinary product moment correlation between systolic blood pressure
and weight is 0.65, the partial correlation coefficient, partialling
age and age2, is reduced to 0.24. Hence, in this pedigree, once
adjustments are made for age, it is unlikely for weight to have much
influence on the results.
Since a major gene is detected for high serum cholesterol, and
one is detected for high systolic blood pressure, and since there is
a correlation between systolic blood pressure and serum G~olesterol,
is it possible that we are observing the pleiotropic effects of only
one gene? In light of the analyses at the end of Chapter III, this
possibility is unlikely. tVhen mixtures of bivariate lognormal dis
tributions were fitted to the systolic blood pressure and serum
cholesterol data (regardless of whether measured by the Sperry method
or the Zak method), two local maxima were found, one corresponding to
serum c11olesterol and the other corresponding to systolic blood
pressure. TIlis result suggests that, if tllere is a major gene for
systolic blood pressure and a major gene for serum cholesterol, there
are two separate genes.
For several reasons, one must be cautious in interpreting the
results of the mixed model analysis. First, the analysis was done
conditional on just one genotypic configuration. The configuration
obtained for Sperry and Zak cholesterol, if they are not the most
probable vector of genotypes, should be very close to them. For
systolic blood pressure, the best that can be said is that the genotypic
configuration used in the ~lalysis is just one of many; hopefully all
of them are clustered near the most probable one.
184
Second, since the configuration was generated using a likelihood
computed assuming an underlying ~1endelian model, the resulting analysis
is biased toward detecting a major gene. It is difficult to determine
to what extent the results are affected by su~~ bias. Comparisons may
have to await more sophisticated numerical methods that will allow the
evaluation of the complete likelihood; the present conditional analysis
could then be replaced by a more general unconditior~l one.
Future research can be focussed in two broad areas: methodologic
and genetic. One methodolic problem has been mentioned above, to
numerically evaluate the appropriate likelihood to perform an uncon
ditional mixed model analysis. Another is to enrich the polygenic and
mixed model by including terms for environmental effects cornmon within
a family. Boyle (1978) has shown in theory how to incorporate an
effect due to common sibling environment, assortative matin~, and an
effect due to common family (parent-offspring) environment. The
challenge is to incorporate the theory intc existing computer
algorithms.
On a more theoretical level, work needs to be done to determine
the asymptotic distribution of the likelihood ratio when some of the
parameters have converged to a boundary value. In addition, the theory
behind testing for a significant fit of a mixture of two or more
univariate (or multivariate) normal distributions over a single normal
distribution should be examined. \volfe (1971) suggests, based on his
Monte-Carlo investigations, that likelihood ratios for mixture problems
are not distributed as a chi square distribution with the degrees of
freedom equal to the number of variables; his studies indicate that
doubling the number of degrees of freedom gives a better fit to the
sampling distribution.
185
In the area of genetics, it would be interesting to compare the
present results with a similar analysis on a pedigree in which it is
more likely for hypertension to be segregating or at least where the
prevalence of hypertension is higher. This might result in a clearer
genetic analysis. At the same time, the search for a biochemical defect
should be continued. However, in the absence of any demonstrable
biochemical defect, the most forceful evidence for a major locus
existing may be derived from linkage analysis. If the presumed locus
for hypertension could be linked, with high probability, to one of the
existing polymorphic genetic markers, then this would be considered
conclusive evidence for a major gene for hypertension.
Appendix 1
Secondary Hypertension: rlypertension OccurringAs a t'<lanifestation of a Known Disease.
1. Disease of the kidneys and urinary tract.
(a) Nephritis; chronic or acute glomerulonephritis.(b) Chronic pyelonephritis
(c) Coarctation of the renal arteries(d) Polycystic kidneys(e) Diabetic glomerulosclerosis
(f) Connective tissue diseases(g) Amyloid contracted kidney(h) Certain tumors
2. Adrenal cortical hyperplasia or tumor
(a) Cushing's syndrome
(b) Aldosteronism
3. Pheochromocytoma
4. Coarctation of the aorta
S. Pre-eclamptic toxemia of pregnancy
6. Post-toxemic hypertension
7. Miscellaneous conditions affecting the nervous system.
8. Oral Contraceptives
Sources: Mend10witz (1961)
Pickering (1968)
186
Appendix 2
List of Variables Observed in 1947
1. Sex
2. Age
3. Total Cholesterol (Bloor and Sperry Method)
4. Cholesterol Esters
5. Phospholipids
6. Total Lipids
7. Systolic Blood Pressure
8. Diastolic Blood Pressure
9. ABO Blood Group
10. !-N Blood Group
11. Rh Blood Group
187
Appendix 3
List of Variables Observed in 1958
l. Sex
2. Age
3. Weight
4. Height
5. Systolic Blood Pressure
6. Diastolic Blood Pressure
7. Pulse
8. Hemoglobin
9. Total Cholesterol Level (Sperry and Zak Hethods)
10. Free Cholesterol
11. Cholesterol Esters
12. Phospholipids
13. Total Fatty Acids
14. Serum Magnesium
15. Uric Acid
16. Ultracentrifuge
Alpha-lipoprotein
Beta-lipoprotein
Prebeta-1ipoprotein
17. ABO Blood Group
18. M~ Blood Group
19 Rh Blood Group
188
189
Appendix 4
Sex. ~e, Hei2ht. Wei¢ht. Systolic Blood Pressure, Diastolic Blood Pressure.Sperry and :&1; Ololesterol, Beta and Prebeulipoprotein Values for ~1el!t>ers
of the PediiTee Observed in 1958. C-l si~ifies data unavailable)
IIGl !It .';,.1 lIE IGH 'I SYS~ liP OIlS BP S PE IiIl Y ZU: BE'U PRE-S!:'! 1i'!F.SC::1 S~X (YPSI (UI (L8S) (~~ H"I (P1~ ilG) (l'lGJ) (''''1 (IlGIlI ('!Glll
L III :: F 73 64.5 172 160 100 207 242 138 213 1;" ~8.V 10; 0 150 80 1<:19 221 118 394 F 63 61.5 146 111J 84 199 229 113 49
L III 7,RlIIlb F' 'is 0;3.0 128 I'll) 110 2214 263 153 30L III 11 " 53 71.0 I~S 165 100 250 27e 1714 29
1.2 F ~6 0.5 157 130 90 300 325 1'77 3713 51 71. v 1<15 120 '10 199 -1 -1 -114 r 46 62.5 172 130 90 189 -1 - 1 -115 w 14 I'! 'i9.5 183 150 95 318 4~J 311 2916 ! 46 65.5 146 132 84 176 -1 -1 -11~
.."'~ 1'';.0 1149 195 1 10 14 18 4 JR 31 Q 38
L III 18.RlII 4 r. 59 6'.5 1'13 130 QO 4 I 8 435 31'1 44RIII ~ ,LIII 19 F 60 61.5 146 134 90 1453 464 JII9 :.16R III i ~ 61 6'.5 1138 130 90 160 21J 105 JIj
8 ,. 59 66.0 145 150 90 222 260 160 209 .. 53 611.~ 175 1110 100 15 J 191 97 20
10 F 52 -l.J 126 124 80 1'19 232 114 3911 F 6" 6 1.0 134 151' 914 200 21,~ 135 2~
1:: " 67 11<:1.0 163 130 05 189 201 11 1 813 ~ 65 lit-.v 195 120 80 165 -1 - 1 -1H F 65 62.5 164 140 80 266 -1 -1 -1
R III lS,LIII 8 w 67 66.0 161 140 85 241 269 148 411RIll 19 r. 50 66.v 155 1140 H 38e 411 280 31
:0 ! 147 614.5 15'\ 130 88 231 323 20 ~ 21\21 :1 69 67.5 194 170 110 19b 222 101 4822 F t9 60.0 leI 190 110 23 I 328 220 31
L IV 1 r 311 1\4.0 135 120 80 163 183 95 12~ 40 71.0 165 130 80 191 -1 -1 -I
3 , III 7l.v 200 130 ~o 1159 1<;5 99 74 F 38 60.5 1114 120 90 119 156 57 105 140 f:7. \) Hl0 1140 100 3106 414 289 246 P' 31; 64.5 173 132 84 196 240 1119 127 F li2 1i"'.0 115 1 HI 78 15 .. 1111 98 7S n 14 I ".5 ,qll 120 90 20 I 231 130 ,.,9 F 39 63.0 h2 140 lOS 191 231 127 13
10 ~ 3<:1 7 0.0 208 130 84 245 231 126 2712 ~: 37 1;9.0 1714 I 10 85 188 209 107 3113 i' 29 65.0 126 114 BO 19 I -, -1 -115 :- 32 62.0 I IS 120 SO 137 176 87 '216 .. 3.. ., 1. 5 175 170 120 212 252 136 26
R IV ~S F ~, 60.5 155 140 104 235 -I -, -I49 w uS '" 8.0 14E: 110 ~O 237 -1 -1 - I50 :1 J9 6e.~ 202 140 90 2145 263 lSI 41151 r 37 1>14.0 173 1214 all 153 168 100 10~. !1 H 6'.0 190 135 '00 2145 286 149 6033 .,. 34 - 1.0 131 1 18 BO 219 - 1 - I -154 I 30 61.0 130 130 90 217 2115 1119 12
L IV 15 F 3 I -1.0 - I lila loa 1111 215 i 1 5124 F 24 6 1.5 1'4 130 ao 186 205 10'! 19~5 ~ 110 67.v 114 1140 90 158 193 93 2126 F 27 67.5 143 130 88 2113 293 198 37,-
~ 2q - 1.0 ,- I 120 !l0 2'9 260 155 20.,28 ~ 25 7'.5 18' 120 75 114 7 193 92 1829 .,. 23 611.0
" ;2120 80 160 -1 -1 -I
30 - 21> E:9.0 IUS 130 80 11414 - I -, - I31 F 19 - 1.0 -1 120 80 222 161 71 20-, :- 1!\ - 1.0 -I 120 80 , 14 2 178 93 II~-33 H f>6.0 1)5 14;) SO 1144 175 86 Ie3~ ~ III 62.5 98 100 60 147 -, -1 -I
190
.~ u! H!lGi:T W!!CH": SIS! S? :H~S Bll SC'E:!lRY ZAK BlTA PRE-IlET AP!:i'SC~ S:X (ns) (IN) (L1:lS) (~!I HG) ('~ MG) ( ~G') llllO') (l~(" ) (llG\)
L IV 35" 20 7:1. V 200 1110 90 )110 - 1 - 1 - 1
R rv ~ 110 6).v 17~ Ill; SO 4)11 1162 296 9)l' 36 f, ~. 0 175 120 '311 207 260 156 10
3 l' )q 62.0 171 11111 911 388 11118 - I 224 ~ 112 66.0 160 1115 110 2::5 249 1211 30S !' )11 63.5 1118 111l 80 ) I II 260 - 1 -I9 ~ )7 6°.5 175 130 90 227 JOEl -1 -1
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BIBLIOGRAPHY
Acheson, R.H. and G.B. Fowler (1967). On the Inheritance of Statureand Blood Pressure. Journal of Chronic Diseases 20:731-745.
Adlersberg, D., A. Parets, E. Boas (1949). Genetics of Atherosclerosis. Studies of Families with Coronary Artery Disease Under theAge of Fifty Years. Journal of the American Medical Association.141:246-253.
Alvord, R.M. (1949). Coronary Heart Disease and Xanthoma TuberosumAssociated with Hereditary tlyperlipemia. Archives of InternalMedicine 84: 1002-1019.
Berg, K. and A. Heiberg (1977). Confirmation of Linkage Between Familial Hypercholesterolemia with Xanthomatosis and the C3 Polymorphism.IVth International Workshop on Human Gene Mapping~ Winnipeg, ~·1ani
toba, August 14-l~.
Borhani, N.O., M. Feinlieb, R.J. Garrison, J .C. Christian, R.H. Rosenman (1976). Genetic Variance in Blood Pressure. Acta GeneticaeMedicaeet GemeUologiae 25:137-144.
Boyle, C.R. (1978). The Likelihood of a Sibship Under Some PolygenicM>dels with Familial Correlation. Unpublished Hanuscript.
Burns, F.S. (1920). A Contribution to the Study of the Etiology ofXanthoma Multiplex. Archives of Dermatology and Syphilogy 2:415.
Cassel, J.C. (ed.) (1971). Evans COlUlty Cardiovascular and Cerebrovascular Epidemiologic Study. Archives of Internal Medicine 128:883-986.
Chakraborty, R., W.J. Schull, E. Harburg, H.A. Schork, P. Roeper (1977).Heredity, Stress, and Blood Pressure, A Family Set Method. V. Heritability Estimates. Journal of Chronic Diseases 30:683-699.
Chauffard, A. and G. LaRoche (1910). Pathogenie du Xanthelasma. SerrrlineMed 30:241.
Cruz-Coke, R. (1959). The Hereditary Factor in Hypertension. ActaGenetica et Statistica Medica (Basel) 9:207-212.
Cruz-Coke, R. (1960). Environmental Influences and Arterial BloodPressure. Lancet ii:885-886 .
194
Downie, W.W., J.A. Boyle, \'I.R. Grieg, W.W. Buchanan, F.P. A1epa (1969).Relative Roles of Genetic and EnviroI1r.1ental Factors in Control ofBlood Pressure in Normotensive Subjects. British Heart Journal 31:21-25.
Elston, R.C. and C.E. Bok1age (1978). An Examination of FundamentalAssumptions of the Twin Method. in International Twin Conference2nd, 1977, Washington, D.C. New York: Alan Liss, in press.
Elston, R.C., K.K. Namboodiri, C.J. Glueck, R. Fa11at, R. Tsang,V. Leuba (1975). Study of the Genetic Transmission of Hypercholesterolemia and liypertrig1yceridemia in a 195 Member Kindred. Annalsof Human Genetics 39:67-87.
Elston, R.C., K.K. Namboodiri, R.C.P. Go, R.H. Siervoge1, C.J. Glueck(1976) . Probable Linkage Between Familial Hypercholesterolemia andThird Complement Component (C3). Baltimore Conference (1975): ThirdInternational Workshop on Human Gene ~~apping. Birth Defects: Original Article Series, XII, 7, 1976, pp. 294-297, The National Foundation, New York.
Elston, R.C., K.K. Namboodiri, E.B. Kaplan (1978). Resolution of HajorLoci for Quantitative Traits. Submitted for Publication.
Elston, R.C. and J. Stewart (1971). A General ~bde1 for the GeneticAnalysis of Pedigree Data. Human Heredity 21:523-543.
Epstein, F.H., W.D. Block, E.A. Hand, T. Francis, Jr. (1959). FamilialHypercholesterolemia, Xanthomatosis, and Coronary Heart Disease.American Journal of Medicine 26: 39-53.
Falconer, D.S. (1960). Introduction to Quantitative Genetics. NewYork: The Ronald Press Company.
Feinlieb, ~1., M. Halperin, R. Garrison (1969). Relationship BetweenBlood Pressure and Age: Regression Analysis of Longitudinal Data.Presented at the Ninety-seventh Annual Meeting of the AmericanPublic Health Association, Philadelphia, Pennsylvania, November1969.
Frederickson, D.S. and R.S. Lees (1965). A System for PhenotypingHyperlipoproteinemia. Circulation 31: 321- 327 .
Frederickson, D.S. and R.S. Lees (1966). Familial Hyper1ipoproteinemia. In The Metabolic Basis of Inherited Disease, J. Stanburg,J. wyngaarden, and D.S. Frederickson (eds.) , Second Edition,pp. 429-485, New York: McGraw-Hill, Inc.
Frederickson, D.S. and R.S. Lees (1972). Familial Hyperlipoproteinemia. In The Metabolic Basis of Inherited Disease, J. Stanburg, J._wyngaarden, and D.S. Frederickson (eds.) , Third Edition, pp. 545614, New York: HcGraw-Hill, Inc.
195
Gearing, F.R., E.G. Clark, G.A. Perera, M.D. Schweitzer (1962). Hypertension Among Relatives of Hypertensives: Progress Report of aFamily Study. American Journal of Public Health 52:2058-2065.
Go, R.C.P., R.C. Elston, E.B. Kaplan (1978). Efficiency and Robustness of Pedigree Segregation Analysis. American Journal of HumanGenetics 30:28-37.
Godal, H.C., J.C. Lund, E. Sivertssen (1956). Lipids and Lipoproteinsin Serum in Cases of Familial Essential Hypercholesterolemia andXanthomatosis. Acta Medica Scandinavica~ supplement 319: 125-133.
Goldstein, J. and ~'I.S. Brown (1974). Expression of the Familial Hypercholesterolemia Gene in Heterozygotes: ~bdel for a Dominant Disorderin Han. Transactions of the Association of American Physicians 87:120-131.
Goldstein, J. and M.S. Brown (1975). Familial Hypercholesterolemia.A Genetic Regulatory Defect in Cholesterol Metabolism. AmericanJournal of Medicine 58:147-150.
Green, P . (1972). Personal Communications.
Guravich, J.L. and J. Venegas (1962). Familial Hypercholesterolemia.Federation Proceedings 21 (Part II): 44- 51.
Hall, E.C. (1966). Analysis and Decomposition of Blood Pressure Levelin a Population By Age, Race, and Sex. Unpublished Ph.D. Dissertation, University of North Carolina, Chapel Hill.
Hamilton, ~1., G.W. Pickering, J.A. Fraser Roberts, G.S.C. Sowry (1954a).The Etiology of Essential Hypertension. 1. The Arterial Pressurein the General Population. Clinical Sciences 13:11-35.
Hamilton, M., G.W. Pickering, J.A. Fraser Roberts, G.S.C. Sowry (1954b'.The Etiology of Essential Hypertension. 2. Scores for Arterial BloodPressures Adjusted for Differences in Age and Sex. Clinical Sciences13:37-50.
Hamilton, ~1., G.W. Pickering, J.A. Fraser Roberts, G.S.C. Sowry (1954c'.The Etiology of Essential Hypertension. 4. The Role of Inheritance.Clinical Sciences 13:273-304.
Harburg, E., J.C. Erfurt, C. Chape, L.S. Hauenstein, W.J. Schull,M.A. Schork (1973). Socio-ecological Stressor Areas and Black-lfuiteBlood Pressure: Detroit. Journal of Chronic Diseases 26:595-611.
Harburg E., J.C. Erfurt, W.J. Schull, M.A. Schork, R. Colman (1977).Heredity, Stress, and Blood Pressure, A Family Set Method - T. StudyAims and Sample Flow. Journal of m,ronic Diseases 30:625-648.
196
Harlan, W., J. Graham, E.H. Estes (1966). Familial Hypercholesterolemia: A Genetic and ~1etabolic Study. Medidne 45: 77 -1l0.
Harlan, W., R.K. Osborne, A. Graybiel (1962). A Longitudinal Studyof Blood Pressure. Circula~ion 26:530-543.
Harris-Jones, J.N., E.G. Jones, P.G. Wells (1957). Xanthomatosis andEssential Hypercholesterolemia. Lancet i:855-857.
Herndon, C.N. (1954). Genetics of the Lipidoses. Research Publications - .4ssociarion for Research in Nervous and Mental Disease33:239-258.
Hirschhorn, K. and C.F. Wilkinson (1959). The Hode of Inheritance inEssential Fami] ial Hypercholesterolemia. American Journal of Medicine 26:60-67.
Jensen, J. and D.H. Blankenhorn (1972). The Inheritance of FamilialHypercholesterolemia. American Journal of Medicine 52: 499- 516.
Jensen, J., D.H. Blankenhorn, V. Kornerup (1967). Coronary Disease inFamilial Hypercholesterolemia. Circulation 36:77-82.
Karme1, W. B. and T. Gordon (eds.) (1970). The Framingham Study. AnEpidemiological Investigation of Cardiovascular Disease. Washington,D.C.: U.S. Government Printing Office, Section 27.
Kaplan, E.B. and R.C. Elston (1972). A Subroutine Package for MaximumLikelihood Estimation (MAXLIK). Institute of Statistics Mimeo SeriesNo. 823, University of North Carolina.
Kaplan, E.B. and R.C. Elston (1975). GENPED - A General PedigreeAnalysis Package. Unpublished Program Write-up.
Kendall, M.G. and A. Stuart (1973). The Advanced Theory of Statistics J
Volume 2J Third Edition. New York: Hafner Publishing Company.
Khachadurian, A. (1964). Inheritance of Essential Familial Hypercholesterolemia. American Journal of Medicr~ne 37: 402-407.
Khachadurian, A. (1969). Lack of Inhibition of Hepatic Synthesis byDietary Cholesterol in Cases of Familial Hypercholesterolemia.Lancet ii:778-780.
Langer, T., W. Strober, R. Levy (1969). Familial Type I I Hyperlipoproteinemia: A Defect of Beta Lipoprotein Apoprotein Catabolism.Jeurnal of Clinical Investigarion 48:49a.
Leonard, J.C. (1956). Hereditary Hypercholesterolaemic Xanthomatosis.Lancet ii:1239-l242.
197
Levy, R. (1971). Classification and Etiology of Hyperlipoproteinemias. Federation Proaeedings 30: 829- 834.
Li, C.C. (1975). Assortative Mating in Man. Memorias de 1a IIRetmi6n Nacional de 1a Sociedad Mexicana de Genetica; Hazat1an,Sin., marzo, 1974. pp. 48-108. Ed. SMG, con el patrocinio deFornento Cultural ~X, Mex., D.F. November 1975.
Lilienfeld, A.(1959). A Methodological Problem in Testing a RecessiveGenetic Hypothesis in Human Disease. Ameriaan Journal of PubliaHealth 49: 199-204.
Lowe, C. R. and T. McKeown (.1g62). Arterial Pressure in an IndustrialPopulation and Its Bearing on the Problem of Essential Hypertension.Lancet i: 1086.
~lcDonough, J.R., G.E. Garrison, C.G. Hames (1964). Blood Pressure andHypertensive Disease Among Negroes and Whites, A Study in EvansCounty, Georgia. Annals of Internal Mediaine 61: 208-228.
McGinley, J., H. Jones, J. Gofman (1952). Lipoproteins and Xanthomatous Diseases. Journal of Investigative Dermatology 19: 7182.
McIlhaney, M.L., J. Shaffer, E.A. Hines, Jr. (1975). The Heritabilityof Blood Pressure: An Investigation of 200 Pairs of Twins Using theCold Pressor Test. John8 Hopkin8 Medical Journal 136: 57-64.
rlcKusick, V. (1960a). Genetics and the Nature of Essential Hypertension. Circulation 22: 857-863.
McKusick, V. (1960b). Medical Genetics - Hypertension. Journal ofCh~onic Di8eases 12: 112-114.
~mcLean, C.J., N.E. Morton, R. Lew (1975). Analysis of FamilyResemblance. IV. Operational Characteristics of SegregationAnalysis. Ameriaan Journal of Human Genetics 27: 365-384.
Meilman, E., C. Ho1t~, P. Samuel (1964). Familial Hypercholesterolemia and Xanthomatosis - Report on a Rare Type of Kindred. AmeriaanJournal of Mediaine 36: 277-280.
Mendlmvitz, M. (1961). Symposium on Hypertension. Introduction.Journal of the Mount Sinai Hospital~ New York. 28: 97-100.
Mendlowitz, ~f., S. Gitlow, N. Naftchi (1959). The Cause of EssentialHypertension. Perspeatives in Biology and Mediaine 2: 354-361.
;"lendlowitz, M., S. Gitlow, R. Wolf, N.E. Naftchi (1964). On the Integration of Factors in Essential Hypertension. Ameriaan HeartJournal 67: 397-409.
Mendlowitz, M., R.L. Wolf, S.E. Gitlow (1970). Catecholamine Metabolismin Essential Hypertension. Ameriaan HeG.I't Journal 79: 401-407.
198
Miall, W.E., P. Heneage, T. Khosla, H.G. Lovell, F. Moore (1967).Factors Influencing the Degree of Resemblance in Arterial Pressureof Close Relatives. CZiniaal Saienaes 33: 271-283.
~tiall, W.E. and H.G. Lovell (1967). The Relationship Between Changeof Blood Pressure and Age. British Mediaal Journal i: 660-664.
Miall, W.E. and P.D. Oldham (1963). The Heredity Factor in ArterialBlood Pressure. British Mediaal Journali: 75-80.
rbrrison, S.L. and J.N. Morris (1959). Epidemiological Observationson High Blood Pressure Witllout Evident Cause. Lanaet ii: 864-870.
~'brrison, S.L. and J.N. Mbrris (1960). Nature of Essential Hypertension. Lanaet ii: 829-832.
Morton, N.E. (1967). The Detection of Major Genes Under AdditiveContinuous Variation. Ameriaan JournaZ of Human Genetias 19: 2324.
Murphy, E. (1964). One Cause? Hany Causes? The Argument from theBinomial Distribution. Journal of Chronia Diseases 17: 301-324.
~urphy, E.A., C.B. Thomas, D.R. Bolling (1966). Genetic Factors inBlood Pressure Control? Evidence From the Distributions.Ciraulation 34 (Supplem. 3): 23.
National Center for Health Statistics (1966). Hypertension and Hypertensive Heart Disease in Adults. United States. 1960-1962. PubZiaHealth Serviae Publiaation No. 1000, Series 11, Number 13.
j~ational Center for nealth Statistics (1969). Blood Pressure, As itRelates to Physique, Blood Glucose, and Serum Cholesterol. Uni~ed
States. 1960-1962. Publia Health Serviae Publiaation No. 1000,Series 11, Number 34.
Neyman, J. and E.S. Pearson (1928). On the Use and Interpretation ofCertain Test Criteria for the Purposes of Statistical Inference.Biometrika 20A: 175 and 263.
Osborne, R.H., F.V. DeGeorge, J.A.L. Mathers (1963). The Variabilityof Blood Pressure: Basal and Casual Measure~ents in Adult Twins.Ameriaan Heart Journal 66: 176-183 .
Ostfeld, A. and O. Paul (1963). The Inheritance of Hypertension.Lanaet i: 575-579.
Ott, J. (1978). The Maxbnum Likelihood Estimates of the VarianceComponents in numan Pedigrees Under Polygenic and Hixed ~1odels ofInheritance. Submitted for Publication.
Ott, J., H.G. Schrott, J.L. Goldstein, W.R. Hazzard, F.H. Allen, Jr.,C.T. Falk, A.G. Motulsky (1974). Linkage Studies in a Large Kindredwith Familial Hypercholesterolemia. Ameriaan Journal of HumanGenetias 26: 598-603.
199
Pearson, E.S. and H.O. Hartley (Editors) (1954). Biometrika Tables fopStati8ticians,Volume I.Cambridge University Press.
Perera, G.A. (1960). Nature of Hypertension. Lancet ii: 153.
Pickering, G. (1961). The Natupe of Essential Hypeptension. Gruneanci Stratton, Inc.
Pickering, G. (1967). The Inheritance of Arterial Pressure. In TheEpidemiology of Hypertension, J. Stamler (ed.), :~ew York and London:Grune and Stratton, Inc., pp. 18-27.
Pickering, G. (1968). High Blood Ppessure. New York: Grune andStratton, Inc.
Piper, J. and L. Orrild (1956). Essential Familial Hypercholesterolemiaand Xanthomatosis. American Journal of Medicine 21: 34-46.
Platt, R. (1959). Nature of Essential Hypertension. Lancet ii: 55-57.
Platt, R. (1961). Essential Hypertension. Incidence, Course, andheredity. Annals of Internal lfedicine. 55: 1-11.
Platt, R. (1963). Heredity in Hypertension. Lancet i: 899-904.
Platt, R. (1967). The Influence of Heredity. In The Epidemiology ofHypertension, J. Stamler (ed.), New York and London: Grune andStratton, Inc., pp. 9-17.
Pollack, H. and D.E. Krueger (ed.) (1960). Epidemiology of Cardiovascular Diseases. Methodology. American Journal of Public Health(supplement) 50: 1-124.
Rao, C.R. (1962). Advanced Stati8tical Method8 in Biometric Re8earch.;~e\"" York: John Wiley and Sons, Inc.
Rao, D.C., C.J. ~1acLean, N.E. Morton, S. Yee (1975). Analysis ofFamily Resemblance. V Height and Weight in Northeastern Brazil.American Journal of Human Genetic8 27: 509-520.
Rayer, P.F.O. (1836). Traite Theorique et FPatique8 des !1aladies dela Peau. Paris.
Report of Inter-Society Commission for Heart Disease Resources (1970).Primary Prevention of Hypertension. Circulation 42: A39-A4l.
Schrott, H.G., J.L. Goldstein, W.R. Hazzard, M.M. ~~Goodwin, A.Motulsky (1972). Familial io/Percholesterolemia in a Large Kindred.Evidences for a Monogenic Mechanism. Annals of Internal Medicine76: 711-720.
Schull, W.J., E. Harburg, M.A. Schork, J. Weener, C. Chape .(1977).Heredity, Stress and Blood Pressure, A Family Set Method - IIIFamily Aggregation of Hypertension. Journal of Chronic Diseases30: 659-669.
BIOMATHEMATICS TRAINING PROGRAM 200
Slack, J. and N.C. Nevin (1968). Hyper1ipidaemic Xanthomatosis.Increased Risk of Death From Ischaemic Heart Disease in FirstRelatives of 53 Patients i~ith Essential Hyper1ipidaemia andXanthomatosis. Journal of MedicaZ Genetics 5: 4-8.
I.Degree
Snedecor, G.W. and W.G. Cochran (1967). Statistical Methods) SixthEdition. Ames, Imva: The Imva State University Press.
Solbye, P. (1948). Heredity of Essential Hypertension and nephrosclerosis: Genetic CZinical Study of 200 Propositi Suffering FromNephrosclerosis. CopenhageIl, A. Busck, Nyt Nordisk.
Sperry, W.~'1. (1945). The Schoenheimer-Sperry Nethod of the Determination of Cholesterol, Revised Edition. Department of Biochemistry,New York Psychiatric Institute.
Stecher, R.M. and A.H. Hersh (1949). Note on the Genetics of Hypercholesterolemia. Science 109: 61-62.
Stone, N. and R. I. Levy (1972). Hyper1ipoproteinemia and CoronaryIieart Disease. Progress in Cardiovascular Diseases XIV: 341-359.
Svendsen, ~l. (1940). Are Supernormal Cholesterol Values in SerumCaused by a Dominantly- Inheri ted Factor? Acta t1edica Scandinavica104: 235-244.
Thomas, C.B. and B.H. Cohen (1955). The Familial Occurrence of Hypertension and Coronary Artery Disease, With Observations ConcerningObesity and Diabetes. Annals of Internal Medicine. 42: 90-127.
Todd, J.C. and A.H. Sanford (1943). Clinical Diagnosis 10th Edition)Philadelphia: W.B. Satmders, Co., p. 380.
\~ee1er, E.O. (1957). The Genetic Aspects of Atherosclerosis. AmericanJournal of Medicine 23: 653-660.
~~ilkinson, C. F., E. Hand, ~1. F1iegelman (1948). Essential FamilialHypercholesterolemia. Annals of Internal Medicine 29: 671-686.
\'Iilks, S.S. (1938). The Large Sample Distribution of the LikelihoodRatio for Testing Con~osite ~iypotheses. Annals of MathematicalStatistics 9: 60-62.
Wolfe, J. (1971). A Monte Carlo Study of the Sampling Distribution ofthe Likelihood Ratio for Mixture of Mu1tinormal Distributions. NavalPersonnel and Training Research Laboratory, Technical Bulletin STB72-2.
Nright, S. (J934). Results of Crosses Between Inbred Strains of GuineaPigs, Differing in ;~lD1lber of Digits. Genetics 19: 537-551.
Zak, B., R.C. Dickenman, E.G. \~ite, H.H. Burnett, P.J. Cherney (1954).Rapid Estimation of Free and Total Cholesterol. American Journalof Clinical Pathology 24: 1307-1315.
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