Abstract
Several statistical methods have been developed for adjusting the Odds Ratio of the relation between two dichotomous variables X and Y for some confounders Z. With the exception of the Mantel-Haenszel method, commonly used methods, notably binary logistic regression, are not symmetrical in X and Y. The classical Mantel-Haenszel method however only works for confounders with a limited number of discrete strata, which limits its utility, and appears to have no basis in statistical models. Here we revisit the Mantel-Haenszel method and propose an extension to continuous and vector valued Z. The idea is to replace the observed cell entries in strata of the Mantel-Haenszel procedure by subject specific classification probabilities for the four possible values of (X,Y) predicted by a suitable statistical model. For situations where X and Y can be treated symmetrically we propose and explore the multinomial logistic model. Under the homogeneity hypothesis, which states that the odds ratio does not depend on Z, the logarithm of the odds ratio estimator can be expressed as a simple linear combination of three parameters of this model. Methods for testing the homogeneity hypothesis are proposed. The relationship between this method and binary logistic regression is explored. A numerical example using survey data is presented.
Highlights
The practice of exploring residual association between two variables X and Y after adjusting for other, confounding, variables Z is at the heart of much of statistical and epidemiological analysis
Classification probabilities pxyi can be obtained from (3) using maximum likelihood (ML) estimates of axy and bxy obtained with standard software, such as SPSS, STATA, R and SAS, and the OR estimate y^ probcan be readily computed using (2)
We proposed a method to adjust an Odds Ratio between two dichotomous variables X and Y for other, ‘confounding’, variables Z, that is symmetrical in X and Y
Summary
The practice of exploring residual association between two variables X and Y after adjusting for other, confounding, variables Z is at the heart of much of statistical and epidemiological analysis. If we want to present, for example, the residual relationship between two cardiovascular risk factors or disorders, with no direct causal link between the two but both potentially influenced by common factors such as gender, the MH-method would seem a more attractive choice than logistic regression. We here propose a very simple method to extend the MH odds ratio to a general case of Z being an m-dimensional vector of covariates, some of which may be continuous.
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