Saddlepoint Approximations For Linear Rank Models


Ehab F. Abdelfattah, Ph.D. Candidate, Department of Statistics, Colorado State University

Tuesday, 17 May 2005
1:00 PM
Room 006, Statistics Building


Linear rank tests are often used to test the effectiveness of a treatment as compares to a control in the two independent samples context. In the survival comparisons that occur in clinical trials, the time to event responses are typically right censored and modified rank tests are needed to accommodate the censoring.

This thesis proposes the use of saddlepoint approximations as a means for determining the mid-p-values for linear rank tests that involve right censoring. The saddlepoint approximations are for the exact permutation distributions of the rank test statistics under the null hypothesis of no treatment effect. Normal approximations are often used by programs such as SAS to approximate the mid-p-values for these permutation distributions. These approximations are shown to be much less accurate in the simulations of this thesis. This thesis handles the weighted log-rank class, which contains log-rank, Gehan, Peto-Prentice generalized Wilcoxon, Tarone-Ware, and Harrington-Fleming tests.

The replacement of an analytical saddlepoint computation for the simulation effort required to determine percentiles of the permutation distribution makes the computation of mid-p-values simple and efficient. This speed of computation also allows for the inversion of the ranks tests to determine 95% confidence intervals from the tests. In an uncensored case that uses the Wilcoxon rank scores, the inversion leads to a confidence interval based around the Hodges-Lehmann estimate. In the censored case, Kalbfleish and Prentice (2002, §7.4) have recognized the need to invert such tests in determining confidence intervals. At the same time, they have also pointed out that the computational burden of such inversion currently prohibits the routine use of such methods. The executable files that accompany this thesis now makes the inversion of such tests routine. They deliver confidence intervals whose nominal level is, for all practical purposes, the exact intended level.

A second objective of the thesis is to deal with the test for trend when there are k?3 treatment levels whose dosage level provides an ordering of the treatment groups. Saddlepoint approximations for the mid-p-values for the permutation distributions of the weighted log-rank type trend tests for testing ordered alternative shown to be more accurate compare to the normal approximations.

Saddlepoint approximations also provide fast and extremely accurate methods to approximate the mid-p-values for linear rank tests for independence and symmetry. Such methods deal with the commonly used Spearman, and weighted Mann tests for independence, and Wilcoxon one sample and Wilcoxon matched pair signed rank tests for symmetry.

Advisory Committee: R.W. Butler (Advisor), M.M. Siddiqui, P.W. Mielke, Jr., J. D. Salas



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