GLM with Gamma(link = "log") in R

Question

I have the output (shown below) from the GLM with Gamma(link = "log"). The outcome (dependent variable) is strictly greater than 0, and the group variable (predictor) is binary (either 0 or 1).

In this case, is it right to conclude as follows?

• Group 1 reduces the mean outcome by a factor of exp(-0.04) = 0.96.
  
• The expected mean ratio of Group 1 to Group 0 is 0.96.

Looking forward to hearing from you!!

Call:
glm(formula = Outcome ~ group, family = Gamma(link = "log"), 
    data = d2)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-0.49019  -0.21677  -0.11818   0.02478   0.96391  

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.85327    0.04017   46.13   <2e-16 ***
group1      -0.04309    0.06844   -0.63     0.53    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for Gamma family taken to be 0.1258647)

    Null deviance: 11.146  on 118  degrees of freedom
Residual deviance: 11.097  on 117  degrees of freedom
AIC: 489.68

Number of Fisher Scoring iterations: 4
```

Answer 1

"Group 1 reduces the mean outcome by a factor of exp(-0.04) = 0.96" needs something like "from the mean of group 0" or "from the baseline mean" in there, or at least implied (e.g. by having been mentioned immediately before this part)

On the second form, beware the distinction between "mean ratio" (which seems to be implying $E(Y_{1i}/Y_{0j})$) and "ratio of means". ($E(Y_{1i})/E(Y_{0j})=\mu_1/\mu_0$), which is what you intend. The first will be larger than the second (e.g. by Jensen's inequality, though there are simpler arguments for this specific case), and might not even be finite.

If you correct for both issues, choose whichever seems to best suit what you're trying to say at the time.

(I would also avoid reducing reporting estimates to a single significant figure.)