Poisson Regression: questions about tests of assumptions

I would like to test in R what regression fits my data best. My dependent
variable is a count, and has a lot of zeros.

And I would need some help to determine what model and family to use
(poisson or quasipoisson, or zero-inflated poisson regression), and how to
test the assumptions.

1) Poisson Regression: as far as I understand, the strong assumption is
that dependent variable mean = variance. How do you test this? How close
together do they have to be? Are unconditional or conditional mean and
variance used for this? What do I do if this assumption does not hold?

2) I read that if variance is greater than mean we have overdispersion, 
and a potential way to deal with this is including more independent 
variables, or family=quasipoisson. Does this distribution have any other 
requirements or assumptions? What test do I use to see whether 1) or 2) 
fits better - simply anova(m1,m2)?

3) I also read that negative-binomial distribution can be used when 
overdispersion appears. How do I do this in R?

What is the difference to quasipoisson?

4) Zero-inflated Poisson Regression: I read that using the vuong test
checks what models fits better.

vuong (model.poisson, model.zero.poisson)

Is that correct?

5) ats.ucla.edu has a section about zero-inflated Poisson Regressions, and
test the zeroinflated model (a) against the standard poisson model (b):

m.a <- zeroinfl(count ~ child + camper | persons, data = zinb)
m.b <- glm(count ~ child + camper, family = poisson, data = zinb)
vuong(m.a, m.b)

I don't understand what the "| persons" part of the first model does, and
why you can compare these models if. I had expected the regression to be
the same and just use a different family.

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

        [[alternative HTML version deleted]]

I would like to test in R what regression fits my data best. My dependent
variable is a count, and has a lot of zeros.

And I would need some help to determine what model and family to use
(poisson or quasipoisson, or zero-inflated poisson regression), and how to
test the assumptions.

1) Poisson Regression: as far as I understand, the strong assumption is
that dependent variable mean = variance. How do you test this? How close
together do they have to be? Are unconditional or conditional mean and
variance used for this? What do I do if this assumption does not hold?

2) I read that if variance is greater than mean we have overdispersion, and
a potential way to deal with this is including more independent variables,
or family=quasipoisson. Does this distribution have any other requirements
or assumptions? What test do I use to see whether 1) or 2) fits better -
simply anova(m1,m2)?

3) I also read that negative-binomial distribution can be used when
overdispersion appears. How do I do this in R? What is the difference to
quasipoisson?

4) Zero-inflated Poisson Regression: I read that using the vuong test
checks what models fits better.

vuong (model.poisson, model.zero.poisson)

Is that correct?

5) ats.ucla.edu has a section about zero-inflated Poisson Regressions, and
test the zeroinflated model (a) against the standard poisson model (b):

m.a <- zeroinfl(count ~ child + camper | persons, data = zinb)
m.b <- glm(count ~ child + camper, family = poisson, data = zinb)
vuong(m.a, m.b)

I don't understand what the "| persons" part of the first model does, and
why you can compare these models if. I had expected the regression to be
the same and just use a different family.

Thank you
T

        [[alternative HTML version deleted]]

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