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# Estimators and Tests in the Analysis of Multiple Nonindependent 2 X 2 Tables with Partially Missing Observations

John M. Lachin and L. J. Wei
Biometrics
Vol. 44, No. 2 (Jun., 1988), pp. 513-528
DOI: 10.2307/2531864
Stable URL: http://www.jstor.org/stable/2531864
Page Count: 16
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## Abstract

We present methods for the analysis of a K-variate binary measure for two independent groups where some observations may be incomplete, as in the case of K repeated measures in a comparative trial. For the K 2 X 2 tables, let θ = (θ1, ..., θK) be a vector of association parameters where θk is a measure of association that is a continuous function of the probabilities πik in each group (i = 1, 2; k = 1, ..., K), such as the log odds ratio or log relative risk. The asymptotic distribution of the estimates θ = ($\hat\theta_1, \ldots,\hat\theta_K$) is derived. Under the assumption that θk = θ for all k, we describe the maximally efficient linear estimator θ of the common parameter θ. Tests of contrasts on the θ are presented which provide a test of homogeneity Ha: θk = θl for all k ≠ l. We then present maximally efficient tests of aggregate as sociation Hb: θ = θ0, where θ0 is a given value. It is shown that the test of aggregate association Hb is asymptotically independent of the preliminary test of homogeneity Ha. These methods generalize the efficient estimators of Gart (1962, Biometrics 18, 601-610), and the Cochran (1954, Biometrics 10, 417-451), Mantel-Haenszel (1959, Journal of the National Cancer Institute 22, 719-748), and Radhakrishna (1965, Biometrics 21, 86-98) tests to nonindependent tables. The methods are illustrated with an analysis of repeated morphologic evaluations of liver biopsies obtained in the National Cooperative Gallstone Study.

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