Random effects model

About: Random effects model is a research topic. Over the lifetime, 8388 publications have been published within this topic receiving 438823 citations. The topic is also known as: random effects & random effect.

Linear Mixed-Effects Model

Abstract: In Chap.10, we presented linear models (LMs) models with fixed effects for correlated data. They are examples of population-averaged models, because their mean-structure parameters can be interpreted as effects of covariates on the mean value of the dependent variable in the entire population. The association between the observations in a dataset was a result of a grouping of the observations sharing the same level of a grouping factor(s). In this chapter, we consider the analysis of continuous, hierarchical data using a different class of models, namely, linear mixed-effects models (LMMs). They allow to take into account the correlation of observations contained in a dataset. Moreover, they allow to effectively partition the overall variation of the dependent variable into components corresponding to different levels of data hierarchy. The models are examples of subject-specific models, because they include subject-specific coefficients. In particular, in Sects.13.2–13.4, we describe the formulation of the model. Sections13.5,13.6, and13.7 are devoted to, respectively, the estimation approaches, diagnostic tools, and inferential methods used for the LMMs, in which the (conditional) residual variance-covariance matrix is independent of the mean value. This is the most common type of LMMs used in practice. In Sect.13.8, we focus on the LMMs, in which the (conditional) residual variance-covariance matrix depends on the mean value. Section13.9 summarizes the contents of this chapter and offers some general concluding comments.

Some algebra and geometry for hierarchical models, applied to diagnostics

Abstract: Recent advances in computing make it practical to use complex hierarchical models. However, the complexity makes it difficult to see how features of the data determine the fitted model. This paper describes an approach to diagnostics for hierarchical models, specifically linear hierarchical models with additive normal or t-errors. The key is to express hierarchical models in the form of ordinary linear models by adding artificial `cases' to the data set corresponding to the higher levels of the hierarchy. The error term of this linear model is not homoscedastic, but its covariance structure is much simpler than that usually used in variance component or random effects models. The re-expression has several advantages. First, it is extremely general, covering dynamic linear models, random effect and mixed effect models, and pairwise difference models, among others. Second, it makes more explicit the geometry of hierarchical models, by analogy with the geometry of linear models. Third, the analogy with linear models provides a rich source of ideas for diagnostics for all the parts of hierarchical models. This paper gives diagnostics to examine candidate added variables, transformations, collinearity, case influence and residuals.

Meta-analysis of ordinal outcomes using individual patient data.

Abstract: Meta-analyses are being undertaken in an increasing diversity of diseases and conditions, some of which involve outcomes measured on an ordered categorical scale. We consider methodology for undertaking a meta-analysis on individual patient data for an ordinal response. The approach is based on the proportional odds model, in which the treatment effect is represented by the log-odds ratio. A general framework is proposed for fixed and random effect models. Tests of the validity of the various assumptions made in the meta-analysis models, such as a global test of the assumption of proportional odds between treatments, are presented. The combination of studies with different definitions or numbers of response categories is discussed. The methods are illustrated on two data sets, in a classical framework using SAS and MLn and in a Bayesian framework using BUGS. The relative merits of the three software packages for such meta-analyses are discussed. Copyright © 2001 John Wiley & Sons, Ltd.

A tutorial on frailty models.

Abstract: The hazard function plays a central role in survival analysis. In a homogeneous population, the distribution of the time to event, described by the hazard, is the same for each individual. Heterogeneity in the distributions can be accounted for by including covariates in a model for the hazard, for instance a proportional hazards model. In this model, individuals with the same value of the covariates will have the same distribution. It is natural to think that not all covariates that are thought to influence the distribution of the survival outcome are included in the model. This implies that there is unobserved heterogeneity; individuals with the same value of the covariates may have different distributions. One way of accounting for this unobserved heterogeneity is to include random effects in the model. In the context of hazard models for time to event outcomes, such random effects are called frailties, and the resulting models are called frailty models. In this tutorial, we study frailty models for survival outcomes. We illustrate how frailties induce selection of healthier individuals among survivors, and show how shared frailties can be used to model positively dependent survival outcomes in clustered data. The Laplace transform of the frailty distribution plays a central role in relating the hazards, conditional on the frailty, to hazards and survival functions observed in a population. Available software, mainly in R, will be discussed, and the use of frailty models is illustrated in two different applications, one on center effects and the other on recurrent events.