Empirical decomposition of the explained variation in the variance components form of the mixed model

Abstract

The coefficient of determination is a standard characteristic in linear models. It is widely used to assess the proportion of variation explained, to determine the goodness-of-fit and to compare models with different covariates. However, there has not been an agreement on a similar quantity for the class of linear mixed models yet. We introduce a natural extension of the well-known adjusted coefficient of determination in linear models to the variance components form of the linear mixed model. We propose a novel coefficient of determination which is dimensionless, has an intuitive and simple definition in terms of variance explained, is additive for several random effects and reduces to the adjusted coefficient of determination in the linear model. To this end, we prove a full decomposition of the sum of squares of the independent variable into the explained and residual variance. Based on the restricted maximum likelihood equations, we introduce a novel measure for the explained variation which we allocate specifically to the contribution of the fixed and the random covariates of the model. We illustrate on two genomic datasets that the novel coefficient of determination can be applied as an improved estimator of the heritability of complex traits. Exemplarily, we allocate the explained variation to the five chromosomes of the model plant Arabidopsis thaliana and determine the contribution of each chromosome as well as crosscorrelations between the chromosomes to the trait heritability.

Spectral organ fingerprints for intraoperative tissue classification with hyperspectral imaging

“…The analysis was performed at first for all organs ( Figure 3 ) and subsequently stratified by organ ( Figure 4 ). In the analysis for all organs, at each wavelength the proportion of explained variation [11] in observed reflectance was decomposed into the components “organ”, “pig”, “angle”, “image” and “repetition”, where “angle” describes the proportion of variation explained by the angle between the organ surface and the camera optical axis, “image” describes the proportion of variation explained by different measurements taken from different organ positions in the same individual or variations in the annotated areas, and “repetition” describes the proportion of explained variation by multiple recordings of the same image under identical measurement conditions.…”

“…Independent linear mixed models were used for an explained variation analysis in order to evaluate the effect of the influencing factors on changes in the spectrum. The (proportion of) explained variance was obtained using the empirical decomposition of the explained variation in the variance components form of the mixed model [11].…”

“…The proportion of variability in reflectance explained by each factor was derived as in [11]. "Repetition" depicts the residual variability, which is here the within image variability (i.e.…”

“…For the second approach with stratification by organ, independent linear mixed models were fitted for each organ and wavelength with fixed effects for "angle" as well as a random effect for "pig" and "image", i.e for each organ and wavelength the same model as given above was fitted excluding covariate "organ". The explained variation of each factor was depicted [11]. "Repetition" depicts the residual variation, which is here the within image variability (i.e.…”

Spectral Organ Fingerprints For Machine Learning-Based Intraoperative Tissue Classification With Hyperspectral Imaging

“…The analysis was performed at rst for all organs (Figure 3) and subsequently strati ed by organ (Figure 4). In the analysis for all organs, at each wavelength the proportion of explained variation 11 in observed re ectance was decomposed into the components "organ", "pig", "angle", "image" and "repetition", where "angle" describes the proportion of variation explained by the angle between the organ surface and the camera optical axis, "image" describes the proportion of variation explained by different measurements taken from different organ positions in the same individual or variations in the annotated areas, and "repetition" describes the proportion of explained variation by multiple recordings of the same image under identical measurement conditions.…”