Understanding the mixed effect model thru creating my own optimization-based model...

Hi all,

Following Macro's suggestion, I am creating a new thread for my questions. (It's somehow related to the original question,

which had been discussed here:

Understanding the linear mixed effects model equation and fitting a random effects model with weights in R

Specifically, I am trying to understand a random intercept model in the form of y=Xbeta + Zb + eps.

(Following Macro's notation for y_i, I stack the group formulas together vertically and obtained one big formula for the whole dataset with all groups).

My questions are:

a. In a random intercept model (all group sharing the same beta), will the following situation arise? All data points in a group reside on one side (the same side) of the regression line of that group?

Here is my thinking: even though for a certain group, the overall beta is a very crappy one for that group. By adjusting the intercept for that group, I at least still can get the regression line passing the cloud, am I right?

Here by "passing the cloud" I mean not all points will reside on the same side of the regression line.

b. In a random intercept model, for each group, which point does the regression line pass thru? In the general OLS, we knew that it's the point (xbar, ybar) that is passed thru by the regression line. But for each group in a random intercept model, which point is that "central" point?

c. Moreover, in R, what's a convenient way to visualize what's happening within a certain group in a mixed model?

d. How do I understand the b's? What are these b's?

May I say they are simply the ybar_i - ybar, i.e. the vertical axis of the group centers - the vertical axis of the whole-data center?

e. Why does LME explicitly impose that the sum of b's =0 ?

f. To understand the LME models better, I am creating my own toy models and doing some comparative studies:

To cook my own model, I decide to solve the following least-square problem:

minimize sum(w1kb_k^2)+sum(w2ieps_i^2)

here b_k: k from 1 to K, where K is the number of groups. eps_i: i from 1 to N, where N is the number of individuals.

w1 is a set of weights on the individuals and w2 is a set of weights on the groups at the group level.

My purpose is to create a model that resembles the random intercept model, but ditching the Gaussian distribution assumption for the b's.

And I would like to have the regression line in each group(sharing the same beta) passing the data cloud for that group...

(Please ref. Q1 above).

Do you think my model makes sense?

Could you please help me critique my model vs. the LME random intercept model?

Shall I add more constraints such as sum(b_k)=0 and sum(eps_i)=0?

Thanks a lot for your help!