Fit methods in LME and GLME

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Alexis on 9 Jun 2020
Edited: Alexis on 10 Jun 2020
There is little guidance or explanation in any of the documentation for choosing the appropriate fit method when using fitlme or fitglme. The terminology also differs from the other mainly used implementations (in R, SAS, etc.), making it difficult to ascertain what exactly MATLAB's approach even is or how closely results should reflect those produced by other software.
Can anyone here explain in which situations, for example, I would use 'ApproximateLaplace' fit method over 'Laplace'?

the cyclist on 9 Jun 2020
I'm not an expert in this type of modeling. However, this section of the documentation for fitglme states that the difference is whether or not the fixed effects are profiled out.
As explained on the wikipedia page for likelihood functions, profile likelihoods are a way of eliminating nuisance parameters.
If you are more accustomed to using R, maybe reading something like the documentation for the ProfileLikelihood R package might be more enlightening.
Laplace approximations to likelihood functions are pretty common, from what I recall or can glean from the internet.
All in all, I'm not sure which specific terminology you are finding non-standard, but I hope this helps point you in the right direction.
Alexis on 10 Jun 2020
Thanks for responding. I think what I am unclear on is whether Laplace and Apx. Laplace are equivalent to ML and restricted ML, since these are what I am accustomed to seeing everywhere else.
Edit: so it's clear what my goals are, I am concerned about a small sample of level-2 (random effect) grouping variable, so I would like to use a fit method that is relatively conservative and less likely to underestimate variance components.
In the literature, from my understanding, I should be using both restricted ML and adjusted degrees of freedom (kenward-rogers). Unfortunatley, it doesn't look like matlab supports adjusting DF for small samples in GLME.
Edit again: I just found this https://uk.mathworks.com/help/stats/estimating-parameters-in-linear-mixed-effects-models.html?s_tid=srchtitle It explains ML and REML in LME (I think I understand why the 'pseduo' only applies to generalized case for non-normal data). But Laplace/apx. Laplace are not described.

R2015b

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