Treatment–covariate interaction is often used in clinical trials to assess the homogeneity of treatment effects over these subgroups defined by a baseline covariate, which is frequently conducted after primary analysis including all patients is completed. When the endpoint is the time to an event, as in the cancer clinical trials, the Cox proportional hazard model with an interaction term has been used exclusively to test the significance of treatment-covariate interaction in oncology literature. But the proportional hazards assumption may not be satisfied by the data from clinical trials. Although there are several procedures proposed in statistical literature to assess the interaction based on a nonparametric measure of interaction or nonparametric models, some of these procedures do not take into the account of the nature of the data well, while some are very complicated which may have limited their applications in practice. In this thesis, a non-parametric procedure based on the smoothed estimate of Patel–Hoel measure is first derived to test the interaction between the treatment and a binary covariate with censored data. The theoretical distribution of the test statistic of the proposed procedure is derived. The proposed procedure is also evaluated through Monte-Carlo simulations and applications to data from a cancer clinical trial. Jackknifed versions of two test statistics based on nonparametric models are then derived by simplifying these test statistics and applying the jackknife method to estimate their variances. These jackknifed tests are also compared with the smoothed test and other related tests. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-09-27 11:09:28.449

在生態學與生物學上，物種數常作為生物多樣性的指標，以估計單一群體物種數為例，較知名的方法首推Good (1953)以在樣本中出現一次的物種為基礎，提出的物種數估計方法堪稱的先驅，隨後許多文獻延伸Good的想法，發展出許多的估計方法，例如Burham and Overton (1978)的摺刀估計法，Chao and Lee (1992)則以涵蓋機率方式估計。相對而言，兩群體的共有物種數的研究少有人探討，目前以Chao et al. (2000)的估計式較為知名。 本研究參考Good (1953)提出估計未發現物種出現機率的想法，估計未發現共有物種的機率，並以Burham and Overton (1978)中應用摺刀法估計物種數的概念，建立一階摺刀估計式與變異數，且另行以多項分配公式推導變異數估計式，進行電腦模擬與實際資料驗證並與Chao et al. (2000)提出的共有物種估計式比較。最後根據Rasmussen and Starr (1979)以抽樣成本建立最適停止規則的概念，應用於本研究所提出的估計式，並經由電腦模擬找出抽樣成本與物種分佈均勻程度的關聯，可作為設定停止規則的依據。 / The number of species is often used to measure the biodiversity of a population in ecology and biology. Good (1953) proposed a famous estimate for the number of species based on the probability of unseen species. Subsequently, many studies applied Good’s idea to create new estimation methods, For example, the Jackknife estimate by Burham and Overton (1978), and the estimate by using the sample coverage probability in Chao and Lee (1992) are two famous examples. However, not many studies focus on estimating the number of shared species of two populations, except the method by Chao et al. (2000). In this study, we modify Good’s idea and extend the Jackknife method of Burham and Overton (1978) to develop the estimate for the number of shared species of two populations. In addition, we also establish the variance formula of the estimator by using the multinomial distribution. Subsequently, we use computer simulation and real data sets to evaluate the proposed method, and compare them with the estimator by Chao et al. (2000). Finally, we adapt the idea of optimal stopping rule by Rasmussen and Starr (1979) and combine it with the proposed jackknife estimate. We found that using the sampling cost as the stopping rule is a feasible approach for estimating the number of shared species.