5 Relationship between mixed model conditional modes (aka BLUPS) and OLS estimates

Last time I introduced the idea of regularization and how we can see the regularization if we extract conditional modes from mixed models. In this video I show the exact relationship between these two things, which is instrumental in assessing whether the two stage summary statistics approach will differ greatly from a full mixed model.

*NEW* (12/31/2020) Bookdown version of R materials. This video's chapter can be found here:

[jonahfoong]

This was EXACTLY what I was looking for: an intuitive explanation to how BLUPs of MLMs are derived with examples and none of the complicated jargon in textbooks. Thank you sooo much!!!!

[joshwillingham]

lme4::ranef documentation says it's extracting modes. The lme4 vignette says in the description for the `fitted` function, "Fitted values given conditional modes" and for the `coef` function, "Sum of the random and fixed effects for each level" all of which is confusing. Michael Clark's Mixed Models With R has a section that calls the `coef` output random intercepts and slopes which are the intercept or slope + random effects which he says is the output of `ranef`. The vignette implies that the conditional modes are either the random intercepts/slopes or the random effects and this series describes them as the fitted values. So, in conclusion — ¯\_(ツ)_/¯.

[andrelopes]

Hi, please could you show an example on how to calculate the BLUPS for random intercept and slope model? What is the formula and how do you derive it?

[tcdizz]

1) Is weighting by the number of observations pretty standard? There are some recommendations for applying weight based on variance (people with more variability are given a lower weight) - but it doesn't seem like there is much guidance on what should be standard practice in the field (not mentioned explicitly in the recent Meteyard & Davies 2020 paper). 2) I've run a 2SSS on a sample that didn't converge as a mixed model (so it's not simple to pull G or sigma2). I'd like to determine the weights for each participant, but I'm not entirely sure how to compute a single w/n subject variance term for all subjects, especially since I have multiple predictors. Is there guidance on estimating a single variance term in 2SSS, when mixed models are not available? I was trying to go back to basics, by computing variances, beta's etc. via their formulas (e.g., (X'X)^-1*X'Y for beta). - but got stuck trying to determine how to compute a single term for all participants.