Empirical likelihood-based inference for multiple regression and treatment comparison

Parameter estimation and statistical inference are generally used in the analysis of epidemiological and biomedical data. Traditional parametric methods often impose the assumption of normality on the data. When this assumption is violated, methods based on the normal approximation can give biased results. Furthermore, normal approximation-based inference methods require estimation of the asymptotic variance, which may be difficult in semi-parametric or non-linear models. Empirical likelihood is a good alternative to make statistical inference for the parameters when the distribution of the data is unspecified. To derive statistical inference for the parameter of interest, we develop empirical likelihood-based methods along with the Bartlett correction to improve the coverage probability of the parameter.;The contributions we make to the existing literature in this dissertation contain two parts. In the first part, we develop an empirical likelihood-based inference for multiple regression models and show that the empirical likelihood ratio statistic follows a chi-square limiting distribution for several model settings. For high- dimensional parameter vectors, we can theoretically use the traditional approach to obtain confidence regions which, however, may not yield helpful information. The profile empirical likelihood method can obtain the confidence interval for each element of the parameter vector. However, its application is limited by the expensive computational cost. The existing empirical likelihood-based method does not scale well for multiple regression models. We suggest doing a projection to erase nuisance parameters for dimension reduction and then using the profile empirical likelihood principle. This mixture reduces the computation burden for multiple regression models and gives better inference in terms of coverage probabilities compared to the normal approximation-based method. Our proposed method can handle many types of multiple regression models including linear, linear mixed-effects, partially linear, longitudinal partial linear and additive models. Also, it does not require any restrictions on the data distribution and computations are simple for nonparametric and semi-parametric models.;In the second part, we develop more efficient methods for treatment comparisons. In epidemiologic and biomedical studies, a common concern is how to evaluate the difference between two treatments. Many methods have been proposed for evaluating this difference for special cases such as under normality assumption. We consider comparing two treatment effects, which can be described as the difference of the parameters in two linear models, and develop a more efficient approach based on empirical likelihood. We first derive an empirical likelihood-based statistic for the difference between the two regression coefficients, and show that the statistic is asymptotically chi-squared, leading to tests and confidence intervals for this situation. The Bartlett correction is applied to obtain the adjusted confidence interval. This method can be extended to handle multi-treatment comparisons and is free of assumptions of normality, homogenous errors and equal sample sizes.;We also consider the comparison of two treatments in a censored data setting by using the hazard formulation for two sample semi-parametric hybrid models. We demonstrate that a proper empirical likelihood definition that takes into account censoring leads to an empirical likelihood ratio test with a standard chi-square limiting distribution under the null hypothesis, which is different from the published results, a sum of chi-squared variables with unknown weights. The proposed test can be used to test hypothesis of the ROC curve with censored data. A version of the log-rank test for this hybrid setup is also derived using the proposed test.