Doctor of Philosophy / Department of Statistics / Haiyan Wang / Statistical tools to detect nonlinear relationship between variables are commonly needed in various practices. The first part of the dissertation presents a test of independence between a response variable, either discrete or continuous, and a continuous covariate after adjusting for heteroscedastic treatment effects. The method first involves augmenting each pair of the data for all treatments with a fixed number of nearest neighbors as pseudo-replicates. A test statistic is then constructed by taking the difference of two quadratic forms. Using such differences eliminate the need to estimate any nonlinear regression function, reducing the computational time. Although using a fixed number of nearest neighbors poses significant difficulty in the inference compared to when the number of nearest neighbors goes to infinity, the parametric standardizing rate is obtained for the asymptotic distribution of the proposed test statistics. Numerical studies show that the new test procedure maintains the intended type I error rate and has robust power to detect nonlinear dependency in the presence of outliers. The second part of the dissertation discusses the theory and numerical studies for testing the nonparametric effects of no covariate-treatment interaction and no main covariate based on the decomposition of the conditional mean of regression function that is potentially nonlinear. A similar test was discussed in Wang and Akritas (2006) for the effects defined through the decomposition of the conditional distribution function, but with the number of pseudo-replicates going to infinity. Consequently, their test statistics have slow convergence rates and computational speeds. Both test limitations are overcome using new model and tests. The last part of the dissertation develops theory and numerical studies to test for no covariate-treatment interaction, no simple covariate and no main covariate effects for cases when the number of factor levels and the number of covariate values are large.

Master of Arts / Department of Statistics / Paul Nelson / There are two important statistical models for multivariate survival analysis, proportional hazards(PH) models and accelerated failure time(AFT) model. PH analysis is most commonly used multivariate approach for analysing survival time data. For example, in clinical investigations where several (known) quantities or covariates, potentially affect patient prognosis, it is often desirable to investigate one factor effect adjust for the impact of others. This report offered a solution to choose appropriate model in testing covariate effects under different situations. In real life, we are very likely to just have limited sample size and censoring rates(people dropping off), which cause difficulty in statistical analysis. In this report, each dataset is randomly repeated 1000 times from three different distributions (Weibull, Lognormal and Loglogistc) with combination of sample sizes and censoring rates. Then both models are evaluated by hypothesis testing of covariate effect using the simulated data using the derived statistics, power, type I error rate and covergence rate for each situation. We would recommend PH method when sample size is small(n<20) and censoring rate is high(p>0.8). In this case, both PH and AFT analyses may not be suitable for hypothesis testing, but PH analysis is more robust and consistent than AFT analysis. And when sample size is 20 or above and censoring rate is 0.8 or below, AFT analysis will have slight higher convergence rate and power than PH, but not much improvement in Type I error rates when sample size is big(n>50) and censoring rate is low(p<0.3). Considering the privilege of not requiring knowledge of distribution for PH analysis, we concluded that PH analysis is robust in hypothesis testing for covariate effects using data generated from an AFT model.

Factor mixture modeling (FMM) has been increasingly used to investigate unobserved population heterogeneity. This Monte Carlo simulation study examined the issue of measurement invariance testing with FMM when there are covariate effects. Specifically, this study investigated the impact of excluding and misspecifying covariate effects on the class enumeration and measurement invariance testing with FMM. Data were generated based on three FMM models where the covariate had impact on the latent class membership only (population model 1), both the latent class membership and the factor (population model 2), and the latent class membership, the factor, and one item (population model 3). The number of latent classes was fixed at two. These two latent classes were distinguished by factor mean difference for conditions where measurement invariance held in the population, and by both factor mean difference and intercept differences across classes when measurement invariance did not hold in the population. For each of the population models, different analysis models that excluded or misspecified covariate effects were fitted to data. Analyses consisted of two parts. First, for each analysis model, class enumeration rates were examined by comparing the fit of seven solutions: 1-class, 2-class configural, metric, and scalar, and 3-class configural, metric, and scalar. Second, assuming the correct solution was selected, the fit of analysis models with the same solution was compared to identify a best-fitting model. Results showed that completely excluding the covariate from the model (i.e., the unconditional model) would lead to under-extraction of latent classes, except when the class separation was large. Therefore, it is recommended to include covariate in FMM when the focus is to identify the number of latent classes and the level of invariance. Specifically, the covariate effect on the latent class membership can be included if there is no priori hypothesis regarding whether measurement invariance might hold or not. Then fit of this model can be compared with other models that included covariate effects in different ways but with the same number of latent classes and the same level of invariance to identify a best-fitting model.

The clinical trial is the most import study for the development of successful novel drugs. The aim of this dissertation is to develop innovative statistical methods to overcome the three main obstacles in clinical trials: (1) lengthy trial duration and inaccurate maximum tolerated dose (MTD) in phase I trials; (2) heterogeneity in drug effect when patients are given the same prescription and same dose; and (3) high failure rates of expensive phase III confirmatory trials due to the discrepancy in the endpoints adopted in phase II and III trials. Towards overcoming the first obstacle, we originally develop a hybrid design for the time-to-event dose escalation method with overdose control using a normalized equivalent toxicity score (NETS) system. This hybrid design can substantially reduce sample size, shorten study length, and estimate accurate MTD by employing a parametric model and adaptive Bayesian approach. Toward overcoming the second obstacle, we propose a new approach to incorporate patients’ characteristic using our proposed design in phase I clinical trials which considers the personalized information for patients who participant in the trials. To conquer the third obstacle, we propose a novel two-stage screening design for phase II trials whereby the endpoint of percent change in of tumor size is used in an initial screening to select potentially effective agents within a short time interval followed by a second screening stage where progression free survival is estimated to confirm the efficacy of agents. These research projects will substantially benefit both cancer patients and researchers by improving clinical trial efficiency and reducing cost and trial duration. Moreover, they are of great practical meaning since cancer medicine development is of paramount importance to human health care.