Lecture 18 Matched Case Control Studies BMTRY 701 Biostatistical Methods II

Matched case control studies References: Hosmer and Lemeshow, Applied Logistic Regression http://staff.pubhealth.ku.dk/~bxc/SPE.2002/Slides/mcc.pdf http://staff.pubhealth.ku.dk/~bxc/Talks/Nested-Matched-CC.pdf http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap49/sect35.htm http://www.ats.ucla.edu/stat/sas/library/logistic.pdf (beginning page 5)

Matched design Matching on important factors is common OP cancer: Why? age gender Why? forces the distribution to be the same on those variables removes any effects of those variables on the outcome eliminates confounding

1-to-M matching For each ‘case’, there is a matched ‘control Process usually dictates that the case is enrolled, then a control is identified For particularly rare diseases or when large N is required, often use more than one control per case

Logistic regression for matched case control studies Recall independence But, if cases and controls are matched, are they still independent?

Solution: treat each matched set as a stratum one-to-one matching: 1 case and 1 control per stratum one-to-M matching: 1 case and M controls per stratum Logistic model per stratum: within stratum, independence holds. We assume that the OR for x and y is constant across strata

How many parameters is that? Assume sample size is 2n and we have 1-to-1 matching: n strata + p covariates = n+p parameters This is problematic: as n gets large, so does the number of parameters too many parameters to estimate and a problem of precision but, do we really care about the strata-specific intercepts? “NUISANCE PARAMETERS”

Conditional logistic regression To avoid estimation of the intercepts, we can condition on the study design. Huh? Think about each stratum: how many cases and controls? what is the probability that the case is the case and the control is the control? what is the probability that the control is the case and the case the control? For each stratum, the likelihood contribution is based on this conditional probability

Conditioning For 1 to 1 matching: with two individuals in stratum k where y indicates case status (1 = case, 0 = control) Write as a likelihood contribution for stratum k:

Likelihood function for CLR Substitute in our logistic representation of p and simplify:

Likelihood function for CLR Now, take the product over all the strata for the full likelihood This is the likelihood for the matched case-control design Notice: there are no strata-specific parameters cases are defined by subscript ‘1’ and controls by subscript ‘2’ Theory for 1-to-M follows similarly (but not shown here)

Interpretation of β Same as in ‘standard’ logistic regression β represents the log odds ratio comparing the risk of disease by a one unit difference in x

When to use matched vs. unmatched? Some papers use both for a matched design Tradeoffs: bias precision Sometimes matched design to ensure balance, but then unmatched analysis They WILL give you different answers Gillison paper

Another approach to matched data use random effects models CLR is elegant and simple can identify the estimates using a ‘transformation’ of logistic regression results But, with new age of computing, we have other approaches Random effects models: allow strata specific intercepts not problematic estimation process additional assumptions: intercepts follow normal distribution Will NOT give identical results

. xi: clogit control hpv16ser, group(strata) or Iteration 0: log likelihood = -72.072957 Iteration 1: log likelihood = -71.803221 Iteration 2: log likelihood = -71.798737 Iteration 3: log likelihood = -71.798736 Conditional (fixed-effects) logistic regression Number of obs = 300 LR chi2(1) = 76.12 Prob > chi2 = 0.0000 Log likelihood = -71.798736 Pseudo R2 = 0.3465 ------------------------------------------------------------------------------ control | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- hpv16ser | 13.16616 4.988492 6.80 0.000 6.26541 27.66742

. xi: logistic control hpv16ser Logistic regression Number of obs = 300 LR chi2(1) = 90.21 Prob > chi2 = 0.0000 Log likelihood = -145.8514 Pseudo R2 = 0.2362 ------------------------------------------------------------------------------ control | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- hpv16ser | 17.6113 6.039532 8.36 0.000 8.992582 34.4904

OR = 17.63 . xi: gllamm control hpv16ser, i(strata) family(binomial) number of level 1 units = 300 number of level 2 units = 100 Condition Number = 2.4968508 gllamm model log likelihood = -145.8514 ------------------------------------------------------------------------------ control | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- hpv16ser | 2.868541 .3429353 8.36 0.000 2.1964 3.540681 _cons | -1.464547 .1692104 -8.66 0.000 -1.796193 -1.1329 Variances and covariances of random effects ***level 2 (strata) var(1): 4.210e-21 (2.231e-11) OR = 17.63