rdf.merMod: Approximate residual degrees of freedom

In GabrielHoffman/variancePartition: Quantify and interpret drivers of variation in multilevel gene expression experiments

Approximate residual degrees of freedom

Description

For a linear model with n samples and p covariates, RSS/\sigma^2 \sim \chi^2_{\nu} where \nu = n-p is the residual degrees of freedom. In the case of a linear mixed model, the distribution is no longer exactly a chi-square distribution, but can be approximated with a chi-square distribution.

Given the hat matrix, H, that maps between observed and fitted responses, the approximate residual degrees of freedom is \nu = tr((I-H)^T(I-H)). For a linear model, this simplifies to the well known form \nu = n - p. In the more general case, such as a linear mixed model, the original form simplifies only to n - 2tr(H) + tr(HH) and is an approximation rather than being exact. The third term here is quadratic time in the number of samples, n, and can be computationally expensive to evaluate for larger datasets. Here we develop a linear time algorithm that takes advantage of the fact that H is low rank.

H is computed as A^TA + B^TB for A=CL and B=CR defined in the code. Since A and B are low rank, there is no need to compute H directly. Instead, the terms tr(H) and tr(HH) can be computed using the eigen decompositions of AA^T and BB^T which is linear time in the number of samples.

Usage

rdf.merMod(model, method = c("linear", "quadratic"))

Arguments

model	An object of class merMod
method	Use algorithm that is "linear" (default) or quadratic time in the number of samples

Details

Compute the approximate residual degrees of freedom from a linear mixed model.

Value

residual degrees of freedom

See Also

rdf_from_matrices

Examples

library(lme4)

# Fit linear mixed model
fit <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)

# Evaluate the approximate residual degrees of freedom
rdf.merMod(fit)

GabrielHoffman/variancePartition documentation built on Nov. 8, 2024, 11:17 a.m.