Samuel Iddi

A combined overdispersed and marginalized multilevel model

Overdispersion and correlation are two features often encountered when modeling non-Gaussian dependent data, usually as a function of known covariates. Methods that ignore the presence of these phenomena are often in jeopardy of leading to biased assessment of covariate effects. The beta-binomial and negative binomial models are well known in dealing with overdispersed data for binary and count data, respectively. Similarly, generalized estimating equations (GEE) and the generalized linear mixed models (GLMM) are popular choices when analyzing correlated data. A so-called combined model simultaneously acknowledges the presence of dependency and overdispersion by way of two separate sets of random effects. A marginally specified logistic-normal model for longitudinal binary data which combines the strength of the marginal and hierarchical models has been previously proposed. These two are brought together to produce a marginalized longitudinal model which brings together the comfort of marginally meaningful parameters and the ease of allowing for overdispersion and correlation. Apart from model formulation, estimation methods are discussed. The proposed model is applied to two clinical studies and compared to the existing approach. It turns out that by explicitly allowing for overdispersion random effect, the model significantly improves.

A joint marginalized multilevel model for longitudinal outcomes

The shared-parameter model and its so-called hierarchical or random-effects extension are widely used joint modeling approaches for a combination of longitudinal continuous, binary, count, missing, and survival outcomes that naturally occurs in many clinical and other studies. A random effect is introduced and shared or allowed to differ between two or more repeated measures or longitudinal outcomes, thereby acting as a vehicle to capture association between the outcomes in these joint models. It is generally known that parameter estimates in a linear mixed model (LMM) for continuous repeated measures or longitudinal outcomes allow for a marginal interpretation, even though a hierarchical formulation is employed. This is not the case for the generalized linear mixed model (GLMM), that is, for non-Gaussian outcomes. The aforementioned joint models formulated for continuous and binary or two longitudinal binomial outcomes, using the LMM and GLMM, will naturally have marginal interpretation for parameters associated with the continuous outcome but a subject-specific interpretation for the fixed effects parameters relating covariates to binary outcomes. To derive marginally meaningful parameters for the binary models in a joint model, we adopt the marginal multilevel model (MMM) due to Heagerty [13] and Heagerty and Zeger [14] and formulate a joint MMM for two longitudinal responses. This enables to (1) capture association between the two responses and (2) obtain parameter estimates that have a population-averaged interpretation for both outcomes. The model is applied to two sets of data. The results are compared with those obtained from the existing approaches such as generalized estimating equations, GLMM, and the model of Heagerty [13]. Estimates were found to be very close to those from single analysis per outcome but the joint model yields higher precision and allows for quantifying the association between outcomes. Parameters were estimated using maximum likelihood. The model is easy to fit using available tools such as the SAS NLMIXED procedure.

Associations between pesticide use and respiratory symptoms: A cross-sectional study in Southern Ghana.

BACKGROUND Indiscriminate use of pesticides is a common practice amongst farmers in Low and Middle Income Countries (LMIC) across the globe. However, there is little evidence defining whether pesticide use is associated with respiratory symptoms. OBJECTIVES This cross-sectional study was conducted with 300 vegetable farmers in southern Ghana (Akumadan). Data on pesticide use was collected with an interviewed-administered questionnaire. The concentration of seven organochlorine pesticides and 3 pyrethroid pesticides was assayed in urine collected from a sub-population of 100 vegetable farmers by a gas chromatograph equipped with an electron capture detector (GC-ECD). RESULTS A statistically significant exposure-response relationship of years per day spent mixing/applying fumigant with wheezing [30-60 days/year: prevalence ratio (PR)=1.80 (95% CI 1.30, 2.50); >60days/year: 3.25 (1.70-6.33), p for trend=0.003] and hours per day spent mixing/applying fumigant with wheezing [1-2h/day: 1.20 (1.02-1.41), 3-5h/day: 1.45 (1.05-1.99), >5h/day: 1.74 (1.07-2.81), p for trend=0.0225]; days per year spent mixing/applying fungicide with wheezing [30-60 days/year: 2.04 (1.31-3.17); >60days/year: 4.16 (1.72-10.08), p for trend=0.0017] and h per day spent mixing/applying fungicide with phlegm production [1-2h/day: 1.25 (1.05-1.47), 3-5h/day: 1.55 (1.11-2.17), >5h/day: 1.93 (1.17-3.19), p for trend=0.0028] and with wheezing [1-2h/day: 1.10 (1.00-1.50), 3-5h/day: 1.20 (1.11-1.72), >5h/day: 1.32 (1.09-2.53), p for trend=0.0088]; h per day spent mixing/applying insecticide with phlegm production [1-2h/day: 1.23 (1.09-1.62), 3-5h/day: 1.51 (1.20-2.58), >5h/day: 1.85 (1.31-4.15), p for trend=0.0387] and wheezing [1-2h/day: 1.22 (1.02-1.46), 3-5h/day: 1.49 (1.04-2.12), >5h/day: 1.81 (1.07-3.08), p for trend=0.0185] were observed. Statistically significant exposure-response association was also observed for a combination of activities that exposes farmers to pesticide with all 3 respiratory symptoms. Furthermore, significant exposure-response associations for 3 organochlorine insecticides: beta-HCH, heptachlor and endosulfan sulfate were noted. CONCLUSIONS In conclusion, vegetable farmers in Ghana may be at increased risk for respiratory symptoms as a result of exposure to pesticides.

A combined beta and normal random-effects model for repeated, overdispersed binary and binomial data

Non-Gaussian outcomes are often modeled using members of the so-called exponential family. Notorious members are the Bernoulli model for binary data, leading to logistic regression, and the Poisson model for count data, leading to Poisson regression. Two of the main reasons for extending this family are (1) the occurrence of overdispersion, meaning that the variability in the data is not adequately described by the models, which often exhibit a prescribed mean-variance link, and (2) the accommodation of hierarchical structure in the data, stemming from clustering in the data which, in turn, may result from repeatedly measuring the outcome, for various members of the same family, etc. The first issue is dealt with through a variety of overdispersion models, such as, for example, the beta-binomial model for grouped binary data and the negative-binomial model for counts. Clustering is often accommodated through the inclusion of random subject-specific effects. Though not always, one conventionally assumes such random effects to be normally distributed. While both of these phenomena may occur simultaneously, models combining them are uncommon. This paper starts from the broad class of generalized linear models accommodating overdispersion and clustering through two separate sets of random effects. We place particular emphasis on so-called conjugate random effects at the level of the mean for the first aspect and normal random effects embedded within the linear predictor for the second aspect, even though our family is more general. The binary and binomial cases are our focus. Apart from model formulation, we present an overview of estimation methods, and then settle for maximum likelihood estimation with analytic-numerical integration. The methodology is applied to two datasets of which the outcomes are binary and binomial, respectively.

The Combined Model: A Tool for Simulating Correlated Counts with Overdispersion

The combined model as introduced by Molenberghs et al. (2007, 2010) has been shown to be an appealing tool for modeling not only correlated or overdispersed data but also for data that exhibit both these features. Unlike techniques available in the literature prior to the combined model, which use a single random-effects vector to capture correlation and/or overdispersion, the combined model allows for the correlation and overdispersion features to be modeled by two sets of random effects. In the context of count data, for example, the combined model naturally reduces to the Poisson-normal model, an instance of the generalized linear mixed model in the absence of overdispersion and it also reduces to the negative-binomial model in the absence of correlation. Here, a Poisson model is specified as the parent distribution of the data conditional on a normally distributed random effect at the subject or cluster level and/or a gamma distribution at observation level. Importantly, the development of the combined model and surrounding derivations have relevance well beyond mere data analysis. It so happens that the combined model can also be used to simulate correlated data. If a researcher is interested in comparing marginal models via Monte Carlo simulations, a necessity to generate suitable correlated count data arises. One option is to induce correlation via random effects but calculation of such quantities as the bias is then not straightforward. Since overdispersion and correlation are simultaneous features of longitudinal count data, the combined model presents an appealing framework for generating data to evaluate statistical properties, through a pre-specification of the desired marginal mean (possibly in terms of the covariates and marginal parameters) and a marginal variance-covariance structure. By comparing the marginal mean and variance of the combined model to the desired or pre-specified marginal mean and variance, respectively, the implied hierarchical parameters and the variance-covariance matrices of the normal and Gamma random effects are then derived from which correlated Poisson data are generated. We explore data generation when a random intercept or random intercept and slope model is specified to induce correlation. The data generator, however, allows for any dimension of the random effects although an increase in the random-effects dimension increases the sensitivity of the derived random effects variance-covariance matrix to deviations from positive-definiteness. A simulation study is carried out for the random-intercept model and for the random intercept and slope model, with or without the normal and Gamma random effects. We also pay specific attention to the case of serial correlation.

Mixed Models with Emphasis on Large Data Sets

In many contexts, hierarchical, multilevel, or clustered data are collected. Examples are longitudinal studies in which subjects are measured repeatedly at various time points (measurements within subject), surveys in which all members of a sample of families are questioned (members within families), educational data in which students from various schools are tested (students within schools), etc. From a statistical perspective, the challenge is to account for the fact that the measurements within clusters are not necessarily independent anymore, implying that standard models such as linear regression or generalized linear regression are no longer applicable. Mixed models are currently amongst the most flexible models for the analysis of such data. They can be interpreted as standard linear, generalized linear, or non-linear models, with cluster-specific random effects shared by all measurements within the cluster, hereby implicitly accounting for within-cluster associations. In this chapter, mixed models will be introduced with special attention for the correct interpretation of the parameters in the models. Also, examples will be given of situations in which results obtained from fitting mixed models are incorrectly interpreted. Many commercial software packages nowadays include mixed model procedures. However, when (extremely) large data sets are to be analyzed, standard likelihood based inference is no longer feasible. Examples include data sets with crossed random effects, with many clusters, with many observations per cluster, or contexts where mixed models are used to build a joint model for high-dimensional multivariate responses. In such cases, pseudo-likelihood techniques provide good alternatives. Various versions will be presented and illustrated. All concepts will be introduced and extensively illustrated using data sets from various contexts.

Empirical Bayes estimates for correlated hierarchical data with overdispersion

An extension of the generalized linear mixed model was constructed to simultaneously accommodate overdispersion and hierarchies present in longitudinal or clustered data. This so-called combined model includes conjugate random effects at observation level for overdispersion and normal random effects at subject level to handle correlation, respectively. A variety of data types can be handled in this way, using different members of the exponential family. Both maximum likelihood and Bayesian estimation for covariate effects and variance components were proposed. The focus of this paper is the development of an estimation procedure for the two sets of random effects. These are necessary when making predictions for future responses or their associated probabilities. Such (empirical) Bayes estimates will also be helpful in model diagnosis, both when checking the fit of the model as well as when investigating outlying observations. The proposed procedure is applied to three datasets of different outcome types.

A Marginalized Combined Gamma Frailty and Normal Random-effects Model for Repeated, Overdispersed, Time-to-event Outcomes

This article proposes a marginalized model for repeated or otherwise hierarchical, overdispersed time-to-event outcomes, adapting the so-called combined model for time-to-event outcomes of Molenberghs et al. (in press), who combined gamma and normal random effects. The two sets of random effects are used to accommodate simultaneously correlation between repeated measures and overdispersion. The proposed version allows for a direct marginal interpretation of all model parameters. The outcomes are allowed to be censored. Two estimation methods are proposed: full likelihood and pairwise likelihood. The proposed model is applied to data from a so-called comet assay and to data from recurrent asthma attacks in children. Both estimation methods perform very well. From simulation results, it follows that the marginalized combined model behaves similarly to the ordinary combined model in terms of point estimation and precision. It is also observed that the pairwise likelihood required more computation time on the one hand but is less sensitive to starting values and stabler in terms of bias with increasing sample size and censoring percentage than full likelihood, on the other, leaving room for both in practice.