# Unexpected estimate in Gamma GLM summary output

I have a question. How on earth is it possible to have a negative estimate for a form of a nominal variable (two forms: "HM" and "LM") when it should be positive? I'm modelling a positive continuous variable (rarefied species richness) by using a single nominal predictor. See the output summary below and the box plot. How is this even possible?

P.S. This is my first gamma glm ever

Call:
glm(formula = Rarified.Richness ~ Macrophyte, family = "Gamma",
data = beysehir.diversity.indices.macr)

Deviance Residuals:
Min        1Q    Median        3Q       Max
-0.39143  -0.09652  -0.00911   0.07579   0.43107

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   0.082461   0.006693  12.320 2.28e-07 ***
MacrophyteLM -0.024028   0.010597  -2.267   0.0468 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for Gamma family taken to be 0.05929918)

Null deviance: 0.87647  on 11  degrees of freedom
Residual deviance: 0.59465  on 10  degrees of freedom
AIC: 65.46

Number of Fisher Scoring iterations: 4


"LM" does have a higher mean value for the response. Shouldn't it has a positive estimate then?

P.S. KS-test and chisq tests both report a good fit for this model.

BONUS QUESTION: Do you think there is any other problem with this model?

• I’m guessing that you’re using the inverse link function here, which is maybe not what you intended. Nov 5, 2018 at 17:10
• Inverse link is canonical for the gamma family. If you want to use log or identity link, you need to specify that explicitly. Nov 5, 2018 at 17:12
• Thank you all for your answers. It worked. I thought log link was the default one but it was not. Problem solved. Nov 5, 2018 at 19:23

You’re using the inverse link function here, which is maybe not what you intended. That explains why you get a negative coefficient for LM (inverse of a smaller linear predictor yields a larger expected value).

Inverse link is canonical for the gamma family. If you want to use log or identity link, you need to specify that explicitly.