Date of Award
Doctor of Philosophy
In many research settings, the effect of interest cannot be characterized by a singleoutcome, but instead multiple outcomes need to be measured on each individual under study. The problem of analyzing multiple outcomes arises frequently in many fields of bio-medical research. To handle such data a stronger model like Linear Mixed Model (LMM) is required. A global measure of the exposure effects on the original scales of the outcomes could be misleading.So, we consider scaling the responses as a preprocessing step. Situation in which the parameters of a model are subject to some restrictions or constraints arises naturally in applied statistical research. In many practical settings, the fixed effects may be subject to constraints. So, in case of scaled linear mixed model (SLMM), we propose an estimation procedure which accounts for constraints on the fixed effect parameter. In this research, we discuss ECME (Expectation-Conditional Maximization Either) Algorithm in the presence of constraints to find maximum likelihood estimators (MLE). A well developed algorithm has been used to estimate different parameters in the linear mixed model. We have programmed simulated data in R to do so, and found relative MSE (mean square error) of different estimators. Simulation results show that the proposed methods improve in terms of MSE (mean square error) on the unconstrained estimators. We have implemented LR, Wald, and global score tests on the fixed effect parameter under constraints. Our simulation shows that the empirical significance levels of all tests approach the nominal level 0.05 as the sample size grows. Finally, we have compared by simulation the power of LR, Wald, and score tests. We have found that incorporating the constraints improves the power of the tests. Improvement is substantial up to 15 $\%$ in situations where the power is low and constrained LR, Wald, and global score tests have comparable power.
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