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I have data on the number of hospitalizations over one year in patients with chronic obstructive pulmonary disease, and I need to estimate the difference in the number of hospitalization between two groups. I only have the number of hospitalizations, no information on hospitalizations before the study. About one third of participants had no events during the study. Each event increases the probability of having a further event, therefore a Poisson distribution does not seem to be appropriate. How would you suggest to model these data also taking into account age and sex as covariates?

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    $\begingroup$ Do you have times between events, or just the number of total events over the year? Also, when you say "model these data," what do you have in mind? Do you have other covariates besides membership in one of your 2 groups? Do your groups include patients who had 0 hospitalizations over the year, or is having at least 1 hospitalization required for inclusion into the study? What do you know about hospitalizations in prior years, which might thus affect the probability of hospitalization during the year in question? $\endgroup$ – EdM Oct 6 '19 at 16:35
  • $\begingroup$ If you not only know the number of patients but also the arrival times you could use a Markov process to model the data. $\endgroup$ – Sebastian Oct 6 '19 at 17:05
  • $\begingroup$ No information on previous hospitalizations, and I only have the total number of hosp. I would like to have sex and age as covariates. I edited the post with this information. $\endgroup$ – Claudio Oct 6 '19 at 21:39
  • $\begingroup$ You may find Chapter 8 of this book interesting, particularly if you use Stata. $\endgroup$ – Dimitriy V. Masterov Oct 8 '19 at 22:20
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Although it might seem unlikely that a Poisson generalized linear model will work, the first step is to try it anyway, incorporating your group membership and covariates. Then see if the variances of predicted counts are significantly different from corresponding predicted means; they should be equal if the Poisson model holds.* Several such tests are described on this page.

If the variance is greater than the mean (over-dispersion) a negative binomial model would be the next thing to try; it adds a single extra parameter to allow for variance to increase greater than the mean.

Under-dispersed (variance less than mean) count data can be handled by other approaches, described for example on this page and this page.


*A nonlinear relationship between age and predicted number of events might confuse this relationship; you might want to start by examining just sex and group membership as predictors within some stratified age groupings.

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