Efficient regression modeling for correlated and overdispersed count data

Abstract

The objective of this paper is to propose an efficient estimation procedure in a marginal mean regression model for longitudinal count data and to develop a hypothesis test for detecting the presence of overdispersion. We extend the matrix expansion idea of quadratic inference functions to the negative binomial regression framework that entails accommodating both the within-subject correlation and overdispersion issue. Theoretical and numerical results show that the proposed procedure yields a more efficient estimator asymptotically than the one ignoring either the within-subject correlation or overdispersion. When the overdispersion is absent in data, the proposed method might hinder the estimation efficiency in practice, yet the Poisson regression based regression model is fitted to the data sufficiently well. Therefore, we construct the hypothesis test that recommends an appropriate model for the analysis of the correlated count data. Extensive simulation studies indicate that the proposed test can identify the effective model consistently. The proposed procedure is also applied to a transportation safety study and recommends the proposed negative binomial regression model.     

An extended random-effects approach to modeling repeated, overdispersed count data

Non-Gaussian outcomes are often modeled using members of the so-called exponential family. The Poisson model for count data falls within this tradition. The family in general, and the Poisson model in particular, are at the same time convenient since mathematically elegant, but in need of extension since often somewhat restrictive. Two of the main rationales for existing extensions are (1) the occurrence of overdispersion, in the sense that the variability in the data is not adequately captured by the model's prescribed mean-variance link, and (2) the accommodation of data hierarchies owing to, for example, repeatedly measuring the outcome on the same subject, recording information from various members of the same family, etc. There is a variety of overdispersion models for count data, such as, for example, the negative-binomial model. Hierarchies are often accommodated through the inclusion of subject-specific, random effects. Though not always, one conventionally assumes such random effects to be normally distributed. While both of these issues may occur simultaneously, models accommodating them at once are less than common. This paper proposes a generalized linear model, accommodating overdispersion and clustering through two separate sets of random effects, of gamma and normal type, respectively. This is in line with the proposal by Booth et al. (Stat Model 3:179-181, 2003). The model extends both classical overdispersion models for count data (Breslow, Appl Stat 33:38-44, 1984), in particular the negative binomial model, as well as the generalized linear mixed model (Breslow and Clayton, J Am Stat Assoc 88:9-25, 1993). Apart from model formulation, we briefly discuss several estimation options, and then settle for maximum likelihood estimation with both fully analytic integration as well as hybrid between analytic and numerical integration. The latter is implemented in the SAS procedure NLMIXED. The methodology is applied to data from a study in epileptic seizures.     

Analysis of hypoglycemic events using negative binomial models

Negative binomial regression is a standard model to analyze hypoglycemic events in diabetes clinical trials. Adjusting for baseline covariates could potentially increase the estimation efficiency of negative binomial regression. However, adjusting for covariates raises concerns about model misspecification, in which the negative binomial regression is not robust because of its requirement for strong model assumptions. In some literature, it was suggested to correct the standard error of the maximum likelihood estimator through introducing overdispersion, which can be estimated by the Deviance or Pearson Chi‐square. We proposed to conduct the negative binomial regression using Sandwich estimation to calculate the covariance matrix of the parameter estimates together with Pearson overdispersion correction (denoted by NBSP). In this research, we compared several commonly used negative binomial model options with our proposed NBSP. Simulations and real data analyses showed that NBSP is the most robust to model misspecification, and the estimation efficiency will be improved by adjusting for baseline hypoglycemia. Copyright © 2013 John Wiley & Sons, Ltd.     

Fast two-stage estimator for clustered count data with overdispersion

Clustered count data are commonly analysed by the generalized linear mixed model (GLMM). Here, the correlation due to clustering and some overdispersion is captured by the inclusion of cluster-specific normally distributed random effects. Often, the model does not capture the variability completely. Therefore, the GLMM can be extended by including a set of gamma random effects. Routinely, the GLMM is fitted by maximizing the marginal likelihood. However, this process is computationally intensive. Although feasible with medium to large data, it can be too time-consuming or computationally intractable with very large data. Therefore, a fast two-stage estimator for correlated, overdispersed count data is proposed. It is rooted in the split-sample methodology. Based on a simulation study, it shows good statistical properties. Furthermore, it is computationally much faster than the full maximum likelihood estimator. The approach is illustrated using a large dataset belonging to a network of Belgian general practices.     

Score tests for heterogeneity and overdispersion in zero‐inflated Poisson and binomial regression models

Hall (2000) has described zero‐inflated Poisson and binomial regression models that include random effects to account for excess zeros and additional sources of heterogeneity in the data. The authors of the present paper propose a general score test for the null hypothesis that variance components associated with these random effects are zero. For a zero‐inflated Poisson model with random intercept, the new test reduces to an alternative to the overdispersion test of Ridout, Demério & Hinde (2001). The authors also examine their general test in the special case of the zero‐inflated binomial model with random intercept and propose an overdispersion test in that context which is based on a beta‐binomial alternative.     

On the Correlation Structure of Gaussian Copula Models for Geostatistical Count Data

We describe a class of random field models for geostatistical count data based on Gaussian copulas. Unlike hierarchical Poisson models often used to describe this type of data, Gaussian copula models allow a more direct modelling of the marginal distributions and association structure of the count data. We study in detail the correlation structure of these random fields when the family of marginal distributions is either negative binomial or zero‐inflated Poisson; these represent two types of overdispersion often encountered in geostatistical count data. We also contrast the correlation structure of one of these Gaussian copula models with that of a hierarchical Poisson model having the same family of marginal distributions, and show that the former is more flexible than the latter in terms of range of feasible correlation, sensitivity to the mean function and modelling of isotropy. An exploratory analysis of a dataset of Japanese beetle larvae counts illustrate some of the findings. All of these investigations show that Gaussian copula models are useful alternatives to hierarchical Poisson models, specially for geostatistical count data that display substantial correlation and small overdispersion.     

Testing overdispersion in the zero-inflated Poisson model

The zero-inflated negative binomial (ZINB) model is used to account for commonly occurring overdispersion detected in data that are initially analyzed under the zero-inflated Poisson (ZIP) model. Tests for overdispersion (Wald test, likelihood ratio test [LRT], and score test) based on ZINB model for use in ZIP regression models have been developed. Due to similarity to the ZINB model, we consider the zero-inflated generalized Poisson (ZIGP) model as an alternate model for overdispersed zero-inflated 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 score tests for overdispersion based on the ZIGP model and illustrates that the derived score statistics are exactly the same as the score statistics under the ZINB model. A simulation study indicates the proposed score statistics are preferred to other tests for higher empirical power. In practice, based on the approximate mean–variance relationship in the data, the ZINB or ZIGP model can be considered, and a formal score test based on asymptotic standard normal distribution can be employed for assessing overdispersion in the ZIP model. We provide an example to illustrate the procedures for data analysis.     

Impact of missing data on type 1 error rates in non‐inferiority trials

In this paper, a simulation study is conducted to systematically investigate the impact of different types of missing data on six different statistical analyses: four different likelihood‐based linear mixed effects models and analysis of covariance (ANCOVA) using two different data sets, in non‐inferiority trial settings for the analysis of longitudinal continuous data. ANCOVA is valid when the missing data are completely at random. Likelihood‐based linear mixed effects model approaches are valid when the missing data are at random. Pattern‐mixture model (PMM) was developed to incorporate non‐random missing mechanism. Our simulations suggest that two linear mixed effects models using unstructured covariance matrix for within‐subject correlation with no random effects or first‐order autoregressive covariance matrix for within‐subject correlation with random coefficient effects provide well control of type 1 error (T1E) rate when the missing data are completely at random or at random. ANCOVA using last observation carried forward imputed data set is the worst method in terms of bias and T1E rate. PMM does not show much improvement on controlling T1E rate compared with other linear mixed effects models when the missing data are not at random but is markedly inferior when the missing data are at random. Copyright © 2009 John Wiley & Sons, Ltd.     

Score Tests for Zero-Inflation in Overdispersed Count Data

The negative binomial (NB) model and the generalized Poisson (GP) model are common alternatives to Poisson models when overdispersion is present in the data. Having accounted for initial overdispersion, we may require further investigation as to whether there is evidence for zero-inflation in the data. Two score statistics are derived from the GP model for testing zero-inflation. These statistics, unlike Wald-type test statistics, do not require that we fit the more complex zero-inflated overdispersed models to evaluate zero-inflation. A simulation study illustrates that the developed score statistics reasonably follow a χ² distribution and maintain the nominal level. Extensive simulation results also indicate the power behavior is different for including a continuous variable than a binary variable in the zero-inflation (ZI) part of the model. These differences are the basis from which suggestions are provided for real data analysis. Two practical examples are presented in this article. Results from these examples along with practical experience lead us to suggest performing the developed score test before fitting a zero-inflated NB model to the data.     

Modelling count data with overdispersion and spatial effects

In this paper we consider regression models for count data allowing for overdispersion in a Bayesian framework. We account for unobserved heterogeneity in the data in two ways. On the one hand, we consider more flexible models than a common Poisson model allowing for overdispersion in different ways. In particular, the negative binomial and the generalized Poisson (GP) distribution are addressed where overdispersion is modelled by an additional model parameter. Further, zero-inflated models in which overdispersion is assumed to be caused by an excessive number of zeros are discussed. On the other hand, extra spatial variability in the data is taken into account by adding correlated spatial random effects to the models. This approach allows for an underlying spatial dependency structure which is modelled using a conditional autoregressive prior based on Pettitt et al. in Stat Comput 12(4):353–367, (2002). In an application the presented models are used to analyse the number of invasive meningococcal disease cases in Germany in the year 2004. Models are compared according to the deviance information criterion (DIC) suggested by Spiegelhalter et al. in J R Stat Soc B64(4):583–640, (2002) and using proper scoring rules, see for example Gneiting and Raftery in Technical Report no. 463, University of Washington, (2004). We observe a rather high degree of overdispersion in the data which is captured best by the GP model when spatial effects are neglected. While the addition of spatial effects to the models allowing for overdispersion gives no or only little improvement, spatial Poisson models with spatially correlated or uncorrelated random effects are to be preferred over all other models according to the considered criteria.     

On the use of corrections for overdispersion

In studying fluctuations in the size of a blackgrouse ( Tetrao tetrix ) population, an autoregressive model using climatic conditions appears to follow the change quite well. However, the deviance of the model is considerably larger than its number of degrees of freedom. A widely used statistical rule of thumb holds that overdispersion is present in such situations, but model selection based on a direct likelihood approach can produce opposing results. Two further examples, of binomial and of Poisson data, have models with deviances that are almost twice the degrees of freedom and yet various overdispersion models do not fit better than the standard model for independent data. This can arise because the rule of thumb only considers a point estimate of dispersion, without regard for any measure of its precision. A reasonable criterion for detecting overdispersion is that the deviance be at least twice the number of degrees of freedom, the familiar Akaike information criterion, but the actual presence of overdispersion should then be checked by some appropriate modelling procedure.     

A Semi-Nonparametric Approach to Model Panel Count Data

In count data models, overdispersion of the dependent variable can be incorporated into the model if a heterogeneity term is added into the mean parameter of the Poisson distribution. We use a nonparametric estimation for the heterogeneity density based on a squared Kth-order polynomial expansion, that we generalize for panel data. A numerical illustration using an insurance dataset is discussed. Even if some statistical analyses showed no clear differences between these new models and the standard Poisson with gamma random effects, we show that the choice of the random effects distribution has a significant influence for interpreting our results.     

Multivariate models for correlated count data

In this study, we deal with the problem of overdispersion beyond extra zeros for a collection of counts that can be correlated. Poisson, negative binomial, zero-inflated Poisson and zero-inflated negative binomial distributions have been considered. First, we propose a multivariate count model in which all counts follow the same distribution and are correlated. Then we extend this model in a sense that correlated counts may follow different distributions. To accommodate correlation among counts, we have considered correlated random effects for each individual in the mean structure, thus inducing dependency among common observations to an individual. The method is applied to real data to investigate variation in food resources use in a species of marsupial in a locality of the Brazilian Cerrado biome.     

Bivariate zero-inflated negative binomial regression model with applications

Count data often display excessive number of zero outcomes than are expected in the Poisson regression model. The zero-inflated Poisson regression model has been suggested to handle zero-inflated data, whereas the zero-inflated negative binomial (ZINB) regression model has been fitted for zero-inflated data with additional overdispersion. For bivariate and zero-inflated cases, several regression models such as the bivariate zero-inflated Poisson (BZIP) and bivariate zero-inflated negative binomial (BZINB) have been considered. This paper introduces several forms of nested BZINB regression model which can be fitted to bivariate and zero-inflated count data. The mean–variance approach is used for comparing the BZIP and our forms of BZINB regression model in this study. A similar approach was also used by past researchers for defining several negative binomial and zero-inflated negative binomial regression models based on the appearance of linear and quadratic terms of the variance function. The nested BZINB regression models proposed in this study have several advantages; the likelihood ratio tests can be performed for choosing the best model, the models have flexible forms of marginal mean–variance relationship, the models can be fitted to bivariate zero-inflated count data with positive or negative correlations, and the models allow additional overdispersion of the two dependent variables.     

Empirical Bayes estimates of finite mixture of negative binomial regression models and its application to highway safety

The empirical Bayes (EB) method is commonly used by transportation safety analysts for conducting different types of safety analyses, such as before–after studies and hotspot analyses. To date, most implementations of the EB method have been applied using a negative binomial (NB) model, as it can easily accommodate the overdispersion commonly observed in crash data. Recent studies have shown that a generalized finite mixture of NB models with K mixture components (GFMNB-K) can also be used to model crash data subjected to overdispersion and generally offers better statistical performance than the traditional NB model. So far, nobody has developed how the EB method could be used with finite mixtures of NB models. The main objective of this study is therefore to use a GFMNB-K model in the calculation of EB estimates. Specifically, GFMNB-K models with varying weight parameters are developed to analyze crash data from Indiana and Texas. The main finding shows that the rankings produced by the NB and GFMNB-2 models for hotspot identification are often quite different, and this was especially noticeable with the Texas dataset. Finally, a simulation study designed to examine which model formulation can better identify the hotspot is recommended as our future research.     

Semiparametric Negative Binomial Regression Models

Negative-binomial (NB) regression models have been widely used for analysis of count data displaying substantial overdispersion (extra-Poisson variation). However, no formal lack-of-fit tests for a postulated parametric model for a covariate effect have been proposed. Therefore, a flexible parametric procedure is used to model the covariate effect as a linear combination of fixed-knot cubic basis splines or B-splines. Within the proposed modeling framework, a log-likelihood ratio test is constructed to evaluate the adequacy of a postulated parametric form of the covariate effect. Simulation experiments are conducted to study the power performance of the proposed test.     

Overdispersed poisson regression models for studies of air pollution and human health

This paper presents results from a simulation study motivated by a recent study of the relationships between ambient levels of air pollution and human health in the community of Prince George, British Columbia. The simulation study was designed to evaluate the performance of methods based on overdispersed Poisson regression models for the analysis of series of count data. Aspects addressed include estimation of the dispersion parameter, estimation of regression coefficients and their standard errors, and the performance of model selection tests. The effects of varying amounts of overdispersion and differing underlying variance structure on this performance were of particular interest. This study is related to work reported by Breslow (1990) although the context is quite different. Preliminary work led to the conclusion that estimation of the dispersion parameter should be based on Pearson's chi-square statistic rather than the Poisson deviance. Regression coefficients are well estimated, even in the présence of substantial overdispersion and when the model for the variance function is incorrectly specified. Despite potential greater variability, the empirical estimator of the covariance matrix is preferred because the model-based estimator is unreliable in general. When the model for the variance function is incorrect, model-based test statistics may perform poorly, in sharp contrast to empirical test statistics, which performed very well in this study.     

Estimation of the additive hazards model with interval-censored data and missing covariates

The additive hazards model is one of the most commonly used regression models in the analysis of failure time data and many methods have been developed for its inference in various situations. However, no established estimation procedure exists when there are covariates with missing values and the observed responses are interval-censored; both types of complications arise in various settings including demographic, epidemiological, financial, medical and sociological studies. To address this deficiency, we propose several inverse probability weight-based and reweighting-based estimation procedures for the situation where covariate values are missing at random. The resulting estimators of regression model parameters are shown to be consistent and asymptotically normal. The numerical results that we report from a simulation study suggest that the proposed methods work well in practical situations. An application to a childhood cancer survival study is provided. The Canadian Journal of Statistics 48: 499–517; 2020 © 2020 Statistical Society of Canada     

Conditional Akaike Information Criteria for a Class of Poisson Mixture Models with Random Effects

Focusing on the model selection problems in the family of Poisson mixture models (including the Poisson mixture regression model with random effects and zero‐inflated Poisson regression model with random effects), the current paper derives two conditional Akaike information criteria. The criteria are the unbiased estimators of the conditional Akaike information based on the conditional log‐likelihood and the conditional Akaike information based on the joint log‐likelihood, respectively. The derivation is free from the specific parametric assumptions about the conditional mean of the true data‐generating model and applies to different types of estimation methods. Additionally, the derivation is not based on the asymptotic argument. Simulations show that the proposed criteria have promising estimation accuracy. In addition, it is found that the criterion based on the conditional log‐likelihood demonstrates good model selection performance under different scenarios. Two sets of real data are used to illustrate the proposed method.