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筆者在論文中引入一個名為加權距離檢驗的雙變項統計。此檢驗方法用於比較兩個隨機變數的集中趨勢。其優勢在於在有偏度的數據中，仍能穩定地控制第一型錯誤，並同時提供可觀的統計檢力。加權距離檢驗利用冪函數修正在偏度數據中的不對稱現象。與一般的冪函數轉換法不同，加權距離檢驗將冪值限制在0和1之間。文中亦提供了一個有效決定冪值的方法，以方便在實際運算中使用。 / 筆者總結了四個主流的雙變項統計方法，並利用蒙地卡羅模擬法在正態分佈、同程度偏度分佈以及不同程度偏度分佈三個情況中比較了它們與加權距離檢驗的表現。結果顯示，加權距離檢驗雖然沒有在任何一個情況中勝出，但卻於兩方面表現了其優勢。首先，它在任何情況下都能把第一型錯誤控制在合理水平之下；其次，它在任何情況下都不至於表現得太差。反觀其他四個檢驗方法總會在某些情況下表現失敗。由此可見，加權距離檢驗比起其他檢驗方法更能提供一個穩定而簡單的方法去比較集中趨勢。 / A new bivariant statistics, namely the weighted distance test, for comparing two groups were introduced. The test aims at providing reliable type I error control and reasonable statistical power across different types of skewed data. It corrects the skewness of the data by applying power transformation with power index ranged between 0 to 1. I also proposed in this thesis a possible way of deciding the power index by considering the skewness difference between the two groups under comparison. / I reviewed 4 commonly used inferential statistics for two-group comparison and compared their performances with the weighted distance test under 1) normal distribution, 2) skewed distribution with equal skewness across groups, and 3) skewed distribution with unequal skewness across groups. Monte Carlo simulations were ran to evaluate the 5 tests. Results showed that the weighted distance test was not the best test in any particular situation, but was the most stable test in the sense that 1) it provided accurate type I error control and 2) it did not produce catastrophically small power in any scenario. All other 4 tests failed in some of the simulated scenario for either inflated type I error, or unsatisfactory power. Therefore, I suggested that the weighted distance test could be one easy-to-use test that works fairly well across a wide range of situation. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Lee, Yung Ho. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 31-33). / Abstracts also in Chinese. / Chapter Chapter One --- : Introduction --- p.1 / Common methods in comparing central tendency --- p.2 / T-test --- p.2 / Median and rank --- p.3 / Trimming --- p.3 / Power transformation --- p.4 / Chapter Chapter Two --- : Weighted distance statistic --- p.5 / Definition --- p.5 / Statistical properties --- p.5 / Specification of Lambda λ --- p.7 / Estimation and inference --- p.7 / Chapter Chapter Three --- : Simulation --- p.10 / Study 1 --- p.12 / Study 2 --- p.15 / Study 3 --- p.18 / Chapter Chapter Four --- : Discussion --- p.21 / Summary --- p.21 / Limitation --- p.22 / Further development --- p.23 / Chapter Appendix I --- : Proofs of theorems of weighted distance statistic --- p.24 / Chapter Appendix II --- : Table of numerical results of simulations --- p.26 / Bibliography --- p.30

published_or_final_version / abstract / Dentistry / Doctoral / Doctor of Philosophy

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

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published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

There are several methods of analysing time-to-event data. These include nonparametric approaches such as Kaplan-Meier estimation and parametric approaches such as regression modeling. Parametric regression modeling involves specifying the distribution of the survival time of the individuals, which are commonly chosen to be either exponential, Weibull, log- normal, log-logistic or gamma distributed. Another well known model that does not require assumptions about the hazard function to be made is the Cox proportional hazards model. However, there may be deviations from proportional hazards which may be explained by unaccounted random heterogeneity. In the early 1980s, a series of studies showed concern with the possible bias in the estimated treatment e®ect when important covariates are omitted. Other problems may be encountered with the traditional proportional hazards model when there is a possibility of correlated data, for instance when there is clustering. A method of handling these types of problems is by making use of frailty modeling. Frailty modeling is a method whereby a random e®ect is incorporated in the Cox pro- portional hazards model. While this concept is fairly simple to understand, the method of estimation of the ¯xed and random e®ects becomes complicated. Various methods have been explored by several authors, including the Expectation-Maximisation (EM) algorithm, pe- nalized partial likelihood approach, Markov Chain Monte Carlo (MCMC) methods, Monte Carlo EM approach and di®erent methods using Laplace approximation. The lack of available software is problematic for ¯tting frailty models. These models are usually computationally extensive and may have long processing times. However, frailty modeling is an important aspect to consider, particularly if the Cox proportional hazards model does not adequately describe the distribution of survival time. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2006.

[by] Y.W. Lee. / Bibliography: p. 39. / Army Signal Corps Contract No. W-36-039 sc-32037, Project No. 102B. Dept. of the Army Project No. 3-99-10-022.

published_or_final_version / Community Medicine / Doctoral / Doctor of Philosophy