Analysis of pair matched dataAnalysis of pair matched data

HRP 261 2/24/03 10HRP 261 2/24/03 10--11 am11 am

Pair MatchingPair MatchingRecall question 7 on homework 2—dependent proportions. We’re returning to this concept today.

Pairing can control for extraneous sources of variability and increase the power of a statistical test.Match 1 control to 1 case based on potential confounders, such as age and gender.

ExampleExampleJohnson and Johnson (NEJM 287: 1122-1125, 1972) selected 85 Hodgkin’s patients who had a sibling of the same sex who was free of the disease and whose age was within 5 years of the patient’s…they presented the data as….

Hodgkin’s

Sib control

Tonsillectomy None

41 44

33 52

OR=1.47; chi-square=1.53 (NS)

From John A. Rice, “Mathematical Statistics and Data Analysis.

ExampleExampleBut several letters to the editor pointed out that those investigators had made an error by ignoring the pairings. These are not independent samples because the sibs are paired…better to analyze data like this:

OR=2.14; chi-square=2.91 (p=.09)

Tonsillectomy

None

Tonsillectomy None

37 7

15 26

CaseControl

From John A. Rice, “Mathematical Statistics and Data Analysis.

Pair Matching: Pair Matching: Agresti Agresti exampleexample

Match each MI case to an MI control based on age and gender.

Ask about history of diabetes to find out if diabetes increases your risk for MI.

Pair Matching: Pair Matching: AgrestiAgresti exampleexampleJust the discordant cells are

informative!

Diabetes

No diabetes

25 119

Diabetes No Diabetes

9 37

16 82

46

98

144

MI cases

MI controls

Which cells are informative?

Pair MatchingPair Matching

Diabetes

No diabetes

25 119

Diabetes No Diabetes

9 37

16 82

46

98

144

MI cases

MI controls

OR estimate comes only from discordant pairs!

The question is: among the discordant pairs, what proportion are discordant in the direction of the case vs. the control. If more discordant pairs “favor” the case, this indicates OR>1.

Diabetes

No diabetes

25 119

Diabetes No Diabetes

9 37

16 82

46

98

144

MI cases

MI controls

P(“favors” case/discordant pair) =

)~/(*)/(~)~/(~*)/()~/(~*)/(

DEPDEPDEPDEPDEPDEP

+

=the probability of observing a case-control pair with only the control exposed

=the probability of observing a case-control pair with only the case exposed

Diabetes

No diabetes

25 119

Diabetes No Diabetes

9 37

16 82

46

98

144

MI cases

MI controls

P(“favors” case/discordant pair) =

5337

163737ˆ =

+=

+=

cbbp

Diabetes

No diabetes

25 119

Diabetes No Diabetes

9 37

16 82

46

98

144

MI cases

MI controls

odds(“favors” case/discordant pair) =

1637

==cbOR

Diabetes

No diabetes

25 119

Diabetes No Diabetes

9 37

16 82

46

98

144

MI cases

MI controls

OR estimate comes only from discordant pairs!!

OR= 37/16 = 2.31

Makes Sense!

McNemar’s McNemar’s TestTest

Diabetes

No diabetes

Diabetes No Diabetes

9 37

16 82

MI casesMI controls

Null hypothesis: P(“favors” case / discordant pair) = .5(note: equivalent to OR=1.0 or cell b=cell c)

...)5(.)5(.3953

)5(.)5(.3853

)5(.)5(.3753 143915381637 +

+

+

=− valuep

By normal approximation to binomial:

01.;88.264.3

5.10)5)(.5(.53

)2

53(37<==

−= pZ

McNemar’s McNemar’s Test: generallyTest: generally

By normal approximation to binomial:

exp

No exp

exp No exp

a b

c d

casescontrols

cbcb

cb

cb

cb

cbbZ

+−

=+

−=

+

+−

=

4

22)5)(.5)(.(

)2

(

Equivalently:

cbcb

cbcb

+−

=+−

=2

221

)()(χ

95% CI for difference in 95% CI for difference in dependent proportionsdependent proportions

Diabetes

No diabetes

25 119

Diabetes No Diabetes

9 37

16 82

46

98

144

MI cases

MI controls

24.05.)0024.(96.115.17.- 32. : CI %95

0024.144

)11.*26.57.*06(.2)83)(.17(.)68)(.32(.

),(2)1()1(

)(),(2)()()(

~//~/~///

~//

212121

−=±=∴

=−−+

=

−−

+−

=

−∴−+=−

==DEDE

controlscases

DEDE

controlscases

DEDE

DEDE

ppCovn

ppn

ppppVar

ppCovpVarpVarppVar

Each pair is it’s own “ageEach pair is it’s own “age--gender” stratumgender” stratum

Diabetes

No diabetes

Case (MI) Control

1 1

0 0

Example: Concordant for

exposure (cell “a” from before)

Diabetes

No diabetes

Case (MI) Control

1 1

0 0

Case (MI) Control

x 9

Diabetes

No diabetes

1 0

0 1x 37

Case (MI) Control

Diabetes

No diabetes

0 1

1 0

Case (MI) Control

x 16

Diabetes

No diabetes

0 0

1 1 x 82

MantelMantel--Haenszel Haenszel for pairfor pair--matched datamatched data

We want to know the relationship between diabetes and MI controlling (very tightly) for age and gender.

Mantel-Haenszel methods apply.

RECALL: The MantelRECALL: The Mantel--HaenszelHaenszelSummary Odds RatioSummary Odds Ratio

∑

∑

=

=k

i i

ii

k

i i

ii

Tcb

Tda

1

1

Exposed

Not Exposed

Case Control

a b

c d

Diabetes

No diabetes

Case (MI) Control

1 1

0 0

Diabetes

No diabetes

Case (MI) Control

1 0

0 1

ad/T = 0

bc/T=0

ad/T=1/2

bc/T=0

Diabetes

No diabetes

Case (MI) Control

0 1

1 0

Diabetes

No diabetes

Case (MI) Control

0 0

1 1

ad/T=0

bc/T=1/2

ad/T=0

bc/T=0

MantelMantel--HaenszelHaenszel Summary ORSummary OR

1637

21*162137

2

2144

1

144

1 ===

∑

∑

=

=x

cb

da

OR

i

ii

i

ii

MH

MantelMantel--HaenszelHaenszel Test StatisticTest Statistic(same as (same as McNemar’sMcNemar’s))

cbcb

cbcbCMH

nVar

nnnnnnVar

nnn

cellsdisc

cellsdisccon cellsdisccase

k

kk

kkkk

k

kk

+−

=+−

=+−

=

=−

===

−=

=

∑∑ ∑

++++

++++

++

++

22

.

..

2

.

21111k

22211

11k

1111k

)()25)(.()](5.)(5[.

25.

]5.5.[41

)12(2)1)(1)(1)(1()(;

21

2)1)(1(

:cells discordant0 contribute cells Concordant

)1()(n

)E(n :recall

µ

Logistic Regression for Logistic Regression for Matched Pairs (1) Matched Pairs (1)

the logisticthe logistic--normal modelnormal modelMixed model; logit=αi+βxWhere αi represents the “stratum effect”– (e.g. different odds of disease for different ages

and genders)– Example of a “random effect”

Allow αi’s to follow a normal distribution with unknown mean and standard deviationGives “marginal ML estimate of β”

Logistic Regression for Matched Logistic Regression for Matched Pairs (2) Pairs (2)

intro. to the conditional likelihoodintro. to the conditional likelihoodThe conditional likelihood is based on….

The conditional probability GIVEN discordant pair =

)/(~*)~/()~/(~*)/()~/(~*)/(

)~/(*)/(~)~/(~*)/()~/(~*)/(

EDPEDPEDPEDPEDPEDP

DEPDEPDEPDEPDEPDEP

+=

+(marginals cancel)

Logistic Regression for Matched Logistic Regression for Matched Pairs (2) Pairs (2)

the conditional likelihoodthe conditional likelihood

∏

∏

+++

++

++

+++

++

++

++

+

+

+

+

+

+

+

casefavor ;discordant stratums

controlfavor ;discordant stratums

1*

11

11*

1

11*

1

11*

11*

11

1*

11

i

i

iii

i

ii

i

ii

i

i

i

i

i

i

i

ee

eeee

eee

x

eee

ee

e

ee

e

α

α

βααβα

βα

αβα

βα

αβα

βα

α

α

βα

α

α

βα

The conditional likelihood=

Conditional Logistic RegressionConditional Logistic Regression

∏∏ ++= +

+

+

casefavor ;discordant stratums

controlfavor ;discordant stratums

1

1

1

1* ii

i

ii

i

eeex

eee

αβα

βα

αβα

α

∏∏ ++=

casefavor ;discordant stratums

controlfavor ;discordant stratums 1

1

1parameter) nuisance of rid (gets !cancel! s' The***

1

1

1 β

β

β

α

eex

e

e i

Conditional Logistic RegressionConditional Logistic Regression

1637

163753)137(

01

53-37dlog(L)

)1log(*5337)log(

1

1

11

1

1

11

=

=

=+

=+

=

+−=

β

β

ββ

β

β

β

β

β

e

eeee

ed

eL