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# Interpretation of Coefficient - GLM with Gamma Link

I have a model that requires a GLM with a log link and gamma distribution. The dependent variable is continuous and the independent variables are all dummies. The code ran for the procedure is:

PROC GENMOD DATA = TEST;

CLASS SEX_CAT BLACK_NH ASIAN_NH HISPANIC;

MODEL PRICE = SEX_CAT BLACK_NH ASIAN_NH HISPANIC / DIST = GAMMA LINK = LOG TYPE1;

WEIGHT SURVEY_WEIGHT;

RUN;

Output is as follows:

``` Intercept               1     6.9972     0.0513     6.8966     7.0978      18588.4       <.0001
SEX_CAT            0    1    -0.2171     0.0152    -0.2469    -0.1872       203.23       <.0001
SEX_CAT            1    0     0.0000     0.0000     0.0000     0.0000          .          .
BLACK_NH           0    1     0.2042     0.0226     0.1599     0.2484        81.84       <.0001
BLACK_NH           1    0     0.0000     0.0000     0.0000     0.0000          .          .
ASIAN_NH           0    1     0.7420     0.0347     0.6740     0.8100       457.63       <.0001
ASIAN_NH           1    0     0.0000     0.0000     0.0000     0.0000          .          .
HISPANIC_NH        0    1     0.7626     0.0200     0.7234     0.8018      1451.48       <.0001
HISPANIC_NH        1    0     0.0000     0.0000     0.0000     0.0000          .          .
Scale                   1     0.3002 ```

How would one interpret the coefficients of sex_cat or black_nh? From another SAS message board I read that:

With a log link and a continuous predictor, you are fitting the model:

ln(mu) = beta0 + beta1*X,

where mu is the expected value. Then, e raised to the left and right sides gives:

mu = exp(beta0 + beta1*X) = exp(beta0)*exp(beta1*X)

So would the interpretation for sex_cat be: exp(6.9972)(exp(-0.2171)?

Additionally, sex_cat eq 1 is female and sex_cat eq 0 equals male - so would the negative be associated with the males in comparison to females? Meaning exp(6.9972) is female, and exp(6.9972)*exp(-0.2171) is for males?

Regular Contributor
Posts: 162

## Re: Interpretation of Coefficient - GLM with Gamma Link

i would guess it's just an estimate statement with the exp option as follows

estimate 'Exp(sex_cat)' sex_cat 1 / exp cl ;

but make sure the results make sense

--------------
blog: papersandprograms.com
SAS Employee
Posts: 386

## Re: Interpretation of Coefficient - GLM with Gamma Link

[ Edited ]

Since the log function is monotonically increasing, you can interpret the parameter signs without thinking in terms of the exponentiated model. That is, the negative parameter for Sex=0 (male) indicates that being male decreases the mean response. If you need an estimate of the effect, then use the LSMEANS statement with the ILINK option:  lsmeans sex / ilink;

BTW, the name of your weight variable indicates that you are attempting to analyze survey data. In general, a proper analysis can only be done using the SURVEY procedures (SURVEYMEANS, SURVEYREG, etc.) since procedures like GENMOD do not incorporate the necessary variance estimators. Unfortunately, there is no SURVEY procedure for fitting a log-linked gamma model.

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