Normal Random Effects Research Articles

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.

Bayesian analysis of longitudinal ordered data with flexible random effects using McMC: application to diabetic macular Edema data

In the analysis of correlated ordered data, mixed-effect models are frequently used to control the subject heterogeneity effects. A common assumption in fitting these models is the normality of random effects. In many cases, this is unrealistic, making the estimation results unreliable. This paper considers several flexible models for random effects and investigates their properties in the model fitting. We adopt a proportional odds logistic regression model and incorporate the skewed version of the normal, Student's t and slash distributions for the effects. Stochastic representations for various flexible distributions are proposed afterwards based on the mixing strategy approach. This reduces the computational burden being performed by the McMC technique. Furthermore, this paper addresses the identifiability restrictions and suggests a procedure to handle this issue. We analyze a real data set taken from an ophthalmic clinical trial. Model selection is performed by suitable Bayesian model selection criteria.

Modeling overdispersed longitudinal binary data using a combined beta and normal random-effects model

BackgroundIn medical and biomedical areas, binary and binomial outcomes are very common. Such data are often collected longitudinally from a given subject repeatedly overtime, which result in clustering of the observations within subjects, leading to correlation, on the one hand. The repeated binary outcomes from a given subject, on the other hand, constitute a binomial outcome, where the prescribed mean-variance relationship is often violated, leading to the so-called overdispersion.MethodsTwo longitudinal binary data sets, collected in south western Ethiopia: the Jimma infant growth study, where the child’s early growth is studied, and the Jimma longitudinal family survey of youth where the adolescent’s school attendance is studied over time, are considered. A new model which combines both overdispersion, and correlation simultaneously, also known as the combined model is applied. In addition, the commonly used methods for binary and binomial data, such as the simple logistic, which accounts neither for the overdispersion nor the correlation, the beta-binomial model, and the logistic-normal model, which accommodate only for the overdispersion, and correlation, respectively, are also considered for comparison purpose. As an alternative estimation technique, a Bayesian implementation of the combined model is also presented.ResultsThe combined model results in model improvement in fit, and hence the preferred one, based on likelihood comparison, and DIC criterion. Further, the two estimation approaches result in fairly similar parameter estimates and inferences in both of our case studies. Early initiation of breastfeeding has a protective effect against the risk of overweight in late infancy (p = 0.001), while proportion of overweight seems to be invariant among males and females overtime (p = 0.66). Gender is significantly associated with school attendance, where girls have a lower rate of attendance (p = 0.001) as compared to boys.ConclusionWe applied a flexible modeling framework to analyze binary and binomial longitudinal data. Instead of accounting for overdispersion, and correlation separately, both can be accommodated simultaneously, by allowing two separate sets of the beta, and the normal random effects at once.

A Seminonparametric Approach to Joint Modeling of A Primary Binary Outcome and Longitudinal Data Measured at Discrete Informative Times.

In a study conducted at the New York University Fertility Center, one of the scientific objectives is to investigate the relationship between the final pregnancy outcomes of participants receiving an in vitro fertilization (IVF) treatment and their β-human chorionic gonadotrophin (β-hCG) profiles. A common joint modeling approach to this objective is to use subject-specific normal random effects in a linear mixed model for longitudinal β-hCG data as predictors in a model (e.g., logistic model) for the final pregnancy outcome. Empirical data exploration indicates that the observation times for longitudinal β-hCG data may be informative and the distribution of random effects for longitudinal β-hCG data may not be normally distributed. We propose to introduce a third model in the joint model for the informative β-hCG observation times, and relax the normality distributional assumption of random effects using the semi-nonparametric (SNP) approach of Gallant and Nychka (1987) [8]. An EM algorithm is developed for parameter estimation. Extensive simulation designed to evaluate the proposed method indicates that ignoring either informative observation times or distributional assumption of the random effects would lead to invalid and/or inefficient inference. Applying our new approach to the data reveals some interesting findings the traditional approach failed to discover.

Statistical analysis of data from dilution assays with censored correlated counts

Frequently, count data obtained from dilution assays are subject to an upper detection limit, and as such, data obtained from these assays are usually censored. Also, counts from the same subject at different dilution levels are correlated. Ignoring the censoring and the correlation may provide unreliable and misleading results. Therefore, any meaningful data modeling requires that the censoring and the correlation be simultaneously addressed. Such comprehensive approaches of modeling censoring and correlation are not widely used in the analysis of dilution assays data. Traditionally, these data are analyzed using a general linear model on a logarithmic-transformed average count per subject. However, this traditional approach ignores the between-subject variability and risks, providing inconsistent results and unreliable conclusions. In this paper, we propose the use of a censored negative binomial model with normal random effects to analyze such data. This model addresses, in addition to the censoring and the correlation, any overdispersion that may be present in count data. The model is shown to be widely accessible through the use of several modern statistical software.

A generalized Poisson-gamma model for spatially overdispersed data

Modern disease mapping commonly uses hierarchical Bayesian methods to model overdispersion and spatial correlation. Classical random-effects based solutions include the Poisson-gamma model, which uses the conjugacy between the Poisson and gamma distributions, but which does not model spatial correlation, on the one hand, and the more advanced CAR model, which also introduces a spatial autocorrelation term but without a closed-form posterior distribution on the other. In this paper, a combined model is proposed: an alternative convolution model accounting for both overdispersion and spatial correlation in the data by combining the Poisson-gamma model with a spatially-structured normal CAR random effect. The Limburg Cancer Registry data on kidney and prostate cancer in Limburg were used to compare the conventional and new models. A simulation study confirmed results and interpretations coming from the real datasets. Relative risk maps showed that the combined model provides an intermediate between the non-patterned negative binomial and the sometimes oversmoothed CAR convolution model.

A Family of Generalized Linear Models for Repeated Measures with Normal and Conjugate Random Effects

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 proposes a 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, count and time-to-event cases are given particular emphasis. Apart from model formulation, we present an overview of estimation methods, and then settle for maximum likelihood estimation with analytic--numerical integration. Implications for the derivation of marginal correlations functions are discussed. The methodology is applied to data from a study in epileptic seizures, a clinical trial in toenail infection named onychomycosis and survival data in children with asthma.

The (Mis)Specification of Discrete Duration Models with Unobserved Heterogeneity: A Monte Carlo Study

Empirical researchers usually prefer statistical models that can be easily estimated using standard software packages. One such model is the sequential binary model with or without normal random effects; such models can be adopted to estimate discrete duration models with unobserved heterogeneity. But ease of estimation may come at a cost. In this paper we conduct a Monte Carlo simulation to evaluate the consequences of omitting or misspecifying the unobserved heterogeneity distribution in single-spell discrete duration models.

The (mis)specification of discrete duration models with unobserved heterogeneity: A Monte Carlo study

Empirical researchers usually prefer statistical models that can be easily estimated with the help of commonly available software packages. Sequential binary models with or without normal random effects are an example of such models that can be adopted to estimate discrete duration models with unobserved heterogeneity. But an easy-to-implement estimation may incur a cost. In this paper we conduct a Monte Carlo simulation to evaluate the consequences of omitting or misspecifying the unobserved heterogeneity distribution in single-spell discrete duration models.

Robust non‐linear mixed modelling of longitudinal PSA levels after prostate cancer treatment

The objective of this study was to develop a robust non-linear mixed model for prostate-specific antigen (PSA) measurements after a high-intensity focused ultrasound (HIFU) treatment for prostate cancer. The characteristics of these data are the presence of outlying values and non-normal random effects. A numerical study proved that parameter estimates can be biased if these characteristics are not taken into account. The intra-patient variability was described by a Student-t distribution and Dirichlet process priors were assumed for non-normal random effects; a process that limited the bias and provided more efficient parameter estimates than a classical mixed model with normal residuals and random effects. It was applied to the determination of the best dynamic PSA criterion for the diagnosis of prostate cancer recurrence, but could be used in studies that rely on PSA data to improve prognosis or compare treatment efficiencies and also with other longitudinal biomarkers that, such as PSA, present outlying values and non-normal random effects.

Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions

Skew-normal and skew-t distributions have proved to be useful for capturing skewness and kurtosis in data directly without transformation. Recently, finite mixtures of such distributions have been considered as a more general tool for handling heterogeneous data involving asymmetric behaviors across subpopulations. We consider such mixture models for both univariate as well as multivariate data. This allows robust modeling of high-dimensional multimodal and asymmetric data generated by popular biotechnological platforms such as flow cytometry. We develop Bayesian inference based on data augmentation and Markov chain Monte Carlo (MCMC) sampling. In addition to the latent allocations, data augmentation is based on a stochastic representation of the skew-normal distribution in terms of a random-effects model with truncated normal random effects. For finite mixtures of skew normals, this leads to a Gibbs sampling scheme that draws from standard densities only. This MCMC scheme is extended to mixtures of skew-t distributions based on representing the skew-t distribution as a scale mixture of skew normals. As an important application of our new method, we demonstrate how it provides a new computational framework for automated analysis of high-dimensional flow cytometric data. Using multivariate skew-normal and skew-t mixture models, we could model non-Gaussian cell populations rigorously and directly without transformation or projection to lower dimensions.

Relative Efficiency of the Fuzzy p‐Value Approach to Hypothesis Testing

Summary In missing data problems, it is often the case that there is a natural test statistic for testing a statistical hypothesis had all the data been observed. A fuzzy p‐value approach to hypothesis testing has recently been proposed which is implemented by imputing the missing values in the “complete data” test statistic by values simulated from the conditional null distribution given the observed data. We argue that imputing data in this way will inevitably lead to loss in power. For the case of scalar parameter, we show that the asymptotic efficiency of the score test based on the imputed “complete data” relative to the score test based on the observed data is given by the ratio of the observed data information to the complete data information. Three examples involving probit regression, normal random effects model, and unidentified paired data are used for illustration. For testing linkage disequilibrium based on pooled genotype data, simulation results show that the imputed Neyman Pearson and Fisher exact tests are less powerful than a Wald‐type test based on the observed data maximum likelihood estimator. In conclusion, we caution against the routine use of the fuzzy p‐value approach in latent variable or missing data problems and suggest some viable alternatives.

Characterizing the variance improvement in linear Dirichlet random effects models

An alternative to the classical mixed model with normal random effects is to use a Dirichlet process to model the random effects. Such models have proven useful in practice, and we have observed a noticeable variance reduction, in the estimation of the fixed effects, when the Dirichlet process is used instead of the normal. In this paper we formalize this notion, and give a theoretical justification for the expected variance reduction. We show that for almost all data vectors, the posterior variance from the Dirichlet random effects model is smaller than that from the normal random effects model.

Comments on: Goodness-of-fit-tests in mixed models

I am very pleased to have the opportunity to discuss this interesting paper. The authors deal with the problem of assessing normality of random effects and/or error distributions in the context of mixed models. As clearly stated in Sect. 1 of the manuscript, this is not only a question of theoretical importance, but it has practical implications as well. In Sects. 3 and 4 of the manuscript there is a very exhaustive overview of available methods. There are essentially two kinds of procedures, graphical methods, which are of no inferential nature, that try to mimic the use of normal probability plots in the context of regression but take into account that the residuals of the mixed model are correlated. Another alternative is the use of classical tests, such as Shapiro Wilk, after trying to “independize” the residuals. A test specifically designed for checking normality in mixed models is that of Jiang (2001), where a χ2 type of tests is proposed, and some power studies are made. These results are compared with the ones proposed in Sect. 6 of the paper, and we present here a related simulation study. The authors address the problem in a very particular fashion. They rather estimate the density of the random effects (and the error terms when needed) instead of the actual effects, and propose two frameworks: order selection/smooth tests and minimum distance estimators. As for the order selection methods, the authors base their proposal on a smooth test based on (truncated) expansions of ( fU(u)

Relapse–remission and remission–relapse switches in rheumatoid arthritis patients were modeled by random effects

Objective To compare statistical models for the analysis of two-state disease processes. Study Design and Setting A two-armed randomized trial of patients with early rheumatoid arthritis (RA) treated by either combination therapy (sulfasazaline, methotrextate, prednisolone) or monotherapy (sulfasazaline). Disease activity (remission or relapse) was analyzed with the logistic regression model, the proportional hazards regression model, and the continuous-time Markov process model for panel data. The dependence among the switching times was studied by (1) including correlated normal random patient effects for the relapse–remission and remission–relapse switching probabilities; (2) assuming the population to be a mixture of patients responsive and nonresponsive to therapy; (3) including separate parameters for the first and subsequent relapse–remission switch; and (4) combining (1) and (3). The four approaches were compared using parametric bootstrap checks. Results The logistic regression model, the proportional hazards regression model, and the continuous-time Markov process model for panel data yielded similar combination therapy effects. The inclusion of random patient effects (approaches 1 and 4) gave the best fit to the observed disease activity pattern. Conclusion Models with correlated random effects can provide a satisfactory fit to two-state disease patterns.

A Parametric Model for Analyzing Anticipation in Genetically Predisposed Families

Anticipation, i.e. a decreasing age-at-onset in subsequent generations has been observed in a number of genetically triggered diseases. The impact of anticipation is generally studied in affected parent-child pairs. These analyses are restricted to pairs in which both individuals have been affected and are sensitive to right truncation of the data. We propose a normal random effects model that allows for right-censored observations and includes covariates, and draw statistical inference based on the likelihood function. We applied the model to the hereditary nonpolyposis colorectal cancer (HNPCC)/Lynch syndrome family cohort from the national Danish HNPCC register. Age-at-onset was analyzed in 824 individuals from 2-4 generations in 125 families with proved disease-predisposing mutations. A significant effect from anticipation was identified with a mean of 3 years earlier age-at-onset per generation. The suggested model corrects for incomplete observations and considers families rather than affected pairs and thereby allows for studies of large sample sets, facilitates subgroup analyses and provides generation effect estimates.

Assessing Treatments Effects in Multi-Center Clinical Trials with Application To Scleroderma Lung Study: A Semiparametric Bayesian Approach

In this article we explore the testing of non-inferiority and equivalence hypotheses arising from multiple centers when the assumption of normality is violated. In a multicenter study, the trials are typically conducted at different centers which vary in terms of location, environment, demographics among others, leading to substantial amount of heterogeneity in the patient population. This unexplained variation in a multi-center clinical study is usually modeled using a random effects model, where the centers are assumed to be a random sample from the population of centers. Most research in this direction uses a parametric normal distribution which can be restrictive and may lead to biased result if the actual distribution is nonnormal. In this article, we overcome this parametric assumption by considering a broader class of random effects distribution for the centers. In particular, we develop a novel nested Dirichlet process (nDP) model to explore the sensitivity of the fixed treatment effects under various hypotheses, in the presence of nonnormality. Additional advantage of our proposed method is that it facilitates a hierarchical clustering structure. At one hand it clusters the centers according to their effects, and hence outlying centers can be identified. Simultaneously, subjects from the clustered centers are again clustered together enabling a borrowing of information across similar centers. Further, we present the methodology to test between the models with nDP versus a normal random center effects models. We discuss the results of our proposed methodology in a real example of a multi-center clinical trial on Scleroderma lung study. The results of the analysis along with the extensive simulation study show the advantage of our method when the center effects distribution is not normal.

Skew random effects in multilevel binomial models

Compared to modelling observable data, it is more difficult to choose a suitable distribution to describe latent variables since no prior knowledge or observable information can be used and only normal or nonparametric distributions are mainly applied to random effects for generalized linear mixed models (GLMMs) in the literature. To enhance the modelling toolkit, this article investigates a class of parametric skew elliptical random effects in multilevel binomial regression models using a Bayesian approach; the class includes skew normal, skew Students’ t-distributions and others. Skewness mechanism is considered through multiplying skewness parameter Δ by standardized folded elliptical random variables, and the posterior sampling is realized by working on a binary skewness indicator (BSI) instead of continuous Δ for parameter identifiability. Simulation study shows that the original continuous skewness parameter Δ and the posterior mean of BSI may have dichotomous signs to describe the directional (right/left) skewness; thus we address the importance of assuming specific random effects distribution and interpreting the skewness carefully. The methodology is exemplified through reanalyzing a teratogenic activity study of two niacin analogs published in the biological literature, and sampling-based model comparison shows that the parametric skew normal random effects model works largely better than nonparametric Dirichlet process mixture models for this data set.

Linear mixed models with skew-elliptical distributions: A Bayesian approach

Normality of random effects and error terms is a routine assumption for linear mixed models. However, such an assumption may be unrealistic, obscuring important features of within- and among-unit variation. A simple and robust Bayesian parametric approach that relaxes this assumption by using a multivariate skew-elliptical distribution, which includes the Skew- t , Skew-normal, t -Student, and Normal distributions as special cases and provides flexibility in capturing a broad range of non-normal and asymmetric behavior is presented. An appropriate posterior simulation scheme is developed and the methods are illustrated with an application to a longitudinal data example.

Stochastic Simulation of Forest Regeneration Establishment Using a Multilevel Multivariate Model

Abstract Here we present a method for simulating high unexplained variation in establishment of tree seedlings in planted Norway spruce (Picea abies [L.] Karst.) stands. The simulation method is based on an existing hierarchical multilevel multivariate model for establishment of regeneration (Miina, J., and T. Saksa. 2006. New For. 32:265–283). The model includes seven simultaneously estimated models for numbers and heights of tree seedlings on 20-m2 plots within a regeneration area. The variation in plot-level expectations is described by fixed effects and two nested multivariate normal random effects. Conditionally, given the expected values, one of the count variables is underdispersed and four are overdispersed relative to Poisson law. The conditional distributions of the square roots of the height variables are normal. For the conditional joint distribution, the model defines the product-moment cross-correlation matrix, which includes both positive and negative cross-correlations. The first two moments do not, however, uniquely define the marginal distributions of the count variables or the joint distribution of all seven variables. We applied binomial distribution for the underdispersed Poisson variation and negative binomial distribution for the overdispersed Poisson variation. The conditional joint distribution was defined by a multivariate normal copula and the marginal distributions. Despite relatively low cross-correlations among responses, if the cross-correlations among random components of the responses were ignored in stochastic simulations, the amount and costs of precommercial thinning (i.e., cleaning out competing broadleaves) were greatly underestimated. If all random effect components in model parameters were dropped, the error increased markedly.