Statistical Inference Based On Partially Validated Data With An Imperfect Gold Standard

Double sampling is usually applied to one classifier without erroneous judgement and the other classifier with low accuracy.However,it could happen in practice that such infallible classifier or gold standard does not exist or very expensive.In this article,we consider hypothesis testing and the determination of sample size when both classifiers are fallible under two models for prevalence rate of disease.For hypothesis test of disease prevalence,we propose asymptotic and approximate unconditional test procedures based on six test statistics that Wald statistic,Wald statistic with the variance being estimated under the null hypothesis,Likelihood rate statistic,Score statistic,based on Log-and Logit-transformation statistics.We consider the determination of sample size from two aspects of empirical power and confidence interval,that can achieve a pre-specified power of a statistical test at a chosen significance level and control the width of a confidence interval at a specified width with a pre-specified confidence level.Through the simulation study,it is found that both asymptotic and approximate unconditional procedures based on the Score statistic perform satisfactorily in the two models,so highly recommended in practice.When sample size is moderate or large,asymptotic procedures based on the Wald statistic with the variance being estimated under the null hypothesis,Likelihood rate statistic,log-and logit-transformation statistics under both models generally perform well and are hence recommended.The approximate unconditional procedures based on the log-transformation statistic under Model I,Wald statistic with the variance being estimated under the null hypothesis,log-and logit transformation statistics under Model II are recommended when sample size is small.For sample size determination from aspect of empirical power,the simulation shows that the sample size decreases with the increase of the parameters.In particular,the parameter ? in Model II has no effect on the sample size.In general,sample size formulae based on Wald statistic with the variance estimated under the null hypothesis,Likelihood rate statistic and Score statistic are more accurate and recommended to use them in practice.For sample size determination from aspect of confidence interval,the simulation shows that sample size formulae based on Wald statistic with the variance estimated under the null hypothesis,Likelihood rate statistic and Score statistic are more accurate and recommended to use them in practice.The applicability of the proposed methods is illustrated by a real-data example.