Asymptotic And Exact Tests For Response Rates Of Paired Binary Data Under Dallal’s Model

Paired binary data naturally arises when paired body parts are investigated in clinical trials.In ophthalmic studies,it is necessary to analyze data from paired eyes.However,it is often highly correlated between the eyes.If the correlation is not considered,it will lead to the misleading results.For this reason,many probability models have been proposed,including Ronser’s,Donner’s and Dallal’s models.The first two models have been studied a lot.The researches on Dallal’s model are still relatively few.Therefore,this paper will further study statistical inference for large and small samples under Dallal’s model.There are two cases in Dallal’s model:the conditional probabilities of each group are equal or unequal.For these two cases,the homogeneity and common of the response rates of multiple groups are tested.In the homogeneity test,NewtonRaphson algorithm was used when solving the parameters by maximum likelihood estimation.Eight test statistics are constructed,and their explicit expressions are given.The simulation results show that the likelihood ratio and Wald-type statistics are not robust in the type I error rates(TIEs).Score and Ronser-type statistics can produce satisfactory TIEs and power,so they are recommended.A real example is given to illustrate the proposed methods.On the basis of homogeneity test,common test is proposed to test response rate of each group.Similar to homogeneity test,estimates of parameters are solved first,and then six test statistics are constructed.The simulation results show that score test produces good results in TIEs and power.Although asymptotic methods have been widely used in large sample settings,they poorly work for controlling TIEs when sample size is relatively small.Therefore,we propose some exact methods to deal with the small sample size.Three statistics are constructed under Dallal’s model:likelihood ratio,Wald-type and score statistics.Based on these statistics,three exact methods are constructed:E,M and E+M methods.The simulation results show that the performance of the exact methods are better than the asymptotic methods on small samples.A real example is given to further illustrate how these methods work.