Compute fraction of variation attributable to each variable in regression model. Also interpretable as the intra-class correlation after correcting for all other variables in the model.
Usage
calcVarPart(fit, returnFractions = TRUE, ...)
# S4 method for lm
calcVarPart(fit, returnFractions = TRUE, ...)
# S4 method for lmerMod
calcVarPart(fit, returnFractions = TRUE, ...)
# S4 method for glm
calcVarPart(fit, returnFractions = TRUE, ...)
# S4 method for negbin
calcVarPart(fit, returnFractions = TRUE, ...)
# S4 method for glmerMod
calcVarPart(fit, returnFractions = TRUE, ...)Arguments
- fit
model fit from lm() or lmer()
- returnFractions
default: TRUE. If TRUE return fractions that sum to 1. Else return unscaled variance components.
- ...
additional arguments (not currently used)
Details
For linear model, variance fractions are computed based on the sum of squares explained by each component. For the linear mixed model, the variance fractions are computed by variance component estimates for random effects and sum of squares for fixed effects.
For a generalized linear model, the variance fraction also includes the contribution of the link function so that fractions are reported on the linear (i.e. link) scale rather than the observed (i.e. response) scale. For linear regression with an identity link, fractions are the same on both scales. But for logit or probit links, the fractions are not well defined on the observed scale due to the transformation imposed by the link function.
The variance implied by the link function is the variance of the corresponding distribution:
logit -> logistic distribution -> variance is pi^2/3
probit -> standard normal distribution -> variance is 1
For the Poisson distribution with rate \(\lambda\), the variance is \(log(1 + 1/\lambda)\).
For the negative binomial distribution with rate \(\lambda\) and shape \(\theta\), the variance is \(log(1 + 1/\lambda + 1/\theta)\).
Variance decomposition is reviewed by Nakagawa and Schielzeth (2012), and expanded to other GLMs by Nakagawa, Johnson and Schielzeth (2017). See McKelvey and Zavoina (1975) for early work on applying to GLMs. Also see DeMaris (2002)
We note that Nagelkerke's pseudo R^2 evaluates the variance explained by the full model. Instead, a variance partitioning approach evaluates the variance explained by each term in the model, so that the sum of each systematic plus random term sums to 1 (Hoffman and Schadt, 2016; Nakagawa and Schielzeth, 2012).
References
Nakagawa S, Johnson PC, Schielzeth H (2017). “The coefficient of determination R 2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded.” Journal of the Royal Society Interface, 14(134), 20170213. Nakagawa S, Schielzeth H (2013). “A general and simple method for obtaining R2 from generalized linear mixed-effects models.” Methods in ecology and evolution, 4(2), 133--142. McKelvey RD, Zavoina W (1975). “A statistical model for the analysis of ordinal level dependent variables.” Journal of mathematical sociology, 4(1), 103--120. DeMaris A (2002). “Explained variance in logistic regression: A Monte Carlo study of proposed measures.” Sociological Methods & Research, 31(1), 27--74. Hoffman GE, Schadt EE (2016). “variancePartition: interpreting drivers of variation in complex gene expression studies.” BMC bioinformatics, 17(1), 1--13.
Examples
library(lme4)
data(varPartData)
# Linear mixed model
fit <- lmer(geneExpr[1, ] ~ (1 | Tissue) + Age, info)
calcVarPart(fit)
#> Tissue Age Residuals
#> 0.08672790 0.00093212 0.91233998
# Linear model
# Note that the two models produce slightly different results
# This is expected: they are different statistical estimates
# of the same underlying value
fit <- lm(geneExpr[1, ] ~ Tissue + Age, info)
calcVarPart(fit)
#> Tissue Age Residuals
#> 0.08080435 0.00101471 0.91818094