I doubt how to treat my outcome variable, and consequently, which regression analysis I should apply. I'm working with a count variable, namely the times a person said "I don't know" on a total of 45 questions. The distribution is very positively skewed, with a high number of zero's. And there is proof of severe overdispersion. I believe this is because when you say "I don't know" to one question, you are more likely to do this on the other questions, because they are all on the same topic. So the 45 'trials' are not independent.
I could recalculate to a proportion, and apply an extra binomial regression (with number of trials=45) and the estimators I get are the log of the difference in probability of scoring 1 (=don't know, DK) on a particular question (odds ratio).
I could treat it as a count of which the distribution will approximate an extra negative binomial distribution (because the average probability of saying "I don't know" in one trial is low, i.e. 0.05). Then the regression coefficients give me the log of the number of difference in times the respondent says DK (incident rate ratio).
Now my question is, will the extra negative binomial regression correct better for the overdispersion than the extra binomial regression? I feel like I allow the variance to be random twice, once for the 'negative binomial instead of poisson' and once for allowing an 'extra parameter' to estimate variation. Or will the two methods be the same, and can I freely choose based on which interpretation I prefer (odds ratio vs incidence rates ratio)?
I understand from the answer of Glen_b that for the theoretical negative binomial distribution the variance increases more quickly than the mean. How could I evaluate this in my empirical distribution, since I only have one measurement of the mean and the variance? Should I compare this for different groups of respondents maybe, say men and women, and compare their increase in mean and variance? Would performing GOF tests for Bin and Negbin give me a conclusive anwer?