Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Social Sciences 110

_____________________________ _____________________________

Essays on the Assumptionof Proportional Hazards in

Cox Regression

BY

INGER PERSSON

ACTA UNIVERSITATIS UPSALIENSISUPPSALA 2002

Dissertation for the Degree of Doctor of Philosophy in Statistics presented at UppsalaUniversity in 2002

Abstract

Persson, I. 2002. Essays on the Assumption of Proportional Hazards in Cox Regression.Acta Universitatis Upsaliensis. Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Social Sciences 110. 36 pp. Uppsala. ISBN 91-554-5208-6.

This thesis consists of four papers about the assumption of proportional hazards for theCox model in survival analysis.

The first paper compares the hazard ratio estimated from the Cox model to an exactcalculation of the geometric average of the hazard ratio when the underlying assumptionof proportional hazards is false, i.e. when the hazards are not proportional. The estimatesare evaluated in a simulation study.

The second paper describes and compares six of the most common numericalprocedures to check the assumption of proportional hazards for the Cox model. Thetests are evaluated in a simulation study.

Six graphical procedures to check the same assumption of proportional hazards for theCox model are described and compared in the third paper. A criterion for rejection isderived for each procedure, to make it possible to compare the results of the differentmethods. The procedures are evaluated in a simulation study.

In the fourth paper the effects of covariate measurement error on testing theassumption of proportional hazards is investigated. Three of the most commonnumerical procedures to check the assumption of proportional hazards for the Coxmodel are evaluated in a simulation study.

Inger Persson, Uppsala University, Department of Information Science, Division ofStatistics, P.O. Box 513, SE 751 20 Uppsala, Sweden

Inger Persson 2002

ISSN 0282-7492ISBN 91-554-5208-6

Printed in Sweden by Uppsala University, Tryck & Medier, Uppsala 2002

Papers summarized in the dissertation

I Persson, I.S. (2001). The Behavior of the Cox Model Hazard Ratio underViolation of the Proportional Hazards Assumption. Revised version ofResearch Report 2001:11, Department of Information Science, Division ofStatistics, Uppsala University.

II. Persson, I.S. (2001). A Comparison of Statistical Tests for Assessing theProportional Hazards Assumption for the Cox Model. Revised version ofResearch Report 2001:1, Department of Information Science, Division ofStatistics, Uppsala University.

III. Persson, I.S. (2001). A Comparison of Graphical Methods for Assessing theProportional Hazards Assumption of the Cox Model. Revised version ofResearch Report 2001:8, Department of Information Science, Division ofStatistics, Uppsala University.

IV. Persson, I.S. (2001). A Comparison of Statistical Tests for Assessing theProportional Hazards Assumption of the Cox Model When Covariates AreMeasured with Error. Revised version of Research Report 2001:12,Department of Information Science, Division of Statistics, UppsalaUniversity.

7

Contents

Acknowledgements..........................................................................9

1. Introduction .................................................................................11

1.1 Survival analysis .............................................................................111.2 The Cox proportional hazards model..............................................151.3 The assumption of proportional hazards.........................................171.4 The need for simulations.................................................................181.5 The choice of distributions and parameter values for the generation of survival times..............................................................................18

2. Summary of the papers..........................................................23

2.1 The Behavior of the Cox Model Hazard Ratio under Violation of the Proportional Hazards Assumption. ...........................................232.2 A Comparison of Statistical Tests for Assessing the Proportional

Hazards Assumption for the Cox Model........................................242.3 A Comparison of Graphical Methods for Assessing the

Proportional Hazards Assumption of the Cox Model. ...................262.4 A Comparison of Statistical Tests for Assessing the Proportional Hazards Assumption of the Cox Model When Covariates Are Measured with Error. ......................................................................29

3. Conclusions...................................................................................30

References...........................................................................................31

9

Acknowledgements

I would first and foremost like to thank my advisor professor Harry Khamis forall the help and guidance he has provided during the work on this thesis. He hasnot only assisted me with his excellent advice, but he also on a personal levelhelped me enjoy the period I stayed in Dayton, Ohio, USA.

My thanks also go to professor Reinhold Bergström, who was my advisor whenI first started my PhD studies. When I was seized with dread and worry aboutnever being able to achieve anything of substance he encouraged me by tellingme “being a doctorate student isn’t that hard, all you have to do is write a fewpapers”. He was right.

I am also grateful to everyone at the department for adding some pleasure tothe work. I would especially like to thank docent Anders Ågren who enticedme to apply to the PhD program in the first place; professor AndersChristoffersson who inspired me by letting me know how good and well paidjobs I will get once I’m done; PhD students Tomas Pettersson and LarsForsberg who friendly shared their knowledge and an office with me the firstcouple of years; PhD Anna Gunsjö who helped me through teaching; PhDLisbeth Hansson whose knocking on the door and company made me lookforward to the coffee brakes and appreciate her more like a friend than acolleague; and professor Adam Taube for not only strengthening my interest inbiostatistics but also for being a devoted fan to the choir I sing in.

My family has been great during this period of time. Marcus, my beloved, hasgiven me all the love and support I could possibly ask for. My siblings, Ola,Gunvor, and Leif, and their children have been a great source of distraction;whenever I spend time with them it’s practically impossible to think aboutwork. Ola recently asked me if I even understand what I’m doing myself. Dad,thank you for giving me my first calculator at the age of two, and Mom, I thinkyou’re right; I did inherit my mother’s intelligence.

My friends and former fellow students Anne Nikolaisen and Elin Waldenvik;thank you for making the study of statistics more entertaining. I’m still lookingforward to spending my first statistician’s salary with you.

11

1. Introduction

1.1 Survival analysis

The term “survival data” has been used in a broad sense for data involving timeto the occurrence of a certain event. This event may be death, the appearance ofa tumor, the development of some disease, recurrence of a disease, conception,cessation of smoking, and so forth. In the past decades, applications of thestatistical methods for survival data analysis have been extended beyondbiomedical and reliability research to other fields, for example, lifetime ofelectronic devices, components or systems (reliability engineering), felons’time to parole (criminology), duration of first marriage (sociology), length ofnewspaper or magazine subscription (marketing), workmen’s compensationclaims (insurance), health insurance practice, business and economics. Thestudy of survival data has previously focused on predicting the probability ofresponse, survival, or mean lifetime, and comparing the survival distributionsof experimental animals or of human patients. In recent years, the identificationof risk and/or prognostic factors related to response, survival, and thedevelopment of a disease has become equally important (see e.g. Lee (1992)Ch. 1).

The analysis of survival data is complicated by issues of censoring andtruncation. Censored data arises when an individual’s life length only is knownto occur in a certain period of time. Possible types of censoring are rightcensoring, where all that is known is that the individual is still alive at a giventime, left censoring is when all that is known is that the individual experiencedthe event of interest prior to the start of the study, or interval censoring, wherethe only information is that the event occurs within some interval of time. Onetype of right censoring that is very common is Type I censoring, where theevent is observed only if it occurs prior to some prespecified time, e.g. at theclosing of a study. A second type of right censoring is Type II censoring inwhich the study continues until the failure of the first r individuals, where r issome predetermined integer. Experiments involving Type II censoring are oftenused in testing of equipment life. Most methods used in survival analysis areproper for right censored data.

Truncation is a condition which screens certain subjects so that the investigatorwill not be aware of their existence, only individuals who meet some conditionare observed. Types of truncation are (1) left truncation, where only individualswho survive a certain time before the study starts are included and (2) righttruncation, where only individuals who have experienced the event by a certaintime are included in the study.

12

A central quantity in survival analysis is the hazard function, or the hazard rate.It is defined by

[ ]t

tTttTtPtt

| lim)( 0 ∆

≥∆+<≤=

→∆λ (1.1)

Thus, tt )( ∆λ can be seen as the conditional probability that the event ofinterest occurs in the interval [ )ttt , ∆+ , given that it has not occurred beforetime t. This function is particularly useful in determining the appropriate failuredistributions utilizing qualitative information about the mechanism of failureand for describing the way in which the chance of experiencing the eventchanges with time. There are many general shapes for the hazard rate. The onlyrestriction on )( tλ is that it be nonnegative, i.e. )( tλ ≥ 0. Models withincreasing hazard rates may arise when there is natural aging or wear.Decreasing hazard functions are much less common but find occasional usewhen there is a very early likelihood of failure, such as in certain types ofelectronic devices or in patients experiencing certain types of transplants. Mostoften, a bathtub-shaped hazard is appropriate in populations followed frombirth. During an early period deaths result, primarily, from infant diseases, afterwhich the death rate stabilizes, followed by an increasing hazard rate due to thenatural aging process. Finally, if the hazard rate is increasing early andeventually begins declining, then, the hazard is termed hump-shaped. This typeof hazard rate is often used in modelling survival after successful surgerywhere there is an initial increase in risk due to infection or other complicationsjust after the procedure, followed by a steady decline in risk as the patientrecovers (see e.g. Klein and Moeschberger (1997) pp. 27-28).

A quantity related to the hazard rate is the probability of an individualsurviving beyond time t, the survival function. The survival function, )(tS , isthe exponential of the negative of the cumulative hazard function, i.e.

[ ] [ ]∫−=−= t

0 )( exp)(exp)( dsstHtS λ (1.2)

where H(t) is the cumulative hazard function. Equivalently, the hazard function

can be expressed in terms of the survival function, ( ))(ln)( tSdtdt −=λ .

Many types of survival curves can be shown but the important point to note isthat they all have the same basic properties. They are monotone, nonincreasingfunctions equal to one at time zero and equal to zero as the time approachesinfinity. Their rate of decline, of course, varies according to the risk ofexperiencing the event at time t but it is difficult to determine the essence of afailure pattern by simply looking at the survival curve. The hazard function isusually more informative about the underlying mechanism of failure than the

13

survival function. Nevertheless, this quantity continues to be a populardescription of survival in the applied literature and can be very useful incomparing two or more mortality patterns (Klein and Moeschberger (1997)p. 23).

The standard estimator of the survival function is the Kaplan-Meier productlimit estimate (Kaplan and Meier (1958)). Assume that we have a sample of nindependent observations denoted (tj, δj), j = 1, … , n where tj is the time onstudy for the jth individual, and δj is the event indicator (δj = 1 if the event hasoccurred and δj = 0 if the lifetime is censored). Assume that among the n

observations there are D ≤ n recorded times of failure, D =∑=

n

jj

1δ . Denote the

rank-ordered failure times as t1 < t2 < … < tD. Let the number at risk at time tibe denoted Yi (the number of individuals who are alive at ti or experience theevent of interest at ti) and let the observed number of events at time ti bedenoted di, di =∑

itjδ . The quantity di /Yi provides an estimate of the conditional

probability that an individual who survives to just prior to time ti experiencesthe event at time ti. The Kaplan-Meier estimator of the survival function at timet in the range where there is data is obtained from the equation

≥

−

<

=∏≤tt i

i

i

ttYd

tttS

1

1

if 1

if 1 )(ˆ (1.3)

Intervals defined by censored observations are not considered, the formula(1.3) only uses the points at which the value of the estimator changes. Thus, theKaplan-Meier estimator is a step function with jumps at the observed eventtimes. The size of these jumps depends not only on the number of eventsobserved at each event time ti, but also on the pattern of the censoredobservations prior to ti. Figure 1 shows an example of a Kaplan-Meier survivalcurve for patients with a certain type of blood cancer called CML, chronicmyeloid leukemia. A more detailed description of the study can be found inPaper 3. The survival is plotted for two groups of patients who receiveddifferent treatments called Busulphane (Bu) and Hydroxyurea (Hu).

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

Years

Sur

viva

lBu n=89Median survival 3.2 years

Hu n=90Median survival 3.5 years

Figure 1

Another quantity of interest in survival analysis is the mean residual life at timet. For individuals of age t, this parameter measures their expected remaininglifetime and is for a continuous random variable defined as

)(

)(

)(

)()( )(mrl

tS

dxxS

tS

dxxftxt tt ∫∫

∞∞

=−

= (1.4)

and the mean lifetime, µ = mrl(0), is the total area under the survival curve,

∫∫∞∞

===

0

0 )( )( )( dxxSdxxfxtEµ . (1.5)

The estimator of the mean lifetime, µ̂ , is obtained by using the Kaplan-Meierestimator (1.3) as an estimator of the survival function S(x) in (1.5). Theestimator does not use the information on survival available in times larger thanthe largest survival time. If those observations that are long and censored hadactually been observed survival times, then the estimated mean survival timewould have been increased substantially. In such circumstances it is moreappropriate to use the median lifetime. If T is a continuous random variablethen the pth quantile is found by solving the equation S(tp) = 1 - p. The medianlifetime is the value t0.5 so that S(t0.5) = 0.5.

15

1.2 The Cox proportional hazards model

The most common approach to model covariate effects on survival is the Coxproportional hazards model by Cox (1972), which takes into account the effectof censored observations.

The data, based on a sample of size n, consists of (tj, δj, zj), j = 1, … , n where tjis the time on study for the jth individual, δj is the event indicator (δj = 1 if theevent has occurred and δj = 0 if the lifetime is censored) and zj is the vector ofcovariates or risk factors for the jth individual (zj may be a function of time)which may affect the survival distribution of T, the time to event.

The relation between the distribution of event time and the covariates or riskfactors z (z is a 1 x p vector) can be described in terms of a model according toCox, in which the hazard rate at time t for an individual is

)exp()();( 0 zβz tt λλ = (1.6)

where λ0(t) is the baseline hazard rate, an unknown (arbitrary) function givingthe hazard function for the standard set of conditions z = 0, and β is a p x 1vector of unknown parameters. The factor exp(zβ) describes the hazard for anindividual with covariates z relative to the hazard at a standard z = 0.

Making special assumptions about λ0(t) leads to parametric models, e.g. theexponential and Weibull distributions. But the advantage of Cox’s model is thefact that such assumptions can be avoided. His approach is said to be semi-parametric.

The Cox model is called a proportional hazards model since the ratio of thehazard rates of two individuals with covariate values z and z* is

βz-zzz ** )exp()|(/)|( =tt λλ , an expression that does not depend on t.

Estimates of the unknowns λ0(t) and β are obtained in the following way.Let t1 < t2 < … < tD denote the ordered distinct event times (suppose there areno ties) and let z(i)k be the kth covariate associated with the individual whosefailure time is ti, k = 1, … , p. Further, define the risk set at time ti, R(ti), as theset of all individuals who are still under study at a time just prior to ti. Thepartial likelihood according to Cox, based on the hazard function (1.6), isexpressed by

16

( )∏ ∑

∑=

∈

=

=D

itRj

j

p

kki

i

L1

)(

1)(

exp

exp)(

βz

βzβ (1.7)

Even though this is not a likelihood in the traditional sense, it is treated as one,and inference is carried out by usual means. The partial maximum likelihoodestimates are found by maximizing (1.7) or, equivalently, the logarithm of(1.7):

∑∑ ∑ ∑ ∑= = = ∈ =

−=

D

i

p

k

D

i tRj

p

kjkkkik

i

zzLL1 1 1 )( 1

)( expln)( βββ (1.8)

The efficient score equations are found by taking partial derivatives of (1.8)with respect to the β ’s as follows. Let Uh(β) = δLL(β)/δβh , h = 1, … , p. Then,

∑ ∑∑ ∑

∑ ∑= =

∈ =

∈ =

−==D

i

D

i

tRj

p

kjkk

tRj

p

kjkkjh

hihh

i

i

z

zzzULL

1 1

)( 1

)( 1)(

exp

exp)(

)(

β

β

βδδ ββ (1.9)

The information matrix is the negative of the matrix of second derivatives ofthe log likelihood and is given by I(β) = [Igh(β)]p×p with the (g, h)th elementgiven by

∑∑ ∑

∑ ∑=

∈ =

∈ =

=D

i

tRj

p

kjkk

tRj

p

kjkkjhjg

gh

i

i

z

zzzI

1

)( 1

)( 1

exp

exp)(

β

ββ

∑∑ ∑

∑ ∑

∑ ∑

∑ ∑=

∈ =

∈ =

∈ =

∈ =

−D

i

tRj

p

kjkk

tRj

p

kjkkjh

tRj

p

kjkk

tRj

p

kjkkjg

i

i

i

i

z

zz

z

zz

1

)( 1

)( 1

)( 1

)( 1

exp

exp

exp

exp

β

β

β

β (1.10)

The (partial) maximum likelihood estimates are found by solving the set of pnonlinear equations Uh(β) = 0, h = 1, … , p. It is not possible to perform thismaximization analytically, so, numerical methods must be employed (see e.g.Klein and Moeschberger (1997) p. 423 ff.). The maximization can be doneusing a Newton-Raphson technique (or some other iterative method for

17

optimization), with (1.9) and (1.10). Algorithms for the estimation of β areavailable in many statistical packages.

When there are ties present between event times, which often occurs inpractice, there are several suggestions of how to modify the model (1.6) tocorrect for this (see e.g. Breslow (1974); Efron (1977); Cox (1972)).

1.3 The assumption of proportional hazards

Since the Cox proportional hazards model relies on the hazards to beproportional, i.e. that the effect of a given covariate does not change over time,it is very important to verify that the covariates satisfy the assumption ofproportionality. If this assumption is violated, the simple Cox model is invalid,and more sophisticated analyses are required. If the interest centers upon abinary covariate, z1, whose relative risk changes over time, one approach is tointroduce a time-dependent covariate as follows. Let

z2(t) = z1 × g(t) = g(t) if the covariate z1 takes on the value 1 = 0 if the covariate z1 takes on the value 0,

where g(t) is a known function of time. One difficulty with this approach is thatthe function g(t) is usually unknown. In such cases, it may be preferable to usea procedure that would allow the function g(t) to be estimated from the data.One approach to this problem is to fit a model with an indicator function forg(t). In the simplest approach, define a time-dependent covariate

≤>

=τ tτ tz

tzif 0if

)( 12

To determine the optimal value of τ, the model including the new covariatez2(t) is fit for a set of τ values, and the value of the maximized log partiallikelihood is recorded. The value of τ which yields the largest log partiallikelihood is the optimal value to use. Proportional hazards can, then, be testedfor each region and if it fails, for t on either side of τ, then this process can berepeated in that region.

The assessment of the proportional hazards assumption can be donenumerically or graphically, a great number of procedures have been proposedover the years. Some of the procedures require partitioning of failure time,some require categorization of covariates, some include a spline function, andsome can be applied to the untransformed dataset. None of the methods, neithernumerical nor graphical, are today known to be better than the others in finding

18

out whether the hazards are proportional or not. Some authors recommendusing numerical tests (e.g. Hosmer and Lemeshow (1999)) and othersrecommend graphical procedures since they believe that the proportionalhazards assumption only approximates the correct model for a covariate andthat any formal test, based on a large enough sample, will reject the nullhypothesis of proportionality (Klein and Moeschberger (1997) p. 354). Powerstudies to compare some numerical tests have been performed, the most recentones by Ng’andu (1997), Quantin et al (1996), and Song and Lee (2000), butnone so far of graphical methods or of numerical tests under certain conditionslike measurement error. That is the main reason for the choice of topic in thisthesis, to try to find some better guidelines for the practitioner.

1.4 The need for simulations

The (partial) maximum likelihood estimates of the parameter β or,equivalently, the hazard ratio exp(zβ) are found by solving a set of nonlinearequations simultaneously, using a Newton-Raphson technique or some otheriterative method. It is not possible to perform this maximization analytically,so, numerical methods must be employed.

In this thesis, a number of tests and graphical methods to assess theproportional hazards assumption are compared and the primary criterion forcomparison is statistical power. Obtaining power can be done analytically insimple tests, e.g. the t-test, but that is not feasible in more complex tests such asthe ones used in this research. The standard, universal practice for comparingpower in any but the most simple testing situations is Monte Carlo simulation.Thus, simulations are performed throughout this thesis.

1.5 The choice of distributions and parameter values for thegeneration of survival times

The simplest but still an important distribution in survival studies is theexponential distribution. Davis (1952) gives a number of examples to describethe life pattern of electronic systems, including bank statement and ledgererrors, payroll check errors, automatic calculating machine failure, and radarset component failure, in which the failure data are well described by theexponential distribution. Epstein and Sobel (1953) report why they select theexponential distribution over the popular normal distribution and show how toestimate the parameter when data are singly censored. Epstein (1958) alsodiscusses in some detail the justification for the assumption of an exponentialdistribution.

19

The exponential distribution plays a role in lifetime studies analogous to that ofthe normal distribution in other areas of statistics. Applications in animal andhuman studies of chronic and infectious diseases can be found in Zelen (1966),Feigl and Zelen (1965), Zippin and Armitage (1966), Byar et al. (1974),DeWals and Bouckaert (1985), Shulz et al. (1986), Walle et al. (1987), andEggermont (1988).

The exponential distribution is often referred to as a purely random failurepattern (Lee (1992) p. 132). It is famous for its unique “lack of memory”,which requires that the age of the animal or individual does not affect futuresurvival, in the following way:

)()|( xTPtTxtTP ≥=≥+≥ (1.11)

The exponential distribution is characterized by a constant hazard rate α, itsonly parameter. A large α indicates high risk and short survival while a small αindicates the opposite. When a survival time T follows the exponentialdistribution with a parameter α, the probability density function is defined as

<>≥

=−

0 00,0

)(

tte

tft αα α

(1.12)

and the survival function is )( tetS α−= which yields the hazard functionαλ =)( t , a constant, independent of t.

Although the exponential distribution has been historically very popular, itsconstant hazard rate appears too restrictive in both health and industrialapplications (Klein and Moeschberger (1997) p. 38).

The Weibull distribution is a generalization of the exponential distribution,proposed by Weibull (1939, 1951) for describing the life length of materials.Unlike the exponential distribution, it does not assume a constant hazard rateand therefore has broader application. The Weibull distribution has been used,among other applications, to investigate the fatigue life of deep-groove ballbearings by Lieblein and Zelen (1956), to describe electron tube failures byKao (1958), to analyze carcinogenesis experiments by Pike (1966), Peto et al.(1972), Peto and Lee (1973), and Williams (1978), to characterize earlyradiation response probabilities by Scott and Hahn (1980) and to model humandisease-specific mortality by Juckett and Rosenberg (1990).

The Weibull distribution is characterized by two parameters, γ and α. Thevalue of γ determines the shape of the distribution curve and the value of αdetermines its scaling. Its probability density function is defined as

2

0 , 0 ) ( )( ) (1 >≥= −− αααγγαγ γ,tettf t (1.13)

The survival function is ( ){ }γα exp)( ttS −= and the hazard function has thefairly flexible form 1) ()( −= γααγλ tt . The exponential distribution is a specialcase when γ = 1. The relationship between the value of γ and hazard rate of theWeibull distribution with scale parameter α = 1 can be seen in Figure 4.

43210

5

4

3

2

1

0

Figure 4. Hazard functions of W

The Weibull distribution is, due to its fapplications as earlier described, thdistribution in the simulations performesituation of nonmonotonic hazards, the

The lognormal distribution in its sdistribution of a variable whose logaGaddum (1945a, b) gave a review of Boag’s (1949) applications in cancer reas Osgood (1958), Feinleib and Maobserved that the distribution of survHodgkin’s disease and chronic leukemby a lognormal distribution since they alogarithms of survival times are appro(1987) showed that the distribution ofollowed the lognormal distribution.

e

)5

5

tim

λ (t

γ = 4

0

eibu

lexibe prid in logno

implerithmits apsearccMahival ia coure mxima

f age

γ = 2

ll di

ility marythis trmal

st f follplich. Seon timeld b

arkedtely at

γ = 1.

γ = 1

γ = 0.

stribution with α = 1.

and close connection to many choice of data generatinghesis. In addition, to cover the distribution is also used.

orm can be defined as theows the normal distribution.ation in biology, followed byveral other investigators such(1960), and Feinleib (1960) of several diseases such ase rather closely approximatedly skewed to the right and thenormally distributed. Horner

onset of Alzheimer’s disease

21

Like the normal distribution, the lognormal distribution is completely specifiedby two parameters µ and σ, the mean and variance of T. Its density function isexpressed by

0 ,0 /ln)2(

ln21exp

)( 2/1

2

>>

−

=

−

−

= σσ

µφσπσ

µ

tttt

t

tf (1.14)

and its survival function is given by

−

Φ−=σ

µttS ln1)( , where ( ) φΦ is the

cumulative distribution function of a standard normal variable. The hazard rateof the lognormal distribution is hump-shaped, it increases initially to amaximum and then decreases (almost as soon as the median is passed) to zeroas time approaches infinity.

The parameter values, sample sizes and censoring proportions are chosen tocreate realistic data sets, different types of hazards alternatives as one is likelyto encounter in practical applications (see e.g. Copelan et al. (1991), Freireichet al. (1963), Ichida et al. (1993), Sedmak et al. (1989), Nahman et al. (1992),Kardaun (1983), Tsuang and Woolson (1977), Lagakos et al. (1988), Beadle etal. (1984a and b), Bonadonna et al. (1976), Frei et al. (1973), Kirk et al. (1980),Oliver et al. (1988), Kashiwagi et al. (1985), Geller et al. (1989) and Aaby etal. (1990)). The types of deviations from proportional hazards used in thisthesis are increasing, decreasing, crossing, diverging, converging, andnonmonotonic hazards.

Besides the survival times, censoring times also need to be generated. Severalauthors (e.g. Breslow et al. (1974), Grambsch and Therneau (1994), Quantin etal (1996), Ng’andu (1997), and Song and Lee(2000)) have generated censoringtimes distributed uniformly over [0, T] with the length of the distribution, T,adjusted to achieve a specific average proportion of censored observations.That alternative has been considered here as well, but the length of thedistribution would have to be extremely short compared to the survival times toachieve higher proportions of censoring, leading to situations where alllifetimes longer than T are censored. Thus, the exponential distribution is usedinstead as the distribution of censoring times, creating more realistic datasets.

Random samples of survival times, ts, are generated from the Weibulldistribution in all cases except the nonmonotonic hazards, where the lognormaldistribution is used. Censoring times, tc, are generated from the exponentialdistribution with hazard function λ(t) = β, where the value of the parameter βis adjusted to achieve the desired censoring proportions. The time on study, t, is

22

defined as t = ts if ts ≤ tc or t = tc if ts > tc. The event indicator is denoted by d,d = 1 if the event has occurred and d = 0 if the time is censored.

The number of repetitions used in each simulation is set to a relatively largenumber, depending on how extensive the calculations are. Previously, thenumber of repetitions in these kinds of studies have typically been 1000 (seee.g. Grambsch and Therneau (1994), Quantin et al. (1996), Song and Lee(2000), Breslow et al. (1984), and Ng’andu (1997)). When there are time-dependent covariates in the Cox model, the calculations are very extensive andthe number of repetitions used in this study is 5000, otherwise it is 10000.

The simulations are done by using the simul procedure in Stata 7.0, with theinitial value for the random-number seed set to be the same for all randomnumber generations (StataCorp. (1999)).

23

2. Summary of the papers

2.1 The Behavior of the Cox Model Hazard Ratio under Violation ofthe Proportional Hazards Assumption.

Unfortunately, there are a lot of Cox regression analyses performed withoutany model checking, and the proportional hazards assumption may not beaccurate. In many occasions the main interest is to see if a certain covariate orrisk factor affects the survival, i.e. to estimate the hazard ratio for thatcovariate. The behavior of the hazard ratio estimated with the Cox model whenthe underlying assumption of proportional hazards is false, i.e. when thehazards are not proportional, is investigated in this paper.

To see what kind of estimates the Cox model provides under nonproportionalhazards, the estimates are compared to an exact calculation of the geometricaverage of the hazard ratio θ (W) that can be defined as

( ) ( )( )∫

∞−=

0 2

1 )( tdWttW

λλθ (Kalbfleisch and Prentice (1981)) (2.1)

where λ1(t) and λ2(t) are the hazard functions of two groups and W(t) is asurvivor or weighting function,

)( )()( 21 tStStW εε= (2.2)

where S1(t) and S2(t) are the survival functions defined by the distributions ofsurvival times for the two groups and ε > 0. The value ε = ½ weights the hazardratio at time t according to the geometric average of the two survivor functions.

An average hazard ratio is not exactly describing the truth since the hazardratio is changing with time when the proportionality assumption is violated, butit still gives a picture of what kind of estimates the Cox model provides atsituations with nonproportional hazards

The estimates of the hazard ratio are evaluated in a simulation study, underproportional hazards and five different types of departure from proportionalhazards; increasing hazards, decreasing hazards, crossing hazards, diverginghazards and converging hazards. The methods are compared only in the two-sample case, corresponding to two groups with different hazard functions.Type I censoring is considered along with early and late censoring. Differentsample sizes and censoring proportions are used. Random samples of survivaltimes, ts, are generated from the Weibull distribution.

24

The results indicate that the hazard ratio estimated from the Cox regression isfairly close to the exact calculation of the average hazard ratio in mostsituations. Generally, increasing the sample size produces estimates closer tothe exact value for constant hazards, but it produces estimates farther from theexact value for increasing and decreasing hazards. Increasing the censoringproportion generally leads to estimates farther from the exact value.

2.2 A Comparison of Statistical Tests for Assessing the ProportionalHazards Assumption for the Cox Model.

There have been several procedures proposed to check the assumption ofproportional hazards, both numerical and graphical, and this paper describesand compares six of the most common numerical procedures. The tests in thisstudy have not been compared all at the same time before, even though othershave compared some of the tests in separate studies (Ng’andu (1997), Quantinet al. (1996), and Song and Lee (2000)). A short summary of the includedprocedures follows.

Cox (1972) described a way to test the proportional hazards assumption byincluding a time-dependent covariate in the model (1.6). The extended modelbecomes

))(''exp()(),( 210 ttt z γ zβz += λλ (2.3)

where z2j(t) = z1j×gj(t) for the jth covariate and g(t) is a known function of thetime t, e.g. g(t) = log(t). The null hypothesis of proportional hazards, i.e.H0: γ = 0 is tested by fitting a proportional hazards model to z1 and z2(t) andobtain estimates of β and γ along with the local tests of the null hypothesis.

Grambsch and Therneau (1994) developed a test based on the Schoenfeldpartial residuals (Schoenfeld (1982)), a measure of the difference between theobserved and expected value of the covariate at each time. This is a test whichis relatively easy to apply. All that is needed are the Schoenfeld residuals alongwith the estimated coefficients and covariance matrix from a standard timeindependent Cox model fit. The residuals are weighted with among other thingsa time-function, as in the test by Cox, and the same hypothesis is tested.

Harrell (1986) also developed a test of the proportional hazards assumptionbased on the Schoenfeld partial residuals, a test that is very easy to compute.The test is based on Fisher’s z-transform of the Pearson correlation between thepartial residuals and the rank order of the failure times and one tests if thatcorrelation differs from 0.

25

Gill and Schumacher (1987) proposed a test of the proportional hazardsassumption based on a comparison of different generalized rank estimatorsof the relative risk. The hypothesis of proportional hazards isH0: )|(/)|( *zz tt λλ = θ for some positive constant θ, where z and z* are thecovariates of two different groups. Two different weight functions are used toestimate θ, then the difference is calculated and simply tested if it parts from 0.

Breslow, Edler and Berger (1984) suggested a test also based on the Cox modelwith a time-dependent covariate (2.3), where the probability that an event(death) comes from a certain group is calculated. Then those probabilities areweighted with either ranks of time or with the cumulative hazard and the samehypothesis is tested as in the first couple of tests.

Quantin et al. (1996) proposed a global test of the proportional hazardsassumption based on a semi-parametric generalization of the proportionalhazards regression model. The hazard function corresponding to a covariatevector z is defined as the extended Cox model (2.3), with the time functiondefined as

{ })( log1)( 0 ttg Λ+= (2.4)

where )(0 tΛ is the cumulative baseline hazard function which can be replacedby Breslow’s maximum likelihood estimator under H0 (Breslow (1974)). Thehypothesis of proportional hazards, H0: γ = 0, is tested by using a score statisticderived from the partial likelihood.

All the compared tests are fairly easy to use, none of them require partitioningof the survival time axis or the covariates; such tests are deliberately notincluded since the choice of partitioning is rather arbitrary and can lead tovarying results.

A number of other available tests of the assumption of proportional hazardsthat are not used in the comparison are also summarized, tests that eitherrequire partitioning of the survival time axis or the covariates, or tests that havebeen investigated by other authors and showed inferior power properties.

The six test statistics are evaluated under proportional hazards and fivedifferent types of departure from proportional hazards; increasing hazards,decreasing hazards, crossing hazards, diverging hazards and nonmonotonichazards. Only Type I censoring is considered. The tests are compared only inthe two-sample case, corresponding to two groups with different hazardfunctions. Different realistic sample sizes and censoring proportions are used.Random samples of survival times, ts, are generated from the Weibull

26

distribution in all cases except for the nonmonotonic hazards, where thelognormal distribution is used.

The results indicate that the time-dependent covariate test by Cox (1972) andthe weighted Schoenfeld residual score test by Grambsch and Therneau (1994)have relatively good power to detect nonproportional hazards in manysituations. The regression survival model by Quantin et al. (1996) has goodpower properties in situations with low censoring proportions, and the linearcorrelation test by Harrell (1986) as well as the generalized rank estimators byGill and Schumacher (1987) are good where the other tests fail, even thoughthe latter one is oversized. Interestingly, the time-dependent covariate test byCox (1972) with squared time as the choice of time function, the weightedSchoenfeld residual score test by Grambsch and Therneau (1994) withuntransformed time as the time function and the rank test for acceleration byBreslow, Edler and Berger (1984) using the rank score have extremely badpower properties in all evaluated situations.

2.3 A Comparison of Graphical Methods for Assessing theProportional Hazards Assumption of the Cox Model.

A number of graphical methods to check if the assumption of proportionality ofthe hazard rates holds are described and compared in this paper. The methodsincluded do not require partitioning of the time axis, which is rather arbitraryand leads to different conclusions depending on the partition used.

Comparing graphical methods can be somewhat arbitrary since there are noclear guidelines for how to interpret the plots. The conclusions are highlydependent on the subjectivity of the viewer. However, to make it possible tocompare the results of the different methods, a new way of comparinggraphical methods is proposed, by using a maximum deviation (Kolmogorov-Smirnov like) criterion for rejection derived for each procedure. A number ofthe graphs are of course also viewed to make sure that the rejection decisionsbased on the criteria coincide with the visual conclusions. The criteria are onlyused to compare the graphical methods to each other, not to assess the power ofany tests since there are no actual tests. The purpose of this paper is mainly toinvestigate if any of the graphical methods is more effective in findingnonproportionality.

The investigated graphical methods are:1) a plot of survival curves based on the Cox model and Kaplan-Meierestimates for each group,2) a plot of cumulative baseline hazards in different groups (the so-calledAndersen (1982) plot),

27

3) a plot of the difference of the log cumulative baseline hazard versus time, 4) a smoothed plot of the ratio of log cumulative baseline hazard rates versustime, 5) a smoothed plot of scaled Schoenfeld residuals versus time, and 6) a plot of estimated cumulative hazard versus number of failures (the so-called Arjas (1988) plot).

The six graphical methods are evaluated in a simulation study, underproportional hazards and five different types of departure from proportionalhazards; increasing hazards, decreasing hazards, crossing hazards, diverginghazards and nonmonotonic hazards. The methods are compared only in thetwo-sample case, corresponding to two groups with different hazard functions.Only Type I censoring is considered. Variations of sample sizes and censoringproportions are used. Random samples of survival times, ts, are generated fromthe Weibull distribution in all cases except for the nonmonotonic hazards,where the lognormal distribution is used.

By using a maximum deviation criterion upon which to base comparisons ofdifferent graphical procedures, this simulation study shows that method 6, theArjas plot, is generally the most effective at identifying nonproportionality ofhazards, especially for crossing and nonmonotonic hazards, and for decreasingand diverging hazards where the censoring proportion is high. Also, the Arjasplot was never the worst among the six procedures. It is also interesting to notethat the effectiveness of the Arjas plot at identifying nonproportionality ofhazards remains relatively constant across the censoring proportions while forall other methods (except perhaps method 2, the Andersen plot) the proportionof rejections tends to decrease with the censoring proportion. The so-calledAndersen plot is inferior in practically every situation. When averaged oversample sizes and censoring proportions, the Andersen plot performs the worstunder decreasing and diverging hazards, and the Arjas plot performs best undercrossing and nonmonotonic hazards.

The six graphical procedures are also applied to the CML 84 study. CML,chronic myeloid leukemia, is a cancer of the blood cells where the patient has ahigh number of white blood cells, granulocytes, in bone marrow and blood.The treatment of this disease essentially aims to reduce the number of whiteblood cells. In 1984 a multicenter study of the disease CML started at theUniversity Hospital in Uppsala, Sweden. Patients were recruited from allhospitals in Sweden. All patients older than five years and willing toparticipate, diagnosed with CML from January 1st 1984 until December 31st

1988, were included in the study. The patients were randomized to one of twotreatments at the date of diagnosis. All patients younger than approximately 45years of age with a compatible donor, only full brothers or sisters, were offeredbone marrow transplantation, which today is known to be the only curable

treatment of CML. The last patient was included in the study in May 1988, andall patients were followed until February 1998. Consequently, the patients werefollowed for a period of up to 14 years and two months. A total of 63 patients50 years and younger were included in the study, 26 of which received bonemarrow transplantation. The transplantation covariate, equal to 1 if the patientreceived a transplantation and 0 if not, was believed to be time-dependent andmight therefore not fulfill the assumption of proportional hazards. Figure 1shows the hazard rates for transplanted and not transplanted patients.

hazard0 hazard1

00

1

Figure 1. Hazard rates for tr

The hazard rates for the two diverge. The results of the simbest graphical method to use twhen the hazard rates cross.graphical methods find nonpexcept the Andersen plot whic

Assessing graphs for the puassumption violations is verinvolved. To the untrained eviolation of the proportionalgenerated from models having

Generally, it is recommended be checked in the Cox model,helpful tool, it should not beconcerning the proportional ha

Nottransplanted

28

14

ansplanted

groups of ulations so assess th The Arjaroportionah is not eas

rpose of y difficultye, severa hazards proportion

that the pr and that w the only zards assu

Transplanted

tidtime

and not transplanted CML patients.

patients cross at early times and thenhow that the Arjas plot should be thee assumption of proportional hazards

s plot along with most of the otherlity for the transplantation covariate,y to interpret.

determining the severity of model because of the lack of objectivityl of the plots may appear to reflect aassumption, even though they wereal hazards.

oportional hazards assumption alwayshile a plot such as the Arjas plot is a

basis upon which a decision is mademption.

29

2.4 A Comparison of Statistical Tests for Assessing the ProportionalHazards Assumption of the Cox Model When Covariates AreMeasured with Error.

Many survival studies involve covariates that are measured with error and notaccurately observable. Instead, the covariates z may be observed through anadditive error model w = z + u where the measurement error u is independentof z for each individual and has mean zero and variance 2

uσ ; i.e. only w isobserved on each subject.

A naive approach to inference in such situations is to use the observed valuesof the covariates, w, in the Cox model (1.6) instead of the true values z. Thisapproach, however, usually produces biased estimates.

This paper investigates the effects of covariate measurement error on testingthe assumption of proportional hazards. The possible influence of measurementerror on the ability to find nonproportionality has not to the author’s knowledgebeen previously investigated. Three tests are evaluated, some of the mostcommon numerical procedures to check the assumption of proportional hazardsfor the Cox model, namely the tests by Cox, Grambsch and Therneau, andHarrell, are briefly described in section 2.1.

The tests are evaluated in a simulation study, under proportional hazards andfive different types of departure from proportional hazards; increasing relativehazards, decreasing relative hazards, diverging hazards, converging/crossinghazards and nonmonotonic hazards. The methods are compared only in theone-covariate case. Only Type I censoring is considered. Sample sizes andcensoring proportions are varied. The observed values w are generated byadding two random variables z and u, where the true values z are generatedfrom the normal distribution, with mean µ z and variance σz

2, z ~ N (µz = 0, σz2).

The error variable u is generated from a normal distribution with mean 0 andvariance σu

2, u ~ N (µu = 0, σu2). Random samples of survival times, ts, given z,

are generated from the Weibull distribution in all cases but the nonmonotonichazards where the lognormal distribution is used.

The size or power of each test is considered along with the sensitivity andconcordance. The sensitivity is the percentage of rejections using the naiveapproach (w) given that the null hypothesis was rejected using the trueapproach (z), and the concordance is the frequency with which the twomethods agree (both reject or both accept). A significance level of 5% is used.

The results indicate that measurement error lowers the ability for theinvestigated tests to find nonproportionality. A relatively small measurementerror does not seem to have a very big impact on the power of the tests,

30

especially not for increasing, diverging and nonmonotonic hazards. A largemeasurement error, however, lowers the power substantially for practicallyevery kind of departure from proportionality. The size is not influenced at all,neither by a small nor large error term. The sensitivity and concordance arevery high for the smaller error term, generally 90% or higher, so the naiveapproach and the true approach agree very well. For the larger error term,though, the sensitivity usually is around 40% which means that the naiveapproach does not find the nonproportionality in more than 40% of the caseswhere the exact approach finds it. The concordance is also lower for the largermeasurement error.

3. Conclusions

This research provides new information about a number of things; a shortsummary of the topics (not previously investigated) follows:

• The Cox model provides a hazard ratio under nonproportional hazardsthat is fairly close to the exact calculation of the average hazard ratioproposed by Kalbfleisch and Prentice (1981).

• A new way of comparing graphical methods is proposed, by using amaximum deviation (Kolmogorov-Smirnov like) criterion for rejectionderived for each procedure.

• The so-called Arjas plot (Arjas (1988)) is generally the most effective atidentifying nonproportionality of hazards, compared to a number ofother graphical methods.

• Measurement error lowers the ability to find nonproportionality for allinvestigated statistical tests except for the nonmonotonic hazards case.

This research also improves on what has been done before on comparingdifferent test statistics to assess the assumption of proportional hazards; ahigher number of Monte Carlo simulations is used, more tests are included inthe comparison, a wider range of sample sizes and censoring proportions areused, and a more practical mechanism for simulating the censoring is used(exponentially instead of uniformly distributed censoring times).

31

References

Aaby, P., Bukh, J., Kronborg, D., Lisse, I.M., and Da Silva, M.C. (1990).Delayed Excess Mortality after Exposure to Measles During the First SixMonths of Life. American Journal of Epidemiology, 132, 211-219.

Andersen, P.K. (1982). Testing Goodness of Fit of Cox’s Regression and LifeModel. Biometrics, 38, 67-77. Correction (1984): 40, 1217.

Arjas, E. (1988). A Graphical Method for Assessing Goodness of Fit in Cox’s

Proporional Hazards Model. American Statistical Association 83, 204-212.

Beadle, G.F., Come, S., Henderson, C., Silver, B., and Hellman, S.A.H.(1984a). The Effect of Adjuvant Chemotherapy on the Cosmetic Resultsafter Primary Radiation Treatment for Early Stage Breast Cancer.International Journal of Radiation Oncology, Biology and Physics, 10,2131-2137.

Beadle, G.F., Harris, J.R., Silver, B., Botnick, L., and Hellman, S.A.H.(1984b). Cosmetic Results Following Primary Radiation Therapy for EarlyStage Breast Cancer. Cancer, 54, 2911-2918.

Boag, J.W. (1949). Maximum Likelihood Estimates of Proportion of PatientsCured by Cancer Therapy. Journal of the Royal Statistical Society, Series B,11, 15.

Bonadonna, G. et al. (1976). Combination Chemotherapy as an AdjuvantTreatment in Operable Breast Cancer. The New England Journal ofMedicine, 294, 405-410.

Breslow, N.E. (1974). Covariance Analysis of Censored Survival Data.Biometrics, 30, 89-99.

Breslow, N. E., Edler, L. and Berger, J. (1984). A Two-Sample Censored-DataRank Test for Acceleration. Biometrics 40, 1049-1062.

Byar, D.P, Huse, R., and Bailar, J.C. III, and the Veterans AdministrationCooperative Urological Research Group (1974). An exponential ModelRelating Censored Survival Data and Concomitant Information for ProstaticCancer Patients. Journal of the National Cancer Institute, 52, 321-326.

32

Copelan, E.A., Biggs, J.C., Thompson, J.M., Crilley, P., Szer, J., Klein, J.P.,Kapoor, N., Avalos, B.R., Cunningham, I., Atkinson, K., Downs, K.,Harmon, G.S., Daly, M.B., Brodsky, I., Bulova, S.I., and Tutschka, P.J.(1991). Treatment for Acute Myelocytic Leukemia with Allogeneic BoneMarrow Transplantation Following Preparation with Bu/Cy. Blood 78,838-843.

Cox, D.R. (1972). Regression Models and Life-Tables (with discussion).Journal of the Royal Statistical Society – Series B, 34, 187-220.

Davis, D.J. (1952). An Analysis of Some Failure Date. Journal of the AmericanStatistical Association, 47, 113-150.

DeWals, P., and Bouckaert, A. (1985). Methods for Estimating the Duration ofBacterial Carriage. International Journal of Epidemiology, 14, 628-634.

Efron, B. (1977). The Efficiency of Cox’s Likelihood Function for CensoredData. Journal of the American Statistical Association 72, 557-565.

Eggermont, J.J. (1988). On the Rate of Maturation of Sensory EvokedPotentials. Electroencephalography and Clinical Neurophysiology, 70,293-305.

Epstein, B. (1958). The Exponential Distribution and Its Role in Life Testing.Industrial Quality Control, 15, 2-7.

Epstein, B. and Sobel, M. (1953). Life Testing. Journal of the AmericanStatistical Association, 48, 486-502.

Feigl, P., and Zelen, M. (1965). Estimation of Exponential SurvivalProbabilities with Concomitant Information. Biometrics, 21, 826-838.

Feinleib, M. (1960). A Method of Analyzing Log-Normally DistributedSurvival Data with Incomplete Follow-up. Journal of the AmericanStatistical Association, 55, 534-545.

Feinleib and MacMahon (1960). Variation in the Duration of Survival ofPatients with Chronic Leukemias. Blood, 17, 332-349.

Frei, E. III, Luce, J.K., Gamble, J.F., et al. (1973). Combination Chemotherapyin Advanced Hodgkin’s Disease: Induction and Maintenance of Remission.Annals of Internal Medicine, 79, 376-382.

33

Freireich, E.J., gehan, E., Frei, E., Schroeder, L.R., Wolman, I.J., Anbari, R.,Burgert, E.O., Mills, S.D., Pinkel, D., Selawry, O.S., Moon, J.H., Gendel,B.R., Spurr, CL., Storrs, R., Haurani, F., Hoogstraten, B., and Lee, S.(1963). The Effect on the Duration of Steroid-Induced Remissions in AcuteLeukemia: A Model for Evaluation of Other Potentially Useful Therapy.Blood 21, 699-716.

Gaddum, J.H. (1945a). Log Normal Distributions. Nature, London, 156, 463.

Gaddum, J.H. (1945b). Log Normal Distributions. Nature, London, 156, 747.

Geller, N., Bosl, G.J., and Chan, E.Y.W. (1989). Prognostic Factors forRelapse after Complete Response in Patients with Metastatic Germ CellTumors. Cancer, 63, 440-445.

Gill, R. and Schumacher, M. (1987). A simple test of the proportional hazardsassumption. Biometrika 74, 2, 289-300.

Grambsch, P.M. and Therneau, T.M. (1994). Proportional hazards tests anddiagnostics based on weighted residuals. Biometrika 81, 3, 515-526.

Harrell, F.E. (1986). The PHGLM procedure, SAS supplemental Library User’sGuide, Version 5 Edition, SAS Institute, Cary, N.C.

Horner, R.D. (1987). Age at Onset of Alzheimer’s Disease: Clue to theRelative Importance of Etiologic Factors? American Journal ofEpidemiology, 126, 409-414.

Hosmer, D.W., and Lemeshow, S. (1999). Regression Modeling of Time ToEvent Data. New York, John Wiley & Sons, Inc, 207.

Ichida, J.M., Wassell, J.T., Keller, M.D., and Ayers, L.W. (1993). Evaluationof Protocol Change in Burn-Care Management Using the Cox ProportionalHazards Model with Time-Dependent Covariates. Statistics in Medicine 12,301-310.

Juckett, D.A., and Rosenberg, B. (1990). Periodic Clustering of HumanDisease-Specific Mortality Distributions by Shape and Time Position, and aNew Integer-based Law of Mortality. Mechanisms of Ageing andDevelopment, 55, 255-291.

Kalbfleisch, J.D. and Prentice , R.L. (1981). Estimation of the average hazardratio. Biometrika, 68, 105-112.

34

Kao, J.H.K. (1958).Computer Methods for Estimating Weibull Parameters inLife-Testing of Electron Tubes. Technometrics, 1, 389-407.

Kaplan, E.L. and Meier, P. (1958). Nonparametric estimation from incompleteobservations. Journal of the American Statistical Association 53, 457-481.

Kardaun, O. (1983). Statistical Analysis of Male Larynx-Cancer Patients – ACase Study. Statistical Nederlandica, 37, 103-126.

Kashiwagi, S., et al. (1985). Prevalence of Immunologic Markers of HepatitisA and B Infection in Hospital Personnel in Miyazaki Prefecture, Japan.American Journal of Epidemiology, 122, 960-969.

Kirk, A.P. et al. (1980). Late Results of the Royal Free Hospital ProspectiveControlled Trial of Predimione Therapy in Hepatitis B Surface AntigenNegative Chronic Active Hepatitis. Gut, 21- 78-83.

Klein, J.P. and Moeschberger, M.L. (1997). Survival Analysis: Techniques forCensored and Truncated Data. New York, Springer.

Lagakos, S.W., Barraj, L.M., and Degruttola, V. (1988). NonparametricAnalysis of Truncated Survival Data, with Application to AIDS. Biometrika,75, 525-523.

Lee, E.T. (1992a). Statistical Methods for Survival Data Analysis, 2nd Ed.Oklahoma City, John Wiley & Sons, Inc.

Lieblein, J., and Zelen, M. (1956). Statistical Investigation of the Fatigue Lifeof Deep-Groove Ball Bearings. Journal of Research, National Bureau ofStandards, 57, 273-316.

Nahman, N.S., Middendorf, D.F., Bay, W.H., McElligott, R., Powell, S., andAnderson, J. (1992). Modification of the Percutaneous Approach toPeritoneal Dialysis Catheter Placement Under PeritoneoscopicVisualization: Clinical Results in 78 Patients. Journal of The AmericanSociety of Nephrology, 3, 103-107.

Ng’andu, N.H. (1997). An empirical comparison of statistical tests forassessing the proportional hazards assumption of Cox’s model. Statistics inMedicine, vol. 16, 611-626.

Oliver, I.N., et al. (1988). Nitrogen Mustard, Vincristine, Procarbazine, and forRelapse after Radiation in Hodgkin’s Disease. Cancer, 62 233-239.

35

Osgood, E.W. (1958). Methods for Analyzing Survival Data, Illustrated byHodgkin’s Disease. American Journal of Medicine, 24, 40-47.

Peto, R., Lee, P.N., and Paige, W.S. (1972). Statistical Analysis of the Bioassayof Continuous Carcinogens. British Journal of Cancer, 26, 258-261.

Peto, R., and Lee, P.N. (1973). Weibull Distributions for continuousCarcinogenesis Experiments. Biometrics, 29, 457-470.

Pike, M.C. (1966). A Method of Analysis of a Certain Class of Experiments inCarcinogenesis. Biometrics, 22, 142-161.

Quantin, C., Moreau, T., Asselain, B., Maccario, J. and Lellouch, J. (1996). ARegression Survival Model for Testing the Proportional HazardsHypothesis. Biometrics 52, 874-885.

Schoenfeld, D. (1980). Chi-squared goodness-of-fit tests for the proportionalhazards regression model. Biometrika, 67, 1, 145-153.

Scott, B.R., and Hahn, F.F. (1980). A Model that Leads to the WeibullDistribution Function to Characterize Early Radiation ResponseProbabilities. Health Physics, 39(3), 521-530.

Sedmak, D.D., Meineke, T.A., Knechtges, D.S., and Anderson, J. (1989).Prognostic Significance of Cytokeratin-Positive Breast Cancer Metastases.Modern Pathology, 2, 516-520.

Shulz, D., Chernichovsky, D., and Allweis, C. (1986). A Novel Method forQuantifying Passive-Avoidance Behavior Based on the ExponentialDistribution of Step-Through Latencies. Pharmacology, Biochemistry andBehavior, 25, 979-983.

Song, H.H. and Lee, S. (2000). Comparison of goodness of fit tests for the Coxproportional hazards model. Communications in Statistics – Simulation andComputation, 29(1), 187-206.

StataCorp. (1999). Stata Reference Manual, Release 6, College Station, TX:Stata Corporation, Vol.1, 520.

Tsuang, M.T., and Woolson, R.F. (1977). Mortality in Patients withSchizophrenia, Mania and Depression. British Journal of Psychiatry, 130,162-166.

36

Walle, A.J., Al-Katib, A., Wong, G.Y., Jhanwar, S.C., Chaganti, R.S.K., andKoziner, B. (1987). Mulitparameter Characterization of L3 Leukemia CellPopulations. Leukemia Research, 11, 73-83.

Weibull, W. (1939). A Statistical Theory of the Strength of Materials.Ingeniors Vetenskaps Akademien Handlingar, 151, The Phenomenon ofRupture in Solids, 293-297.

Weibull, W. (1951) . A Statistical Distribution of Wide Applicability. Journalof Applied Mechanics, 18, 293-297.

Williams, J.S. (1978). Efficient Analysis of Weibull Survival Data fromExperiments on Heterogeneous Patient Populations. Biometrics, 34,209-222.

Zelen, M. (1966). Applications of Exponential Models to Problems in CancerResearch. Journal of the Royal Statistical Society, Series A, 129, 368-398.

Zippin, C., and Armitage, P. (1966). Use of Concomitant Variables andIncomplete Survival Information in the Estimation of an ExponentialSurvival Parameter. Biometrics, 22, 665-672.