The Calculation Of Sample Size In The Homogeneity Test Of Finite Mixed Model

The conclusion of a hypothesis test can generally result in two types of errors.One is when a valid null hypothesis is rejected,while the other is when the alternative hypothesis is true but the null is not rejected.When the error rate of the first kind is under control,we wish to make the error rate of the second type as low as possible.Given a test,the error rate of the second type generally decreases when the amount of data increases.Therefore,we may choose to collect more data to ensure a low type II error or a high power of the test.In real-world,however,data-collection comes with costs in both time and money.In some cases,there are natural restrictions that prevent us to get more data ultimately.These issues lead to an important research problem in statistics: how to judge whether we have collected a sufficient amount of data to have large enough power to test a scientific hypothesis? This thesis discusses this problem under the finite mixture models for homogeneity.This thesis first reviews two commonly used test methods for homogeneity under the finite mixture models,namely,EM-test and C(?)test.We will provide the background of these two tests,some theoretical results,and some proofs.We focus on two types of alternative hypotheses,the limiting distributions of the above two tests under these hypotheses when the sample size increases.Based on these results,we learn the required sample size in order to attain a specific level of power in various situations.That is,we give a sample-size calculation formula.Under a number of specific distributions such as normal,Poisson,Cauchy,geometric,we use computer simulation to check the agreement between the predicted power based on the asymptotic result and the power observed by simulation.In applications,the sample sizes(the amount of data)are often lower than the sizes required for the asymptotic theory becoming useful.This undermines the unsatisfactory precision of the sample-size formulas.We study the various factors behind the unsatisfactory precision and used computer simulation to verify these suggestions.Based on these results,we suggest several approximate distributions for the test statistic under the alternative hypothesis.More appropriate sample-size formulas are given.Finally,the thesis also discusses the influence of the form of parameterization on the sample size formula.Although different parameterization under the same distribution gives the same asymptotic sample-size formula,they have some influence on the sample-size formula.