Before I dove too deeply into modeling the covariance structure of the repeated measures, I first would look into whether the Poisson distribution is the best choice. From your output, I suspect not and that something like the negative binomial or a generalized Poisson might suit the data better. I'm not seeing a lot of evidence of heterogeneity of variance in the current results either, so a relatively simple covariance structure might work well enough, once other issues are resolved. This paper by Walt Stroup dates to 2011, and I know that Walt is continuing to refine his understanding of GLMMs (and recommendations for use) but I think the paper might still be quite helpful.
There is a distinction between a GLMM and a GEE-type model that focuses on whether or how you model "residual" (the R-side stuff). For an example of the latter, see Example 38.12 Fitting a Marginal (GEE-Type) Model. I usually take the GLMM approach because I think it is more "natural" (Walt Stroup addresses this concept in his writings), but a GEE-type model could do the job as well.
What is DAP? From the output, it looks like a continuous covariate measured on each tree (ARV). Are you comfortable with assuming a linear relationship between DAP and log(PROD)? A misspecified mean model could contribute to overdispersion. I assume that LOCAL is site, and ANO is year.
What is the range of values for PROD? Are values equal to or close to zero, or large? I'd ponder zero-inflation if counts are small, or even a normal or lognormal distribution (probably with heterogeneous variances) if counts are large.
I would use method=laplace or quad, unless results were pathological (which sometimes happens). These methods allow information criteria (e.g., AIC) which you could use to compare models. The default pseudo-likelihood method does not.
I would start with
model prod = local|ano|dap / dist=poisson;
random intercept / subject=arv(local);
and see whether the Generalized chisq/df indicated overdispersion. (Pretty sure it will.)
If so, then I'd try
model prod = local|ano|dap / dist=poisson;
random intercept ano / subject=arv(local);
And then I'd try either a negative binomial distribution or a generalized Poisson distribution (see Example 38.14 Generalized Poisson Mixed Model for Overdispersed Count Data). As far as I know, it is not possible to model a covariance structure among repeated measures when you move to a two-parameter distribution (e.g., negative binomial) from a one-parameter distribution (e.g., Poisson); SAS tech support might be able to weigh in on that.
And then I'd look to see whether the fruit production story changed with the model. It's a comfort when results point you in the same direction, regardless 🙂
I hope this helps.
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