Showing papers on "Random effects model published in 2000"

Methods for Meta-Analysis in Medical Research

Abstract: PART A: META-ANALYSIS METHODOLOGY: THE BASICS Introduction: Meta-analysis: Its Development and Uses Defining Outcome Measures used for Combining via Meta-analysis Random Effects Models for Combining Study Estimates Exploring Between Study Heterogeneity Publication Bias Study Quality Sensitivity Analysis Reporting the Results of a Meta-analysis Fixed Effects Methods for Combining Study Estimates PART B: ADVANCED AND SPECIALIZED META-ANALYSIS TOPICS Bayesian Methods in Meta-analysis Meta Regression Meta-analysis of Different Types of Data Incorporating Study Quality into a Meta-analysis Meta-analysis of Multiple and Correlated Outcome Measures Meta-analysis of Epidemiological and other Observational Studies Generalised Synthesis of Evidence - Combining Different Sources of Evidence Meta-analysis of Survival Data Cumulative Meta-analysis Miscellaneous and Developing Areas of Applications in Meta-Analysis Appendix I: Software Used for the Examples in this Book

Modelling covariance structure in the analysis of repeated measures data.

Abstract: The term 'repeated measures' refers to data with multiple observations on the same sampling unit. In most cases, the multiple observations are taken over time, but they could be over space. It is usually plausible to assume that observations on the same unit are correlated. Hence, statistical analysis of repeated measures data must address the issue of covariation between measures on the same unit. Until recently, analysis techniques available in computer software only offered the user limited and inadequate choices. One choice was to ignore covariance structure and make invalid assumptions. Another was to avoid the covariance structure issue by analysing transformed data or making adjustments to otherwise inadequate analyses. Ignoring covariance structure may result in erroneous inference, and avoiding it may result in inefficient inference. Recently available mixed model methodology permits the covariance structure to be incorporated into the statistical model. The MIXED procedure of the SAS((R)) System provides a rich selection of covariance structures through the RANDOM and REPEATED statements. Modelling the covariance structure is a major hurdle in the use of PROC MIXED. However, once the covariance structure is modelled, inference about fixed effects proceeds essentially as when using PROC GLM. An example from the pharmaceutical industry is used to illustrate how to choose a covariance structure. The example also illustrates the effects of choice of covariance structure on tests and estimates of fixed effects. In many situations, estimates of linear combinations are invariant with respect to covariance structure, yet standard errors of the estimates may still depend on the covariance structure.

Fixed Effects vs. Random Effects Meta‐Analysis Models: Implications for Cumulative Research Knowledge

Abstract: Research conclusions in the social sciences are increasingly based on meta-analysis, making questions of the accuracy of meta-analysis critical to the integrity of the base of cumulative knowledge. Both fixed effects (FE) and random effects (RE) meta-analysis models have been used widely in published meta-analyses. This article shows that FE models typically manifest a substantial Type I bias in significance tests for mean effect sizes and for moderator variables (interactions), while RE models do not. Likewise, FE models, but not RE models, yield confidence intervals for mean effect sizes that are narrower than their nominal width, thereby overstating the degree of precision in meta-analysis findings. This article demonstrates analytically that these biases in FE procedures are large enough to create serious distortions in conclusions about cumulative knowledge in the research literature. We therefore recommend that RE methods routinely be employed in meta-analysis in preference to FE methods.

Estimation of Disease Rates in Small Areas: A new Mixed Model for Spatial Dependence

Abstract: In this paper, a new model is proposed for spatial dependence that includes separate parameters for overdispersion and the strength of spatial dependence The new dependence structure is incorporated into a generalized linear mixed model useful for the estimation of disease incidence rates in small geographic regions The mixed model allows for log-linear covariate adjustment and local smoothing of rates through estimation of the spatially correlated random effects Computer simulation studies compare the new model with the following sub-models: intrinsic autoregression, an independence model, and a model with no random effects The major finding was that regression coefficient estimates based on fitting intrinsic autoregression to independent data can have very low precision compared with estimates based on the full model Additional simulation studies demonstrate that penalized quasi-likelihood (PQL) estimation generally performs very well although the estimates are slightly biased for very small counts

Interpreting Parameters in the Logistic Regression Model with Random Effects

Abstract: Logistic regression with random effects is used to study the relationship between explanatory variables and a binary outcome in cases with nonindependent outcomes. In this paper, we examine in detail the interpretation of both fixed effects and random effects parameters. As heterogeneity measures, the random effects parameters included in the model are not easily interpreted. We discuss different alternative measures of heterogeneity and suggest using a median odds ratio measure that is a function of the original random effects parameters. The measure allows a simple interpretation, in terms of well-known odds ratios, that greatly facilitates communication between the data analyst and the subject-matter researcher. Three examples from different subject areas, mainly taken from our own experience, serve to motivate and illustrate different aspects of parameter interpretation in these models.

Maximum Likelihood for Generalized Linear Models with Nested Random Effects via High-Order, Multivariate Laplace Approximation

Abstract: Nested random effects models are often used to represent similar processes occurring in each of many clusters. Suppose that, given cluster-specific random effects b, the data y are distributed according to f(y|b, Θ), while b follows a density p(b|Θ). Likelihood inference requires maximization of ∫ f(y|b, Θ)p(b|Θdb with respect to Θ. Evaluation of this integral often proves difficult, making likelihood inference difficult to obtain. We propose a multivariate Taylor series approximation of the log of the integrand that can be made as accurate as desired if the integrand and all its partial derivatives with respect to b are continuous in the neighborhood of the posterior mode of b|Θ,y. We then apply a Laplace approximation to the integral and maximize the approximate integrated likelihood via Fisher scoring. We develop computational formulas that implement this approach for two-level generalized linear models with canonical link and multivariate normal random effects. A comparison with approximation...

Estimation of multivariate frailty models using penalized partial likelihood.

Abstract: Summary. There exists a growing literature on the estimation of gamma distributed multiplicative shared frailty models. There is, however, often a need to model more complicated frailty structures, but attempts to extend gamma frailties run into complications. Motivated by hip replacement data with a more complicated dependence structure, we propose a model based on multiplicative frailties with a multivariate log-normal joint distribution. We give a justification and an estimation procedure for this generally structured frailty model, which is a generalization of the one presented by McGilchrist (1993, Biometrics49, 221-225). The estimation is based on Laplace approximation of the likelihood function. This leads to estimating equations based on a penalized fixed effects partial likelihood, where the marginal distribution of the frailty terms determines the penalty term. The tuning parameters of the penalty function, i.e., the frailty variances, are estimated by maximizing an approximate profile likelihood. The performance of the approximation is evaluated by simulation, and the frailty model is fitted to the hip replacement data.

Heterogeneity and statistical significance in meta-analysis: an empirical study of 125 meta-analyses.

Abstract: For meta-analysis, substantial uncertainty remains about the most appropriate statistical methods for combining the results of separate trials. An important issue for meta-analysis is how to incorporate heterogeneity, defined as variation among the results of individual trials beyond that expected from chance, into summary estimates of treatment effect. Another consideration is which 'metric' to use to measure treatment effect; for trials with binary outcomes, there are several possible metrics, including the odds ratio (a relative measure) and risk difference (an absolute measure). To examine empirically how assessment of treatment effect and heterogeneity may differ when different methods are utilized, we studied 125 meta-analyses representative of those performed by clinical investigators. There was no meta-analysis in which the summary risk difference and odds ratio were discrepant to the extent that one indicated significant benefit while the other indicated significant harm. Further, for most meta-analyses, summary odds ratios and risk differences agreed in statistical significance, leading to similar conclusions about whether treatments affected outcome. Heterogeneity was common regardless of whether treatment effects were measured by odds ratios or risk differences. However, risk differences usually displayed more heterogeneity than odds ratios. Random effects estimates, which incorporate heterogeneity, tended to be less precisely estimated than fixed effects estimates. We present two exceptions to these observations, which derive from the weights assigned to individual trial estimates. We discuss the implications of these findings for selection of a metric for meta-analysis and incorporation of heterogeneity into summary estimates. Published in 2000 by John Wiley & Sons, Ltd.

Marginalized Multilevel Models and Likelihood Inference

Abstract: Hierarchical or ‘‘multilevel’’ regression models typically parameterize the mean response conditional on unobserved latent variables or ‘‘random’’ effects and then make simple assumptions regarding their distribution. The interpretation of a regression parameter in such a model is the change in possibly transformed mean response per unit change in a particular predictor having controlled for all conditioning variables including the random effects. An often overlooked limitation of the conditional formulation for nonlinear models is that the interpretation of regression coefficients and their estimates can be highly sensitive to difficult-to-verify assumptions about the distribution of random effects, particularly the dependence of the latent variable distribution on covariates. In this article, we present an alternative parameterization for the multilevel model in which the marginal mean, rather than the conditional mean given random effects, is regressed on covariates. The impact of random effects model violations on the marginal and more traditional conditional parameters is compared through calculation of asymptotic relative biases. A simple two-level example from a study of teratogenicity is presented where the binomial overdispersion depends on the binary treatment assignment and greatly influences likelihood-based estimates of the treatment effect in the conditional model. A second example considers a three-level structure where attitudes toward abortion over time are correlated with person and district level covariates. We observe that regression parameters in conditionally specified models are more sensitive to random effects assumptions than their counterparts in the marginal formulation.

A multilevel model framework for meta-analysis of clinical trials with binary outcomes

Abstract: In this paper we explore the potential of multilevel models for meta-analysis of trials with binary outcomes for both summary data, such as log-odds ratios, and individual patient data. Conventional fixed effect and random effects models are put into a multilevel model framework, which provides maximum likelihood or restricted maximum likelihood estimation. To exemplify the methods, we use the results from 22 trials to prevent respiratory tract infections; we also make comparisons with a second example data set comprising fewer trials. Within summary data methods, confidence intervals for the overall treatment effect and for the between-trial variance may be derived from likelihood based methods or a parametric bootstrap as well as from Wald methods; the bootstrap intervals are preferred because they relax the assumptions required by the other two methods. When modelling individual patient data, a bias corrected bootstrap may be used to provide unbiased estimation and correctly located confidence intervals; this method is particularly valuable for the between-trial variance. The trial effects may be modelled as either fixed or random within individual data models, and we discuss the corresponding assumptions and implications. If random trial effects are used, the covariance between these and the random treatment effects should be included; the resulting model is equivalent to a bivariate approach to meta-analysis. Having implemented these techniques, the flexibility of multilevel modelling may be exploited in facilitating extensions to standard meta-analysis methods.

Proportional hazards model with random effects.

Abstract: We propose a general proportional hazards model with random effects for handling clustered survival data. This generalizes the usual frailty model by allowing a multivariate random effect with arbitrary design matrix in the log relative risk, in a way similar to the modelling of random effects in linear, generalized linear and non-linear mixed models. The distribution of the random effects is generally assumed to be multivariate normal, but other (preferably symmetrical) distributions are also possible. Maximum likelihood estimates of the regression parameters, the variance components and the baseline hazard function are obtained via the EM algorithm. The E-step of the algorithm involves computation of the conditional expectations of functions of the random effects, for which we use Markov chain Monte Carlo (MCMC) methods. Approximate variances of the estimates are computed by Louis' formula, and posterior expectations and variances of the individual random effects can be obtained as a by-product of the estimation. The inference procedure is exemplified on two data sets.

Summarizing the goodness of fit of generalized linear models for longitudinal data

Abstract: This paper extends four goodness-of-fit measures of a generalized linear model (GLM) to random effects and marginal models for longitudinal data. The four measures are the proportional reduction in entropy measure, the proportional reduction in deviance measure, the concordance correlation coefficient and the concordance index. The extended measures satisfy the basic requirements for measures of association. Two examples illustrate their use in model selection.

Random-Effects Modeling of Categorical Response Data

Abstract: In many applications observations have some type of clustering, with observations within clusters tending to be correlated. A common instance of this occurs when each subject in the sample undergoes repeated measurement, in which case a cluster consists of the set of observations for the subject. One approach to modeling clustered data introduces cluster-level random effects into the model. The use of random effects in linear models for normal responses is well established. By contrast, random effects have only recently seen much use in models for categorical data. This chapter surveys a variety of potential social science applications of random effects modeling of categorical data. Applications discussed include repeated measurement for binary or ordinal responses, shrinkage to improve multiparameter estimation of a set of proportions or rates, multivariate latent variable modeling, hierarchically structured modeling, and cluster sampling. The models discussed belong to the class of generalized linear mixed models (GLMMs), an extension of ordinary linear models that permits nonnormal response variables and both fixed and random effects in the predictor term. The models are GLMMs for either binomial or Poisson response variables, although we also present extensions to multicategory (nominal or ordinal) responses. We also summarize some of the technical issues of model-fitting that complicate the fitting of GLMMs even with existing software.

Bayesian inference for finite mixtures of generalized linear models with random effects

Abstract: We present an hierarchical Bayes approach to modeling parameter heterogeneity in generalized linear models. The model assumes that there are relevant subpopulations and that within each subpopulation the individual-level regression coefficients have a multivariate normal distribution. However, class membership is not known a priori, so the heterogeneity in the regression coefficients becomes a finite mixture of normal distributions. This approach combines the flexibility of semiparametric, latent class models that assume common parameters for each sub-population and the parsimony of random effects models that assume normal distributions for the regression parameters. The number of subpopulations is selected to maximize the posterior probability of the model being true. Simulations are presented which document the performance of the methodology for synthetic data with known heterogeneity and number of sub-populations. An application is presented concerning preferences for various aspects of personal computers.

Analysis of left-censored longitudinal data with application to viral load in HIV infection.

Abstract: The classical model for the analysis of progression of markers in HIV-infected patients is the mixed effects linear model. However, longitudinal studies of viral load are complicated by left censoring of the measures due to a lower quantification limit. We propose a full likelihood approach to estimate parameters from the linear mixed effects model for left-censored Gaussian data. For each subject, the contribution to the likelihood is the product of the density for the vector of the completely observed outcome and of the conditional distribution function of the vector of the censored outcome, given the observed outcomes. Values of the distribution function were computed by numerical integration. The maximization is performed by a combination of the Simplex algorithm and the Marquardt algorithm. Subject-specific deviations and random effects are estimated by modified empirical Bayes replacing censored measures by their conditional expectations given the data. A simulation study showed that the proposed estimators are less biased than those obtained by imputing the quantification limit to censored data. Moreover, for models with complex covariance structures, they are less biased than Monte Carlo expectation maximization (MCEM) estimators developed by Hughes (1999) Mixed effects models with censored data with application to HIV RNA Levels. Biometrics 55, 625-629. The method was then applied to the data of the ALBI-ANRS 070 clinical trial for which HIV-1 RNA levels were measured with an ultrasensitive assay (quantification limit 50 copies/ml). Using the proposed method, estimates obtained with data artificially censored at 500 copies/ml were close to those obtained with the real data set.

Using a New Statistical Model for Testlets to Score TOEFL

Abstract: Standard item response theory (IRT) models fit to examination responses ignore the fact that sets of items (testlets) often are matched with a single common stimulus (e.g., a reading comprehension passage). In this setting, all items given to an examinee are unlikely to be conditionally independent (given examinee proficiency). Models that assume conditional independence will overestimate the precision with which examinee proficiency is measured. Overstatement of precision may lead to inaccurate inferences as well as prematurely ended examinations in which the stopping rule is based on the estimated standard error of examinee proficiency (e.g., an adaptive test). The standard three parameter IRT model was modified to include an additional random effect for items nested within the same testlet (Wainer, Bradlow, & Du, 2000). This parameter, γ characterizes the amount of local dependence in a testlet. We fit 86 TOEFL testlets (50 reading comprehension and 36 listening comprehension) with the new model, and obtained a value for the variance of γ for each testlet. We compared the standard parameters (discrimination (a), difficulty (b) and guessing (c)) with what is obtained through traditional modeling. We found that difficulties were well estimated either way, but estimates of both a and c were biased if conditional independence is incorrectly assumed. Of greater import, we found that test information was substantially over-estimated when conditional independence was incorrectly assumed.

Analysis of a cluster randomized trial with binary outcome data using a multi‐level model

Abstract: The use of multi-level logistic regression models was explored for the analysis of data from a cluster randomized trial investigating whether a training programme for general practitioners' reception staff could improve women's attendance at breast screening. Twenty-six general practices were randomized with women nested within them, requiring a two-level model which allowed for between-practice variability. Comparisons were made with fixed effect (FE) and random effects (RE) cluster summary statistic methods, ordinary logistic regression and a marginal model based on generalized estimating equations with robust variance estimates. An FE summary statistic method and ordinary logistic regression considerably understated the variance of the intervention effect, thus overstating its statistical significance. The marginal model produced a higher statistical significance for the intervention effect compared to that obtained from the RE summary statistic method and the multi-level model. Because there was only a moderate number of practices and these had unbalanced cluster sizes, reliable asymptotic properties for the robust standard errors used in the marginal model may not have been achieved. While the RE summary statistic method cannot handle multiple covariates easily, marginal and multi-level models can do so. In contrast to multi-level models however, marginal models do not provide direct estimates of variance components, but treat these as nuisance parameters. Estimates of the variance components were of particular interest in this example. Additionally, parametric bootstrap methods within the multi-level model framework provide confidence intervals for these variance components, as well as a confidence interval for the effect of intervention which allows for the imprecision in the estimated variance components. The assumption of normality of the random effects can be checked, and the models extended to investigate multiple sources of variability.

Random-effects regression analysis of correlated grouped-time survival data.

Abstract: Random-effects regression modelling is proposed for analysis of correlated grouped-time survival data Two analysis approaches are considered The first treats survival time as an ordinal outcome, which is either right-censored or not The second approach treats survival time as a set of dichotomous indicators of whether the event occurred for time periods up to the period of the event or censor For either approach both proportional hazards and proportional odds versions of the random-effects model are developed, while partial proportional hazards and odds generalizations are described for the latter approach For estimation, a full-information maximum marginal likelihood solution is implemented using numerical quadrature to integrate over the distribution of multiple random effects The quadrature solution allows some flexibility in the choice of distributions for the random effects; both normal and rectangular distributions are considered in this article An analysis of a dataset where students are clustered within schools is used to illustrate features of random-effects analysis of clustered grouped-time survival data

A Structural Modeling Approach to a Multilevel Random Coefficients Model

Abstract: A method for estimating the random coefficients model using covariance structure modeling is presented. This method allows one to estimate both fixed and random effects. A way of translating the general linear mixed model into a structural equation modeling (SEM) format is presented. In particular, a LISREL setup for the multiple group linear latent growth curve model is illustrated with suggestions on ways to parameterize more complex models. To illustrate the procedure, we apply the method to both simulated and real data. The method is shown to recover the simulated parameter values. Results and interpretation for the Belsky and Rovine (1990) marriage data are presented. Other applications of the more general model are suggested.

Modeling Repeated Count Data Subject to Informative Dropout

Abstract: Summary. In certain diseases, outcome is the number of morbid events over the course of follow-up. In epilepsy, e.g., daily seizure counts are often used to reflect disease severity. Follow-up of patients in clinical trials of such diseases is often subject to censoring due to patients dying or dropping out. If the sicker patients tend to be censored in such trials, estimates of the treatment effect that do not incorporate the censoring process may be misleading. We extend the shared random effects approach of Wu and Carroll (1988, Biometrics44, 175–188) to the setting of repeated counts of events. Three strategies are developed. The first is a likelihood-based approach for jointly modeling the count and censoring processes. A shared random effect is incorporated to introduce dependence between the two processes. The second is a likelihood-based approach that conditions on the dropout times in adjusting for informative dropout. The third is a generalized estimating equations (GEE) approach, which also conditions on the dropout times but makes fewer assumptions about the distribution of the count process. Estimation procedures for each of the approaches are discussed, and the approaches are applied to data from an epilepsy clinical trial. A simulation study is also conducted to compare the various approaches. Through analyses and simulations, we demonstrate the flexibility of the likelihood-based conditional model for analyzing data from the epilepsy trial.

Mixed model analysis of quantitative trait loci

Abstract: We develop a mixed model approach of quantitative trait locus (QTL) mapping for a hybrid population derived from the crosses of two or more distinguished outbred populations. Under the mixed model, we treat the mean allelic value of each source population as the fixed effect and the allelic deviations from the mean as random effects so that we can partition the total genetic variance into between- and within-population variances. Statistical inference of the QTL parameters is obtained by using the Bayesian method implemented by Markov chain Monte Carlo (MCMC). This unified QTL mapping algorithm treats the fixed and random model approaches as special cases of the general mixed model methodology. Utility and flexibility of the method are demonstrated by using a set of simulated data.

Random regression models for human immunodeficiency virus ribonucleic acid data subject to left censoring and informative drop‐outs

Abstract: Summary. Objectives in many longitudinal studies of individuals infected with the human immunodeficiency virus (HIV) include the estimation of population average trajectories of HIV ribonucleic acid (RNA) over time and tests for differences in trajectory across subgroups. Special features that are often inherent in the underlying data include a tendency for some HIV RNA levels to be below an assay detection limit, and for individuals with high initial levels or high rates of change to drop out of the study early because of illness or death. We develop a likelihood for the observed data that incorporates both of these features. Informative drop-outs are handled by means of an approach previously published by Schluchter. Using data from the HIV Epidemiology Research Study, we implement a maximum likelihood procedure to estimate initial HIV RNA levels and slopes within a population, compare these parameters across subgroups of HIV-infected women and illustrate the importance of appropriate treatment of left censoring and informative drop-outs. We also assess model assumptions and consider the prediction of random intercepts and slopes in this setting. The results suggest that marked bias in estimates of fixed effects, variance components and standard errors in the analysis of HIV RNA data might be avoided by the use of methods like those illustrated.

Diagnostic test analyses in search of their gold standard: latent class analyses with random effects

Abstract: We review methods for analysing the performance of several diagnostic tests when patients must be classified as having a disease or not, when no gold standard is available. For latent class analysis (LCA) to provide consistent estimates of sensitivity, specificity and prevalence, traditionally 'independent errors conditional on disease status' have been assumed. Recent approaches derive estimators under more flexible assumptions. However, all likelihood-based approaches suffer from the sparseness of tables generated by this type of data; an issue which is often ignored. In light of this, we examine the potential and limitations of LCAs of diagnostic tests. We are guided by a data set of visceral leishmaniasis tests. In the example, LCA estimates suggest that the traditional reference test, parasitology, has poor sensitivity and underestimates prevalence. From a technical standpoint, including more test results in one analysis yields increasing degrees of sparseness in the table which are seen to lead to discordant values of asymptotically equivalent test statistics and eventually lack of convergence of the LCA algorithm. We suggest some strategies to cope with this.

Methods and systems for plant performance analysis

Abstract: Method and systems for assessing wide area performance of a crop variety using a linear mixed model that incorporates geologic statistical components and includes parameters for fixed effects, random effects and covariance (J20) A wide area database is constructed that includes covariate data, spatial coordinates of testing locations of one or more crop varieties, geographic areas in which the testing location reside, and performance trait values of one or more crop varieties (H25, H30) The parameters for the fixed effects, random effects and covariances are estimated by fitting the linear mixed model with data in the wide area database Long term expected performance of the crop variety may be estimated using the parameter estimates Average performance of the crop variety for a given time period may be predicted using the parameter estimates

Variance components analysis for pedigree-based censored survival data using generalized linear mixed models (GLMMs) and Gibbs sampling in BUGS.

Abstract: Complex human diseases are an increasingly important focus of genetic research. Many of the determinants of these diseases are unknown and there is often a strong residual covariance between relatives even when all known genetic and environmental factors have been taken into account. This must be modeled correctly whether scientific interest is focused on fixed effects, as in an association analysis, or on the covariance structure itself. Analysis is straightforward for multivariate normally distributed traits, but difficulties arise with other types of trait. Generalized linear mixed models (GLMMs) offer a potentially unifying approach to analysis for many classes of phenotype including right censored survival times. This includes age-at-onset and age-at-death data and a variety of other censored traits. Markov chain Monte Carlo (MCMC) methods, including Gibbs sampling, provide a convenient framework within which such GLMMs may be fitted. In this paper, we use BUGS ("Bayesian inference using Gibbs sampling": a readily available, generic Gibbs sampler) to fit GLMMs for right-censored survival times in nuclear and extended families. We discuss parameter interpretation and statistical inference, and show how to circumvent a number of important theoretical and practical problems. Using simulated data, we show that model parameters are consistent. We further illustrate our methods using data from an ongoing cohort study. Finally, we propose that the random effects associated with a genetic component of variance (e.g., sigma(2)(A)) in a GLMM may be regarded as an adjusted "phenotype" and used as input to a conventional model-based or model-free linkage analysis. This provides a simple way to conduct a linkage analysis for a trait reflected in a right-censored survival time while comprehensively adjusting for observed confounders at the level of the individual and latent environmental effects shared across families.