I'm interested in fitting a model using a GLMM with quasi-Poisson distribution (my data is event occurrence and there was an overdispersion). I'm looking at the different options available, but I feel confused as to which one to choose.
So far, what I encountered are the following:
glmmPQL: family = quasipoisson. Used when a) the data is not normally distributed, b) the Poisson of mean >= 5, and c) overdispersion Bolker et al., 2009.
glmmTMB: family = nbinom1 @D A Wells, 20017
glmer with quasi_table: family = poisson, then use quasi_table function provided by @Ben Bolker, 2021.
What are the differences in these three methods, when to use one and when not to, or are there any other ways to go about it?
Thank you so much in advance.
All three of these methods use a GLMM estimation algorithm to estimate a model with the mean-variance relationship mean = phi*var, phi>0, but they do three different things:
glmmPQL uses penalized quasi-likelihood to fit a quasi-Poisson model. PQL is a fast method, but suffers when the effective sample size per group (i.e. expected number of counts per group) is low (Breslow 2004).glmer uses Laplace approximation (by default) or adaptive Gauss-Hermite quadrature (if you set the nAGQ argument to a value >1) to fit a Poisson model, then quasi_table to adjust the standard errors and p-values based on a squared-Pearson-residual estimate of dispersion. Laplace (which corresponds to nAGQ=1) is a little slower but more accurate than PQL; AGHQ with nAGQ>1 is slower still and more accurate.glmmTMB uses Laplace approximation to fit a negative binomial model that is parameterized so that the variance is proportional to the mean (i.e. it uses a mean-dependent dispersion parameter theta(mu) = mu/(phi-1) so that var = mu*(1+mu/theta(mu)) = mu*phi: Hardin and Hilbe 2007).I would expect all three of these methods to give fairly similar, but not identical results: if they gave qualitatively different answers I would worry about the sensitivity of my model (except in the case where effective sample size per group was very low, where I would steer away from PQL). I might default to the glmmTMB/nbinom1 approach just because you'd be on slightly firmer theoretical ground (i.e. this is a proper MLE, not a quasi-likelihood estimate).
Breslow, N. E. 2004. “Whither PQL?” In Proceedings of the Second Seattle Symposium in Biostatistics: Analysis of Correlated Data, edited by Danyu Y. Lin and P. J. Heagerty, 1–22. Springer. http://www.bepress.com/uwbiostat/paper192/.
Hardin, James William, and Joseph Hilbe. 2007. Generalized Linear Models and Extensions. Stata Press.