Interpreting random effect variance in glmer

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I’m revising a paper on pollination, where the data are binomially distributed (fruit matures or does not). So I used glmer with one random effect (individual plant) and one fixed effect (treatment). A reviewer wants to know whether plant had an effect on fruit set — but I’m having trouble interpreting the glmer results.

I’ve read around the web and it seems there can be issues with directly comparing glm and glmer models, so I’m not doing that. I figured the most straightforward way to answer the question would be to compare the random effect variance (1.449, below) to the total variance, or the variance explained by treatment. But how do I calculate these other variances? They don’t seem to be included in the output below. I read something about residual variances not being included for binomial glmer — how do I interpret the relative importance of the random effect?

> summary(exclusionM_stem)
Generalized linear mixed model fit by maximum likelihood (Laplace
  Approximation) [glmerMod]
 Family: binomial  ( logit )
Formula: cbind(Fruit_1, Fruit_0) ~ Treatment + (1 | PlantID)

     AIC      BIC   logLik deviance df.resid 
   125.9    131.5    -59.0    117.9       26 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-2.0793 -0.8021 -0.0603  0.6544  1.9216 

Random effects:
 Groups  Name        Variance Std.Dev.
 PlantID (Intercept) 1.449    1.204   
Number of obs: 30, groups:  PlantID, 10

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)   
(Intercept)  -0.5480     0.4623  -1.185   0.2359   
TreatmentD   -1.1838     0.3811  -3.106   0.0019 **
TreatmentN   -0.3555     0.3313  -1.073   0.2832   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
           (Intr) TrtmnD
TreatmentD -0.338       
TreatmentN -0.399  0.509

Answer

While getting an analogue of “proportion variance explained by each effect” is in principle possible for GLMMs, there are several complicating factors (which levels of the model do you consider “total variance”, and how do you quantify the sampling variation due to the lowest-level [Binomial in this case] sampling distribution)? Nakagawa and Schielzeth (doi:10.1111/j.2041-210x.2012.00261.x) present a general approach to calculating R^2 (proportion of total variance explained) for (G)LMMs which has gotten pretty popular in ecology; Xu et al 2003 take a similar approach. In principle this approach could probably be extended to consider proportion of variance explained by different terms [but note that the ‘proportion of variance’ of all the terms in the model considered in this way would probably not add up to 100% — it could be either more or less].

However, if your reviewer isn’t hung up on statistical details and would be satisfied with a more heuristic explanation of “importance”, you could point out that the estimated among-plant standard deviation is 1.20, very close to the magnitude of the largest treatment effect (-1.18); this means that the plants vary quite a bit, relative to the magnitude of the treatment effects (e.g., the 95% range of the plant effects is approximately $4\sigma$, from $-1.96 \sigma$ to $+1.96\sigma$).

Visually:

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Attribution
Source : Link , Question Author : jwb4 , Answer Author : Ben Bolker

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