Risk Analysis in Cohort Studies with Heterogeneous Strata. A Global χ²-Test for Dose-Response Relationship, Generalizing the Mantel-Haenszel Procedure

Abstract

In cohort studies the Mantel-Haenszel estimator OR̂ ^MH is computed from sample data and is used as a point estimator of relative risk. Test-based confidence intervals are estimated with the help of the asymptotic chi-squared distributed MH-statistic χ² _MHS . The Mantel-exten-sion-chi-squared is used as a test statistic for a dose-response relationship. Both test statistics – the Mantel-Haenszel-chi as well as the Mantel-extension-chi – assume homogeneity of risk across strata, which is rarely present. Also an extended nonparametric statistic, proposed by Terpstra, which is based on the Mann-Whitney-statistics assumes homogeneity of risk across strata.

We have earlier defined four risk measures RR ^k _j (k = 1,2,...,4) in the population and considered their estimates and the corresponding asymptotic distributions. In order to overcome the homogeneity assumption we use the δ-method to get “test-based” confidence intervals. Because the four risk measures RR ^k _j are presented as functions of four weights g_ik we give, consequently, the asymptotic variances of these risk estimators also as functions of the weights g _ik in a closed form. Approximations to these variances are given.

For testing a dose-response relationship we propose a new class of χ²(1)-distributed global measures Ĝ^k and the corresponding global χ²-test. In contrast to the Mantel-extension-chi homogeneity of risk across strata must not be assumed. These global test statistics are of the Wald type for composite hypotheses. The Mantel-extension-chi is a special case of the global-chi statistic and it is further shown that the Mantel-extension-chi can be expressed as a special weighted Terpstra statistic. Formulas for computing estimators of the global measures are provided. Three elaborated examples with hypothetical data of varying structure show, that the Mantel-extension-chi is systematically biased, generally overestimates the dose-response relation and leads to wrong conclusions, when heterogeneity is present. This is consistent with our theoretical considerations. In case of heterogeneity, when one wants to test an association between exposure and effect, or a dose-response relationship, the new global-χ²-test should be used in cohort studies.

Key-Words

Cohort Studies - Risk Measures - Heterogeneous Strata - Mantel-Haenszel-chi - Global-χ²-Test - Dose-Response Relation

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