# Interpretation Beta coefficient regression gamma distribution

I am currently working on a panel data model of 30 companies over 10 years where the dependent variable is a score (decimal bounded between 0 and 1, continuous) while the independent are dummies and their lags. I ran different models on Stata, but the only one fitting my data is generalized estimating equations (GEE) with family Gamma and link reciprocal, for which I get significant result.

My question is: how do I interpret the coefficients? they are very big (eg -21, 18) and I know I can't interpret them as in the linear regression.

• You may request the program to exponentiate the beta coefficient in Gamma regression if your analysis program had that option, the exponentiated beta coefficient is interpreted just like a risk rate, or literally a rate intepreted just like an odds ratio. – MOHAMMAD AL-KHATEEB Nov 23 '20 at 17:22

The documentation in xtxtgee stata is quite specific as to what this is.

The reciprocal link is regression with the target (A.K.A., dependent variable linked to) being $\frac{1}{y}$, as contrasted to the usual $y$. In that software, the family(gamma) has link(reciprocal) as its default. The gamma referred to appears to be the gamma distribution PDF, which in stata has the form $$\operatorname{gammaden}(a,b,g,x)= \frac{(x-g)^{a-1} e^{-\frac{x-g}{b}}}{\Gamma (a)b^a}\,, \quad x>0 \,,$$ which is often used setting $g=0$ to become a two parameter distribution.

Now since $a>0$ by definition, I think $a=-21$ says something is very wrong. It might help your cause if you were to show some I/O. It would not be too unusual to write the gamma distribution parameters as $\beta$ and $\theta$ but I cannot confirms this without more information.

The reason I say that this is probably the gamma distribution is because the table in the xtxtgee file lists

family                  Description
______________________________________________________________________
gaussian                Gaussian (normal); family(normal) is a synonym
igaussian               inverse Gaussian
binomial [# | varname]  Bernoulli/binomial
poisson                 Poisson
nbinomial [#]           negative binomial
gamma                   gamma


That is, everything else in the table besides "gamma" is a probability distribution.

• Dear @Carl , thank you very much for your answer. So, in the GEE with gamma distribution and reciprocal link all the regression beta coefficients should be greater than zero? And what would be the interpretation? Would it be as in normal linear regression, ie. one unit increase in X1 leads to Beta1 increase in the dependent variable? From my results my regression beta coefficients are both positive and negative and are big, they oscillate between -21 to +18 depending on the independent variable. Thank you very much in advance! – Davide L Mar 26 '17 at 11:42
• Dear @Carl I just noticed that probably I have not presented my question in the right way: I am interested in understanding the interpretation of the Beta coefficient in a regression where I use GEE family(gamma) link(reciprocal), not in estimating the two parameters of the Gamma function. May you please help me out with that? Thank you!! – Davide L Mar 26 '17 at 14:06
• @DavideL If you show a plot of the fit, and the values in table form, it would be a lot easier to comment. Otherwise, I am just reading stata documentation, which has me somewhat at a disadvantage (although slight) since I do not use that particular program, so that I cannot test my guesses as to what they mean when the documentation is inexact. – Carl Mar 27 '17 at 4:06
• @DavideL Can't be absolutely sure but what you have is probably not the gamma function, $\Gamma (a)$, nor is it likely to be the incomplete upper gamma function, symbolized $\Gamma (a,b)$. Rather, from the context it is likely the two parameter gamma distribution density function. – Carl Mar 27 '17 at 4:50
• Thank you very much! I used in the end a simple Gaussian as my variable di not have excessive skewness to justify a gamma – Davide L Mar 30 '17 at 20:17