Is an overdispersion parameter of 5.17 for GLMM with Beta family too high to yield reliable results?

Question

I'm running a generalised linear mixed model with beta family on the effect of overhead cover (proportion ∈ (0,1)) on the proportion of birds scavenging from carrion left out in nature (proportion ∈ (0,1)), with Area as random factor (factor w/ 6 levels).

> myglmm <- glmmTMB(ProportionBirdsScavenging ~ OverheadCover + (1|Area), data = df_prop_birds_eating, beta_family(link = "logit"), weights = pointWeight_scaled)
> summary(myglmm)
Family: beta  ( logit )
Formula:          ProportionBirdsScavenging ~ OverheadCover + (1 | Area)
Data: df_prop_birds_eating
Weights: pointWeight_scaled

     AIC      BIC   logLik deviance df.resid 
    -5.3      0.8      6.7    -13.3       30 

Random effects:

Conditional model:
 Groups Name        Variance  Std.Dev. 
 Area   (Intercept) 1.198e-10 1.094e-05
Number of obs: 34, groups:  Area, 6

Overdispersion parameter for beta family (): 5.17 

Conditional model:
              Estimate Std. Error z value Pr(>|z|)    
(Intercept)     1.7869     0.7196   2.483 0.013017 *  
OverheadCover  -4.7387     1.2661  -3.743 0.000182 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The overdispersion parameter is 5.17. I have tried to find some information about this parameter for beta models, but I could not find much. Most of what I found was about the poisson or binomial distribution, and tests about significance e.g. DHARMa::testOverdispersion, performance::check_overdispersion, and AER::dispersiontest only test for Poisson GLMs. My question is whether this overdispersion parameter value of 5.17 is too high? Does this mean that the model assumptions are not met and the output can not be trusted? If so, is there a way of fixing this, so that my model yields reliable results?

About overdispersion in Poisson models I frequently read that adding a dispersion parameter would 'fix' the overdispersion, but in the beta model I am using there already is a dispersion parameter. Can somebody elaborate on this?

Answer 1

Using the word "overdispersion parameter" is maybe not an ideal choice by the glmmTMB developers - "dispersion parameter" may be more appropriate.

Overdispersion is when your residual variance is larger than what your fitted model assumes. It mainly only occurs in models that have a fixed dispersion (e.g. Poisson)

When you fit a model with variable dispersion (as you do here), your model adjusts the expected residual variance (=dispersion) during the fit. The dispersion parameter gives your feedback about this adjustment, but relative to this fitted dispersion, the model isn't overdispersed.

A dispersion test for a model with a dispersion parameter > 1 should therefore not indicate overdispersion (at least assuming that the residuals scatter with the estimated dispersion). See also https://github.com/florianhartig/DHARMa/issues/143. Also, There is no reason to put less trust in a model with a large estimated dispersion parameter.

Btw., DHARMa::testOverdispersion should work for the beta family. I have just tried this out, and we see exactly the behavior I describe above: estimated dispersion is around 4, but the dispersion test ist negative

x = runif(100,-1,1)
y = plogis(x + rnorm(100))

library(glmmTMB)
library(DHARMa)

fit <- glmmTMB(y~x, family = beta_family())

summary(fit)
res <-simulateResiduals(fit, plot = T)