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How is AR(1) autocorrelation defined / parameterised when using a R {glmmTMB} GLM? I have read through Kristensen and McGillycuddy's vignette (https://cran.r-project.org/web/packages/glmmTMB/vignettes/covstruct.html) and understand the Gaussian case, but I don't understand how it's implemented within a generalised linear model structure. In particular, I want to be able to understand it when implemented in a logistic regression, but more general theory is fine (I'd also be interested in being able to apply it to Poisson etc.).

If I could have a link to a reference paper, that would be greatly appreciated. Or, some equations describing how it's done. Thank you!

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  • $\begingroup$ don't have time to answer right now, but: the autocorrelation is always defined over latent Gaussian variables, which would underlie the observed binary responses. You have to be careful in this case about identifiability constraints (i.e. the variance of the autocorrelated latent variable might need to be fixed with the map argument) $\endgroup$
    – Ben Bolker
    Sep 30, 2022 at 0:06
  • $\begingroup$ @BenBolker Using bbolker.github.io/mixedmodels-misc/notes/corr_braindump.html I was able to set up a simple logistic regression intercept + AR(1) model. It was three parameter: intercept $\beta_0$, and AR1 covariance matrix parameters $\rho$ and $\sigma^2$. I was always able to recover the former two parameters from the model, but modelled $\sigma^2$ never matched my input parameter. Is this what you mean by identifiability? $\endgroup$
    – Alex J
    Oct 3, 2022 at 21:40

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