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I have two glm, one with a gaussian distribution and identity link and one with gamma family and log link. The predictors are the same, the only thing that change is the response that in the gaussian glm is log-transformed and in the gamma glm is not.

The deviance is lower in the second model, and when plotting the residuals the second model looks much better than the first. However, when looking and the AIC, the first model has half the AIC of the second! How is that possible? Is it wrong to compare AIC of glm with different distribution families?

summary(mod1)

Call:
glm(formula = log_RS ~ DIET + log_Disp + log_II + log_LS + log_SM + 
    log_AS + log_LONG, family = gaussian)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-5.1050  -0.2672   0.1063   0.3945   1.3012  

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.061888   0.008574 123.856  < 2e-16 ***
DIETOH      -0.077913   0.012762  -6.105 1.07e-09 ***
log_Disp     1.529989   0.006960 219.822  < 2e-16 ***
log_II      -0.422808   0.006786 -62.310  < 2e-16 ***
log_LS       0.325270   0.006093  53.382  < 2e-16 ***
log_SM      -0.268292   0.008372 -32.048  < 2e-16 ***
log_AS       0.185077   0.010166  18.205  < 2e-16 ***
log_LONG     0.067347   0.010494   6.418 1.45e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for gaussian family taken to be 0.3246263)

    Null deviance: 24060.5  on 9345  degrees of freedom
Residual deviance:  3031.4  on 9338  degrees of freedom
AIC: 16018

Number of Fisher Scoring iterations: 2

> summary(mod2)

Call:
glm(formula = RS ~ DIET + log_Disp + log_II + log_LS + log_SM + 
    log_AS + log_LONG, family = Gamma(link = log))

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.94338  -0.35264  -0.00239   0.27669   1.40915  

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.186181   0.006435 184.332  < 2e-16 ***
DIETOH      -0.075002   0.009579  -7.830 5.41e-15 ***
log_Disp     1.536406   0.005224 294.104  < 2e-16 ***
log_II      -0.364580   0.005093 -71.585  < 2e-16 ***
log_LS       0.306213   0.004573  66.955  < 2e-16 ***
log_SM      -0.233847   0.006283 -37.216  < 2e-16 ***
log_AS       0.156994   0.007630  20.575  < 2e-16 ***
log_LONG     0.053129   0.007876   6.745 1.62e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for Gamma family taken to be 0.1828775)

    Null deviance: 23484.5  on 9345  degrees of freedom
Residual deviance:  2349.8  on 9338  degrees of freedom
AIC: 33167

Number of Fisher Scoring iterations: 8

> 
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    $\begingroup$ You can't use the AIC to compare models with different response variables; the likelihood is in different units as explained in answers to the linked post. $\endgroup$ Commented Jan 23, 2014 at 19:42
  • $\begingroup$ Ok I see, thanks. So I suppose I should rely on model checking by plotting the residuals, is that right? $\endgroup$ Commented Jan 23, 2014 at 20:00
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    $\begingroup$ AIC is like temperature in Celsius or Fahrenheit: its origin (the value 0) has little inherent meaning. Characterizing one AIC as half another is just like characterizing 20 degrees C as being ten times as hot as 2 degrees C. Unlike temperature (with its absolute zero), AIC does not even have an intrinsic origin. Compare your AICs (when that is valid) by looking at their differences. $\endgroup$
    – whuber
    Commented Jan 23, 2014 at 20:38
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    $\begingroup$ If you are wondering how to choose between a model w/ Y as the response variable vs ln(Y), you don't usually do it based on statistical criteria. These answer different questions. To understand what they mean, try reading this excellent CV thread: Interpretation of log transformed predictor. $\endgroup$ Commented Jan 23, 2014 at 20:56
  • $\begingroup$ A better duplicate: stats.stackexchange.com/questions/190839/… $\endgroup$ Commented Aug 9, 2020 at 17:48

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