Say we're interested in how student exam grades are affected by the number of hours that those students study. We sample students from several different schools. We run the following mixed effects model: $$ \text{exam.grades}_i = a + \beta_1 \times \text{hours.studied}_i + \text{school}_j + e_i $$

Am I right in saying that, in this model, each school is assumed to have been picked from a larger population of schools, and that the effect of school is normally distributed? Therefore, can we do all the 'usual' normal distribution-type procedures for the group effect of school? Can we say things like 68% of schools will be within 1 standard deviation of mean group effect of school? And can we calculate a 95% confidence interval for the overall mean group effect of school?

Am I also right in saying that linear regression with fixed effect of school cannot calculate these normal distribution statistics because they use a reference group and dummy variables?


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You are correct in saying that in standard linear mixed effects models, the random effects are assumed to be normally distributed. Thus, if this assumption holds (at least approximately), we can use what we know about normal distributions to help describe the distribution of the random effects, such as 95% of the random effects should be within two standard deviations of 0 (since random effects are centered around 0).

That being said, it's important to check these assumptions, and it's not always so easy! If you have a good deal of data about each cluster, you can do something like a stratafied analysis and plot the confidence intervals for each cluster. This can still be a little difficult; suppose you have one extreme outlier, i.e. a tight confidence interval several standard deviations away from 0. Is this because this random effect is really huge and we are very certain about this? Or is this because we don't have a lot of data about this random effect and we've underestimated the variance due to small sample size?

As for the difference between simple linear regression and mixed effects models, the answer is that the mixed effects model is considerably more complicated. The random effects are assumed to have all been generated from the same (typically normal) distribution. As such, the estimate of a random effect is actually pulled in toward 0 (remember that random effects are centered at 0) compared if you had just fit a simple linear regression model with all fixed effects.

Also, another difference is that the random effects are fixed to have mean 0, allowing full identifiability of the model: if you tried to fit the main effect AND all the random effects in a simple linear model, your model would not be identifiable. This is because adding 1 to the main effect and subtracting 1 from the "random" effects (quotes used because you would be fitting them as fixed effects) would lead to the exact same predicted values. This issue is not so important though: one could easily just exclude the main effect from the model, and then if we were interested in examining the main effect, we would just take the average of all the "random" effects. However, as noted above, the estimated "random" effects would be much noisier than if they had been fit by a mixed effects model: the estimate would be based only on that cluster's information, rather than also borrowing off the information provided about the distribution of cluster effects.


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