Empirical likelihood, which was pioneered by Thomas and Grunkemeier (1975) and Owen (1988), is a powerful nonparametric method of statistical inference that has been widely used in the statistical literature. In this thesis, we investigate the merits of empirical likelihood for various problems arising in ratio estimation. First, motivated by the smooth empirical likelihood (SEL) approach proposed by Zhou & Jing (2003), we develop empirical likelihood estimators for diagnostic test likelihood ratios (DLRs), and derive the asymptotic distributions for suitable likelihood ratio statistics under certain regularity conditions. To skirt the bandwidth selection problem that arises in smooth estimation, we propose an empirical likelihood estimator for the same DLRs that is based on non-smooth estimating equations (NEL). Via simulation studies, we compare the statistical properties of these empirical likelihood estimators (SEL, NEL) to certain natural competitors, and identify situations in which SEL and NEL provide superior estimation capabilities. Next, we focus on deriving an empirical likelihood estimator of a baseline cumulative hazard ratio with respect to covariate adjustments under two nonproportional hazard model assumptions. Under typical regularity conditions, we show that suitable empirical likelihood ratio statistics each converge in distribution to a 2 random variable. Through simulation studies, we investigate the advantages of this empirical likelihood approach compared to use of the usual normal approximation. Two examples from previously published clinical studies illustrate the use of the empirical likelihood methods we have described. Empirical likelihood has obvious appeal in deriving point and interval estimators for time-to-event data. However, when we use this method and its asymptotic critical value to construct simultaneous confidence bands for survival or cumulative hazard functions, it typically necessitates very large sample sizes to achieve reliable coverage accuracy. We propose using a bootstrap method to recalibrate the critical value of the sampling distribution of the sample log-likelihood ratios. Via simulation studies, we compare our EL-based bootstrap estimator for the survival function with EL-HW and EL-EP bands proposed by Hollander et al. (1997) and apply this method to obtain a simultaneous confidence band for the cumulative hazard ratios in the two clinical studies that we mentioned above. While copulas have been a popular statistical tool for modeling dependent data in recent years, selecting a parametric copula is a nontrivial task that may lead to model misspecification because different copula families involve different correlation structures. This observation motivates us to use empirical likelihood to estimate a copula nonparametrically. With this EL-based estimator of a copula, we derive a goodness-of-fit test for assessing a specific parametric copula model. By means of simulations, we demonstrate the merits of our EL-based testing procedure. We demonstrate this method using the data from Wieand et al. (1989). In the final chapter of the thesis, we provide a brief introduction to several areas for future research involving the empirical likelihood approach.

Empirical likelihood, which was pioneered by Thomas and Grunkemeier (1975) and Owen (1988), is a powerful nonparametric method of statistical inference that has been widely used in the statistical literature. In this thesis, we investigate the merits of empirical likelihood for various problems arising in ratio estimation. First, motivated by the smooth empirical likelihood (SEL) approach proposed by Zhou & Jing (2003), we develop empirical likelihood estimators for diagnostic test likelihood ratios (DLRs), and derive the asymptotic distributions for suitable likelihood ratio statistics under certain regularity conditions. To skirt the bandwidth selection problem that arises in smooth estimation, we propose an empirical likelihood estimator for the same DLRs that is based on non-smooth estimating equations (NEL). Via simulation studies, we compare the statistical properties of these empirical likelihood estimators (SEL, NEL) to certain natural competitors, and identify situations in which SEL and NEL provide superior estimation capabilities. Next, we focus on deriving an empirical likelihood estimator of a baseline cumulative hazard ratio with respect to covariate adjustments under two nonproportional hazard model assumptions. Under typical regularity conditions, we show that suitable empirical likelihood ratio statistics each converge in distribution to a 2 random variable. Through simulation studies, we investigate the advantages of this empirical likelihood approach compared to use of the usual normal approximation. Two examples from previously published clinical studies illustrate the use of the empirical likelihood methods we have described. Empirical likelihood has obvious appeal in deriving point and interval estimators for time-to-event data. However, when we use this method and its asymptotic critical value to construct simultaneous confidence bands for survival or cumulative hazard functions, it typically necessitates very large sample sizes to achieve reliable coverage accuracy. We propose using a bootstrap method to recalibrate the critical value of the sampling distribution of the sample log-likelihood ratios. Via simulation studies, we compare our EL-based bootstrap estimator for the survival function with EL-HW and EL-EP bands proposed by Hollander et al. (1997) and apply this method to obtain a simultaneous confidence band for the cumulative hazard ratios in the two clinical studies that we mentioned above. While copulas have been a popular statistical tool for modeling dependent data in recent years, selecting a parametric copula is a nontrivial task that may lead to model misspecification because different copula families involve different correlation structures. This observation motivates us to use empirical likelihood to estimate a copula nonparametrically. With this EL-based estimator of a copula, we derive a goodness-of-fit test for assessing a specific parametric copula model. By means of simulations, we demonstrate the merits of our EL-based testing procedure. We demonstrate this method using the data from Wieand et al. (1989). In the final chapter of the thesis, we provide a brief introduction to several areas for future research involving the empirical likelihood approach.

The mechanisms that control growth processes in biology tissues have attracted continuous research interest despite their complexity. With the emergence of big data experimental approaches there is an urgent need to develop statistical and computational models to fit the experimental data and that can be used to make predictions to guide future research. In this work we apply statistical methods on growth process of different biological tissues, focusing on development of neuron dendrites and tumor cells. We first examine the neuron cell growth process, which has implications in neural tissue regenerations, by using a computational model with uniform branching probability and a maximum overall length constraint. One crucial outcome is that we can relate the parameter fits from our model to real data from our experimental collaborators, in order to examine the usefulness of our model under different biological conditions. Our methods can now directly compare branching probabilities of different experimental conditions and provide confidence intervals for these population-level measures. In addition, we have obtained analytical results that show that the underlying probability distribution for this process follows a geometrical progression increase at nearby distances and an approximately geometrical series decrease for far away regions, which can be used to estimate the spatial location of the maximum of the probability distribution. This result is important, since we would expect maximum number of dendrites in this region; this estimate is related to the probability of success for finding a neural target at that distance during a blind search. We then examined tumor growth processes which have similar evolutional evolution in the sense that they have an initial rapid growth that eventually becomes limited by the resource constraint. For the tumor cells evolution, we found an exponential growth model best describes the experimental data, based on the accuracy and robustness of models. Furthermore, we incorporated this growth rate model into logistic regression models that predict the growth rate of each patient with biomarkers; this formulation can be very useful for clinical trials. Overall, this study aimed to assess the molecular and clinic pathological determinants of breast cancer (BC) growth rate in vivo.

<p>This study extends the existing literature on how social networks enhance the performance and sustainability of small enterprises. More specifically, the study isolates and investigates the mechanisms through which social capital helps with the growth and survival of MSMEs. The evidence presented in this study strongly suggests that an indigenous social network widely practiced in Ethiopia, the &ldquo / iqqub&rdquo / , contributes significantly to the start-up, survival and development of urban MSMEs.</p>

<p>This study extends the existing literature on how social networks enhance the performance and sustainability of small enterprises. More specifically, the study isolates and investigates the mechanisms through which social capital helps with the growth and survival of MSMEs. The evidence presented in this study strongly suggests that an indigenous social network widely practiced in Ethiopia, the &ldquo / iqqub&rdquo / , contributes significantly to the start-up, survival and development of urban MSMEs.</p>

Philosophiae Doctor - PhD / This study extends the existing literature on how social networks enhance the performance and sustainability of small enterprises. More specifically, the study isolates and investigates the mechanisms through which social capital helps with the growth and survival of MSMEs. The evidence presented in this study strongly suggests that an indigenous social network widely practiced in Ethiopia, the "iqqub", contributes significantly to the start-up, survival and development of urban MSMEs. / South Africa

Indiana University-Purdue University Indianapolis (IUPUI) / Background: The hazard ratio (HR), representing the quantified estimate of treatment effect in survival analysis, measures the instantaneous relative difference of failure risk between two groups. The HR is typically assumed to be independent of time; however, this assumption is usually violated in practice. If the proportionality assumption holds, HR can be validly with the popular Cox proportional hazards model. When not proportional, the Wilcoxon-Gehan has been proposed to test the hypothesis of no difference. These have been recently generalized to evaluate differences in survival time for more than zero survival differences (the “net survival benefit”). Method: In this thesis, an attempt is made to illustrate the properties of generalized Wilcoxon Gehan tests as proposed by Buyse (2009). We use the concept of net survival benefit to re-analyze the trial by the Gastrointestinal Tumor Study Group (1982) by comparing chemotherapy versus combined chemotherapy and radiation in the treatment of locally unresectable gastric cancer. Survival times in days, for the 45 patients were recorded in each treatment arm. In that trial, a delayed treatment effect was observed, thus the HR is non-proportional. To provide a flexible assessment of the treatment effect, the net survival benefit was computed using datasets simulated under typical scenarios of proportional hazards, such as delayed treatment effect. Results: The generalized Wilcoxon statistic U, favored not adding radiation to chemotherapy, but only for survival up to 12 months. At Δ=0, U (0) = 491. In the simulated data sets, the confidence interval under the null hypothesis U (0) is (-152, 388). The test statistic 491 is outside this interval indicating radiation treatment might be beneficial. At U(12) = 219, it is inside the confidence interval of no treatment effect (-154,268) indicating the benefit of Chemo only is gone after 12 months. Conclusions: The net survival benefit measured via Buyse’s generalized Wilcoxon statistic is a measure of treatment effect that is meaningful whether or not hazards are proportional. The associated statistical test is more powerful than the standard log-rank test when a delayed treatment effect is anticipated.

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