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Quartz | Level 8

## generalized linear mixed model 'proc glimmix'

Hello,

I appreciate for any guidance that help me understand how to check the unconditional (marginal) distribution which is assumed to be known for maximum likelihood estimation in generalized linear mixed model (proc glimmix) for multilevel repeated measures data with random effect. If I'm right one assumption for proc glimmix is that the random effect variables should follow normal distribution, so does the variable that has random effect play the role of (marginal) distribution ? Is there any article or any source that explains how to check the normality of marginal distribution and assumptions in proc glimmix and what is the concept of marginal distribution in generalized linear mixed model ?

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SAS Super FREQ

## Re: generalized linear mixed model 'proc glimmix'

The first sentence of that article says that the discussion is about "how to check the assumptions of an ordinary least squares linear regression model." Some of the discussion is relevant to the fixed effects in generalized linear models.

> The variables in a least squares regression model do not have to be normally distributed. Do the variables here mean independent variables and dependent variable ( response )?

Yes, that is correct. Neither the explanatory variables nor the response variable is required to be normal in OLS. As discussed in the article, the residuals of the model are assumed to be normal

> Does this rule apply to mixed model( proc mixed and proc glimmix )

There is not a requirement that any of the fixed-effect variables (or response variable) in mixed models be normal. The estimation method does not matter.  Generalized models do not require the normality of residuals.

The random effects in mixed models are assumed to be normally distributed with mean zero and a correctly specified covariance  matrix. We usually estimate the variance-covariance components.

You can search the internet for details about the model assumptions for linear mixed models. Some of the course notes from professors (such as Haelwig's Linear Mixed-Effects Regression (umn.edu)) are good resources.

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Jade | Level 19

## Re: generalized linear mixed model 'proc glimmix'

Marginal: population averaged

Conditional: subject 'averaged"

Marginal distribution: In GLIMMIX, this is specified as an R-side distribution of the residuals. It can be whatever you choose from the available distributions. The variance components are assumed to be normal.

Conditional distribution: In GLIMMIX, this is specified as a G-side distribution.  Again, you may choose an appropriate distribution, but take care that it is appropriate at the subject level.

The Introduction to Mixed Models in the SAS/STAT documentation ought to help get you started.

SteveDenham

SAS Super FREQ

## Re: generalized linear mixed model 'proc glimmix'

@SteveDenham can correct me if I'm wrong, but I don't see many people trying to "verify the assumptions" for random effects. The normality assumptions for random effects can't be checked as easily as the assumptions for fixed effects. One reason is that the random effects are often "groups" such as schools or hospitals, and there might only be a dozen groups. Another is that the model will be more heavily influenced by violations in the assumptions of the fixed effects, and those are rarely satisfied exactly (outside of simulation studies).

For a discussion of random effects and what happens if they do not satisfy the assumptions, see Schielzeth et all (2020). The paper concludes that "Our simulation analysis shows that the effect of violations of distributional assumptions of random effect variances and residuals is surprisingly small." They conclude that "mixed-effects models are largely robust even to quite severe violations of model assumptions."

Quartz | Level 8

## Re: generalized linear mixed model 'proc glimmix'

Thank you all for your brilliant thoughts and responses,
according to this article " https://blogs.sas.com/content/iml/2018/08/27/on-the-assumptions-and-misconceptions-of-linear-regress... " ,The variables in a least squares regression model do not have to be normally distributed. Do the variables here mean independent variables and dependent variable ( response )?  Dose this rule apply to mixed model( proc mixed and proc glimmix ) also, since mixed model is multilevel regression lines but estimation method can be maximum likelihood or (restricted) maximum likelihood?

SAS Super FREQ

## Re: generalized linear mixed model 'proc glimmix'

The first sentence of that article says that the discussion is about "how to check the assumptions of an ordinary least squares linear regression model." Some of the discussion is relevant to the fixed effects in generalized linear models.

> The variables in a least squares regression model do not have to be normally distributed. Do the variables here mean independent variables and dependent variable ( response )?

Yes, that is correct. Neither the explanatory variables nor the response variable is required to be normal in OLS. As discussed in the article, the residuals of the model are assumed to be normal

> Does this rule apply to mixed model( proc mixed and proc glimmix )

There is not a requirement that any of the fixed-effect variables (or response variable) in mixed models be normal. The estimation method does not matter.  Generalized models do not require the normality of residuals.

The random effects in mixed models are assumed to be normally distributed with mean zero and a correctly specified covariance  matrix. We usually estimate the variance-covariance components.

You can search the internet for details about the model assumptions for linear mixed models. Some of the course notes from professors (such as Haelwig's Linear Mixed-Effects Regression (umn.edu)) are good resources.

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