Compute variance statistics

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 class 'lm'
calcVarPart(fit, returnFractions = TRUE, ...)

# S4 method for class 'lmerMod'
calcVarPart(fit, returnFractions = TRUE, ...)

# S4 method for class 'glm'
calcVarPart(fit, returnFractions = TRUE, ...)

# S4 method for class 'negbin'
calcVarPart(fit, returnFractions = TRUE, ...)

# S4 method for class '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)

Value

fraction of variance explained / ICC for each variable in the regression model

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

nakagawa2017coefficientvariancePartition nakagawa2013generalvariancePartition mckelvey1975statisticalvariancePartition demaris2002explainedvariancePartition hoffman2016variancepartitionvariancePartition

Examples

library(lme4)
data(varPartData)

# Linear mixed model
fit <- lmer(geneExpr[1, ] ~ (1 | Tissue) + Age, info)
calcVarPart(fit)
#>     Tissue        Age  Residuals 
#> 0.08672786 0.00093212 0.91234002 

# 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