model checking and test of overdispersion for glmer - r

Question

I am testing differences on the number of pollen grains loading on plant stigmas in different habitats and stigma types.
My sample design comprises two habitats, with 10 sites each habitat.
In each site, I have up to 3 stigma types (wet, dry and semidry), and for each stigma stype, I have different number of plant species, with different number of individuals per plant species (code).
So, I ended up with nested design as follow: habitat/site/stigmatype/stigmaspecies/code
As it is a descriptive study, stigmatype, stigmaspecies and code vary between sites.
My response variable (n) is the number of pollengrains (log10+1)per stigma per plant, average because i collected 3 stigmas per plant.
Data doesnt fit Poisson distribution because (i) is not integers, and (ii) variance much higher than the mean (ratio = 911.0756). So, I fitted as negative.binomial.
After model selection, I have:
m4a <- glmer(n ~ habitat*stigmatype + (1|stigmaspecies/code),
family=negative.binomial(2))
> summary(m4a)
Generalized linear mixed model fit by maximum likelihood ['glmerMod']
Family: Negative Binomial(2) ( log )
Formula: n ~ habitat * stigmatype + (1 | stigmaspecies/code)
AIC BIC logLik deviance
993.9713 1030.6079 -487.9856 975.9713
Random effects:
Groups Name Variance Std.Dev.
code:stigmaspecies (Intercept) 1.034e-12 1.017e-06
stigmaspecies (Intercept) 4.144e-02 2.036e-01
Residual 2.515e-01 5.015e-01
Number of obs: 433, groups: code:stigmaspecies, 433; stigmaspecies, 41
Fixed effects:
Estimate Std. Error t value Pr(>|z|)
(Intercept) -0.31641 0.08896 -3.557 0.000375 ***
habitatnon-invaded -0.67714 0.10060 -6.731 1.68e-11 ***
stigmatypesemidry -0.24193 0.15975 -1.514 0.129905
stigmatypewet -0.07195 0.18665 -0.385 0.699885
habitatnon-invaded:stigmatypesemidry 0.60479 0.22310 2.711 0.006712 **
habitatnon-invaded:stigmatypewet 0.16653 0.34119 0.488 0.625491
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Correlation of Fixed Effects:
(Intr) hbttn- stgmtyps stgmtypw hbttnn-nvdd:stgmtyps
hbttnn-nvdd -0.335
stgmtypsmdr -0.557 0.186
stigmatypwt -0.477 0.160 0.265
hbttnn-nvdd:stgmtyps 0.151 -0.451 -0.458 -0.072
hbttnn-nvdd:stgmtypw 0.099 -0.295 -0.055 -0.403 0.133
Two questions:
How do I check for overdispersion from this output?
What is the best way to go through model validation here?
I have been using:
qqnorm(resid(m4a))
hist(resid(m4a))
plot(fitted(m4a),resid(m4a))
While qqnorm() and hist() seem ok, and there is a tendency of heteroscedasticity on the 3rd graph. And here is my final question:
Can I go through model validation with this graph in glmer? or is there a better way to do it? if not, how much should I worry about the 3rd graph?

Answer 1

a simple way to check for overdispersion in glmer is:
> library("blmeco")
> dispersion_glmer(your_model) #it shouldn't be over
> 1.4
To solve overdispersion I usually add an observation level random factor
For model validation I usually start from these plots...but then depends on your specific model...
par(mfrow=c(2,2))
qqnorm(resid(your_model), main="normal qq-plot, residuals")
qqline(resid(your_model))
qqnorm(ranef(your_model)$id[,1])
qqline(ranef(your_model)$id[,1])
plot(fitted(your_model), resid(your_model)) #residuals vs fitted
abline(h=0)
dat_kackle$fitted <- fitted(your_model) #fitted vs observed
plot(your_data$fitted, jitter(your_data$total,0.1))
abline(0,1)
hope this helps a little....
cheers

Answer 2

Just an addition to Q1 for those who might find this by googling: the blmco dispersion_glmer function appears to be outdated. It is better to use #Ben_Bolker's function for this purpose:
overdisp_fun <- function(model) {
rdf <- df.residual(model)
rp <- residuals(model,type="pearson")
Pearson.chisq <- sum(rp^2)
prat <- Pearson.chisq/rdf
pval <- pchisq(Pearson.chisq, df=rdf, lower.tail=FALSE)
c(chisq=Pearson.chisq,ratio=prat,rdf=rdf,p=pval)
}
Source: https://bbolker.github.io/mixedmodels-misc/glmmFAQ.html#overdispersion.
With the highlighted notion:
Do PLEASE note the usual, and extra, caveats noted here: this is an APPROXIMATE estimate of an overdispersion parameter.
PS. Why outdated?
The lme4 package includes the residuals function these days, and Pearson residuals are supposedly more robust for this type of calculation than the deviance residuals. The blmeco::dispersion_glmer sums up the deviance residuals together with u cubed, divides by residual degrees of freedom and takes a square root of the value (the function):
dispersion_glmer <- function (modelglmer)
{
n <- length(resid(modelglmer))
return(sqrt(sum(c(resid(modelglmer), modelglmer#u)^2)/n))
}
The blmeco solution gives considerably higher deviance/df ratios than Bolker's function. Since Ben is one of the authors of the lme4 package, I would trust his solution more although I am not qualified to rationalize the statistical reason.
x <- InsectSprays
x$id <- rownames(x)
mod <- lme4::glmer(count ~ spray + (1|id), data = x, family = poisson)
blmeco::dispersion_glmer(mod)
# [1] 1.012649
overdisp_fun(mod)
# chisq ratio rdf p
# 55.7160734 0.8571704 65.0000000 0.7873823