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

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!]

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Well, why exactly are you using a mixed model? If you can explain that to your audience, this could go a long way in terms of providing intuition for the audience. –  Macro May 31 '12 at 12:24