Interval estimation of the common odds ratio from k(2 x 2) tables under cluster sampling

We propose a simple correction factor for the variance of the logarithm of the common odds ratio estimated by the method of Mantel and Haenszel from a series of (2 x 2) tables when data are cluster correlated. The adjustment is applied to the variance estimators proposed by Hauck and by Robins, Breslow and Greenland for the log of the Mantel-Haenszel common odds ratio, and its performance is evaluated in a simulation study. The key features of the proposed adjustment are: (i) it has closed-form; (ii) it can accommodate covariates defined at the cluster-specific level, the site-specific level, or both; and (iii) it does not require the user to specify a particular correlation structure for the response data. The correction derives from Liang and Zeger's generalized estimating equations (GEE) technique for logistic regression modelling. Via simulation, we examine empirical versus nominal coverage probabilities for interval estimation of the common odds ratio using adjusted and unadjusted variance estimates, and we present ratios of observed to estimated variances. Results are compared to those obtained from the fully iterated GEE analysis. The characteristics of the simulation study mimic scenarios common in the periodontal research setting, with small numbers of subjects (N = 25, 50), moderate numbers of sites per cluster (m = 4, 16, 32), and modest intracluster correlation levels (rho = 0.0, 0.1, 0.2, 0.3). Results show that adjusted confidence intervals (applied to the Hauck or the Robins, Breslow, Greenland variance estimate) provide coverage probabilities close to the nominal level for the Mantel--Haenszel common odds ratio over a variety of cluster sizes and levels of correlation.