This dissertation consists of four chapters. The first chapter motivates problems of interest and gives a brief literature review.The second chapter investigates popular methods of testing in linear mixed models. The linear mixed model is prominently used in research involving human and animal subjects. Drawing inferences on model parameters is primarily important for explaining the biological outcomes. Unlike the standard linear regression models, the linear mixed model inference is based on asymptotic theory. Thus a study of finite sample performance of the existing procedure could be of practical interest. This chapter reviews the following popular approaches of testing fixed effects in a linear mixed model: (1) likelihood ratio test, (2) restricted maximum likelihood ratio test (3) Bartlett corrected profile likelihood ratio test (4) Bartlett corrected Cox-Reid likelihood ratio test and (5) Kenward-Roger approximate F-test. The performance of these methods are compared based on Type-I error rate via an extensive simulation study. We conclude that the Kenward-Roger test is the best in preserving Type-I error rate.The third chapter develops a test for fixed effects in small sample linear mixed model with missing covariates. Partially observed variables are common in scientific research. Ignoring the subjects with partial information may lead to biased and or inefficient estimators, and consequently any test based only on the completely observed subjects may inflate the error probabilities. Missing data issue has been extensively considered in the regression model, especially in the independently identically (IID) data setup. Relatively less attention has been paid for handling missing covariate data in the linear mixed effects model-- a dependent data scenario. In case of complete data, Kenward-Roger's F test is a well established method for testing of fixed effects in a linear mixed model. In this chapter, we present a modified Kenward-Roger type test for testing fixed effects in a linear mixed model when the covariates are missing at random. In the proposed method, we attempt to reduce bias from three sources, the small sample bias, the bias due to missing values, and the bias due to estimation of variance components. The operating characteristics of the method is judged and compared with two existing approaches, listwise deletion and mean imputation, via simulation studies.The fourth chapter applies the random effects model to meta-analysis of rare event data. In clinical trials and many other applications, meta-analysis is mainly conducted to summarize the effect size on primary endpoints or on key secondary endpoints. However, there are times when safety endpoints such as risk of complications in pancreatic surgery or the risk of myocardial infarction (MI) is the main point of interest in a drug study. As these types of safety endpoints are rare in nature, most studies report zero such incidences; the general statistical framework of meta-analysis based on large sample theory falls apart to combine the effect size. As a workaround, either trials with both arms having zero events are deleted or a 0.5 correction is applied. In a randomized control trial (RCT) set-up, Cai, Parast and Ryan, 2010, proposed methods based on Poisson random effects models to draw inferences on relative risks for two arms with rare event data. In this chapter, we give the general framework of how their assumption of having RCT can be further relaxed and utilized for non-RCT studies to draw inferences for two treatments. We also develop two new approaches based on zero inflated Poisson random effects models that are more appropriate with excessive zero counts data.
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