Rank Based Methods for Repeated Measurement Data

Abstract

This dissertation considers rank based methods for one sample and two sample repeated measurement data. As a specific example, in Chapter 2, this dissertation considers nonparametric tests for selective predation. \cite{randles06} proposed a nonparametric tests for selective predation using the linear score function. Motivated by this method, general rank tests are given for the case of one predatory species and prey characterized by a binary feature of interest and the case of two predatory species and prey characterized by either a continuous or a categorical feature of interest. The score functions used to construct the test statistics are monotone and hence the test is designed to detect simple ordered alternatives. The results based on the asymptotic Gaussian distribution of the test statistics show that the tests retain nominal Type-I error rates. The results also show that power of the asymptotic test depends on the chosen score function. In Chapter 2, we study the one sample and two sample repeated measurement data with random censoring and the simulation results show that we can take the asymptotic distribution as the underlying distribution of the test statistic even with a high censoring rate and a small sample size.

In Chapter 3, this dissertation considers using rank based methods to test the trend of the difference between two samples for two sample repeated measurement data. As a specific example, we consider nonparametric methods for testing whether the rate of prey feature change in the selection of one species is faster than that of another species. Although the Page test is used in conjunction with a single randomized complete block design, we extend it to the situation where we have two randomized complete block designs. We derive the asymptotic distribution of a general test statistic which includes the Page statistic as a special case. The results based on the asymptotic Gaussian distribution of the test statistics show that the tests retain nominal Type-I error rates.

In Chapter 4, the finite sample performance of the rank estimator of regression coefficients obtained using the iteratively reweighted least squares (IRLS) of Sievers and Abebe (2004) is evaluated. Efficiency comparisons show that the IRLS method does quite well in comparison to least squares or the traditional rank estimates in cases of moderate tailed error distributions; however, the IRLS method does not appear to be suitable for heavy tailed data. Moreover, the results show that the IRLS estimator will have an unbounded influence function even if we use an initial estimator with a bounded influence function.

In Chapter 5, this dissertation study how to test the trend of the difference in the two sample repeated measurement data using two generalized estimating equation (GEE) models: the likelihood based GEE model proposed by Liang and Zeger (1986) and the IRLS Rank-based GEE model proposed by \cite{gee}. The results show that the GEE model proposed by \cite{gee} is robust to outliers in response space and can be used to analyze data with small sample sizes compared to the GEE model proposed by Liang and Zeger (1986).