Subject-specific Random Effects Research Articles

A crossed random effects modeling approach for estimating diagnostic accuracy from ordinal ratings without a gold standard

In diagnostic studies without a gold standard, the assumption on the dependence structure of the multiple tests or raters plays an important role in model performance. In case of binary disease status, both conditional independence and crossed random effects structure have been proposed and their performance investigated. Less attention has been paid to the situation where the true disease status is ordinal. In this paper, we propose crossed subject-specific and rater-specific random effects to account for the dependence structure and assess the robustness of the proposed model to misspecification in the random effects distributions. We applied the models to data from the Physician Reliability Study, which focuses on assessing the diagnostic accuracy in a population of raters for the staging of endometriosis, a gynecological disorder in women. Using this new methodology, we estimate the probability of a correct classification and show that regional experts can more easily classify the intermediate stage than resident physicians.

Bayesian nonparametric centered random effects models with variable selection

In a linear mixed effects model, it is common practice to assume that the random effects follow a parametric distribution such as a normal distribution with mean zero. However, in the case of variable selection, substantial violation of the normality assumption can potentially impact the subset selection and result in poor interpretation and even incorrect results. In nonparametric random effects models, the random effects generally have a nonzero mean, which causes an identifiability problem for the fixed effects that are paired with the random effects. In this article, we focus on a Bayesian method for variable selection. We characterize the subject-specific random effects nonparametrically with a Dirichlet process and resolve the bias simultaneously. In particular, we propose flexible modeling of the conditional distribution of the random effects with changes across the predictor space. The approach is implemented using a stochastic search Gibbs sampler to identify subsets of fixed effects and random effects to be included in the model. Simulations are provided to evaluate and compare the performance of our approach to the existing ones. We then apply the new approach to a real data example, cross-country and interlaboratory rodent uterotrophic bioassay.

Estimating Youth Locomotion Ground Reaction Forces Using an Accelerometer-Based Activity Monitor

To address a variety of questions pertaining to the interactions between physical activity, musculoskeletal loading and musculoskeletal health/injury/adaptation, simple methods are needed to quantify, outside a laboratory setting, the forces acting on the human body during daily activities. The purpose of this study was to develop a statistically based model to estimate peak vertical ground reaction force (pVGRF) during youth gait. 20 girls (10.9±0.9 years) and 15 boys (12.5±0.6 years) wore a Biotrainer AM over their right hip. Six walking and six running trials were completed after a standard warm-up. Average AM intensity (g) and pVGRF (N) during stance were determined. Repeated measures mixed effects regression models to estimate pVGRF from Biotrainer activity monitor acceleration in youth (girls 10–12, boys 12–14 years) while walking and running were developed. Log transformed pVGRF had a statistically significant relationship with activity monitor acceleration, centered mass, sex (girl), type of locomotion (run), and locomotion type-acceleration interaction controlling for subject as a random effect. A generalized regression model without subject specific random effects was also developed. The average absolute differences between the actual and predicted pVGRF were 5.2% (1.6% standard deviation) and 9% (4.2% standard deviation) using the mixed and generalized models, respectively. The results of this study support the use of estimating pVGRF from hip acceleration using a mixed model regression equation.

Editorial: Special issue on time series analysis in the biological sciences

Editorial: Special issue on time series analysis in the biological sciences

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.

Extending the latent variable model for extra correlated longitudinal dichotomous responses

Since generalized nonlinear mixed-effects modeling methodology of ordered categorical data became available in the pharmacokinetic/pharmacodynamic (PK/PD) literature over a decade ago, pharmacometricians have been increasingly performing exposure-response analyses of such data to inform drug development. Also, as experiences with and scrutiny of these data have increased, pharmacometricians have noted fewer transitions (or greater correlations) between response values than predicted by the model. In this paper, we build on the latent variable (LV) approach, which is convenient for incorporating pharmacological concepts such as pharmacodynamic onset of drug effect, and present a PK/PD methodology which we term the multivariate latent variable (MLV) approach. This approach uses correlations between the latent residuals (LR) to address extra correlation or a fewer number of transitions, relative to if the LR were independent. Four approximation methods for handling dichotomous MLV data are formulated and then evaluated for accuracy and computation time using simulation studies. Some analytical results for models linear in the subject-specific random effects are also presented, and these provide insight into modeling such repeated measures data. Also, a case study previously analyzed using the LV approach is revisited using one of the MLV approximation methods and the results are discussed. Overall, consideration of the simulation and analytical results lead us to some conclusions we feel are applicable to many of the models and situations frequently encountered in analysis of such data: the MLV approach is a flexible method that can handle many different extra correlated data structures and therefore can more accurately predict the number of transitions between response values; incorrect modeling of the population covariances by implementing an LV model when extra correlation exists is not likely to (and in many cases does not) influence accuracy of the population (marginal) mean predictions; adequate prediction of the population mean probabilities achieves adequate predictions of the population variances, regardless of the correct specification of the population covariances--that is, if the LV model accurately predicts the means in the presence of extra correlation, it will accurately predict the variances; the between subject random effects component to the model describe the marginal covariances in responses--not the marginal variances as with continuous-type data. From these conclusions we make a general statement that it may not be necessary to model the extra correlation in every case using the MLV model, which requires technical implementation with currently available commercially or publically available software. The LV model may be sufficient for answering many of the typical questions arising during drug development. The MLV approach should be considered however if prediction or simulation of individual level data is an objective of the analysis.

Estimating transformations for repeated measures modeling of continuous bounded outcome data

Continuous bounded outcome data are unlikely to meet the usual assumptions for mixed-effects models of normally distributed and independent subject-specific and residual random effects. Additionally, overly complicated model structures might be necessary to account adequately for non-drug (time-dependent) and drug treatment effects. A transformation strategy with a likelihood component for censoring is developed to promote the simplicity of model structures and to improve the plausibility of assumptions on the random effects. The approach is motivated by Health Assessment Questionnaire Disability Index (HAQ-DI) data from a study in subjects with rheumatoid arthritis and is evaluated using a simulation study.

A Penalized Spline Approach to Functional Mixed Effects Model Analysis

In this article, we propose penalized spline (P-spline)-based methods for functional mixed effects models with varying coefficients. We decompose longitudinal outcomes as a sum of several terms: a population mean function, covariates with time-varying coefficients, functional subject-specific random effects, and residual measurement error processes. Using P-splines, we propose nonparametric estimation of the population mean function, varying coefficient, random subject-specific curves, and the associated covariance function that represents between-subject variation and the variance function of the residual measurement errors which represents within-subject variation. Proposed methods offer flexible estimation of both the population- and subject-level curves. In addition, decomposing variability of the outcomes as a between- and within-subject source is useful in identifying the dominant variance component therefore optimally model a covariance function. We use a likelihood-based method to select multiple smoothing parameters. Furthermore, we study the asymptotics of the baseline P-spline estimator with longitudinal data. We conduct simulation studies to investigate performance of the proposed methods. The benefit of the between- and within-subject covariance decomposition is illustrated through an analysis of Berkeley growth data, where we identified clearly distinct patterns of the between- and within-subject covariance functions of children's heights. We also apply the proposed methods to estimate the effect of antihypertensive treatment from the Framingham Heart Study data.

Sequential Testing with Recurrent Events over Multiple Treatment Periods

Recurrent event outcomes are adopted increasingly often as a basis for evaluating experimental interventions. In clinical trials involving recurrent events, patients are frequently observed for a baseline period while under standard care, and then randomised to receive either an experimental treatment or continue on standard care. When events are generated according to a mixed Poisson model, having baseline data permits a conditional analysis which can eliminate the subject-specific random effect and yield a more efficient analysis regarding treatment effect. When studies are expected to recruit a large number of patients over an extended period of accrual, or if the period of follow-up is long, sequential testing is desirable to ensure the study is stopped as soon as sufficient data have been collected to establish treatment benefits. We describe methods which facilitate sequential analysis of data arising from trials with recurrent event responses observed over two treatment periods where one is a baseline period of observation. Formulae to help schedule analyses at approximately equal increments of information are given. Simulation studies show that the sequential testing procedures have rejection rates compatible with the nominal error rates under the null and alternative hypotheses. Data from a trial of patients with herpes simplex virus infection are analysed to illustrate the utility of these methods.

Bayesian Nonparametric Regression Analysis of Data with Random Effects Covariates from Longitudinal Measurements

We consider nonparametric regression analysis in a generalized linear model (GLM) framework for data with covariates that are the subject-specific random effects of longitudinal measurements. The usual assumption that the effects of the longitudinal covariate processes are linear in the GLM may be unrealistic and if this happens it can cast doubt on the inference of observed covariate effects. Allowing the regression functions to be unknown, we propose to apply Bayesian nonparametric methods including cubic smoothing splines or P-splines for the possible nonlinearity and use an additive model in this complex setting. To improve computational efficiency, we propose the use of data-augmentation schemes. The approach allows flexible covariance structures for the random effects and within-subject measurement errors of the longitudinal processes. The posterior model space is explored through a Markov chain Monte Carlo (MCMC) sampler. The proposed methods are illustrated and compared to other approaches, the "naive" approach and the regression calibration, via simulations and by an application that investigates the relationship between obesity in adulthood and childhood growth curves.

Empirical constrained Bayes predictors accounting for non‐detects among repeated measures

When the prediction of subject-specific random effects is of interest, constrained Bayes predictors (CB) have been shown to reduce the shrinkage of the widely accepted Bayes predictor while still maintaining desirable properties, such as optimizing mean-square error subsequent to matching the first two moments of the random effects of interest. However, occupational exposure and other epidemiologic (e.g. HIV) studies often present a further challenge because data may fall below the measuring instrument's limit of detection. Although methodology exists in the literature to compute Bayes estimates in the presence of non-detects (Bayes(ND)), CB methodology has not been proposed in this setting. By combining methodologies for computing CBs and Bayes(ND), we introduce two novel CBs that accommodate an arbitrary number of observable and non-detectable measurements per subject. Based on application to real data sets (e.g. occupational exposure, HIV RNA) and simulation studies, these CB predictors are markedly superior to the Bayes predictor and to alternative predictors computed using ad hoc methods in terms of meeting the goal of matching the first two moments of the true random effects distribution.

A mixed-effects expectancy-valence model for the Iowa gambling task

The Iowa gambling task (IGT; Bechara, Damasio, Damasio, & Anderson, 1994) was developed to simulate real-life decision making under uncertainty. The task has been widely used to examine possible neurocognitive deficits in normal and clinical populations. Busemeyer and Stout (2002) proposed the expectancy-valence (EV) model to explicitly account for individual participants' repeated choices in the IGT. Parameters of the EV model presumably measure different psychological processes that underlie performance on the task, and their values may be used to differentiate individuals across different populations. In the present article, the EV model is extended to include both fixed effects and subject-specific random effects. The mixed-effects EV model fits the nested structure of observations in the IGT naturally and provides a unified procedure for parameter estimation and comparisons among groups of populations. We illustrate the utility of the mixed-effects approach with an analysis of gender differences using a real data set. A simulation study was conducted to verify the advantages of this approach.

Impact of non-normal random effects on inference by multiple imputation: A simulation assessment

Multivariate extensions of well-known linear mixed-effects models have been increasingly utilized in inference by multiple imputation in the analysis of multilevel incomplete data. The normality assumption for the underlying error terms and random effects plays a crucial role in simulating the posterior predictive distribution from which the multiple imputations are drawn. The plausibility of this normality assumption on the subject-specific random effects is assessed. Specifically, the performance of multiple imputation created under a multivariate linear mixed-effects model is investigated on a diverse set of incomplete data sets simulated under varying distributional characteristics. Under moderate amounts of missing data, the simulation study confirms that the underlying model leads to a well-calibrated procedure with negligible biases and actual coverage rates close to nominal rates in estimates of the regression coefficients. Estimation quality of the random-effect variance and association measures, however, are negatively affected from both the misspecification of the random-effect distribution and number of incompletely-observed variables. Some of the adverse impacts include lower coverage rates and increased biases.

Analysis of Interval-Censored Data from Clustered Multistate Processes: Application to Joint Damage in Psoriatic Arthritis

Summary A conditionally Markov multiplicative intensity model is described for the analysis of clustered progressive multistate processes under intermittent observation. The model is motivated by a long-term prospective study of patients with psoriatic arthritis with the aim of characterizing progression of joint damage via an irreversible four-state model. The model accommodates heterogeneity in transition rates between different individuals and correlation in transition rates within patients. To do this we introduce subject-specific multivariate random effects in which each component acts multiplicatively on a specific transition intensity. Through the association between the components of the random effect, correlations in transition intensities are accommodated. A Monte Carlo EM algorithm is developed for estimation, which features closed form expressions for estimators at each M-step.

A Bayesian approach to functional-based multilevel modeling of longitudinal data: applications to environmental epidemiology.

Flexible multilevel models are proposed to allow for cluster-specific smooth estimation of growth curves in a mixed-effects modeling format that includes subject-specific random effects on the growth parameters. Attention is then focused on models that examine between-cluster comparisons of the effects of an ecologic covariate of interest (e.g. air pollution) on nonlinear functionals of growth curves (e.g. maximum rate of growth). A Gibbs sampling approach is used to get posterior mean estimates of nonlinear functionals along with their uncertainty estimates. A second-stage ecologic random-effects model is used to examine the association between a covariate of interest (e.g. air pollution) and the nonlinear functionals. A unified estimation procedure is presented along with its computational and theoretical details. The models are motivated by, and illustrated with, lung function and air pollution data from the Southern California Children's Health Study.

Analysis of Longitudinal Data with Drop-Out: Objectives, Assumptions and a Proposal

Summary The problem of analysing longitudinal data that are complicated by possibly informative drop-out has received considerable attention in the statistical literature. Most researchers have concentrated on either methodology or application, but we begin this paper by arguing that more attention could be given to study objectives and to the relevant targets for inference. Next we summarize a variety of approaches that have been suggested for dealing with drop-out. A long-standing concern in this subject area is that all methods require untestable assumptions. We discuss circumstances in which we are willing to make such assumptions and we propose a new and computationally efficient modelling and analysis procedure for these situations. We assume a dynamic linear model for the expected increments of a constructed variable, under which subject-specific random effects follow a martingale process in the absence of drop-out. Informal diagnostic procedures to assess the tenability of the assumption are proposed. The paper is completed by simulations and a comparison of our method and several alternatives in the analysis of data from a trial into the treatment of schizophrenia, in which approximately 50% of recruited subjects dropped out before the final scheduled measurement time.

Mixed effects modeling of weight change associated with placebo and pregabalin administration

To characterize change from baseline weight over time for pregabalin and placebo administration. Asymptotic fraction of baseline weight was modeled with a nonmixture model and a mixture model as a function of baseline weight, exposure, time, covariate effects, and subject-specific random effects. Model fit was assessed using standard diagnostic plots. Predictive performance was assessed using both data similar to the original data, and open-label data. The nonmixture model indicated that a typical patient (baseline weight 82 kg) receiving placebo or 300 mg/day pregabalin approached an asymptotic fractional change from baseline weight of [mean (95% prediction interval for typical individual)] 0.7% (-5.5% to 7.4%) or 2.5% (-3.8% to 9.1%), respectively, with a half-life of 17 days. Substantial between-subject variability is observed, with some drug-treated subjects remaining weight neutral or losing weight, at all levels of exposure. Structural fixed effects parameters for the two submodels (mixture model) were in close agreement with each other and with those for the nonmixture model. The mixture model described two subpopulations differing in interindividual variability. No significant interindividual-varying covariates influencing the mixture probabilities were identified other than exposure. Both models had adequate fit; both models performed well during external validation. Predictive performance (nonmixture model) was adequate to ~900 days. The weight of a typical 82-kg patient receiving placebo or pregabalin (300 mg/day) approaches an asymptotic fractional change from baseline weight of 0.7%, or 2.5%, respectively, with a half-life of 17 days. Substantial between-subject variability remains unexplained.

Nonlinear growth generates age changes in the moments of the frequency distribution: the example of height in puberty

Higher moments of the frequency distribution of child height and weight change with age, particularly during puberty, though why is not known. Our aims were to confirm that height skewness and kurtosis change with age during puberty, to devise a model to explain why, and to test the model by analyzing the data longitudinally. Heights of 3245 Christ's Hospital School boys born during 1927-1956 were measured twice termly from 9 to 20 years (n=129508). Treating the data as independent, the mean, standard deviation (SD), skewness, and kurtosis were calculated in 40 age groups and plotted as functions of age t. The data were also analyzed longitudinally using the nonlinear random-effects growth model H(t)=h(t-epsilon )+alpha, with H(t) the cross-sectional data, h(t) the individual mean curve, and epsilon and alpha subject-specific random effects reflecting variability in age and height at peak height velocity (PHV). Mean height increased monotonically with age, while the SD, skewness, and kurtosis changed cyclically with, respectively, 1, 2, and 3 turning points. Surprisingly, their age curves corresponded closely in shape to the first, second, and third derivatives of the mean height curve. The growth model expanded as a Taylor series in epsilon predicted such a pattern, and the longitudinal analysis showed that adjusting for age at PHV on a multiplicative scale largely removed the trends in the higher moments. A nonlinear growth process where subjects grow at different rates, such as in puberty, generates cyclical changes in the higher moments of the frequency distribution.

Joint models for a primary endpoint and multiple longitudinal covariate processes.

Joint models are formulated to investigate the association between a primary endpoint and features of multiple longitudinal processes. In particular, the subject-specific random effects in a multivariate linear random-effects model for multiple longitudinal processes are predictors in a generalized linear model for primary endpoints. Li, Zhang, and Davidian (2004, Biometrics60, 1-7) proposed an estimation procedure that makes no distributional assumption on the random effects but assumes independent within-subject measurement errors in the longitudinal covariate process. Based on an asymptotic bias analysis, we found that their estimators can be biased when random effects do not fully explain the within-subject correlations among longitudinal covariate measurements. Specifically, the existing procedure is fairly sensitive to the independent measurement error assumption. To overcome this limitation, we propose new estimation procedures that require neither a distributional or covariance structural assumption on covariate random effects nor an independence assumption on within-subject measurement errors. These new procedures are more flexible, readily cover scenarios that have multivariate longitudinal covariate processes, and can be implemented using available software. Through simulations and an analysis of data from a hypertension study, we evaluate and illustrate the numerical performances of the new estimators.

Sample Size Calculations Based on Generalized Estimating Equations for Population Pharmacokinetic Experiments

ABSTRACT We present a method for calculating the sample size of a pharmacokinetic study analyzed using a mixed effects model within a hypothesis testing framework. A sample size calculation method for repeated measurement data analyzed using generalized estimating equations has been modified for nonlinear models. The Wald test is used for hypothesis testing of pharmacokinetic parameters. A marginal model for the population pharmacokinetic is obtained by linearizing the structural model around the subject specific random effects. The proposed method is general in that it allows unequal allocation of subjects to the groups and accounts for situations where different blood sampling schedules are required in different groups of patients. The proposed method has been assessed using Monte Carlo simulations under a range of scenarios. NONMEM was used for simulations and data analysis and the results showed good agreement.