Fit methods in LME and GLME
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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.