Conditional Logistic Regression for Conditional Logistic Regression for Matched DataMatched Data

HRP 261 02/25/04HRP 261 02/25/04

reading: Agresti chapter 9.2reading: Agresti chapter 9.2

Recall: MatchingRecall: MatchingMatching can control for extraneous

sources of variability and increase the power of a statistical test.

Match M controls to each case based on potential confounders, such as age and gender.

Recall: Recall: Agresti Agresti example, example, diabetes and MIdiabetes and MI

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.

Diabetes

No diabetes

25 119

Diabetes No Diabetes

9 37

16 82

46

98

144

MI cases

MI controls

P(“favors” case/discordant pair) =

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

)~/(~*)/(

DEPDEPDEPDEP

DEPDEP

=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

odds(“favors” case/discordant pair) =

16

37

c

bOR

Logistic Regression for Logistic Regression for Matched Pairs Matched Pairs option 1:option 1: the logistic-normal modelthe logistic-normal model

Mixed model; logit=i+xWhere i represents the “stratum effect”

– (e.g. different odds of disease for different ages and genders)

– Example of a “random effect”Allow i’s to follow a normal distribution

with unknown mean and standard deviationGives “marginal ML estimate of ”

option 2: option 2: Conditional Logistic RegressionConditional Logistic Regression

The conditional likelihood is based on….

The conditional probability (for pair-matched data):

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

)~/(~*)/(

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

)~/(~*)/(

EDPEDPEDPEDP

EDPEDP

DEPDEPDEPDEP

DEPDEP

or, prospectively:

P(“favors” case/discordant pair) =

The Conditional Likelihood: The Conditional Likelihood: each each discordant discordant stratumstratum (rather than individual) (rather than individual) gets 1 term in the likelihoodgets 1 term in the likelihood

control favor thethat strata discordant

1

case favor thethat strata discordant

1

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

)/(~*)~/(

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

)~/(~*)/(

m

j

n

i

EDPEDPEDPEDP

EDPEDP

xEDPEDPEDPEDP

EDPEDP

Note: the marginal probability of disease may differ in each age-gender stratum, but we assume that the (multiplicative) increase in disease risk due to exposure is constant across strata.

Recall probability terms:Recall probability terms:

e

eEDP

1)/(

e

eEDP

1)~/(

eEDP

1

1)/(~

eEDP

1

1)~/(~

α)0(α))~/(1

)~/(ln(

)1(α))/(1

)/(ln(

EDP

EDP

EDP

EDP

The conditional likelihood=The conditional likelihood=

case favor thethat strata discordant

1i

control favor the that strata discordant

1

1*

1

1

1

1*

1

1

1*

1

1

1*

11*

1

11

*1

1

n

m

j

i

i

iii

i

ii

i

jj

j

j

j

j

j

j

j

e

e

eee

eee

e

x

ee

e

e

e

e

e

e

e

Each age-gender stratum has the same baseline odds of disease; but these

baseline odds may differ across strata

Conditional Logistic RegressionConditional Logistic Regression

case favor thethat

strata discordant

1

control favor thethat strata discordant

1j

n

i

m

ii

i

jj

j

ee

ex

ee

e

nmn

i

m

j e

e

ee

ex

e

e i

)1

()1

1(

1

1

1

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

11

Example: MI and diabetesExample: MI and diabetes

1637 )1

()1

1()L(

e

e

e

Conditional Logistic RegressionConditional Logistic Regression

16

37

1637

53)137(

01

53-37

dlog(L)

)1log(*5337)log(

e

e

ee

e

e

d

eL

Could there be an association between exposure to ultrasound in utero and an increased risk of childhood malignancies?

Previous studies have found no association, but they have had poor statistical power to detect an association.

Swedish researchers performed a nationwide population based case-control study using prospectively assembled data on prenatal exposure to ultrasound.

Example:Example:Prenatal ultrasound examinations and risk Prenatal ultrasound examinations and risk of childhood leukemia: case-control study of childhood leukemia: case-control study

BMJBMJ 2000;320:282-283 2000;320:282-283

Example:Example:Prenatal ultrasound examinations and risk Prenatal ultrasound examinations and risk of childhood leukemia: case-control study of childhood leukemia: case-control study

BMJBMJ 2000;320:282-283 2000;320:282-283

535 cases: all children born and diagnosed as having myeloid leukemia between 1973 and 1989 in Swedish registers of birth, cancer, and causes of death.

535 matched controls: 1 control was randomly selected for each case from the Swedish Birth Registry, matched by sex and year and month of birth.

Ultrasound

No ultrasound

215 320

Ultrasound No Ultrasound

200

335

535

Leukemia cases

Myeloid leukemia controls

235100

115 85

85.100

85

c

bOR

But this type of analysis is limited to single dichotomous exposure…

Used conditional logistic regression to look at dose-response with number of ultrasounds:

Results: Reference OR = 1.0; no ultrasounds OR =.91 for 1-2 ultrasounds OR=.64 for >=3 ultrasounds

Conclusion: no evidence of a positive association between prenatal ultrasound and childhood leukemia; even evidence of inverse association (which could be explained by reasons for frequent ultrasound)

Each term in the likelihood represents a stratum of 1+M individuals

More complicated likelihood expression! See: 02/02/04 lecture

Extension: 1:M matchingExtension: 1:M matching

http://www.stanford.edu/class/hrp223/2003/Lecture15/Lecture15_223_2003.ppt

Available here:-SAS tips, explanations and code-SAS macro that generates automatic logit plots

(under “Lecture 15” at: http://www.stanford.edu/class/hrp223/) to check if predictor is linear in the logit.

Conditional Logistic Regression in Conditional Logistic Regression in SAS: Please read Ray’s slides at:SAS: Please read Ray’s slides at:

M:N Matching SyntaxM:N Matching Syntax

The basic syntax is shown here.proc phreg data=BLAH;model WEIRD*IsOUTCOME(Censor_v)= PREDICTORS /ties=discrete;strata STRATA_VARS;

run; Put the values in the IsOUTCOME variable here that are the controls. Typically this is just the value 0.

This is the switch requesting a m:n CLR.

This is the m:n matching variable.

Courtesy: Ray Balise

Part II: Rater agreement: Part II: Rater agreement: Cohen’s KappaCohen’s Kappa

Agresti, Chapter 9.5 Agresti, Chapter 9.5

Cohen’s KappaCohen’s KappaActual agreement = sum of the proportions found on the diagonals.

ii

Cohen: Compare the actual agreement with the “chance agreement” (which depends on the marginals).

iiii

Normalize by its maximum possible value.

ii

iiii

1

Ex: student teacher ratingsEx: student teacher ratingsRating by supervisor 2

Rating by supervisor 1

 Authoritarian

 Democratic

 Permissive

 Totals

Authoritarian 17 4 8 29

Democratic 5 12 0 17

Permissive 10 3 13 26

         

Totals 32 19 21 72

agreement chance1

agreement chanceagreementˆ

K

Example: student teacher Example: student teacher ratingsratings

Null hypothesis: Kappa=0 (no agreement beyond chance)

**.542 - .182 CI %95

362.347.1

347.583.

)72*72

21*2619*1732*29(1

)72*72

21*2619*1732*29()

72131217

(ˆ

K

Example: student teacher Example: student teacher ratingsratings

Null hypothesis: Kappa=0 (no agreement beyond chance)

Interpretation: achieved 36.2% of maximum possible improvement over

that expected by chance alone