Homogeneity And Common Tests Of Relative Risk Ratios For Stratified Binary Date Under Different Algorithms

In Medical clinical studies about paired body parts,the correlation between responses from paired parts should be taken into account to avoid biased or misleading results.If the trial is multi-center or with confounding factors,we can investigate treatment-by-stratum through comparing the responses in different strata.Thus,homogeneity and common test are necessary for stratified correlated data.When the correlation coefficient of each stratum is different,this paper proposes homogeneity test of relative risk ratios in different strata for stratified correlated binary data under Donner’s model.The homogeneity test aims to test whether relative risk ratios are equal.When the homogeneity of relative risk ratio exists,we further test whether the common relative risk ratio is equal to specific value.There is usually no closed-form solutions for MLEs in stratified model.Iterative algorithms are introduced for calculating MLEs to construct test statistics of homogeneity test.Global MLEs are obtained by Fisher scoring algorithm and the two-step algorithm.Simulation results show,two algorithms have close mean square errors(MSEs).With the same convergence precision,Fisher scoring algorithm needs less iteration number.And it takes less time for the same iteration number.So Fisher scoring algorithm is more efficient for global MLEs.Constrained MLEs under homogeneity assumption are calculated by Fisher scoring algorithm and two-stage procedure.Fisher scoring algorithm can produce constrained MLEs with lower MSEs.In other words,Fisher scoring algorithm has higher accuracy for constrained MLEs under homogeneity assumption.According to the performance of algorithms in terms of MSE and convergence rate,Fisher scoring algorithms for global MLEs and MLEs under homogeneity assumption are chosen to construct test statistics for homogeneity test.Based on global MLEs and MLEs under homogeneity assumption both from Fisher scoring algorithm,the likelihood ratio test TLa,the score test TSCa and the Wald-type test TWa are construct in asymptotic method for homogeneity test.Monte Carlo simulations show,TSCa test is recommended for it has more robust type I error rate(TIE)and satisfactory power.In asymptotic methods,sample size for the specific power can be calculated by the iterative way when the significance level is given.Simulation results show test statistics TLa,TSCa and TWa have robust TIEs and good power with the sample size obtained by the iterative way.Asymptotic method has bad performance when the sample size is small.In order to solve this problem,this paper proposes exact methods including A,E and M approaches for homogeneity test under small size.A approach is close to asymptotic method,which calculates p values according to the asymptotic distribution of TSCa.In E approach,nuisance parameters are replaced by constrained MLEs under homogeneity assumption for calculating values of TSCa.With values of score test,the tail area is defined and p values are obtained.M approach,is to maximize the sum of probabilities of all tables in the tail area over the whole range of nuisance parameters.In exact methods,E approach has better performance in terms of TIE and power.When the homogeneity exists,we further proposed common test,which tests whether common relative risk ratio is equal to specific value when the relative risk ratio of each stratum is same.Common test is similar to homogeneity test.First,constrained MLEs under the assumption of common test are calculated by Fisher scoring algorithm.Then,the likelihood ratio test TLb,the score test TSCb and the Wald-type test TWb are construct based on constrained MLEs.Simulation results show,three tests have close power.TSCb test has more robust TIE.Therefore,TSCb is more recommended.Asymptotic sample size for TLb,TSCb and TWb can be calculated by iterative method too.In simulations,the asymptotic sample size can provide that TLb,TSCb and TWb have satisfactory TIEs and power.Two real examples are used to demonstrate the performance of the proposed homogeneity and common tests.