How to interpret intercept (and coefficients) in a GLMM with gaussian family and log link?

Question

I am working with some bounded continuous data (between 1 and 10), which I think means I should be using a log link for my GLMM. In my reading for interpretation, it look like putting the effects (coefficients) back on the response scale is accomplished by exp(coeff) - 1. Is this also the case for the intercept (i.e., exp(intercept) - 1?

Example in R:

## List of required packages
Pkgs <- c("dplyr","lme4", "lmerTest", "MCMCglmm")

# Load packages
lapply(Pkgs, require, c = T)

Data <- data.frame(expand.grid(Subject = LETTERS[1:10], 
                               Group = factor(c("T", "U")),
                               Cond = factor(c("X", "Y","Z"))) %>%
        mutate(Y = round(rtnorm(nrow(.), 4.5, 2, 1, 10), digits = 0)))

summary(Mod <- glmer(Y ~ Group + Cond + (1|Subject), 
                     data = Data,
                     family = gaussian(link = "log")))

Output:

Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
 Family: gaussian  ( log )
Formula: Y ~ Group + Cond + (1 | Subject)
   Data: Data

     AIC      BIC   logLik deviance df.resid 
   245.8    258.4   -116.9    233.8       54 

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-1.87992 -0.50189 -0.08389  0.77279  2.80705 

Random effects:
 Groups   Name        Variance Std.Dev.
 Subject  (Intercept) 0.00     0.00    
 Residual             2.79     1.67    
Number of obs: 60, groups:  Subject, 10

Fixed effects:
            Estimate Std. Error t value Pr(>|z|)    
(Intercept)  1.55556    0.09086  17.121   <2e-16 ***
GroupU      -0.08988    0.09249  -0.972    0.331    
CondY       -0.09433    0.12229  -0.771    0.441    
CondZ        0.17140    0.10804   1.586    0.113    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Am I correct that transforming the coefficients to the response scale is accomplished by exponentiating and subtracting 1?

For example, the effect of GroupU on the response scale would be GroupU = exp(-0.09) - 1 = -0.08595914

Which is almost the same as the returned coefficient estimate and indicates the effect of a 1 category increase in Group (from T to U) indicated a reduction in Y of 0.09?

Is it correct to also transform the intercept and subtract 1 to interpret on the response scale? Intercept = exp(1.55556) - 1 = 3.737739

Indicating the mean for Group = T in Cond = X, is 3.74?

Edit to add results of predict():

  Group predicted  std.error conf.low conf.high Cond
1     1  4.737733 0.09085592 3.964915  5.661184    X
2     1  4.311264 0.09738051 3.562166  5.217892    Y
3     1  5.623520 0.08441573 4.765993  6.635338    Z
4     2  4.330473 0.09841750 3.570772  5.251803    X
5     2  3.940664 0.10655983 3.197905  4.855940    Y
6     2  5.140117 0.08228950 4.374496  6.039736    Z

It looks like it's incorrect to subtract 1 from the intercept. But, I also don't see how to transform the coefficients back to the response scale.

If the difference in response for Cond X between Group 1 and Group 2 is 4.74 - 4.33 = 0.41, how do I calculate that effect from a coefficient of -0.09?

Answer 1

Briefly explaining, with log link, effects are proportional, not additive. The predicted values are given as

$$\hat{y}_i=\exp(\hat{\beta}_0+\hat{\beta}_1 x_{1i}+\dots+\hat{\beta}_p x_{pi})$$

Taking the case where one effect is different by 1, such as $x_{1j}=x_{1i}+1$ then we obtain

$$\hat{y}_j=\exp(\hat{\beta}_0+\hat{\beta}_1 (x_{1i}+1)+\dots+\hat{\beta}_px_{pi})=\hat{y}_i\times\exp(\hat{\beta}_1)$$

So, this shows the effects are proportional. The coefficient $-0.09$ is roughly $\exp(-0.09)=0.914$ when back-transformed. So the effect for group to is to reduce the fitted value by around 8.6 percent. This is equal to the ratio of predicted values $\frac{4.330}{4.738}=0.914$.

hope this helps!