What is the difference between using random intercepts and slopes instead of separate regressions per subject? I have recorded a DV and IV of 20 participants. The IV is a repeated measure, and my goal is to see how variation in the IV can explain variations in the DV. More specifically, I want a beta coefficient for each participant.
My first thought was to set up a linear mixed effects model with random intercept and random slopes for each subject. But then I asked myself: Why can't I just run 20 separate ordinary linear regressions (only fixed effects)? 
Would I get the same beta coefficients with these two methods? And if not, where would be the difference?
 A: There are two major differences, related to each other.

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*Running separate regressions for each subject takes up many more degrees of freedom, since you have an intercept and slope to estimate for every subject.


*Mixed models make use of partial pooling; random effects are shrunk towards the mean. This basically means that the data from other subjects informs your best estimate of the parameters for any particular subject. If you fit your regressions separately (or via fixed effects for each subject), you will likely get more extreme values than if you make use of random effects. Note that the utility of this approach relies on the assumption that the random effects are drawn from a normal distribution, though I believe that it is robust to deviations from it. This is generally a reasonable assumption, but it could be useful to consider whether it is likely to be true in your case. If you have good reasonable to believe a different distribution would be better for your case, you could specify this in a Bayesian hierarchical model instead of a standard mixed model.
