Why Use REML Instead of ML?

 

• In standard Maximum Likelihood (ML), we estimate both $\beta$ and $\mathrm{\Sigma }$ from the full data.
• In REML, we remove the influence of $\beta$ by transforming the data into residuals — the part of the data left after accounting for the fixed effects.

REML improves estimation by removing the influence of fixed effects from the likelihood. It does this by:

• Transforming the data into residuals,
• Building a likelihood function that depends only on the variance structure.

This leads to more accurate and reliable estimates of variance components, especially in small or unbalanced datasets.

Feature	Maximum Likelihood (ML)	Residual Maximum Likelihood (REML)
What it estimates	Estimates both fixed effects $\beta$ and variance components $\mathrm{\Sigma }$ together	Focuses on estimating variance components $\mathrm{\Sigma }$ only
Bias in variance estimates	Can be biased, especially in small samples, because it doesn't account for the uncertainty in estimating $\beta$	Provides unbiased estimates of variance components by adjusting for the estimation of $\beta$
Likelihood based on	Full data (including fixed effects)	Residuals — the part of the data independent of $\beta$
Use case	Useful when comparing models with different fixed effects	Preferred when comparing models with the same fixed effects but different random structures