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Various X 2 ‐Tests of Homogeneity: Some Comments
Author(s) -
ElandtJohnson Regina C.
Publication year - 1989
Publication title -
biometrical journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.108
H-Index - 63
eISSN - 1521-4036
pISSN - 0323-3847
DOI - 10.1002/bimj.4710310208
Subject(s) - homogeneity (statistics) , statistic , statistics , mathematics , combinatorics , null hypothesis , confounding
Methodological issues in the analysis of incidence rates or prevalence proportions for count data, presented in a form of a sequence of 2×2 tables, corresponding to levels (strata) of a specified variable (risk factor) X , are discussed. Suppose λ 1i and λ 2i are the incidence rates of an event D in the ith stratum for populations 1 and 2, respectively. The homogeneity (null) hypothesis is formulated in the form: H 0:λ1i =λ 2i for all i ( i = 1, 2, …, I ). Three X 2 ‐tests for H 0 and their theoretical bases are discussed: X Total 2 which is sensitive to alternatives H A :λ1 i ± λ2 i for at least some i ; X Comb 2 which is sensitive to alternatives H A : λ1 i λ2 i 2 or < λ 2 i but not both for all i ; and X Diff 2 which is sensitive to alternatives H A :λ1 i >λ2 i 3 for some i and λ 1 i ′ < λ 2 i ′ for some i ′ ( i ≠ i ′). These statistics satisfy the relation X Total 2 = X Comb 2 + X Diff 2 . Also, X ′ 2 ‐statistic for pooled data is calculated, which in conjunction with X Comb 2 can serve for detecting confounding. Although most of these techniques are known, they are rather scattered in the literature, and not always considered jointly, as it is emphasized in the present paper. It is hoped that these comments will be helpful to biostatisticians as well as to epidemiologists and medical researchers in the analysis of mortality and morbidity data. For illustration, two examples with large sets of epidemiological data are given.