Resumen de New multi-sample nonparametric tests for panel count data
This talk would consider the problem of multi-sample nonparametric comparison of point processes with panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. For this problem, I will construct two new classes of nonparametric test statistics based on the accumulated weighted di erences between the rates of increase of the estimated mean functions of the point processes over observation times, where the nonparametric maximum likelihood approach instead of the nonparametric maximum pseudo-likelihood (or the isotonic regression) is used to estimate the mean function. The asymptotic distributions of the proposed statistics will be derived and their nite-sample properties will be examined through Monte Carlo simulations. The simulation results will demonstrate that the proposed methods work quite well. Finally, two real data sets will be analyzed and presented as illustrative examples, and some open problems will be mentioned.
Consider a study that concerns some recurrent event and suppose that each subject in the study gives rise to a point process N(t), denoting the total number of occurrences of the event of interest up to time t. Also suppose that for each subject, observations include only the values of N(t) at discrete observation times or the numbers of occurrences of the event between the observation times. Such data are usually referred to in the survival analysis literature as panel count data. Our focus here will be on the situation when such a study involves k groups, where k 2. Let l(t) denote the mean function of N(t) corresponding to the l-th group for l = 1; ; k. The problem of interest is then to test the hypothesis H0 : 1(t) = = k(t).
A number of authors have discussed the analysis of recurrent event data when each subject in the study is observed continuously over an interval or when the exact times of occurrences of the recurrent event are known. Many of the commonly used statistical methods are available in the literature for the analysis of recurrent event data. In contrast, there exists modest amount of research on the analysis of panel count data. Some limited studies have been carried out on the estimation of the mean function of N(t) and regression analysis for such data. To test the hypothesis H0, one suggestion that has been made is to transform the problem to a multivariate comparison problem and then apply a multivariate Wilcoxon-type rank test. Another nonparametric procedure has been proposed for this problem, but this procedure depends on the assumption that treatment indicators can be regarded as independent and identically distributed random variables, which may not be the case in practice. However, a class of nonparametric tests have been proposed for the two-sample comparison based on the istonic regression estimator of the mean function of point process. Nonparametric tests have also been suggested for the problem based on the nonparametric maximum pseudo-likelihood estimator that is equivalent to the istonic regression estimator. But, it has been proved that the nonparametric maximum likelihood estimator (NPMLE) of the mean function is more ecient than the nonparametric maximum pseudo-likelihood estimator (NPMPLE) by means of Monte Carlo simulations; yet, no nonparametric test has been developed in the literature for panel count data based on the NPMLE. One would naturally expect that the tests based on the NPMLE would be more powerful than the proposed tests based on the NPMPLE.
However, unlike the isotonic regression estimate, the maximum likelihood estimate has no closed form and its computation requires an iterative convex minorant algorithm. Here, we propose nonparametric tests using the maximum likelihood estimator and compare them with the existing tests for the problem of multi-sample nonparametric comparison of point processes with panel count data.
In this talk, I will rst discuss the estimation of the mean function and the existing nonparametric tests for the hypothesis H0 when only panel count data are available. The asymptotic normality of the functional of the NPMLE will then be presented. Next, I will present two classes of nonparametric test statistics. The statistics, motivated by the property of the NPMLE and the idea used in survival analysis, will then be formulated as the integrated weighted di erence between the rates of increase of the estimated mean functions corresponding to the pooled data and each group or two groups. Also, the asymptotic normality of these test statistics will be established. Then, the nite-sample properties of the proposed test statistics will be examined by means of Monte Carlo simulations. For the purpose of illustration of the developed inferential methods, I will apply the proposed methods to two data sets, one from a oating gallstones study and the other from a bladder tumor study, respectively. Finally, I will make some concluding remarks and suggest some research problems of interest for further study.
