Fluorite | Level 6

## PROC GLIMMIX - QUASI LIKELIHOOD REGRESSION

Running a quasi-likelihood beta regression but having difficulty with the intercept term that comes out in my regressions...

Dependent variable can only take on values between 0.10-1; dependent variable is also left-skewed and there is a spike at 1. So using quasi-likelihood regressions with a beta distribution.

The intercept term comes out above 1.30 when I run my models - which doesn't make sense. Wondering if there is something wrong with my code:

PROC GLIMMIX DATA = TEST;
MODEL QOL = income age ...

/LINK = LOGIT S DIST = BETA;
RANDOM _RESIDUAL_;
WHERE QOL GT 0;
weight wtfa_sa;
RUN;

Any suggestions greatly appreciated.

3 REPLIES 3

## Re: PROC GLIMMIX - QUASI LIKELIHOOD REGRESSION

Well, the intercept would be the logit of the value if all of your IV values were zero.  Putting the value back on the original scale, I get intercept (orig scale) = exp(1.3)/(1+exp(1.3)) = 0.7858.  Is that close to what you would expect?

Steve Denham

Fluorite | Level 6

## Re: PROC GLIMMIX - QUASI LIKELIHOOD REGRESSION

Thanks! That is closer to what I am expecting. But would that change the interpretation of the coefficients of the variables that I seek to analyze?

Here is some of my output (in this model the intercept was 1.20, others it's 1.3 or ranges between):

 INTERCEPT 1.2091 0.005849 137E3 206.73 <.0001 LESS_HIGH 0.003781 0.006926 137E3 0.55 0.5852 SOME_COLLEGE 0.02738 0.007094 137E3 3.86 0.0001 INCOME_CAT 0.003528 0.006746 137E3 0.52 0.6010 FEMALE  0.05704 0.005573 137E3 10.24 <.0001 26042 93.2452 . . . 1.3039 . . . .

Say looking at the 'female' dummy variable (coeffiicent is next to the variable name)...would the interpretation be like that of an OLS model - being female leads to a 0.06 higher quality of life? Or would there be a need to transform as you have done with the intercept term?

I do believe dist = beta is the right choice, but am concerned with the interpretation of the coefficients.

## Re: PROC GLIMMIX - QUASI LIKELIHOOD REGRESSION

The coefficient implies that a one unit change in the variable leads to a change in the logit equal to the coefficient, with all other values at their mean.  To calculate a change on the original scale, you would need to plug the coefficients times the mean values for all other variables, and then calculate the logit for female=0 and female=1, using that coefficient.  Then apply the inverse transformation for the logit to get the value on the original scale.  The coefficients can't be interpreted exactly as those in an OLS regression because of the logistic link function implied by the selection of the beta distribution.

Steve Denham

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