Making a REML option

[fhui28]

Can we offer this option like what glmmTMB does? This should be relatively simple to do (in principle!!), because all it requires is adding any fixed effect beta coefficients in the X component as random effects. This then gets TMB to integrate over these as well, thus leading to an integrated likelihood approach to REML for LVMs.

This makes the biggest difference for Gaussian responses, we know REML should need to better (less biased) estimates of any variance component/loading matrices. For non-Gaussian responses, things are less clear, but it can often be beneficial (https://arxiv.org/abs/2402.12719)

Well I say it is easy to implement in principle, but with fourth corner models and all that jazz I am not as positive.

[BertvanderVeen]

What is the difference between a REML argument and explicitly adding covariates as random effects? Standard error calculation and degrees of freedom?

[fhui28]

• In a computational sense: integrated likelihood REML is the same as assuming a flat prior on the fixed effects coefficients, or assuming random effects by setting the variance of the random effects to be infinity.
• In a philosophical sense: the goal of random effects is to induce correlations, account for sampling design etc; the goal of integrated likelihood REML is only about getting less biased estimates of the variance components/loadings etc.

So will this have effects on standard error calculation and degrees of freedom? Yes e.g., actual it is possible for REML estimates to have larger standard errors than ML estimates, and sometimes that is price to pay for less bias. But these are secondary concerns relative to the differences in philosophical aims above.