الفهرس | Only 14 pages are availabe for public view |
Abstract Sometimes, it is required to approximate the distribution of some statistics whose exact distribution cannot conveniently obtain. Saddlepoint approximation methods are mathematical techniques applied to give a highly accurate approximation of the density and cumulative distribution functions for a statistic using its moment generating function. The proposed method is distinguished from the asymptotic method in being very accurate with small and moderate sample sizes, and being very accurate in the tails of the distribution. The saddlepoint approximation is more precise than asymptotic method and computationally faster than the simulation procedure. In general, for a sample of size n, the saddlepoint approximation is accurate up to O(n − 3 2 ), when considered over sets of bounded central tendency, while up to O(n − 1 2 ) for the central limit theory. Here, the saddlepoint approximations are used to approximate the exact p-value of the weighted log-rank tests for left-truncated data. The thesis consists of five chapters: Chapter 1 This chapter is an introduction for the thesis in which basic concepts and information needed for the following chapters are presented. First, we propose the weighted log-rank class of two sample tests. Second, we provide the saddlepoint approximation methods and their applications. Chapter 2 In this chapter, the saddlepoint approximation methods is used to approximate the mid-p-values of the weighted log-rank tests for left-truncated data under the random allocation design. The accuracy of saddlepoint approximation is verified through real data examples and simulation studies. The speed and accuracy of saddlepoint approximation allows us to invert the permutation tests to determine 100(1 − α)% confidence intervals for the treatment effect under Cox proportional hazard model. 1 Summary Chapter 3 In this chapter, we derive the saddlepoint approximation for the exact permutation p-values of the weighted log-rank tests for left-truncated data under Wei’s urn design as a more accurate alternative than the asymptotic approach. Real data examples are analyzed and simulation studies are applied to show the accuracy and efficiency of the saddlepoint approximation. The weighted log-rank tests are inverted to determine nominal 95% confidence intervals for the treatment effect in the presence of left-truncated data. Chapter 4 If the data are collected from different clinics and we need to reduce selection bias and control the imbalance of group sizes, the appropriate design is the randomized block design. In this chapter, the permutation distribution of the weighted log-rank tests for left-truncated data under randomized block design is derived. Furthermore, the saddlepoint approximation is proposed to approximate the exact p-values of the weighted log-rank tests as a possible alternative to the normal approximation method. The performance of the proposed method is shown through real data examples and simulation studies. The test is inverted to calculate the confidence intervals for the treatment effect. Chapter 5 The truncated binomial design is one of the commonly used designs for forcing balance in clinical trials to eliminate experimental bias. In this chapter, we consider the exact distribution of the weighted log-rank class of left-truncated data under the truncated binomial design. A double saddlepoint approximation for the p-value of this class is derived under truncated binomial design. The accuracy of the proposed method over the normal asymptotic facilitate the inversion of the weighted log-rank tests to determine the nominal 95% confidence intervals for the treatment effect in the presence of the left-truncated data. |