Conditional Logistic Regression for Matched Data HRP /25/04 reading: Agresti chapter 9.2

Recall: Matching Matching 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: Agresti example, diabetes 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 DiabetesNo Diabetes MI cases MI controls P(“favors” case/discordant pair) = =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 DiabetesNo Diabetes MI cases MI controls odds(“favors” case/discordant pair) =

Logistic Regression for Matched Pairs option 1: the logistic-normal model Mixed model; logit=  i +  x Where  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 deviation Gives “marginal ML estimate of  ”

option 2: Conditional Logistic Regression The conditional likelihood is based on…. The conditional probability (for pair-matched data): or, prospectively: P(“favors” case/discordant pair) =

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

 The conditional likelihood= Each age-gender stratum has the same baseline odds of disease; but these baseline odds may differ across strata

Conditional Logistic Regression

Example: MI and diabetes

Conditional Logistic Regression

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:Prenatal ultrasound examinations and risk of childhood leukemia: case-control study BMJ 2000;320:

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 UltrasoundNo Ultrasound Leukemia cases Myeloid leukemia controls 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 matching

e15/Lecture15_223_2003.ppt e15/Lecture15_223_2003.ppt Available here: -SAS tips, explanations and code -SAS macro that generates automatic logit plots (under “Lecture 15” at: to check if predictor is linear in the logit. Conditional Logistic Regression in SAS: Please read Ray’s slides at:

M: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: Cohen’s Kappa Agresti, Chapter 9.5

Cohen’s Kappa Actual agreement = sum of the proportions found on the diagonals. Cohen: Compare the actual agreement with the “chance agreement” (which depends on the marginals). Normalize by its maximum possible value.

Ex: student teacher ratings Rating by supervisor 2 Rating by supervisor 1 Authoritarian Democratic Permissive Totals Authoritarian Democratic Permissive Totals

Example: student teacher ratings Null hypothesis: Kappa=0 (no agreement beyond chance)

Example: student teacher ratings Null hypothesis: Kappa=0 (no agreement beyond chance) Interpretation: achieved 36.2% of maximum possible improvement over that expected by chance alone