An R² statistic for fixed effects in the linear mixed model

Abstract

SUMMARY Statisticians most often use the linear mixed model to analyze Gaussian longitudinal data. The value and familiarity of the R2 statistic in the linear univariate model naturally creates great interest in extending it to the linear mixed model. We define and describe how to compute a model R2 statistic for the linear mixed model by using only a single model. The proposed R2 statistic measures multivariate association between the repeated outcomes and the fixed effects in the linear mixed model. The R2 statistic arises as a 1–1 function of an appropriate F statistic for testing all fixed effects (except typically the intercept) in a full model. The statistic compares the full model to a null model with all fixed effects deleted (except typically the intercept) while retaining exactly the same covariance structure. Furthermore, the R2 statistic leads immediately to a natural definition of a partial R2 statistic. A mixed model in which ethnicity gives a very small p-value as a longitudinal predictor of blood pressure compellingly illustrates the value of the statistic. In sharp contrast to the extreme p-value, a very small R2, a measure of statistical and scientific importance, indicates that ethnicity has an almost negligible association with the repeated blood pressure outcomes for the study.

Metabolic responses to adrenocorticotropic hormone (ACTH) vary with life-history stage in adult male northern elephant seals

“…Mixed model linear regressions were used to assess relationships between AUC's. R2 for mixed models were calculated and regressions were visualized using the slope and intercept parameters from the mixed model [ 51 ]. Post-hoc differences between life history stages and time points were evaluated using Tukey's corrected differences in least square means.…”

Caffeinated Alcohol, Sensation Seeking, and Injury Risk

“…Model specification was based on the covariates used previously in O'Brien et al 14 (student gender, age, race, fraternity or sorority status, athlete status, typical number of drinks consumed). R-square statistics (R b 2 ) were estimated for these linear mixed models using the methods from Edwards et al 38 Multivariable mixed-effects logistic regression was used to assess whether the CaffAlc group was associated with alcohol-related consequences and injury outcomes, adjusting for student characteristics, within campus clustering, typical number of drinks consumed, and sensation seeking. A single mean scale score for sensation seeking using the BSSS-8 using the mean of all 8 items was used in all subsequent analyses (Cronbach's a = 0.81).…”

Isolation by resistance across a complex coral reef seascape

“…An alternative approach involves fitting mixed-effects models to account for the correlated structure of regression on distance matrices (maximum-likelihood population effects or MLPE model) [71]; however, parameter optimization is achieved using the restricted/residual maximum-likelihood procedure, raising doubts about the use of traditional information criteria such as AIC for model selection [72]. Summaries of 'variance explained' such as the R 2 b -value [73] have been proposed as suitable alternative statistics for model evaluation; however, several practical and theoretical issues remain, with an overall lack of consensus towards this approach [74]. The goal of our approach was to improve on the IBD model by including a measure of distance that reflected the physical seascape.…”