The Analysis of Life Table and Follow-up Data with Covariates Using Poisson Regression Model

The Poisson regression model for the analysis of life table and follow-up data with covariates is presented. An example is presented to show how this technique can be used to construct a parsimonious model which describes a set of survival data. All parameters in the model, the hazard and survival functions are estimated by maximum likelihood.     

Analysis of Rates Using a Generalized Poisson Regression Model

In this paper a generalization of the Poisson regression model indexed by a shape parameter is proposed for the analysis of life table and follow-up data with concomitant variables. The model is suitable for analysis of extra-Poisson variation data. The model is used to fit the survival data given in Holford (1980). The model parameters, the hazard and survival functions are estimated by the method of maximum likelihood. The results obtained from this study seem to be comparable to those obtained by Chen (1988). Approximate tests of the dispersion and goodness-of-fit of the data to the model are also discussed.     

Zero-Altered and other Regression Models for Count Data with Added Zeros

On occasion, generalized linear models for counts based on Poisson or overdispersed count distributions may encounter lack of fit due to disproportionately large frequencies of zeros. Three alternative types of regression models that utilize all the information and explicitly account for excess zeros are examined and given general formulations. A simple mechanism for added zeros is assumed that directly motivates one type of model, here called the added-zero type, particular forms of which have been proposed independently by D. LAMBERT (1992) and in unpublished work by the author. An original regression formulation (the zero-altered model) is presented as a reduced form of the two-part model for count data, which is also discussed. It is suggested that two-part models be used to aid in development of an added-zero model when the latter is thought to be appropriate.     

广义岭回归在家禽育种值估计中的应用

讨论了岭回归方法应用于混合线性模型方程组中估计家禽育种值的方法,其实质是将传统的混合线性模型方程组理解为一种广义岭回归估计,为确定遗传参数的估计提供了一种途径;同时,以番鸭为例,考虑了一个性状和两个固定效应,采用广义岭回归法对公番鸭育种值进行了估计,并与最佳线性无偏预测法(BLUP 法)进行了比较,结果表明,广义岭回归方法和BLUP 法估计的育种值及其排序非常接近,其相关系数和秩相关系数分别达到了0.998~(**)和0.986~(**),且采用广义岭回归法预测的误差率低(在±10%以内);表明在混合线性模型方程组中使用广义岭回归估计动物育种值的方法具有可行性,并可省去估计遗传参数的过程,使BLUP 法在动物选育中的应用更具实用性.     

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.     

Climate variations and salmonellosis transmission in Adelaide,South Australia: a comparison between regression models

This is the first study to identify appropriate regression models for the association between climate variation and salmonellosis transmission. A comparison between different regression models was conducted using surveillance data in Adelaide, South Australia. By using notified salmonellosis cases and climatic variables from the Adelaide metropolitan area over the period 1990–2003, four regression methods were examined: standard Poisson regression, autoregressive adjusted Poisson regression, multiple linear regression, and a seasonal autoregressive integrated moving average (SARIMA) model. Notified salmonellosis cases in 2004 were used to test the forecasting ability of the four models. Parameter estimation, goodness-of-fit and forecasting ability of the four regression models were compared. Temperatures occurring 2 weeks prior to cases were positively associated with cases of salmonellosis. Rainfall was also inversely related to the number of cases. The comparison of the goodness-of-fit and forecasting ability suggest that the SARIMA model is better than the other three regression models. Temperature and rainfall may be used as climatic predictors of salmonellosis cases in regions with climatic characteristics similar to those of Adelaide. The SARIMA model could, thus, be adopted to quantify the relationship between climate variations and salmonellosis transmission.     

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.     

Loglinear models for the robust design in mark-recapture experiments

The robust design is a method for implementing a mark-recapture experiment featuring a nested sampling structure. The first level consists of primary sampling sessions; the population experiences mortality and immigration between primary sessions so that open population models apply at this level. The second level of sampling has a short mark-recapture study within each primary session. Closed population models are used at this stage to estimate the animal abundance at each primary session. This article suggests a loglinear technique to fit the robust design. Loglinear models for the analysis of mark-recapture data from closed and open populations are first reviewed. These two types of models are then combined to analyze the data from a robust design. The proposed loglinear approach to the robust design allows incorporating parameters for a heterogeneity in the capture probabilities of the units within each primary session. Temporary emigration out of the study area can also be accounted for in the loglinear framework. The analysis is relatively simple; it relies on a large Poisson regression with the vector of frequencies of the capture histories as dependent variable. An example concerned with the estimation of abundance and survival of the red-back vole in an area of southeastern Québec is presented.     

Inference on Collapsibility in Generalized Linear Models

GREENLAND and MICKEY (1988) derived a closed-form collapsibility test and confidence interval for IxJxK contingency tables with qualitative factors, and presented a small simulation study of its performance. We show how their method can be extended to regression models linear in the natural parameter of a one-parameter exponential family, in which the parameter of interest is the difference of “crude” and “adjusted” regression coefficients. A simplification of the method yields a generalization of the test for omitted covariates given by HAUSMAN (1978) for ordinary linear regression. We present an application to a study of coffee use and myocardial infarction, and a simulation study which indicates that the simplified test performs adequately in typical epidemiologic settings.     

Semiparametric analysis of zero-inflated count data

Medical and public health research often involve the analysis of count data that exhibit a substantially large proportion of zeros, such as the number of heart attacks and the number of days of missed primary activities in a given period. A zero-inflated Poisson regression model, which hypothesizes a two-point heterogeneity in the population characterized by a binary random effect, is generally used to model such data. Subjects are broadly categorized into the low-risk group leading to structural zero counts and high-risk (or normal) group so that the counts can be modeled by a Poisson regression model. The main aim is to identify the explanatory variables that have significant effects on (i) the probability that the subject is from the low-risk group by means of a logistic regression formulation; and (ii) the magnitude of the counts, given that the subject is from the high-risk group by means of a Poisson regression where the effects of the covariates are assumed to be linearly related to the natural logarithm of the mean of the counts. In this article we consider a semiparametric zero-inflated Poisson regression model that postulates a possibly nonlinear relationship between the natural logarithm of the mean of the counts and a particular covariate. A sieve maximum likelihood estimation method is proposed. Asymptotic properties of the proposed sieve maximum likelihood estimators are discussed. Under some mild conditions, the estimators are shown to be asymptotically efficient and normally distributed. Simulation studies were carried out to investigate the performance of the proposed method. For illustration purpose, the method is applied to a data set from a public health survey conducted in Indonesia where the variable of interest is the number of days of missed primary activities due to illness in a 4-week period.     

Generalized estimating equations: A pragmatic and flexible approach to the marginal GLM modelling of correlated data in the behavioural sciences

Within behavioural research, non‐normally distributed data with a complicated structure are common. For instance, data can represent repeated observations of quantities on the same individual. The regression analysis of such data is complicated both by the interdependency of the observations (response variables) and by their non‐normal distribution. Over the last decade, such data have been more and more frequently analysed using generalized mixed‐effect models. Some researchers invoke the heavy machinery of mixed‐effect modelling to obtain the desired population‐level (marginal) inference, which can be achieved by using simpler tools—namely by marginal models. This paper highlights marginal modelling (using generalized estimating equations [GEE]) as an alternative method. In various situations, GEE can be based on fewer assumptions and directly generate estimates (population‐level parameters) which are of immediate interest to the behavioural researcher (such as population means). Using four examples from behavioural research, we demonstrate the use, advantages, and limits of the GEE approach as implemented within the functions of the ‘geepack’ package in R.     

Bringing consistency to simulation of population models--Poisson simulation as a bridge between micro and macro simulation

Population models concern collections of discrete entities such as atoms, cells, humans, animals, etc., where the focus is on the number of entities in a population. Because of the complexity of such models, simulation is usually needed to reproduce their complete dynamic and stochastic behaviour. Two main types of simulation models are used for different purposes, namely micro-simulation models, where each individual is described with its particular attributes and behaviour, and macro-simulation models based on stochastic differential equations, where the population is described in aggregated terms by the number of individuals in different states. Consistency between micro- and macro-models is a crucial but often neglected aspect. This paper demonstrates how the Poisson Simulation technique can be used to produce a population macro-model consistent with the corresponding micro-model. This is accomplished by defining Poisson Simulation in strictly mathematical terms as a series of Poisson processes that generate sequences of Poisson distributions with dynamically varying parameters. The method can be applied to any population model. It provides the unique stochastic and dynamic macro-model consistent with a correct micro-model. The paper also presents a general macro form for stochastic and dynamic population models. In an appendix Poisson Simulation is compared with Markov Simulation showing a number of advantages. Especially aggregation into state variables and aggregation of many events per time-step makes Poisson Simulation orders of magnitude faster than Markov Simulation. Furthermore, you can build and execute much larger and more complicated models with Poisson Simulation than is possible with the Markov approach.     

Comparison of Bayesian methods for flexible modeling of spatial risk surfaces in disease mapping

Bayesian hierarchical models usually model the risk surface on the same arbitrary geographical units for all data sources. Poisson/gamma random field models overcome this restriction as the underlying risk surface can be specified independently to the resolution of the data. Moreover, covariates may be considered as either excess or relative risk factors. We compare the performance of the Poisson/gamma random field model to the Markov random field (MRF)‐based ecologic regression model and the Bayesian Detection of Clusters and Discontinuities (BDCD) model, in both a simulation study and a real data example. We find the BDCD model to have advantages in situations dominated by abruptly changing risk while the Poisson/gamma random field model convinces by its flexibility in the estimation of random field structures and by its flexibility incorporating covariates. The MRF‐based ecologic regression model is inferior. WinBUGS code for Poisson/gamma random field models is provided.     

Generalized additive modeling with implicit variable selection by likelihood-based boosting

The use of generalized additive models in statistical data analysis suffers from the restriction to few explanatory variables and the problems of selection of smoothing parameters. Generalized additive model boosting circumvents these problems by means of stagewise fitting of weak learners. A fitting procedure is derived which works for all simple exponential family distributions, including binomial, Poisson, and normal response variables. The procedure combines the selection of variables and the determination of the appropriate amount of smoothing. Penalized regression splines and the newly introduced penalized stumps are considered as weak learners. Estimates of standard deviations and stopping criteria, which are notorious problems in iterative procedures, are based on an approximate hat matrix. The method is shown to be a strong competitor to common procedures for the fitting of generalized additive models. In particular, in high-dimensional settings with many nuisance predictor variables it performs very well.     

Minimum Hellinger distance estimation for finite mixtures of Poisson regression models and its applications

Minimum Hellinger distance estimation (MHDE) has been shown to discount anomalous data points in a smooth manner with first-order efficiency for a correctly specified model. An estimation approach is proposed for finite mixtures of Poisson regression models based on MHDE. Evidence from Monte Carlo simulations suggests that MHDE is a viable alternative to the maximum likelihood estimator when the mixture components are not well separated or the model parameters are near zero. Biometrical applications also illustrate the practical usefulness of the MHDE method.     

Comparison of random-effects meta-analysis models for the relative risk in the case of rare events: A simulation study

Pooling the relative risk (RR) across studies investigating rare events, for example, adverse events, via meta-analytical methods still presents a challenge to researchers. The main reason for this is the high probability of observing no events in treatment or control group or both, resulting in an undefined log RR (the basis of standard meta-analysis). Other technical challenges ensue, for example, the violation of normality assumptions, or bias due to exclusion of studies and application of continuity corrections, leading to poor performance of standard approaches. In the present simulation study, we compared three recently proposed alternative models (random-effects [RE] Poisson regression, RE zero-inflated Poisson [ZIP] regression, binomial regression) to the standard methods in conjunction with different continuity corrections and to different versions of beta-binomial regression. Based on our investigation of the models' performance in 162 different simulation settings informed by meta-analyses from the Cochrane database and distinguished by different underlying true effects, degrees of between-study heterogeneity, numbers of primary studies, group size ratios, and baseline risks, we recommend the use of the RE Poisson regression model. The beta-binomial model recommended by Kuss (2015) also performed well. Decent performance was also exhibited by the ZIP models, but they also had considerable convergence issues. We stress that these recommendations are only valid for meta-analyses with larger numbers of primary studies. All models are applied to data from two Cochrane reviews to illustrate differences between and issues of the models. Limitations as well as practical implications and recommendations are discussed; a flowchart summarizing recommendations is provided.     

A score test for testing a zero-inflated Poisson regression model against zero-inflated negative binomial alternatives

Count data often show a higher incidence of zero counts than would be expected if the data were Poisson distributed. Zero-inflated Poisson regression models are a useful class of models for such data, but parameter estimates may be seriously biased if the nonzero counts are overdispersed in relation to the Poisson distribution. We therefore provide a score test for testing zero-inflated Poisson regression models against zero-inflated negative binomial alternatives.     

前瞻研究中多因素人群归因危险度的Poisson回归模型估计

本文提出多因素前瞻研究中利用Poisson回归发病率预测模型和相对危险度估计调整和综合人群归因危险度的方法,与Bruzzi等和Deubner等提出的多因素人群归因危险度估计方法进行了比较,强调在前瞻资料的人群归因危险度的估计中利用poisson回归模型考虑失访病例和随访时间效应,并能直接估计相对危险度的优势.应用所建立的方法对启东县肝病人群14年前瞻观察资料进行肝癌危险因素的人群归因危险度的估计。     

Bayesian interval mapping of count trait loci based on zero-inflated generalized Poisson regression model

Count phenotypes with excessive zeros are often observed in the biological world. Researchers have studied many statistical methods for mapping the quantitative trait loci (QTLs) of zero-inflated count phenotypes. However, most of the existing methods consist of finding the approximate positions of the QTLs on the chromosome by genome-wide scanning. Additionally, most of the existing methods use the EM algorithm for parameter estimation. In this paper, we propose a Bayesian interval mapping scheme of QTLs for zero-inflated count data. The method takes advantage of a zero-inflated generalized Poisson (ZIGP) regression model to study the influence of QTLs on the zero-inflated count phenotype. The MCMC algorithm is used to estimate the effects and position parameters of QTLs. We use the Haldane map function to realize the conversion between recombination rate and map distance. Monte Carlo simulations are conducted to test the applicability and advantage of the proposed method. The effects of QTLs on the formation of mouse cholesterol gallstones were demonstrated by analyzing an mouse data set.     

Zero-inflated generalized Poisson regression mixture model for mapping quantitative trait loci underlying count trait with many zeros

Phenotypes measured in counts are commonly observed in nature. Statistical methods for mapping quantitative trait loci (QTL) underlying count traits are documented in the literature. The majority of them assume that the count phenotype follows a Poisson distribution with appropriate techniques being applied to handle data dispersion. When a count trait has a genetic basis, “naturally occurring” zero status also reflects the underlying gene effects. Simply ignoring or miss-handling the zero data may lead to wrong QTL inference. In this article, we propose an interval mapping approach for mapping QTL underlying count phenotypes containing many zeros. The effects of QTLs on the zero-inflated count trait are modelled through the zero-inflated generalized Poisson regression mixture model, which can handle the zero inflation and Poisson dispersion in the same distribution. We implement the approach using the EM algorithm with the Newton-Raphson algorithm embedded in the M-step, and provide a genome-wide scan for testing and estimating the QTL effects. The performance of the proposed method is evaluated through extensive simulation studies. Extensions to composite and multiple interval mapping are discussed. The utility of the developed approach is illustrated through a mouse F₂ intercross data set. Significant QTLs are detected to control mouse cholesterol gallstone formation.