Normality Assumption For Random Effects Research Articles

A methodology for improving efficiency estimation based on conditional mix-GEE models in longitudinal studies

Estimating random effects accurately is crucial since it reflects the subject-specific effect in longitudinal studies. In this paper, we develop a new methodology for improving the efficiency of fixed-effects and random-effects estimation based on conditional mix-GEE models. The advantage of our proposed approach is that the serial correlation over time was accommodated in estimating random effects. Meanwhile, the normality assumption for random effects is not required. In addition, according to the estimates of some mixture proportions, the true working correlation matrix can be identified. The feature of our proposed approach is that the estimators of the regression parameters are more efficient than CCQIF, cmix-GEE and CQIF approaches even if the working correlation structure is not correctly specified. In theory, we show that the proposed method yields a consistent and more efficient estimator than the random-effect estimator that ignores correlation information from longitudinal data. We establish asymptotic results for both fixed-effects and random-effects estimators. Simulation studies confirm the performance of our proposed method.

Notes on misspecifying the random effects distribution regarding analysis under the AB/BA crossover trial in dichotomous data – a Monte Carlo evaluation

When patient responses are dichotomous under an AB/BA design, we commonly assume the normal random effects logistic regression model. This normality assumption for random effects is, however, unlikely to hold. Based on the maximum likelihood estimator, we apply Monte Carlo simulations to investigate the impact of misspecification on hypothesis testing and estimation because of the incorrect normal assumption for random effects. We find that Type I error is not affected by misspecifying the normal random effects distributions. We further find that the influence due to misspecifying distributions of random effects on power, bias and mean-squared-error, the coverage probability and the average length of the confidence interval is generally minimal when the variation of responses between patients is small and the number of patients per group is large. This influence can be substantial when the variation of responses between patients is large and the number of patients per group is small. Thus, the estimate of the required sample size or the accuracy of an interval estimator under the normal random effects assumption can be liberal. We use the data comparing a drug with placebo in treating cerebrovascular deficiency to illustrate the potential difference in inference between various random effects distributions considered here.

Conditional mix-GEE models for longitudinal data with unspecified random-effects distributions

ABSTRACTIn the longitudinal studies, the mixture generalized estimation equation (mix-GEE) was proposed to improve the efficiency of the fixed-effects estimator for addressing the working correlation structure misspecification. When the subject-specific effect is one of interests, mixed-effects models were widely used to analyze longitudinal data. However, most of the existing approaches assume a normal distribution for the random effects, and this could affect the efficiency of the fixed-effects estimator. In this article, a conditional mixture generalized estimating equation (cmix-GEE) approach based on the advantage of mix-GEE and conditional quadratic inference function (CQIF) method is developed. The advantage of our new approach is that it does not require the normality assumption for random effects and can accommodate the serial correlation between observations within the same cluster. The feature of our proposed approach is that the estimators of the regression parameters are more efficient than CQIF even if the working correlation structure is not correctly specified. In addition, according to the estimates of some mixture proportions, the true working correlation matrix can be identified. We establish the asymptotic results for the fixed-effects parameter estimators. Simulation studies were conducted to evaluate our proposed method.

Joint modeling of survival time and longitudinal outcomes with flexible random effects.

Joint models with shared Gaussian random effects have been conventionally used in analysis of longitudinal outcome and survival endpoint in biomedical or public health research. However, misspecifying the normality assumption of random effects can lead to serious bias in parameter estimation and future prediction. In this paper, we study joint models of general longitudinal outcomes and survival endpoint but allow the underlying distribution of shared random effect to be completely unknown. For inference, we propose to use a mixture of Gaussian distributions as an approximation to this unknown distribution and adopt an Expectation-Maximization (EM) algorithm for computation. Either AIC and BIC criteria are adopted for selecting the number of mixtures. We demonstrate the proposed method via a number of simulation studies. We illustrate our approach with the data from the Carolina Head and Neck Cancer Study (CHANCE).

Bayesian semiparametric reproductive dispersion mixed models for non-normal longitudinal data: estimation and case influence analysis

ABSTRACTSemiparametric reproductive dispersion mixed model (SPRDMM) is a natural extension of the reproductive dispersion model and the semiparametric mixed model. In this paper, we relax the normality assumption of random effects in SPRDMM and use a truncated and centred Dirichlet process prior to specify random effects, and present the Bayesian P-spline to approximate the smoothing unknown function. A hybrid algorithm combining the block Gibbs sampler and the Metropolis–Hastings algorithm is implemented to sample observations from the posterior distribution. Also, we develop Bayesian case deletion influence measure for SPRDMM based on the φ-divergence and present those computationally feasible formulas. Several simulation studies and a real example are presented to illustrate the proposed methodologies.

Semiparametric Bayesian inference on skew-normal joint modeling of multivariate longitudinal and survival data.

We propose a semiparametric multivariate skew-normal joint model for multivariate longitudinal and multivariate survival data. One main feature of the posited model is that we relax the commonly used normality assumption for random effects and within-subject error by using a centered Dirichlet process prior to specify the random effects distribution and using a multivariate skew-normal distribution to specify the within-subject error distribution and model trajectory functions of longitudinal responses semiparametrically. A Bayesian approach is proposed to simultaneously obtain Bayesian estimates of unknown parameters, random effects and nonparametric functions by combining the Gibbs sampler and the Metropolis-Hastings algorithm. Particularly, a Bayesian local influence approach is developed to assess the effect of minor perturbations to within-subject measurement error and random effects. Several simulation studies and an example are presented to illustrate the proposed methodologies.

A Fast Algorithm for Using Semi-Parametric Random Effects Model for Analyzing Longitudinal Data

Mixed effects models are frequently used for analyzing longitudinal data. Normality assumption of random effects distrbution is a routine assumption for these models, violation of which leads to model misspecifcation and misleading parameter estimates. We propose a semi-parametric approach using gradient function for random effect estimation. In this approach, we relax the normality assumption for random effects by estimating their distribution over a pre-specifed grid. Unknown parameters of the marginal model are estimated using maximum likelihood methods. Some simulation studies and analyzing of a real data set are performed for illustration of the proposed semi-parametric method.

Conditional Inference Functions for Mixed-Effects Models With Unspecified Random-Effects Distribution

In longitudinal studies, mixed-effects models are important for addressing subject-specific effects. However, most existing approaches assume a normal distribution for the random effects, and this could affect the bias and efficiency of the fixed-effects estimator. Even in cases where the estimation of the fixed effects is robust with a misspecified distribution of the random effects, the estimation of the random effects could be invalid. We propose a new approach to estimate fixed and random effects using conditional quadratic inference functions (QIFs). The new approach does not require the specification of likelihood functions or a normality assumption for random effects. It can also accommodate serial correlation between observations within the same cluster, in addition to mixed-effects modeling. Other advantages include not requiring the estimation of the unknown variance components associated with the random effects, or the nuisance parameters associated with the working correlations. We establish asymptotic results for the fixed-effect parameter estimators that do not rely on the consistency of the random-effect estimators. Real data examples and simulations are used to compare the new approach with the penalized quasi-likelihood (PQL) approach, and SAS GLIMMIX and nonlinear mixed-effects model (NLMIXED) procedures. Supplemental materials including technical details are available online.

Joint analysis of bivariate longitudinal ordinal outcomes and competing risks survival times with nonparametric distributions for random effects

We propose a semiparametric joint model for bivariate longitudinal ordinal outcomes and competing risks failure time data. The association between the longitudinal and survival endpoints is captured by latent random effects. This approach generalizes previous joint analysis that considers only one response variable at the longitudinal endpoint. One unique feature of the proposed model is that we relax the commonly used normality assumption for random effects and leave the distribution completely unspecified. We use a modified version of the vertex exchange method in conjunction with an expectation-maximization algorithm to estimate the random effects distribution and model parameters. We show via simulations that robust parameter estimates are obtained from the proposed method under various scenarios. We illustrate the approach using cough severity and frequency data from a scleroderma lung study.

Discriminant analysis using a multivariate linear mixed model with a normal mixture in the random effects distribution

We have developed a method to longitudinally classify subjects into two or more prognostic groups using longitudinally observed values of markers related to the prognosis. We assume the availability of a training data set where the subjects' allocation into the prognostic group is known. The proposed method proceeds in two steps as described earlier in the literature. First, multivariate linear mixed models are fitted in each prognostic group from the training data set to model the dependence of markers on time and on possibly other covariates. Second, fitted mixed models are used to develop a discrimination rule for future subjects. Our method improves upon existing approaches by relaxing the normality assumption of random effects in the underlying mixed models. Namely, we assume a heteroscedastic multivariate normal mixture for random effects. Inference is performed in the Bayesian framework using the Markov chain Monte Carlo methodology. Software has been written for the proposed method and it is freely available. The methodology is applied to data from the Dutch Primary Biliary Cirrhosis Study.

Bayesian Nonparametric Meta‐Analysis Using Polya Tree Mixture Models

Summary. A common goal in meta-analysis is estimation of a single effect measure using data from several studies that are each designed to address the same scientific inquiry. Because studies are typically conducted in geographically disperse locations, recent developments in the statistical analysis of meta-analytic data involve the use of random effects models that account for study-to-study variability attributable to differences in environments, demographics, genetics, and other sources that lead to heterogeneity in populations. Stemming from asymptotic theory, study-specific summary statistics are modeled according to normal distributions with means representing latent true effect measures. A parametric approach subsequently models these latent measures using a normal distribution, which is strictly a convenient modeling assumption absent of theoretical justification. To eliminate the influence of overly restrictive parametric models on inferences, we consider a broader class of random effects distributions. We develop a novel hierarchical Bayesian nonparametric Polya tree mixture (PTM) model. We present methodology for testing the PTM versus a normal random effects model. These methods provide researchers a straightforward approach for conducting a sensitivity analysis of the normality assumption for random effects. An application involving meta-analysis of epidemiologic studies designed to characterize the association between alcohol consumption and breast cancer is presented, which together with results from simulated data highlight the performance of PTMs in the presence of nonnormality of effect measures in the source population.

A unified approach to estimation of nonlinear mixed effects and Berkson measurement error models

AbstractMixed effects models and Berkson measurement error models are widely used. They share features which the author uses to develop a unified estimation framework. He deals with models in which the random effects (or measurement errors) have a general parametric distribution, whereas the random regression coefficients (or unobserved predictor variables) and error terms have nonparametric distributions. He proposes a second‐order least squares estimator and a simulation‐based estimator based on the first two moments of the conditional response variable given the observed covariates. He shows that both estimators are consistent and asymptotically normally distributed under fairly general conditions. The author also reports Monte Carlo simulation studies showing that the proposed estimators perform satisfactorily for relatively small sample sizes. Compared to the likelihood approach, the proposed methods are computationally feasible and do not rely on the normality assumption for random effects or other variables in the model.

A Linear Mixed-Effects Model with Heterogeneity in the Random-Effects Population

Abstract This article investigates the impact of the normality assumption for random effects on their estimates in the linear mixed-effects model. It shows that if the distribution of random effects is a finite mixture of normal distributions, then the random effects may be badly estimated if normality is assumed, and the current methods for inspecting the appropriateness of the model assumptions are not sound. Further, it is argued that a better way to detect the components of the mixture is to build this assumption in the model and then “compare” the fitted model with the Gaussian model. All of this is illustrated on two practical examples.