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I am modeling count data with a non-Poisson distribution and potential zero inflation in glmmTMB. I am using different distributions and zero inflation specifications, including Conway Maxwell Poisson, and comparing AIC to determine the best model specification.

One of my more complex models, with a CMP distribution and a zero inflation that varies over my predictor levels, appears to converge and is providing an AIC value and parameter estimates, but is producing NaN values for the z and p values. What does that mean? Did the model converge properly? Should it be included in my candidate model set?

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  • $\begingroup$ Did you get any warnings when fitting the model? $\endgroup$ Commented Nov 11 at 22:24
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    $\begingroup$ I don't know whether this is causing your problem, but I note that the random-effect variance for your zero-inflation model is very close to 0. Also, I just caught the enormously negative coefficient estimates for the zero-inflation model, as @AlexJ points out. $\endgroup$ Commented Nov 11 at 22:25
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    $\begingroup$ The zero-inflated component looks incredibly complicated. Look at the size of the estimates. I suspect you will have to drastically simplify it $\endgroup$ Commented Nov 11 at 22:26

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The NaN values indicate that glmmTMB can't compute standard errors properly (and therefore can't compute Z statistics or p-values. It's a little bit surprising that you can't get standard errors but you can get a log-likelihood and an AIC; usually in this case glmmTMB will report something about a "non-positive-definite Hessian" and refuse to give you log-likelihood/AIC (although there is a way to get these values if you really want them ...)

In any case, the diagnosis of your model is that the variance of the zero-inflation of the random effect is nearly zero (a singular fit), and there is complete separation in the fixed-effects model (indicated by the extreme values of the Z-I fixed effects parameters).

In theory you could still include this model in your list of candidate models. While the singular fit suggests overfitting (the model is trying to throw away the random effect for ZI by setting the variance near zero), the complete separation suggests that there is at least some signal for a difference between categories in zero-inflation (the model is effectively trying to set the estimated zero-inflation probability for the second and fourth categories to zero, while the others are small (exp(-2) approx 13%, exp(-2 -3) approx 0.5%) but non-zero.

I would be a little surprised if this model ends up near the top of your candidate set ...

Please be aware that AIC-driven model selection may give you reasonable predictions (although with overly narrow confidence intervals), but you absolutely shouldn't try to make inferences on the parameters (e.g. p-values) after selecting a model from a large set of candidates ...

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  • $\begingroup$ Thanks for your response! There were no warnings from the model. My best model is CMP with no Z-I , CMP with constant Z-I is 2nd, and this one is third, which I was surprised by. Maybe the CMP distribution is more important than Z-I for this data. I was also surprised I could get an AIC without standard errors, that was my main concern. The other concern is that the residuals of the best model indicate zero inflation and flag homogeneity of variance. Residuals are best for my 4th best model, negative binomial with no Z-I. Coefficient estimates are very similar in all the models. $\endgroup$ Commented Nov 12 at 3:51

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