Interpretation and estimation of summary ratios under heterogeneity

Of the several types of summary odds ratios in use in epidemiology, only those proposed by Mantel and Haenszel and Miettinen do not assume homogeneity (constancy) of stratum-specific parameters. Under heterogeneity, Miettinen's estimators consistently estimate meaningful epidemiologic parameters, whereas the asymptotic expectation of the Mantel-Haenszel odds ratio varies according to irrelevant features of the sampling design. Although one can modify the Mantel-Haenszel summary to overcome the latter problem, one can also extend the Miettinen estimators to sparse data and thus limit the use of Mantel-Haenszel summaries to situations in which the homogeneity hypothesis is reasonable. This paper provides formulae for the extended Miettinen estimators and large-sample variances and confidence limits for summary ratios under heterogeneity.