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.

Bias and Efficiency of Meta-Analytic Variance Estimators in the Random-Effects Model:

Abstract: The meta-analytic random effects model assumes that the variability in effect size estimates drawn from a set of studies can be decomposed into two parts: heterogeneity due to random population effects and sampling variance. In this context, the usual goal is to estimate the central tendency and the amount of heterogeneity in the population effect sizes. The amount of heterogeneity in a set of effect sizes has implications regarding the interpretation of the meta-analytic findings and often serves as an indicator for the presence of potential moderator variables. Five population heterogeneity estimators were compared in this article analytically and via Monte Carlo simulations with respect to their bias and efficiency.

Extension of Nakagawa & Schielzeth's R2GLMM to random slopes models.

Abstract: Nakagawa & Schielzeth extended the widely used goodness-of-fit statistic R2 to apply to generalized linear mixed models (GLMMs). However, their R2GLMM method is restricted to models with the simplest random effects structure, known as random intercepts models. It is not applicable to another common random effects structure, random slopes models. I show that R2GLMM can be extended to random slopes models using a simple formula that is straightforward to implement in statistical software. This extension substantially widens the potential application of R2GLMM. Keywords: coefficient of determination, generalized linear mixed model, random slopes model, random regression Introduction The coefficient of determination, R2, is a widely used statistic for assessing the goodness-of-fit, on a scale from 0 to 1, of a linear regression model (LM). It is defined as the proportion of variance in the response variable that is explained by the explanatory variables or, equivalently, the proportional reduction in unexplained variance. Unexplained variance can be viewed as variance in model prediction error, so R2 can also be defined in terms of reduction in prediction error variance. Insofar as it is justifiable to make the leap from ‘prediction’ to ‘understanding’, R2 can be intuitively interpreted as a measure of how much better we understand a system once we have measured and modelled some of its components. R2 has been extended to apply to generalized linear models (GLMs) (Maddala 1983) and linear mixed effects models (LMMs) (Snijders & Bosker 1994) [reviewed by (Nakagawa & Schielzeth 2013)]. Nakagawa & Schielzeth (2013) proposed a further generalization of R2 to generalized linear mixed effects models (GLMMs), a useful advance given the ubiquity of GLMMs for data analysis in ecology and evolution (Bolker et al. 2009). A function to estimate this R2GLMM statistic, r.squaredGLMM, has been included in the MuMIn package (Barton 2014) for the R statistical software (R Core Team 2014). However, Nakagawa and Schielzeth's R2GLMM formula is applicable to only a subset of GLMMs known as random intercepts models. Random intercepts models are used to model clustered observations, for example, where multiple observations are taken on each of a sample of individuals. Correlations between clustered observations within individuals are accounted for by allowing each subject to have a different intercept representing the deviation of that subject from the global intercept. Random intercepts are typically modelled as being sampled from a normal distribution with mean zero and a variance parameter that is estimated from the data. Although random intercepts are probably the most popular random effects models in ecology and evolution, other random effect specifications are also common, in particular random slopes models, where not only the intercept but also the slope of the regression line is allowed to vary between individuals. Random intercepts and slopes are typically modelled as normally distributed deviations from the global intercept and slope, respectively. For example, random slopes models, under the name of ‘random regression’ models, are used to investigate individual variation in response to different environments (Nussey, Wilson & Brommer 2007). The aim of this article is to show how Nakagawa and Schielzeth's R2GLMM can be further extended to encompass random slopes models.

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.

Undue reliance on I(2) in assessing heterogeneity may mislead.

Abstract: The heterogeneity statistic I 2, interpreted as the percentage of variability due to heterogeneity between studies rather than sampling error, depends on precision, that is, the size of the studies included. Based on a real meta-analysis, we simulate artificially 'inflating' the sample size under the random effects model. For a given inflation factor M = 1, 2, 3,... and for each trial i, we create a M-inflated trial by drawing a treatment effect estimate from the random effects model, using /M as within-trial sampling variance. As precision increases, while estimates of the heterogeneity variance τ 2 remain unchanged on average, estimates of I 2 increase rapidly to nearly 100%. A similar phenomenon is apparent in a sample of 157 meta-analyses. When deciding whether or not to pool treatment estimates in a meta-analysis, the yard-stick should be the clinical relevance of any heterogeneity present. τ 2, rather than I 2, is the appropriate measure for this purpose.