Models For Longitudinal Count Data Research Articles

The dynamic interdependence in the demand of primary and emergency secondary care: A hidden Markov approach

SummaryThis paper develops an extension of the class of finite mixture models for longitudinal count data to the bivariate case by using a hidden Markov chain approach. The model allows for disentangling unobservable time‐varying heterogeneity from the dynamic effect of utilisation of primary and secondary care and measuring their potential substitution effect. Three points of supports adequately describe the distribution of the latent states suggesting the existence of three profiles of low, medium and high users who shows persistency in their behaviour, but not permanence as some switch to their neighbour's profile.

Competing risks joint models using R-INLA

The methodological advancements made in the field of joint models are numerous. None the less, the case of competing risks joint models has largely been neglected, especially from a practitioner's point of view. In the relevant works on competing risks joint models, the assumptions of a Gaussian linear longitudinal series and proportional cause-specific hazard functions, amongst others, have remained unchallenged. In this article, we provide a framework based on R-INLA to apply competing risks joint models in a unifying way such that non-Gaussian longitudinal data, spatial structures, times-dependent splines and various latent association structures, to mention a few, are all embraced in our approach. Our motivation stems from the SANAD trial which exhibits non-linear longitudinal trajectories and competing risks for failure of treatment. We also present a discrete competing risks joint model for longitudinal count data as well as a spatial competing risks joint model as specific examples.

The dynamic interdependence in the demand of primary and emergency secondary care: A hidden Markov approach

This paper develops an extension of the class of finite mixture models for longitudinal count data to the bivariate case by using a trivariate reduction technique and a hidden Markov chain approach. The model allows for disentangling unobservable time-varying heterogeneity from the dynamic effect of utilisation of primary and secondary care and measuring their potential substitution effect. Three points of supports adequately describe the distribution of the latent states suggesting the existence of three profiles of low, medium and high users who shows persistency in their behaviour, but not permanence as some switch to their neighbour's profile.

Marginal models for longitudinal count data with dropouts

In this article, we investigate marginal models for analyzing incomplete longitudinal count data with dropouts. Specifically, we explore commonly used generalized estimating equations and weighted generalized estimating equations for fitting log-linear models to count data in the presence of monotone missing responses. A series of simulations were carried out to examine the finite-sample properties of the estimators in the presence of both correctly specified and misspecified dropout mechanisms. An application is provided using actual longitudinal survey data from the Health and Retirement Study (HRS) (HRS, 2019)

Competing risks joint models using R-INLA

The methodological advancements made in the field of joint models are numerous. None the less, the case of competing risks joint models has largely been neglected, especially from a practitioner's point of view. In the relevant works on competing risks joint models, the assumptions of a Gaussian linear longitudinal series and proportional cause-specific hazard functions, amongst others, have remained unchallenged. In this article, we provide a framework based on R-INLA to apply competing risks joint models in a unifying way such that non-Gaussian longitudinal data, spatial structures, times-dependent splines and various latent association structures, to mention a few, are all embraced in our approach. Our motivation stems from the SANAD trial which exhibits non-linear longitudinal trajectories and competing risks for failure of treatment. We also present a discrete competing risks joint model for longitudinal count data as well as a spatial competing risks joint model as specific examples.

A new Bayesian joint model for longitudinal count data with many zeros, intermittent missingness, and dropout with applications to HIV prevention trials

In longitudinal clinical trials, it is common that subjects may permanently withdraw from the study (dropout), or return to the study after missing one or more visits (intermittent missingness). It is also routinely encountered in HIV prevention clinical trials that there is a large proportion of zeros in count response data. In this paper, a sequential multinomial model is adopted for dropout and subsequently a conditional model is constructed for intermittent missingness. The new model captures the complex structure of missingness and incorporates dropout and intermittent missingness simultaneously. The model also allows us to easily compute the predictive probabilities of different missing data patterns. A zero-inflated Poisson mixed-effects regression model is assumed for the longitudinal count response data. We also propose an approach to assess the overall treatment effects under the zero-inflated Poisson model. We further show that the joint posterior distribution is improper if uniform priors are specified for the regression coefficients under the proposed model. Variations of the g-prior, Jeffreys prior, and maximally dispersed normal prior are thus established as remedies for the improper posterior distribution. An efficient Gibbs sampling algorithm is developed using a hierarchical centering technique. A modified logarithm of the pseudomarginal likelihood and a concordance based area under the curve criterion are used to compare the models under different missing data mechanisms. We then conduct an extensive simulation study to investigate the empirical performance of the proposed methods and further illustrate the methods using real data from an HIV prevention clinical trial.

Two Stage Cluster Sampling Based Asymptotic Inferences in Survey Population Models for Longitudinal Count and Categorical Data

Sutradhar (2008, Sankhya B, 18-33) has studied a GLMM (generalized linear mixed model) for count data in a finite population setup where a sample of families/clusters were chosen from a finite population using PPS (probability proportional to size) sampling scheme. The properties of the estimators were studied through a simulation study. In this paper we consider (1) an auto-regressive type dynamic (ARTD) model for longitudinal count data, and (2) a MDL (multinomial dynamic logits) model for longitudinal categorical/multinomial data; and develop a practically important two-stage cluster sampling (TSCS) design weights based estimating equations for the regression and dynamic dependence parameters involved in both ARTD and MDL models in a finite population setup. Specifically, the survey weighted generalized quasi-likelihood (WGQL) and the survey weighted maximum likelihood (WML) estimation approaches are used for the count and multinomial models, respectively. We also demonstrate in the paper that the proposed TSCS based WGQL and WML estimators specially for the regression parameters (of main interest) are consistent, whereas the existing ‘working’ longitudinal correlations based GEE (generalized estimating equations) fails to produce consistent estimates and/or consistent variance estimates. The variances of the regression estimates along with their unbiased estimates are derived for both count and multinomial models. Furthermore, the asymptotic normality of the regression estimators are developed based on a large number of independent but non-identical clusters under non-overlapping stratums.

Marginalized models for longitudinal count data

In this paper, we propose two marginalized models for longitudinal count data. The first marginalized model has a Markovian structure to account for the serial correlation of longitudinal outcomes. We also propose another marginalized model with a Markovian structure for serial correlation as well as random effects for both overdispersion and long-term dependence. In these models, along with it being possible to permit likelihood-based estimation, inference is valid under ignorability which distinguishes them from generalized estimating equation (GEE) approaches. Fisher-scoring and Quasi-Newton algorithms are developed for estimation purposes. Monte Carlo studies show that the proposed models perform well in the sense of reducing the bias of marginal mean parameters compared to the misspecification of the dependence model in these models. The models are used to draw inferences from a previously analyzed trial on epileptic seizures.

Efficient regression modeling for correlated and overdispersed count data

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.

A Comparative Study of Observation- and Parameter-driven Zero-inflated Poisson Models for Longitudinal Count Data

Longitudinal count data with excessive zeros frequently occur in social, biological, medical, and health research. To model such data, zero-inflated Poisson (ZIP) models are commonly used, after separating zero and positive responses. As longitudinal count responses are likely to be serially correlated, such separation may destroy the underlying serial correlation structure. To overcome this problem recently observation- and parameter-driven modelling approaches have been proposed. In the observation-driven model, the response at a specific time point is modelled through the responses at previous time points after incorporating serial correlation. One limitation of the observation-driven model is that it fails to accommodate the presence of any possible over-dispersion, which frequently occurs in the count responses. This limitation is overcome in a parameter-driven model, where the serial correlation is captured through the latent process using random effects. We compare the results obtained by the two models. A quasi-likelihood approach has been developed to estimate the model parameters. The methodology is illustrated with analysis of two real life datasets. To examine model performance the models are also compared through a simulation study.

Comparison of Additive and Multiplicative Bayesian Models for Longitudinal Count Data with Overdispersion Parameters: A Simulation Study

In applied statistical data analysis, overdispersion is a common feature. It can be addressed using both multiplicative and additive random effects. A multiplicative model for count data incorporates a gamma random effect as a multiplicative factor into the mean, whereas an additive model assumes a normally distributed random effect, entered into the linear predictor. Using Bayesian principles, these ideas are applied to longitudinal count data, based on the so-called combined model. The performance of the additive and multiplicative approaches is compared using a simulation study.

A Semi-Parametric Model for Longitudinal Count Data

Longitudinal count data analysis has been widely used in various social science and nature science. In this paper, we proposed a semi-parametric negative multinomia model for longitudinal count data. Using local likelihood technique and marginal distribution, we proposed an estimation procedure for our new model. The finite-sample properties are studied with a simulation study.

Modelling Count Responses with Overdispersion

We frequently encounter outcomes of count that have extra variation. This paper considers several alternative models for overdispersed count responses such as a quasi-Poisson model, zero-inflated Poisson model and a negative binomial model with a special focus on a generalized linear mixed model. We also explain various goodness-of-fit criteria by discussing their appropriateness of applicability and cautions on misuses according to the patterns of response categories. The overdispersion models for counts data have been explained through two examples with different response patterns. Poisson distribution is the standard model for count responses, for which the variance of responses is expected to equal mean. But violations in the assumptions characterizing a Poisson distribution may lead to the presence of extra variation in the analysis of counts and rates in longitudinal studies. There may exist a dependence between the elemental units in a study where experimental units are clusters. When responses are observed in clusters they frequently exhibit extra variation than the permitted variance of the assumed model. We often encounter this extra variation in a real data of count or binomial responses. Overdispersion may sometimes be observed when the residual variation obtained is greater than which can be attributed to the sampling variation assumed by the model. McCullagh and Nelder (1989) showed that overdispersion is not uncommon in practice. Overdis- persion should be considered deliberately in modelling count responses. The ordinary Poisson general linear model(GLM) cannot be fitted well in the presence of overdispersion. In the analysis of clus- tered or longitudinal data the generalized linear mixed model(GLMM) and the generalized estimating equations(GEE) approaches are most popular. The GLMM incorporates random subject effects into a GLM by allowing subjects variability; however, the GEE method solves equations that include the correlation structure of repeated responses to estimate regression coefficients. Thall and Vail (1990) discussed covariance models for longitudinal count data with overdisper- sion, and Jowaheer and Sutradhar (2002) applied a GEE method to analyze longitudinal count data; however, Sutradhar et al. (2007) suggested three kinds of mixture models to account for the overdis- persion in binomial data. The beta-binomial, the finite mixture, and the zero-inflated binomial model all belong to the same class of mixture models. They suggested a chi-squared type goodness-of-fit statistic to test the assumed null model against the alternative three kinds of models without con- sidering covariate variables. Recently, Morel and Neerchal (2012) have extensively studied various overdispersion models using SAS.

Likelihood estimation for longitudinal zero-inflated power series regression models

In this paper, a zero-inflated power series regression model for longitudinal count data with excess zeros is presented. We demonstrate how to calculate the likelihood for such data when it is assumed that the increment in the cumulative total follows a discrete distribution with a location parameter that depends on a linear function of explanatory variables. Simulation studies indicate that this method can provide improvements in obtaining standard errors of the estimates. We also calculate the dispersion index for this model. The influence of a small perturbation of the dispersion index of the zero-inflated model on likelihood displacement is also studied. The zero-inflated negative binomial regression model is illustrated on data regarding joint damage in psoriatic arthritis.

On Dynamic Generalized Linear Models with Applications

In this paper we combine the idea of ‘power steady model’, ‘discount factor’ and ‘power prior’, for a general class of filter model, more specifically within a class of dynamic generalized linear models (DGLM). We show an optimality property for our proposed method and present the particle filter algorithm for DGLM as an alternative to Markov chain Monte Carlo method. We also present two applications; one on dynamic Poisson models for hurricane count data in Atlantic ocean and the another on the dynamic Poisson regression model for longitudinal count data.

Hidden Markov models for zero‐inflated Poisson counts with an application to substance use

Paradigms for substance abuse cue-reactivity research involve pharmacological or stressful stimulation designed to elicit stress and craving responses in cocaine-dependent subjects. It is unclear as to whether stress induced from participation in such studies increases drug-seeking behavior. We propose a 2-state Hidden Markov model to model the number of cocaine abuses per week before and after participation in a stress-and cue-reactivity study. The hypothesized latent state corresponds to 'high' or 'low' use. To account for a preponderance of zeros, we assume a zero-inflated Poisson model for the count data. Transition probabilities depend on the prior week's state, fixed demographic variables, and time-varying covariates. We adopt a Bayesian approach to model fitting, and use the conditional predictive ordinate statistic to demonstrate that the zero-inflated Poisson hidden Markov model outperforms other models for longitudinal count data.

A semiparametric approach to hidden Markov models under longitudinal observations

We propose a hidden Markov model for longitudinal count data where sources of unobserved heterogeneity arise, making data overdispersed. The observed process, conditionally on the hidden states, is assumed to follow an inhomogeneous Poisson kernel, where the unobserved heterogeneity is modeled in a generalized linear model (GLM) framework by adding individual-specific random effects in the link function. Due to the complexity of the likelihood within the GLM framework, model parameters may be estimated by numerical maximization of the log-likelihood function or by simulation methods; we propose a more flexible approach based on the Expectation Maximization (EM) algorithm. Parameter estimation is carried out using a non-parametric maximum likelihood (NPML) approach in a finite mixture context. Simulation results and two empirical examples are provided.

A Bayesian Model for Longitudinal Count Data with Non-Ignorable Dropout

Asthma is an important chronic disease of childhood. An intervention programme for managing asthma was designed on principles of self-regulation and was evaluated by a randomized longitudinal study.The study focused on several outcomes, and, typically, missing data remained a pervasive problem. We develop a pattern-mixture model to evaluate the outcome of intervention on the number of hospitalizations with non-ignorable dropouts. Pattern-mixture models are not generally identifiable as no data may be available to estimate a number of model parameters. Sensitivity analyses are performed by imposing structures on the unidentified parameters.We propose a parameterization which permits sensitivity analyses on clustered longitudinal count data that have missing values due to non-ignorable missing data mechanisms. This parameterization is expressed as ratios between event rates across missing data patterns and the observed data pattern and thus measures departures from an ignorable missing data mechanism. Sensitivity analyses are performed within a Bayesian framework by averaging over different prior distributions on the event ratios. This model has the advantage of providing an intuitive and flexible framework for incorporating the uncertainty of the missing data mechanism in the final analysis.

A serially correlated gamma frailty model for longitudinal count data

A Poisson-gamma model is introduced to account for between-subjects heterogeneity and within-subjects serial correlation occurring in longitudinal count data. The model extends the usual time-constant shared frailty approach to allow time-varying serially correlated gamma frailty whilst retaining standard marginal assumptions. A composite likelihood approach to estimation and testing for serial correlation is proposed. The work is motivated by a clinical trial on patient-controlled analgesia where the number of analgesic doses taken by hospital patients in successive time intervals following abdominal surgery is recorded.

Air pollution effects on hospital admissions: a statistical analysis of parallel time series

A statistical model for longitudinal count data is used to examine the potential adverse health effects of ambient air pollution. Daily respiratory admissions to 164 acute care hospitals in Ontario are obtained for the period 1983 to 1988. Estimates of ozone levels in the vicinity of each hospital are determined from air pollution monitoring stations maintained by the Ontario Ministry of the Environment. Generalized estimating equation methods are used to make inferences about the regression and overdispersion parameters. The admission data display little evidence of serial correlation and extra Poisson variation. However, admission rates vary considerably among hospitals. This latter source of variation needs to be taken into account in examining the effects of air pollution on respiratory health status.