Poisson Regression: questions about tests of assumptions

Thank you for the detailed answer, that was really helpful.
I did some excessive reading and calculating in the last hours since your
reply, and have a few (hopefully much more informed) follow up questions.


1) In the vignette("countreg", package = "pscl"), LLH, AIC and BIC values
are listed for the models Negative-binomial (NB), Zero-Inflated (ZI), ZI
NB, Hurdle NB, and Poisson (Standard). And although I found a way to
determine LLH and DF for all models, BIC & AIC values are not displayed by
default, neither using the code given in the vignette. How do I calculate
these? (AIC is given as per default only in 2 models, BIC in none).


2) For the zero-inflated models, the first block of count model
coefficients is only in the output in order to compare the changes,
correct? That is, in my results in a paper I would only report the second
block of (zero-inflation) model coefficients? Or do I misunderstand
something? I am confused because in their large table to compare
coefficients, they primarily compare the first block of coefficients (p. 18)
R> fm <- list("ML-Pois" = fm_pois, "Quasi-Pois" = fm_qpois, "NB" = fm_nbin,
+ "Hurdle-NB" = fm_hurdle, "ZINB" = fm_zinb)
R> sapply(fm, function(x) coef(x)[1:8])


3)

There are various formal tests for this, e.g., dispersiontest() in package
"AER".

I have to run 9 models - I am testing the influence of several predictors
on different individual symptoms of a mental disorder, as "counted" in the
last week (0=never in the last week, to 3 = on all day within the last
week).
So I'm regressing the same predictors onto 9 different symptoms in 9
models.

Dispersiontest() for these 9 models result in 3-4 overdispersed models
(depending if testing one- or two-sided on p=.05 level), 2 underdispersed
models, and 4 non-significant models. The by far largest dispersion value
is 2.1 in a model is not overdispersed according to the test, but that's
the symptom with 80% zeros, maybe that plays a role here.

Does BN also make sense in underdispersed models?

However, overdispersion can already matter before this is detected by a

significance test. Hence, if in doubt, I would simply use an NB model and
you're on the safe side. And if the NB's estimated theta parameter turns
out to be extremely large (say beyond 20 or 30), then you can still switch
back to Poisson if you want.

Out of the 9 models, the 3 overdispersed NB models "worked" with theta
estimates between 0.4 and 8.6. The results look just fine, so I guess NB is
appropriate here.

4 other models (including the 2 underdispersed ones) gave warnings (theta
iteration / alternation limit reached), and although the other values look
ok (estimates, LLH, AIC), theta estimates are between 1.000 and 11.000.
Could you recommend me what to do here?

Thanks
T

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