Test of the validity of the Poisson assumption for analysis of most-probable-number results.

A test of the validity of the Poisson assumption for sample replicates in dilution series of finite length is proposed and its properties are examined by using Monte Carlo simulation. The test is based on an examination of the number of intervals between complete sterility and complete infection in a series. The test is applied to a data set of routine influent coliform samples at the Chicago water supply intake. By this test, the data set is rejected as being drawn from a Poisson replication. Tables for direct application to a 3-dilution, 5-tube decimal series are presented, and their application is illustrated.     

Testing approaches for overdispersion in poisson regression versus the generalized poisson model

Overdispersion is a common phenomenon in Poisson modeling, and the negative binomial (NB) model is frequently used to account for overdispersion. Testing approaches (Wald test, likelihood ratio test (LRT), and score test) for overdispersion in the Poisson regression versus the NB model are available. Because the generalized Poisson (GP) model is similar to the NB model, we consider the former as an alternate model for overdispersed count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes a score test for overdispersion based on the GP model and compares the power of the test with the LRT and Wald tests. A simulation study indicates the score test based on asymptotic standard Normal distribution is more appropriate in practical application for higher empirical power, however, it underestimates the nominal significance level, especially in small sample situations, and examples illustrate the results of comparing the candidate tests between the Poisson and GP models. A bootstrap test is also proposed to adjust the underestimation of nominal level in the score statistic when the sample size is small. The simulation study indicates the bootstrap test has significance level closer to nominal size and has uniformly greater power than the score test based on asymptotic standard Normal distribution. From a practical perspective, we suggest that, if the score test gives even a weak indication that the Poisson model is inappropriate, say at the 0.10 significance level, we advise the more accurate bootstrap procedure as a better test for comparing whether the GP model is more appropriate than Poisson model. Finally, the Vuong test is illustrated to choose between GP and NB2 models for the same dataset.     

On the generalized poisson regression mixture model for mapping quantitative trait loci with count data

Statistical methods for mapping quantitative trait loci (QTL) have been extensively studied. While most existing methods assume normal distribution of the phenotype, the normality assumption could be easily violated when phenotypes are measured in counts. One natural choice to deal with count traits is to apply the classical Poisson regression model. However, conditional on covariates, the Poisson assumption of mean-variance equality may not be valid when data are potentially under- or overdispersed. In this article, we propose an interval-mapping approach for phenotypes measured in counts. We model the effects of QTL through a generalized Poisson regression model and develop efficient likelihood-based inference procedures. This approach, implemented with the EM algorithm, allows for a genomewide scan for the existence of QTL throughout the entire genome. The performance of the proposed method is evaluated through extensive simulation studies along with comparisons with existing approaches such as the Poisson regression and the generalized estimating equation approach. An application to a rice tiller number data set is given. Our approach provides a standard procedure for mapping QTL involved in the genetic control of complex traits measured in counts.     

Graphical representation of a generalized linear model-based statistical test estimating the fit of the single-hit Poisson model to limiting dilution assays.

Standardized statistical and graphical methods for analysis of limiting dilution assays are highly desirable to enable investigators to compare and interpret results and conclusions with greater accuracy and precision. According to these requirements, we present in this work a powerful statistical slope test that estimates the fit of the single-hit Poisson model to limiting dilution experiments. This method is readily amenable to a graphical representation. This slope test is obtained by modeling limiting dilution data according to a linear log-log regression model, which is a generalized linear model specially designed for modeling binary data. The result of the statistical slope test can then be graphed to visualize whether the data are compatible or not with the single-hit Poisson model. We demonstrate this statistical test and its graphical representation by using two examples: a real limiting dilution experiment evaluating the growth frequency of IL-2-responsive tumor-infiltrating T cells in a malignant lymph node involved by a B cell non-Hodgkin's lymphoma, and a simulation of a limiting dilution assay corresponding to a theoretical non-single-hit Poisson model, suppressor two-target Poisson model.     

Spatial modeling of data with excessive zeros applied to reindeer pellet‐group counts

We analyze a real data set pertaining to reindeer fecal pellet‐group counts obtained from a survey conducted in a forest area in northern Sweden. In the data set, over 70% of counts are zeros, and there is high spatial correlation. We use conditionally autoregressive random effects for modeling of spatial correlation in a Poisson generalized linear mixed model (GLMM), quasi‐Poisson hierarchical generalized linear model (HGLM), zero‐inflated Poisson (ZIP), and hurdle models. The quasi‐Poisson HGLM allows for both under‐ and overdispersion with excessive zeros, while the ZIP and hurdle models allow only for overdispersion. In analyzing the real data set, we see that the quasi‐Poisson HGLMs can perform better than the other commonly used models, for example, ordinary Poisson HGLMs, spatial ZIP, and spatial hurdle models, and that the underdispersed Poisson HGLMs with spatial correlation fit the reindeer data best. We develop R codes for fitting these models using a unified algorithm for the HGLMs. Spatial count response with an extremely high proportion of zeros, and underdispersion can be successfully modeled using the quasi‐Poisson HGLM with spatial random effects.     

A Multinomial Model for the Quality Control of Colony Counting Procedures

The so‐called good‐laboratory‐practice (GLP) test provides an experimental design and appropriate statistical analysis for the problem of analyst performance assessment in microbiological laboratories. For a given sample material multiple dilution series are generated yielding colony counts from several dilution levels. Statistical evaluation is based on the assumption of Poisson‐distributed colony forming units. In this paper a new model based on conditional binomial and multinomial distributions is presented and it is shown how it is related to the standard model which assumes Poisson‐distributed colony counts. The effects of common working errors on the statistical evaluation of the GLP‐test are investigated.     

Estimation of the Upper Confidence Limit on the Mean of Datasets with Count-Based Concentration Values

Mathematical approaches are not well established for calculating the upper confidence limit (UCL) of the mean of a set of concentration values that have been measured using a count-based analytical approach such as is commonly used for asbestos in air. This is because the uncertainty around the sample mean is determined not only by the authentic between-sample variation (sampling error), but also by random Poisson variation that occurs in the measurement of sample concentrations (measurement error). This report describes a computer-based application, referred to as CB-UCL, that supports the estimation of UCL values for asbestos and other count-based samples sets, with special attention to datasets with relatively small numbers of samples and relatively low counts (including datasets with all-zero count samples). Evaluation of the performance of the application with a range of test datasets indicates the application is useful for deriving UCL estimates for datasets of this type.     

Smooth tests for the zero-inflated poisson distribution

In this article we construct three smooth goodness-of-fit tests for testing for the zero-inflated Poisson (ZIP) distribution against general smooth alternatives in the sense of Neyman. We apply our tests to a data set previously claimed to be ZIP distributed, and show that the ZIP is not a good model to describe the data. At rejection of the null hypothesis of ZIP, the individual components of the test statistic, which are directly related to interpretable parameters in a smooth model, may be used to gain insight into an alternative distribution.     

基于不同零模型的点格局分析

在种群空间格局研究中,定量分析格局及其形成过程已成为生态学家的主要目标。在量化分析的众多方法中,点格局分析是最常用的方法,而在选择零模型时,完全空间随机模型以外的复杂零模型很少使用,实际上,这些零模型可能有助于认识格局的内在特征。为此,我们在研究实例中,选择完全空间随机模型(complete spatial randomness)、泊松聚块模型(Poisson cluster process)和嵌套双聚块模型(nested double-cluster process)对典型草原处于不同恢复演替阶段的羊草(Leymus chinensis)种群空间格局进行了分析。结果发现:完全空间随机模型仅能检测种群在不同尺度下的格局类型;而通过泊松聚块模型和嵌套双聚块模型检验表明,在恢复演替的初期阶段,羊草种群在小尺度范围内偏离泊松聚块模型,而在整个取样范围内完全符合嵌套双聚块模型;随着恢复演替时间的推移,在恢复演替的后期,在整个取样尺度上,羊草种群与泊松聚块模型相吻合。这是很有意义的生态学现象。这一实例表明在应用点格局分析种群空间格局时,仅通过完全空间随机模型的检验来分析格局特征,或许很难论证复杂的生态过程,而选择一些完全空间随机模型以外的较复杂的零模型,可能发现一些有价值的生态学现象,对揭示格局掩盖下的内在机制有所裨益。     

Some goodness-of-fit methods for the Poisson plus added zeros distribution

Methods for making inferences about the Poisson plus added zeros distribution and the truncated Poisson distribution are presented and illustrated with bacteriological data. Some of the methods are designed for testing the compatibility of the zero frequency with the Poisson distribution, whereas others are given for testing the goodness of fit for the truncated Poisson. In particular, a modified form of the Fisher index of dispersion is presented which is suitable for the truncated case. It is shown that the use of the usual expression of the index of dispersion for testing the adequacy of the truncated Poisson is not correct and leads to accepting inadequate fits more frequently than expected on the basis of test of significance. Furthermore, three test statistics are presented for testing the compatability of the zero frequency with the Poisson distribution. The results of the simulation show that two test statistics, one due to Cochran (W. G. Cochran, Biometrics 10:417-451, 1954) and the other to Rao and Chakravarti (C. R. Rao and I. M. Chakravarti, Biometrics 12:264-282, 1956), are preferable to those from the likelihood ratio test.     

Some goodness-of-fit methods for the Poisson plus added zeros distribution.

Methods for making inferences about the Poisson plus added zeros distribution and the truncated Poisson distribution are presented and illustrated with bacteriological data. Some of the methods are designed for testing the compatibility of the zero frequency with the Poisson distribution, whereas others are given for testing the goodness of fit for the truncated Poisson. In particular, a modified form of the Fisher index of dispersion is presented which is suitable for the truncated case. It is shown that the use of the usual expression of the index of dispersion for testing the adequacy of the truncated Poisson is not correct and leads to accepting inadequate fits more frequently than expected on the basis of test of significance. Furthermore, three test statistics are presented for testing the compatability of the zero frequency with the Poisson distribution. The results of the simulation show that two test statistics, one due to Cochran (W. G. Cochran, Biometrics 10:417-451, 1954) and the other to Rao and Chakravarti (C. R. Rao and I. M. Chakravarti, Biometrics 12:264-282, 1956), are preferable to those from the likelihood ratio test.     

Sample size calculations for noninferiority trials with Poisson distributed count data

Clinical trials with Poisson distributed count data as the primary outcome are common in various medical areas such as relapse counts in multiple sclerosis trials or the number of attacks in trials for the treatment of migraine. In this article, we present approximate sample size formulae for testing noninferiority using asymptotic tests which are based on restricted or unrestricted maximum likelihood estimators of the Poisson rates. The Poisson outcomes are allowed to be observed for unequal follow‐up schemes, and both the situations that the noninferiority margin is expressed in terms of the difference and the ratio are considered. The exact type I error rates and powers of these tests are evaluated and the accuracy of the approximate sample size formulae is examined. The test statistic using the restricted maximum likelihood estimators (for the difference test problem) and the test statistic that is based on the logarithmic transformation and employs the maximum likelihood estimators (for the ratio test problem) show favorable type I error control and can be recommended for practical application. The approximate sample size formulae show high accuracy even for small sample sizes and provide power values identical or close to the aspired ones. The methods are illustrated by a clinical trial example from anesthesia.     

Identification of nonlinear systems using random impulse train inputs

Nonlinear systems that require discrete inputs can be characterized by using random impulse train (Poisson process) inputs. The method is analagous to the Wiener method for continuous input systems, where Gaussian white-noise is the input. In place of the Wiener functional expansion for the output of a continuous input system, a new series for discrete input systems is created by making certain restrictions on the integrals in a Volterra series. The kernels in the new series differ from the Wiener kernels, but also serve to identify a system and are simpler to compute. For systems whose impulse responses vary in amplitude but maintain a similar shape, one argument may be held fixed in each kernel. This simplifies the identification problem. As a test of the theory presented, the output of a hypothetical second order nonlinear system in response to a random impulse train stimulus was computer simulated. Kernels calculated from the simulated data agreed with theoretical predictions. The Poisson impulse train method is applicable to any system whose input can be delivered in discrete pulses. It is particularly suited to neuronal synaptic systems when the pattern of input nerve impulses can be made random.     

Clustered mixed nonhomogeneous Poisson process spline models for the analysis of recurrent event panel data

Summary .   A flexible semiparametric model for analyzing longitudinal panel count data arising from mixtures is presented. Panel count data refers here to count data on recurrent events collected as the number of events that have occurred within specific follow-up periods. The model assumes that the counts for each subject are generated by mixtures of nonhomogeneous Poisson processes with smooth intensity functions modeled with penalized splines. Time-dependent covariate effects are also incorporated into the process intensity using splines. Discrete mixtures of these nonhomogeneous Poisson process spline models extract functional information from underlying clusters representing hidden subpopulations. The motivating application is an experiment to test the effectiveness of pheromones in disrupting the mating pattern of the cherry bark tortrix moth. Mature moths arise from hidden, but distinct, subpopulations and monitoring the subpopulation responses was of interest. Within-cluster random effects are used to account for correlation structures and heterogeneity common to this type of data. An estimating equation approach to inference requiring only low moment assumptions is developed and the finite sample properties of the proposed estimating functions are investigated empirically by simulation.     

Optimal experimental design and sample size for the statistical evaluation of data from somatic mutation and recombination tests (SMART) in Drosophila

In genetic toxicology it is important to know whether chemicals should be regarded as clearly hazardous or whether they can be considered sufficiently safe, which latter would be the case from the genotoxicologist's view if their genotoxic effects are nil or at least significantly below a predefined minimal effect level. A previously presented statistical decision procedure which allows one to make precisely this distinction is now extended to the question of how optimal experimental sample size can be determined in advance for genotoxicity experiments using the somatic mutation and recombination tests (SMART) of Drosophila. Optimally, the statistical tests should have high power to minimise the chance for statistically inconclusive results. Based on the normal test, the statistical principles are explained, and in an application to the wing spot assay, it is shown how the practitioner can proceed to optimise sample size to achieve numerically satisfactory conditions for statistical testing. The somatic genotoxicity assays of Drosophila are in principle based on somatic spots (mutant clones) that are recovered in variable numbers on individual flies. The underlying frequency distributions are expected to be of the Poisson type. However, some care seems indicated with respect to this latter assumption, because pooling of data over individuals, sexes, and experiments, for example, can (but need not) lead to data which are overdispersed, i.e, the data may show more variability than theoretically expected. It is an undesired effect of overdispersion that in comparisons of pooled totals it can lead to statistical testing which is too liberal, because overall it yields too many seemingly significant results. If individual variability considered alone is not contradiction with Poisson expectation, however, experimental planning can help to minimise the undesired effects of overdispersion on statistical testing of pooled totals. The rule for the practice is to avoid disproportionate sampling. It is recalled that for optimal power in statistical testing, it is preferable to use equal total numbers of flies in the control and treated series. Statistical tests which are based on Poisson expectations are too liberal if there is overdispersion in the data due to excess individual variability. In this case we propose to use the U test as a non-parametric two-sample test and to adjust the estimated optimal sample size according to (i) the overdispersion observed in a large historical control and (ii) the relative efficiency of the U test in comparison to the t test and related parametric tests.     

Comparison of mathematical models for the maternal age dependence of Down's syndrome rates

Summary The maternal age dependence of Down's syndrome rates was analyzed by two mathematical models, a discontinuous (DS) slope model which fits different exponential equations to different parts of the 20–49 age interval and a CPE model which fits a function that is the sum of a constant and exponential term over this whole 20–49 range. The CPE model had been considered but rejected by Penrose, who preferred models postulating changes with age assuming either a power function X¹⁰, where X is age or a Poisson model in which accumulation of 17 events was the assumed threshold for the occurrence of Down's syndrome. However, subsequent analyses indicated that the two models preferred by Penrose did not fit recent data sets as well as the DS or CPE model. Here we report analyses of broadened power and Poisson models in which n (the postulated number of independent events) can vary. Five data sets are analyzed. For the power models the range of the optimal n is 11 to 13; for the Poisson it is 17 to 25. The DS, Poisson, and power models each give the best fit to one data set; the CPE, to two sets. No particular model is clearly preferable. It appears unlikely that, with a data set from any single available source, a specific etiologic hypothesis for the maternal age dependence of Down's syndrome can be clearly inferred by the use of these or similar regression models.     

On the estimation of the number of antibody-forming cells

The variability of the number of antibody-forming cells (AbFC) on using the plaque method is discussed. In the first part of the study it is shown on the basis of experimental data that the number of AbFC in a standard number of spleen cells has approximately Poisson distribution in one animal if the number of AbFC in a suspension is small, but that as the number of AbFC rises (even if the number of test cells in the given volume remains the same), variance increases more rapidly than the mean value and negative binomial distribution is a better probability model. The maintenance of conditions for Poisson distribution is evidently also related to the absolute number of test cells in a given volume, however. If the number of spleen cells in a given volume is raised, it is impossible to maintain complete homogeneity of the suspension; the cells form agglomerates, resulting in greater variability of the number of cells in individual drops. Under certain conditions, however, negative binomal distribution tends to Poisson distribution. In the second part, the question of determination of the error of estimation of the number of AbFC, expressed as percentages of the true value, is studied. Presuming Poisson distribution, it is pointed out that it is impossible to predetermine a fixed limit of the experiment, i.e one which will ensure for a given confidence level that the empirically determined value will not differ from the true value by more than a predetermined percentage of the true value; for routine use, however, a nomogram for the approximate estimation of the degree of error with which we work is suggested. Alternatively, the sequential method can be employed for estimating the number of AbFC with a given degree of error, but this statistical method is more exacting as regards both the preparation and the execution of the experiment. Dedicated to Academician Ivan Málek on the occasion of his 60th birthday     

Analyzing multi-environment variety trials using randomization-derived mixed models

Of interest is the analysis of results of a series of experiments repeated at several environments with the same set of plant varieties. Suppose that the experiments, multi-environment variety trials, are all conducted in resolvable incomplete block (IB) designs. Following the randomization approach adopted in Caliński and Kageyama (2000, Lecture Notes in Statistics, 150), two models for analyzing such trial data can be considered. One is derived under a complete additivity assumption, the other takes into account possible different responses of the varieties to variable environmental conditions. The analysis under the first, the standard model, does not provide answers to questions related to the performance of the individual varieties at different environments. These can be considered when using the more general second model. The purpose of this article is to devise interesting parameter estimation and hypothesis testing procedures under that more realistic model. Its application is illustrated by a thorough analysis of a set of data from a winter wheat series of trials.     

Minimum Hellinger distance estimation for k-component poisson mixture with random effects

Summary .   The k-component Poisson regression mixture with random effects is an effective model in describing the heterogeneity for clustered count data arising from several latent subpopulations. However, the residual maximum likelihood estimation (REML) of regression coefficients and variance component parameters tend to be unstable and may result in misleading inferences in the presence of outliers or extreme contamination. In the literature, the minimum Hellinger distance (MHD) estimation has been investigated to obtain robust estimation for finite Poisson mixtures. This article aims to develop a robust MHD estimation approach for k-component Poisson mixtures with normally distributed random effects. By applying the Gaussian quadrature technique to approximate the integrals involved in the marginal distribution, the marginal probability function of the k-component Poisson mixture with random effects can be approximated by the summation of a set of finite Poisson mixtures. Simulation study shows that the MHD estimates perform satisfactorily for data without outlying observation(s), and outperform the REML estimates when data are contaminated. Application to a data set of recurrent urinary tract infections (UTI) with random institution effects demonstrates the practical use of the robust MHD estimation method.