# Using R for GLM with Gamma distribution

I currently have a problem understanding the syntax for R for fitting a GLM using the Gamma distribution.

I have a set of data, where each row contains 3 co-variates ($X_1, X_2, X_3$), a response variable ($Y$), and a shape parameter ($K$). I want to model the scale of the Gamma distribution as a linear function of the 3 covariates, but I don’t understand how to set the shape of the distribution to $K$ for each row of data.

A situation that I think is analogous is that for a binomial distribution, the GLM requires that the number of trials ($N$) is known for each data entry.

The usual gamma GLM contains the assumption that the shape parameter is constant, in the same way that the normal linear model assumes constant variance.

In GLM parlance the dispersion parameter, $\phi$ in $\text{Var}(Y_i)=\phi\text{V}(\mu_i)$ is normally constant.

More generally, you have $a(\phi)$, but that doesn’t help.

It might perhaps be possible to use a weighted Gamma GLM to incorporate this effect of a specified shape parameter, but I haven’t investigated this possibility yet (if it works it is probably the easiest way to do it, but I am not at all sure that it will).

If you had a double GLM you could estimate that parameter as a function of covariates… and if the double glm software let you specify an offset in the variance term you could do this. It looks like the function dglm in the package dglm let you specify an offset. I don’t know if it will let you specify a variance model like (say) ~ offset(<something>) + 0 though.

Another alternative would be to maximize the likelihood directly.

> y <- rgamma(100,10,.1)

> summary(glm(y~1,family=Gamma))

Call:
glm(formula = y ~ 1, family = Gamma)

Deviance Residuals:
Min        1Q    Median        3Q       Max
-0.93768  -0.25371  -0.05188   0.16078   0.81347

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.0103660  0.0003486   29.74   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for Gamma family taken to be 0.1130783)

Null deviance: 11.223  on 99  degrees of freedom
Residual deviance: 11.223  on 99  degrees of freedom
AIC: 973.56

Number of Fisher Scoring iterations: 5


The line where it says:

   (Dispersion parameter for Gamma family taken to be 0.1130783)


is the one you want.

That $\hat\phi$ is related to the shape parameter of the Gamma.