[R-sig-ME] different overdispersion parameter for binomial GLMM in, lme4, glmmADMB and glmmTMB (Muldoon, Ariel)

[John Maindonald]

I have just now posted a document that bears on this at: https://rpubs.com/johnhm/Overdispersed

It explores, with a dataset that relates to work in which I have recently been involved,
the glmmTMB’s package recently added ability to use the betabinomial family to
model what is effectively dispersion, defined as the factor by which the variance
exceeds the binomial variance.  These abilities are clearly relatively undeveloped.
(I had to delve to discover how to extract the estimates of theta, for which see below.)
I was surprised at how easy it was to get models to converge that are some way
sensible.  What have others done to explore these abilities?  Comments will be very
welcome.

The betabinomial implies that the “dispersion” increases with the binomial size n,
whereas quasibinomial models assume a “dispersion" that is independent of n.

[The beta-binomial distribution accounts for a multiplier (effectively, a “dispersion”
estimate) that is the ratio of the beta-binomial variance to the binomial variance.
It is (1 + rho*(n-1)).  In the what I take to be the glmmTMB parameterization,
the variance is n*Pi*(1-Pi)*(1+(n-1)/(theta+1))

NB: Using the notation at,
  https://en.wikipedia.org/wiki/Beta-binomial_distribution
but substituting a=alpha and b=beta

Pi=a/(a+b); rho = 1/(a+b+1); theta=a+b]

John Maindonald             email: john.maindonald at anu.edu.au<mailto:john.maindonald at anu.edu.au>

On 17/03/2018, at 10:04, Highland Statistics Ltd <highstat at highstat.com<mailto:highstat at highstat.com>> wrote:


Ben, sorry but I am bit confused by your answer. If I understand
correctly, the approach you would recommend is to calculate the
dispersion parameter on the binomial model and if there is
overdispersion compare models with different ways to deal with it (e.g
observation-level random effects and beta-binomial) to the binomial one
to find out which ones fits the data better. Is that correct? And so
there would be no point in calculating the dispersion parameter for the
OLRE and beta-binomial model and see how much it goes down?

Cheers,

Correct. Overdispersion/underdispersion is only relevant for distributions in which the variance is determined by the mean. Like the Poisson: mean(Y) = var(Y)  and the binomial: E(Y) = N * pi and var(Y) = Pi * N * (1 - Pi).

No need to check for overdispersion for the normal, Gamma, inverse Gaussian, beta-binomial, and beta distributions. These distributions have an extra parameter (like the variance in the normal distribution) in the variance term. Having said that...I am still confused why the Negative binomial GLM can be overdispersed. I guess that is because the NB GLM is not a real GLM and iterates between two algorithms (when doing frequentist analysis). I guess (again) it is more about whether the functional form of the NB variance is correct...or not.

Instead of using a dispersion statistic based on Pearson residuals (coming from models with fancy random effects) it is perhaps better to simulate data from your model and compare the variation in the simulated data with the variation in the observed data. Or do what Ben Bolker suggested a few days/weeks ago...simulate data and compare the corresponding residuals with the original residuals.

Alain





--

Dr. Alain F. Zuur
Highland Statistics Ltd.
9 St Clair Wynd
AB41 6DZ Newburgh, UK
Email: highstat at highstat.com<mailto:highstat at highstat.com>
URL:   www.highstat.com

And:
NIOZ Royal Netherlands Institute for Sea Research,
Department of Coastal Systems, and Utrecht University,
P.O. Box 59, 1790 AB Den Burg,
Texel, The Netherlands



Author of:
1. Beginner's Guide to Spatial, Temporal and Spatial-Temporal Ecological Data Analysis with R-INLA. (2017).
2. Beginner's Guide to Zero-Inflated Models with R (2016).
3. Beginner's Guide to Data Exploration and Visualisation with R (2015).
4. Beginner's Guide to GAMM with R (2014).
5. Beginner's Guide to GLM and GLMM with R (2013).
6. Beginner's Guide to GAM with R (2012).
7. Zero Inflated Models and GLMM with R (2012).
8. A Beginner's Guide to R (2009).
9. Mixed effects models and extensions in ecology with R (2009).
10. Analysing Ecological Data (2007).

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