Asymptotic and Exact Power for the McNemar Test and Its Analogue with R Controls Per Case

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SUMMARY Miettinen has derived the asymptotic power for testing for a difference between cases and controls with dichotomous response and one-to-one or one-to-R matching. In this paper the accuracy of Miettinen's estimates is assessed by comparison with results obtained by simulation for large samples and by exact computation of power for small samples. The relation between exact and asymptotic power is discussed. It is concluded that the asymptotic power function will suffice except in the rare case where small sample sizes can yield high power. Individual matching of cases with one or more controls, and with a dichotomous response, commonly occurs in epidemiological research and clinical trials. In the present paper it is assumed that matching is appropriate, although this is not invariably so, and the relative merits and demerits of the strategy are not discussed. For the one-to-one case, Bennett and Underwood (1970) compared exact power with the noncentral chi square approximation for sample sizes of 10, 20 and 40, and found the approximation to be adequate. Miettinen's work (1968, 1969), which was extended by Walter (1980) to a variable number of controls per case, seems more generally applicable. Miettinen tabulated four asymptotic estimates of power but did not compare the tabulated values with empirical or exact measures. That omission has given rise to the present work. 2. Statement of Problem and Notation As far as possible, Miettinen's original notation will be used. We have J sets of R + 1 individuals, of whom one is a case and R are controls. The case is matched with the controls on a matching variable M. The J cases (and RJ controls) have underlying probabilities of positive response PI and P2, respectively. Here PI and P2 are random variables with expectations 01 and 02; that is, 01 is the expected probability that a given case has positive response and 02 the corresponding expected probability for a given control. The objective is to test the hypothesis that 01 = 02. The problem dealt with here is the agreement of Miettinen's asymptotic formulae for the power and sample size of the standard test with exact power for small samples and simulation-derived estimates for large samples. The response vector of length R + 1 is