Solved – Calculate (backtransform) coefficients of a Gamma (type inverse) GLMM in R

glmmrregression coefficients

I calculated a Generalized Linear Mixed Model for speech times (DV) with fixed effects congruency (2 levels) and stability (3 levels). I include random effects of type subject and length of word. The equation would be:

gmod<-glmer(time~congruency+stability+(1|SbjID)+1|lengthWord),timedata,family=Gamma())

So the output is:

Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
 Family: Gamma  ( inverse )
Formula: time ~ congruency + stability + (1 | SbjID) + (1 | lengthWord)
   Data: timedata

     AIC      BIC   logLik deviance df.resid 
  1443.4   1482.9   -714.7   1429.4     2093 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-1.3096 -0.6161 -0.2716  0.2094  6.8077 

Random effects:
 Groups     Name        Variance Std.Dev.
 SbjID      (Intercept) 0.13282  0.3644  
 lengthWord (Intercept) 0.05684  0.2384  
 Residual               0.58248  0.7632  
Number of obs: 2100, groups:  SbjID, 24; lengthWord, 18

Fixed effects:
            Estimate Std. Error t value Pr(>|z|)    
(Intercept)  1.05964    0.20255   5.232 1.68e-07 ***

congruency1  1.28503    0.04971  25.852  < 2e-16 ***

stability1  -0.03965    0.15899  -0.249    0.803    
stability2  -0.06091    0.15531  -0.392    0.695    

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr) cngrn1 stblt1
congruency1 -0.072              
stability1  -0.736  0.028       
stability2  -0.763  0.034  0.954

Now my question is how to backtransform the coefficients of my fixed effects to say something about my data; such as 'the speechtime increases by 0.5 seconds depending on congruency' similar to findings of an LMM.

I am completely new to GLMMs and would be grateful for a simplified explanation of how to do this. There is not much out there to explain the intrinsics of such a model (Gamma inverse). So an explanation of how to approach this in general would also be much appreciated.

Best Answer

This is a little tricky. For what it's worth, this isn't specific to mixed models; it would apply to any GLM fitted with an inverse link.

if fitting a log-link model is an alternative (i.e. family=Gamma(link="log")), it might make your life easier. The inverse link is the default for the Gamma because it's mathematically convenient, but unless you have historical/cultural/theoretical reasons to prefer the inverse, I would suggest the log link. Lo and Andrews 2015 Frontiers in Psychology recommend an identity link (which is computationally inconvenient but gives easy-to-understand results), although they mention the possibility of

[o]ther theoretical positions [that] assume a different relationship between RT [reaction time] and mental operations that is most appropriately measured by a transformation such as log or inverse RT. For example, differences calculated on the logarithmic metric reflect proportional change [i.e., log(700 ms)−log(600 ms) = log(700/600 ms)], which aligns with many theories of aging which attribute a causal role to general cognitive slowing (e.g., Salthouse, 1985). However, the vast majority of cognitive theories have been developed and validated on raw RT.

the natural scale for an inverse-link model would measure the speed of reaction (in seconds$^{-1}$) rather than elapsed time of reaction (in seconds). In this case you would quote effects in terms of changes in speed: "the predicted effect of congruency (condition 1) is to increase reaction speed by 1.29 sec$^{-1}$". However, I would guess there's a chance that trying to interpret behavior in terms of speeds rather than times might make your (or your audience's) head explode ...
another way to handle this is to give examples of predictions: e.g. "the mean predicted time (in seconds?) is 1/1.06=0.94 seconds; the predicted effect of congruency (condition 1) is to decrease the mean time to 1/(1.06+1.29)=0.43 seconds; the predicted effect of stability condition 1 is to increase the mean time to 1/(1.06-0.04) = 0.98 seconds ..." etc. (I've used rounded values here.) Note that positive coefficients lead to an increase in reaction speed and hence a decrease in reaction time ...

For comparison, there are fairly established ways to back-transform or understand the effects of parameters on:

the identity scale (i.e., regular linear models); the parameter gives the estimated change in the response given a one-unit change in the predictor
the log scale; $\exp(\beta)$ gives the proportional change in the response given a one-unit change in the predictor
the logit or log-odds scale; $\exp(\beta)$ gives the proportional change in the odds (not the probability) of the response given ...
complementary log-log scale or log-hazard scale; $\exp(\beta)$ gives the proportional change in the hazard (not the probability) of the response ...

Related Solutions

Solved – GLMM in SPSS: what does significance of the fixed coefficients mean

The fixed coefficients in GLMM are subject specific estimates, which mean the association between $X$ (the predictors) and $Y$ (the outcome, refer to the latent one in nonlinear models) holding constant the random effects, and it describes how an individual's mean of outcome depends on the predictors. You can find the clarification of subject specific and population average estimates here.

The importance of fixed coefficients depends on your goal of analysis. The fixed effects are often the focus, unless you are interested in the individual trajectories (random effects) or the scale (variance) of the random effects.

The significance here (mostly based on $z$-test) has the same meaning as that in other tests, which means that you can reject the null hypothesis that fixed coefficient $\beta=0$.

Solved – Backtransform coefficients of a Gamma-log GLMM

Because the fitted response is non-linear, the slope will be constantly varying. visreg will have produced the fitted lines by predicting from the model for a range of values over date.2k for each level of PNRB.

In general you do exp(coef) so exp(-0.74) is right and -exp(0.74) is wrong. For the other groups you add the coefficients for that group (so the date.2k coef, of 0.074, plus 0.03169) and then exponentiate that: exp(-0.074 + 0.03169).

It is how you are interpreting these values as "slopes" that is defeating you. These aren't slopes, certainly not in the sense you seem to be using them. What they are are multiplicative factors. On the log scale the model is (assumed to be) linear or additive. On the original scale, additive terms become multiplicative.

Hence, the values you computed for BSKAS as

> exp(-0.074 + 0.0317)
[1] 0.9585822

indicates that for a unit increase in date.2k, the response is multiplied by ~ 0.96. If you multiply something by this value it gets smaller. By how much depends on what the value of the response was at the value of $x$ before you increased it by 1 unit.

Worked example

We don't have the fitted model (it would have simplified this) but we have sufficient information to approximate it from your output. Let's illustrate how this works for your model, by predicting $y$ for two data points $x$, 2 and 3 (i.e. date.2k = 2, and date.2k = 3) for the level BSKAS. Here's an (inefficient) function to do this but at least I don't need to create a model matrix...

pBSKAS <- function(x, backtf = FALSE) {
  ret <- 0.84997 + (-0.074 * x) + (-0.58319 * 1) + (0.0317 * x)
  if (backtf)
    ret <- exp(ret)
  ret
}

This gives for $x \in {2, 3}$

> pBSKAS(c(2,3), backtf = TRUE)
[1] 1.199830 1.150136

If we now compute the magnitude of change between the two observations

> 1.150136 / 1.199830
[1] 0.9585825

we see that same value that we worked out earlier just by exponentiating the coefficients:

> exp(-0.074 + 0.0317)
[1] 0.9585822

(it's slightly different but that will just be rounding in the other coefs.) Note that all the intercept and the coefficient for PNRBSKAS do to the calculation is put a scale on it (i.e. shift the thing up and down).