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# Assumptions of mixed model poisson regression

Hello all,

In advance, please let me thank whoever reads this entire post.  I have been working very hard to understand and build this model, and stats are not my strength.  Any assistance you can provide in answering these few remaining questions I have will be greatly appreciated.

I am using a mixed model poisson regression to model my data (because I have random effects), and have been working with another statistician to make sure that is the correct model.  However, I still have a few BASIC but not completely resolved questions that I am hoping someone can help me with.

Mostly, I am not entirely certain what ALL of the model assumptions are.  I KNOW that I need to check for overdispersion, and have been using the negative binomial model when that is the case.  However, I am not sure what other assumptions I need to validate.  Namely:

1) Do the model residuals need to be normally distributed?

2) I have read that for mixed models in general, the random effects are assumed to be randomly distributed.  Is this true for poisson as well and how is this assessed in SAS (I am using Proc GLIMMIX)?  I am not necessarily asking for complete code, just a general approach.

3) Assumptions of equal variance:  Is this an assumption of this model ? (please read more detailed questions below)

a. Do I need to look for random scatter when I plot the overall model residuals against the linear predictor values.

b.  Do I need to look for random scatter when I plot the residuals against the individual predictor values (for each independent variable.

c.  If A and/or B are true, what can/should I do when I see a cone shaped pattern indicating homoscedasticity.

THANK YOU!!

Meghan

Posts: 2,655

## Re: Assumptions of mixed model poisson regression

1.  One of the assumptions under a mixed model with a Poisson distribution (without explicit overdispersion fitting) is that the mean and variance are equal.  Consequently, there is really no "residual" in the classic linear model sense.  There is a deviation, but there is no IID residual term that you assume to be normally distributed.

2.  Random effects are assumed to have N(0, Sigma) distributions.  SAS does not test this.  If you really want to get into distributional testing of random effects, I suggest you start with Bates and Pinheiro's work and then pursue all of this in one of the dozen or so R packages (which all seem to give different results) for generalized linear mixed models.

3.  Recall that the variance and the mean are directly related, so cone-shaped plots of the deviance should be expected: As the mean increases, so does the variance.  If you are concerned about equal variance amongst groups, this can be tested using the COVTEST option.  For instance, if your code has:

random _residual_;

You could try:

random _residual_/group=<fixed_effect_of_interest>;

covtest homogeneity;

The results will tell you something important about the deviance by group.

I hope this attracts some comments from and .  Their insights on this are really helpful.

Also, track down a copy of Walt Stroup's Generalized Linear Mixed Models. It will make the assumptions much clearer than what I did here.

Steve Denham

Message was edited by: Steve Denham

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