How to present and compare mixed models vs. regular linear regression models?

Question

The regular linear regression model is easy to present; we are minimizing the sum of the least square errors: $$\sum_{i=1}^{n} ( y_i - \hat{y}_i)^2$$

How do we present the mixed model? More specifically, we are wondering about how to present the following simple model:

lmefit = lmer(MathAch ~ SES + (1 |School) , MathScores)

We need to explain/present the model to some one who are not of statistical background. Our questions are:

(a) What objective function are we minimizing here?

We don't want to show that
$$y = X \beta + Zb + e$$

Because this is not very intuitive, especially the $Zb$ part.Is there a way that we could intuitively see what we are doing via the mixed-model approach?

(b) Is there a presentation-friendly way of comparing the regular linear regression vs. the mixed model above? Hopefully we can visualize them intuitively.

(c) What sorts of model-checking and post-fit-diagnosis could we show for mixed-model? What are the typical goodness-of-fit checking procedures?

(d) How do we show the benefit of using mixed model vs. regular linear regression? Hopefully there are some visualization or intuitive arguments that we can put into one slide?

Could anybody please shed some lights to us? We also appreciate any good books/resources one this.

Thanks a lot!

[c.p. on R-mixed list in case no one sees our questions... Thank you!]

Answer 1

The random effects are effects that we can view as being pick from a population of possible effects. In clincial trials we often have multicenter studies. In analyzing the study results we often use a mixed model wher the centers are random effects. This makes sense because we have a large population of potential sites out of which we selected the few that are in the study. They are not really selected at random but we do want to incorporate the variability associated with the choice of sites. Another choice would probably give different results. So it is much more realistic to think of the site effect as random rather than fixed because the fixed effect assumption would mean that if say we selected 10 sites the effect would not change if we picked a different set of 10. I think this gives some insight in layman's terms to what random effects are. in other applications it may not be so clear.