# When no model comparison, should I use REML vs ML?

I'm running LMM, and I will make no comparison of models. Could I ask which one should I use between REML and ML?

• When there is no model comparison, the difference between restricted (or residual) maximum likelihood (REML) and maximum likelihood (ML) is that, REML can give you unbiased estimates of the variance parameters. Recap that, ML estimates for variance has a term $1/n$, but the unbiased estimate should be $1/(n-p)$, where $n$ is the sample size, $p$ is the number of mean parameters. So REML should be used when you are interested in variance estimates and $n$ is not big enough as compared to $p$.
• When there is model comparison, notice that REML cannot be used to compare mean models, since REML transformes the data thus makes the likelihood incomparable.

The restricted maximum likelihood (REML, aka RML) procedure separates the estimation of fixed and random parameters (Raudenbush & Bryk, 2002; Searle, Casella & McCulloch, 1992). Snijders and Bosker (2012) noted that when J-q-1 is equal or larger than 50 (J is the number of clusters and q is the number of level-2 predictors), the difference between ML and REML estimates are negligible. If J-q-1 is smaller than 50; ML estimates of the variance components are biased, generally downward (Hox, 2010).

Raudenbush and Bryk (2002) posit that ML estimates for level-2 variances and covariances will be smaller than REML by a factor of approximately (J-F)/J where F is the total number of regression coefficients. To my experience, this adjustment works satisfactorily.

However, when deviance tests are the choice to compare models with different fixed effects but the same variance components, the REML deviance should not be used because it is a deviance for the variance components only. Instead the ML deviance should be used (Snijders & Bosker, 2012). If the models differ in both the fixed effects and the variance components, neither deviance can be used to conduct tests on the fixed effects.

• would you provide a reference to better understand the following: "However, when deviance tests are the choice to compare models with different fixed effects but the same variance components, the REML deviance should not be used because it is a deviance for the variance components only. Instead the ML deviance should be used (Snijders & Bosker, 2012). If the models differ in both the fixed effects and the variance components, neither deviance can be used to conduct tests on the fixed effects." I am facing exactly this problem, as JASP outputs Deviance, REML Deviance, and AIC, which is clearly Apr 24, 2021 at 9:39