New Nonparametric Tests for Panel Count Data

Abstract

<p> Statistical analysis of panel count data is an important topic to a number of applied fields including biology, engineering, econometrics, medicine, and public health. Panel count data include observations on subjects over multiple time points where the response variable is a count or recurrent event process when only the numbers of events occurring between observation time points are available. The choice of method for analyzing panel count data usually depends on the relationship between the observation times and the response variable and questions of interest. Most of the previous research was done when the observation times are fixed. If the observation times are random, the data structure becomes more challenging since the observation times for individual subjects vary in addition to the incompleteness of observations. The model-based approach was used to deal with such data. However, this method relies on extra assumptions on the observation scheme and thus is restrictive in practice. In this dissertation, we discuss the problem of multi-sample nonparametric comparison of counting processes with panel count data, which arise naturally when recurrent events are considered. For the problem considered, we develop some new nonparametric tests.</p> <p> First, we construct a class of nonparametric test statistics based on the integrated weighted differences between the estimated mean functions of the count processes, where the isotonic regression estimate is used for the mean functions. The asymptotic distributions of the proposed statistics are derived and their finite-sample properties are examined through Monte Carlo simulations. A panel count data from a cancer study is analyzed and presented as an illustrative example.</p> <p>As shown through Monte Carlo simulations, the nonparametric maximum likelihood estimator (NPMLE) of the mean function is more efficient than the nonparametric maximum pseudo-likelihood estimator (NPMPLE). However, no nonparametric tests have been discussed in the literature for panel count data based on the NPMLE since the NPMLE is more complicated both theoretically and computationally. It is, therefore, particularly important to develop nonparametric tests based on the NPMLE for panel count data.</p> <p> In the second part of the dissertation, we focus on the situation when treatment indicators can be regarded as independent and identically distributed random variables and propose a nonparametric test in this case using the maximum likelihood estimator. The asymptotic property of the test statistic is derived. Simulation studies are carried out which suggest that the proposed method works well for practical situations, and is more powerful than the existing tests based on the NPMPLEs of the mean functions.</p> <p>In the third part of the dissertation, we consider more general situations. We construct a class of nonparametric tests based on the accumulated weighted differences between the rates of increase of the estimated mean functions of the counting processes over observation times, where the nonparametric maximum likelihood approach is used to estimate the mean functions instead of the nonparametric maximum pseudolikelihood. The asymptotic distributions of the proposed statistics are derived and their finite-sample properties are evaluated by means of Monte Carlo simulations. The simulation results show that the proposed methods work quite well and the tests based on NPMLE are more powerful than those based on NPMPLE. Two real data sets are analyzed and presented as illustrative examples.</p> <p>The last part of the dissertation discusses a special type of panel count data, namely, current status or case 1 interval-censored data. Such data often occur in tumorigenicity experiments. For nonparametric two-sample comparison based on censored or interval-censored data, most of the existing methods have focused on testing the hypothesis that specifies the two population distributions to be identical under the assumption that observation or censoring times have the same distribution. We consider the nonparametric Behrens-Fisher hypothesis (NBFH) under this settings. For this purpose, we study the asymptotic property of the nonparametric maximum likelihood estimator of the probability that an observation from the first distribution exceeds an observation from the second distribution. A nonparametric test for the NBFH is proposed and the asymptotic normality of the proposed test is established. The method is evaluated using simulation studies and illustrated by a set of real data from a tumorigenicity experiment.</p>