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.     

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.     

A note on model selection using information criteria for general linear models estimated using REML

It is common practice to compare the fit of non‐nested models using the Akaike (AIC) or Bayesian (BIC) information criteria. The basis of these criteria is the log‐likelihood evaluated at the maximum likelihood estimates of the unknown parameters. For the general linear model (and the linear mixed model, which is a special case), estimation is usually carried out using residual or restricted maximum likelihood (REML). However, for models with different fixed effects, the residual likelihoods are not comparable and hence information criteria based on the residual likelihood cannot be used. For model selection, it is often suggested that the models are refitted using maximum likelihood to enable the criteria to be used. The first aim of this paper is to highlight that both the AIC and BIC can be used for the general linear model by using the full log‐likelihood evaluated at the REML estimates. The second aim is to provide a derivation of the criteria under REML estimation. This aim is achieved by noting that the full likelihood can be decomposed into a marginal (residual) and conditional likelihood and this decomposition then incorporates aspects of both the fixed effects and variance parameters. Using this decomposition, the appropriate information criteria for model selection of models which differ in their fixed effects specification can be derived. An example is presented to illustrate the results and code is available for analyses using the ASReml‐R package.     

The likelihood ratio test for homogeneity in finite mixture models

The authors study the asymptotic behaviour of the likelihood ratio statistic for testing homogeneity in the finite mixture models of a general parametric distribution family. They prove that the limiting distribution of this statistic is the squared supremum of a truncated standard Gaussian process. The autocorrelation function of the Gaussian process is explicitly presented. A re‐sampling procedure is recommended to obtain the asymptotic p‐value. Three kernel functions, normal, binomial and Poisson, are used in a simulation study which illustrates the procedure.     

A general maximum likelihood analysis of overdispersion in generalized linear models

This paper presents an EM algorithm for maximum likelihood estimation in generalized linear models with overdispersion. The algorithm is initially derived as a form of Gaussian quadrature assuming a normal mixing distribution, but with only slight variation it can be used for a completely unknown mixing distribution, giving a straightforward method for the fully non-parametric ML estimation of this distribution. This is of value because the ML estimates of the GLM parameters may be sensitive to the specification of a parametric form for the mixing distribution. A listing of a GLIM4 algorithm for fitting the overdispersed binomial logit model is given in an appendix.A simple method is given for obtaining correct standard errors for parameter estimates when using the EM algorithm.Several examples are discussed.     

A generalization of the beta–binomial distribution

Summary.  The paper describes a distribution generated by the Gaussian hypergeometric function that may be seen as a generalization of the beta–binomial distribution. It is expressed as a generalized beta mixture of a binomial distribution. This new mixing distribution allows the existence of a mode and an antimode, which is very useful for fitting some data sets. Two examples illustrate the greater versatility of the new distribution compared with the beta–binomial distribution.     

A mixture-based approach to robust analysis of generalised linear models

A method for robustness in linear models is to assume that there is a mixture of standard and outlier observations with a different error variance for each class. For generalised linear models (GLMs) the mixture model approach is more difficult as the error variance for many distributions has a fixed relationship to the mean. This model is extended to GLMs by changing the classes to one where the standard class is a standard GLM and the outlier class which is an overdispersed GLM achieved by including a random effect term in the linear predictor. The advantages of this method are it can be extended to any model with a linear predictor, and outlier observations can be easily identified. Using simulation the model is compared to an M-estimator, and found to have improved bias and coverage. The method is demonstrated on three examples.     

Focused Information Criterion for Capture–Recapture Models for Closed Populations

Abstract. We propose a criterion for selecting a capture–recapture model for closed populations, which follows the basic idea of the focused information criterion (FIC) of Claeskens and Hjort. The proposed criterion aims at selecting the model which, among the available models, leads to the smallest mean‐squared error (MSE) of the resulting estimator of the population size and is based on an index which, up to a constant term, is equal to the asymptotic MSE of the estimator. Two alternative approaches to estimate this FIC index are proposed. We also deal with multimodel inference; in this case, the population size is estimated by using a weighted average of the estimates coming from different models, with weights chosen so as to minimize the MSE of the resulting estimator. The proposed model selection approach is compared with more common approaches through a series of simulations. It is also illustrated by an application based on a dataset coming from a live‐trapping experiment.     

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.     

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.     

Improved two-parameter estimators for the negative binomial and Poisson regression models

Negative binomial regression (NBR) and Poisson regression (PR) applications have become very popular in the analysis of count data in recent years. However, if there is a high degree of relationship between the independent variables, the problem of multicollinearity arises in these models. We introduce new two-parameter estimators (TPEs) for the NBR and the PR models by unifying the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [The restricted and unrestricted two-parameter estimators. Commun Stat Theory Methods. 2007;36:2707–2725]. These new estimators are general estimators which include maximum likelihood (ML) estimator, ridge estimator (RE), Liu estimator (LE) and contraction estimator (CE) as special cases. Furthermore, biasing parameters of these estimators are given and a Monte Carlo simulation is done to evaluate the performance of these estimators using mean square error (MSE) criterion. The benefits of the new TPEs are also illustrated in an empirical application. The results show that the new proposed TPEs for the NBR and the PR models are better than the ML estimator, the RE and the LE.     

Generalized additive models for location, scale and shape

Summary.  A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y , as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton–Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.     

Empirical Bayes models of Poisson clinical trials and sample size determination

Bayesian methods are often used to reduce the sample sizes and/or increase the power of clinical trials. The right choice of the prior distribution is a critical step in Bayesian modeling. If the prior not completely specified, historical data may be used to estimate it. In the empirical Bayesian analysis, the resulting prior can be used to produce the posterior distribution. In this paper, we describe a Bayesian Poisson model with a conjugate Gamma prior. The parameters of Gamma distribution are estimated in the empirical Bayesian framework under two estimation schemes. The straightforward numerical search for the maximum likelihood (ML) solution using the marginal negative binomial distribution is unfeasible occasionally. We propose a simplification to the maximization procedure. The Markov Chain Monte Carlo method is used to create a set of Poisson parameters from the historical count data. These Poisson parameters are used to uniquely define the Gamma likelihood function. Easily computable approximation formulae may be used to find the ML estimations for the parameters of gamma distribution. For the sample size calculations, the ML solution is replaced by its upper confidence limit to reflect an incomplete exchangeability of historical trials as opposed to current studies. The exchangeability is measured by the confidence interval for the historical rate of the events. With this prior, the formula for the sample size calculation is completely defined. Published in 2009 by John Wiley & Sons, Ltd.     

A note on the unification of the Akaike information criterion

To measure the distance between a robust function evaluated under the true regression model and under a fitted model, we propose generalized Kullback–Leibler information. Using this generalization we have developed three robust model selection criteria, AICR*, AICCR* and AICCR, that allow the selection of candidate models that not only fit the majority of the data but also take into account non-normally distributed errors. The AICR* and AICCR criteria can unify most existing Akaike information criteria; three examples of such unification are given. Simulation studies are presented to illustrate the relative performance of each criterion.     

Evaluating the relative merits of competing models based on empirical likelihood ratio test

Competing models arise naturally in many research fields, such as survival analysis and economics, when the same phenomenon of interest is explained by different researcher using different theories or according to different experiences. The model selection problem is therefore remarkably important because of its great importance to the subsequent inference; Inference under a misspecified or inappropriate model will be risky. Existing model selection tests such as Vuong's tests [26 Q.H. Vuong, Likelihood ratio test for model selection and non-nested hypothesis, Econometrica 57 (1989), pp. 307–333. doi: 10.2307/1912557[Crossref], [Web of Science ®] , [Google Scholar]] and Shi's non-degenerate tests [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]] suffer from the variance estimation and the departure of the normality of the likelihood ratios. To circumvent these dilemmas, we propose in this paper an empirical likelihood ratio (ELR) tests for model selection. Following Shi [21 X. Shi, A non-degenerate Vuong test, Quant. Econ. 6 (2015), pp. 85–121. doi: 10.3982/QE382[Crossref], [Web of Science ®] , [Google Scholar]], a bias correction method is proposed for the ELR tests to enhance its performance. A simulation study and a real-data analysis are provided to illustrate the performance of the proposed ELR tests.     

A hybrid method to estimate the full parametric hazard model

Abstract

In this paper, we propose a hybrid method to estimate the baseline hazard for Cox proportional hazard model. In the proposed method, the nonparametric estimate of the survival function by Kaplan Meier, and the parametric estimate of the logistic function in the Cox proportional hazard by partial likelihood method are combined to estimate a parametric baseline hazard function. We compare the estimated baseline hazard using the proposed method and the Cox model. The results show that the estimated baseline hazard using hybrid method is improved in comparison with estimated baseline hazard using the Cox model. The performance of each method is measured based on the estimated parameters of the baseline distribution as well as goodness of fit of the model. We have used real data as well as simulation studies to compare performance of both methods. Monte Carlo simulations carried out in order to evaluate the performance of the proposed method. The results show that the proposed hybrid method provided better estimate of the baseline in comparison with the estimated values by the Cox model.     

Variable selection in finite mixture of semi-parametric regression models

Abstract

In this paper we are concerned with variable selection in finite mixture of semiparametric regression models. This task consists of model selection for non parametric component and variable selection for parametric part. Thus, we encountered separate model selections for every non parametric component of each sub model. To overcome this computational burden, we introduced a class of variable selection procedures for finite mixture of semiparametric regression models using penalized approach for variable selection. It is shown that the new method is consistent for variable selection. Simulations show that the performance of proposed method is good, and it consequently improves pervious works in this area and also requires much less computing power than existing methods.     

Modelling trends in groundwater levels by segmented regression with constraints

This paper provides a statistically unified method for modelling trends in groundwater levels for a national project that aims to predict areas at risk from salinity in 2020. It was necessary to characterize the trends in groundwater levels in thousands of boreholes that have been monitored by Agriculture Western Australia throughout the south-west of Western Australia over the last 10 years. The approach investigated in the present paper uses segmented regression with constraints when the number of change points is unknown. For each segment defined by change points, the trend can be described by a linear trend possibly superimposed on a periodic response. Four different types of change point are defined by constraints on the model parameters to cope with different patterns of change in groundwater levels. For a set of candidate change points provided by the user, a modified Akaike information criterion is used for model selection. Model parameters can be estimated by multiple linear regression. Some typical examples are presented to demonstrate the performance of the approach.     

Consistent Bayesian information criterion based on a mixture prior for possibly high-dimensional multivariate linear regression models

In the problem of selecting variables in a multivariate linear regression model, we derive new Bayesian information criteria based on a prior mixing a smooth distribution and a delta distribution. Each of them can be interpreted as a fusion of the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Inheriting their asymptotic properties, our information criteria are consistent in variable selection in both the large-sample and the high-dimensional asymptotic frameworks. In numerical simulations, variable selection methods based on our information criteria choose the true set of variables with high probability in most cases.     

Testing the linearity of negative binomial regression models

The negative binomial (NB) is frequently used to model overdispersed Poisson count data. To study the effect of a continuous covariate of interest in an NB model, a flexible procedure is used to model the covariate effect by fixed-knot cubic basis-splines or B-splines with a second-order difference penalty on the adjacent B-spline coefficients to avoid undersmoothing. A penalized likelihood is used to estimate parameters of the model. A penalized likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the continuous covariate effect. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one degree of freedom. The smoothing parameter value is determined by setting a specified value equal to the asymptotic expectation of the test statistic under the null hypothesis. The power performance of the proposed test is studied with simulation experiments.