PROC GENMMOD: Interpreting Gamma Regression (dist-GAMMA, link=LOG)

[mepidemiology]

Hi there! I am struggling to interpret my SAS output for gamma regression using PROC GENMOD, where the dist=GAMMA and the link=LOG. My dependent and independent variables (and other covariates) are continuous. The dependent variable is measured in weeks.

I am stuck because in the "Contrast Estimate Results" table I get a mean estimate of 1.01, and I'm not sure how to interpret it because when I convert the dependent variable to days (so I can interpret a larger beta estimate), I get an identical mean estimate of 1.01, and the only thing that changes is my intercept.

With that in mind, is the mean estimate not interpreted as the change in y for every unit change in x? I've seen mixed responses and based on my identical answer despite changing the units of y, I wanted to inquire. Thank you!

[StatDave]

You need to show the GENMOD statements that you submitted - including your ESTIMATE statement. I assume that the second run uses identical code except possibly a different response variable name for the variable that multiplies the week-based response variable by 7, yes?

[mepidemiology]

Hi! Yes, that is correct, the code is identical except the variable for days is that of weeks multiplied by 7. I apologize - here are the statements:

proc genmod data=dataset;
model age_days = exposure covariate1 covariate2 covariate3  /dist=GAMMA link=log;
estimate "Age, days with exposure" exposure 1;
ods output estimates = day_exp;
run;

proc genmod data=dataset;
model age_wks = exposure covariate1 covariate2 covariate3  /dist=GAMMA link=log;
estimate "Age, weeks with exposure" exposure 1;
ods output estimates = week_exp;
run;

[StatDave]

Algebraically, the model (just considering the exposure) is
log(y)=b0 + b1*x.
If you multiply y by 7, then the model is
log(7y) = log(7)+log(y) = b0 + b1*x
which can be written
log(y)=(b0-log(7)) + b1*x
So, all that happens is that the intercept gets decreased by log(7).

[mepidemiology]

Thank you! I will look deeper into this to wrap my head around it, but it makes sense. Thanks!

[mepidemiology]

Thank you again! Quick follow up - given this, how would I interpret Beta for days and then weeks? For instance a b1-unit increase in x is associated with ## change in y (in days), and in the other model, a b1-unit increase in x is associated with a ## change in y (weeks). Do I need the intercepts?

[StatDave]

The log response changes by b1 for each unit change in the predictor - whatever the predictor's units.