A nonparametric test for the equality of counting processes with panel count data
Introduction
Consider a study that concerns some recurrent event and suppose that each subject in the study gives rise to a counting process , denoting the total number of occurrences of the event of interest up to time . Also suppose that for each subject, observations include only the values of at discrete observation times or the numbers of occurrences of the event between the observation times. Such data are usually referred to as panel count data (Sun and Kalbfleisch, 1995, Wellner and Zhang, 2000), which frequently occur in medical follow-up studies and reliability experiments, for example. Our focus here will be on the situation when such a study involves () groups. Let denote the mean function of corresponding to the th group for . The problem of interest is then to test the hypothesis , , where is the maximum observation time.
For the analysis of panel count data, Sun and Kalbfleisch (1995) and Wellner and Zhang (2000) studied estimation of the mean function of . Sun and Wei (2000), Zhang (2002), Hu et al. (2003) and Sun et al. (2007) discussed regression analysis for such data. The model-based methods depend on the expression of the model used and model checking is needed but difficult in practice. To test the hypothesis , Thall and Lachin (1988) suggested to transform the problem to a multivariate comparison problem and then apply a multivariate Wilcoxon-type rank test. Sun and Fang (2003) proposed a nonparametric procedure for this problem under the assumption that treatment indicators can be regarded as independent and identically distributed random variables. Park et al. (2007) proposed a class of nonparametric tests for the two-sample comparison based on the isotonic regression estimator of the mean function of counting process. Zhang (2006) also presented nonparametric tests for the problem based on the nonparametric maximum pseudo-likelihood estimator that is equivalent to the isotonic regression estimator (Wellner and Zhang, 2000). Also, Wellner and Zhang (2000) showed through Monte Carlo simulations that 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. However, unlike the isotonic regression estimate, the maximum likelihood estimate has no closed-form expression and its computation requires an iterative convex minorant algorithm. In this paper, for simplicity, we focus on the situation considered by Sun and Fang (2003) and propose a nonparametric test using the maximum likelihood estimator and then compare its power with those of existing tests for the problem of two-sample nonparametric comparison of counting processes with simulated panel count data.
The rest of this paper is organized as follows. Section 2 discusses estimation of the mean function and the existing nonparametric tests for the hypothesis when only panel count data are available. Section 3 presents a new nonparametric test statistic motivated by the property of the NPMLE and the idea used by Sun and Fang (2003). Also, the asymptotic normality of the test statistic is established. In Section 4, finite-sample property of the proposed test statistic is examined through Monte Carlo simulations. In Section 5, we apply the proposed method to a data from a floating gallstone study. Finally, some concluding remarks are made in Section 6.
Section snippets
Nonparametric maximum likelihood estimation of mean function
Wellner and Zhang (2000) studied two estimators of the mean of a counting process with panel count data: the nonparametric maximum pseudo-likelihood estimator and the nonparametric maximum likelihood estimator. To describe the test statistics, we first introduce the NPMLE. Suppose that is a nonhomogeneous Poisson process with the mean function . Also suppose that for each subject, observations include only the values of at discrete observation times
A nonparametric test with panel count data
Consider a longitudinal study that is concerned with some recurrent event and involves independent subjects from different groups. Let denote the group indicator of subject () and assume that group indicator is a scalar variable. For the two-sample comparison problem, and or 1. For the dose-effects problem, is the number of doses tested in the experiment and denotes the dose given to subject (). Let denote the counting process arising from subject
Simulation study
To examine the finite-sample property of the proposed test statistic and compare it with those of the tests presented by Sun and Fang (2003), Park et al. (2007) and Zhang (2006), we carry out a simulation study for the two-sample comparison problem. Let be defined as in Section 3. Let denote the test proposed by Sun and Fang (2003), and let denote the tests presented by Park et al. (2007) and Zhang (2006) with three different weight processes: ,
Illustrative example
To illustrate the proposed method, we consider a floating gallstone study presented by Thall and Lachin (1988). The data comprise the first year follow-up of the patients in two study groups, placebo (48) and high-dose chenodiol (65), from the National Cooperative Gallstone Study. The data include the successive visit times in study weeks and the associated counts of episodes of nausea. The whole study consists of 916 patients who were randomized to placebo, low dose, or high dose group and
Concluding remarks
This paper discusses the problem of the nonparametric comparison of counting processes when only panel count data are available. The nonparametric maximum likelihood estimators are used to estimate the mean functions of counting processes. A new nonparametric test is proposed for the problem and 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
Acknowledgements
The authors are very grateful to the editor and two referees for their helpful comments and suggestions that greatly improved the paper. They thank Drs. Ying Zhang and Minggen Lu for supplying the R code for the modified iterative convex minorant algorithm presented by Wellner and Zhang (2000). The research of Xingqiu Zhao is supported in part by a grant from the Hong Kong Polytechnic University.
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