Tests For Homogeneity Of Odds Ratio In Several 2×2 Tables

In some epidemiological studies,such as case-control studies and medical follow-up studies,stratification analysis was often used in order to control the possible confounding and effect correction.The data can be written into a series of 2×2 tables when grouping variables and exposure factors variables are both binary.Before evaluating the causal relationship between exposure factors and diseases and estimating the common odds ratio,it is necessary to test whether the odds ratio among contingency tables at all levels is the same,that is,homogeneity test.According to the different estimation methods of odds ratio,the test can be divided into M-H method,unconditional likelihood method and conditional likelihood method.Some literatures^[10,33]have found that the conditional likelihood method,which assumes that the two-sides of contingency tables are fixed and the four frequency variables in the tables are all?non-central?hypergeometric distribution,is more effective than the other two methods.At present,there are two main types of test for homogeneity of odds ratio based on conditional likelihood method:the first one is to model the odds ratio itself,that is,to test the hypothesis H₀:?₁=?₂=···=?_K,where K is the number of contingency tables,but it is found that the degree of freedom of test statistics of this method will increase with the increase of the number of contingency tables;the other one is to introduce stratification variables to establish the logarithmic odds ratio regression model,i.e.log??_k?=?+?^?z_k,k=1,2,···,K,where z_kis stratification variables and?is a regression coefficient.The hypothesis is simplified as H₀:?=0.However,we can find that the tests of logarithmic odds ratio regression model all assume that regression coefficient?is a fixed value.The model restriction in the case of fixed?is too strong.If the model errors occur,the corresponding methods will lose the effectiveness of the test.In this paper,considering the heterogeneity of data and improving the generalization ability of the model,we establish a mixed effect model of logarithmic odds ratio by adding the random effect of regression coefficient?.We derive a set of two score statistics,testing the group effect by variable characteristics and the heterogeneity effect.We make a novel modification to these score statistics so that they are independent under the null hypothesis and their asymptotic distributions can be derived.We use the summation method to combine the two independent tests,comparing to the two more commonly used methods:Fisher procedure and Tippett procedure.Furthermore,in contrast to the score test of general conditional likelihood method and Wald test and likelihood ratio test of logarithmic odds ratio regression model,our proposed test has an added capacity to identify which components of variable characteristics and heterogeneity contribute to the non-homogeneity.Simulations under a wide range of scenarios show that the proposed test is valid,robust and powerful.Finally,the existing methods all use limit distribution to determine the test critical value,but this paper and some references^[10,33]find that this method can not control the type I error well in the case of small samples.For this reason,this paper proposes a Bootstrap method.This method can not only control the type I error of the test better,but also overcome the shortcomings of the limit distribution method effectively.It is also applicable to all the tests for homogeneity of odds ratio in literatures.