I was wondering if people had recommendations for modeling severely skewed and/or kurtotic (coefficients more extreme than + or -3) outcome data with negative values. I would traditionally use a Poisson or negative binomial or gamma regression (i.e., generalized linear model), but those models require all scores to be non-negative. Any idea of what you would use to 1) test whether the mean of the non-normal scores is different than zero and 2) predict the non-normal scores with IVs?
I have count data at two timepoints (number of mental health referrals in the past 3 months). The count data at each timepoint are Poisson distributed with moderate overdispersion (75% zeros). I want to look at changes in the counts from pre-intervention to post-intervention. So the outcome variable I am working with is the difference of the two counts (post - pre). The difference scores are very kurtose positive in that most people don't change (difference score = 0), but then it is also skewed positive with more people increasing than decreasing. But there are people who decreased, leading to negative values. I want to estimate a central tendency for the difference scores and predict the difference scores with demographic variables, such as gender and age. I can get the point estimates with the mean and OLS regression, but how do I get accurate measures of uncertainty? The standard error of the mean and the standard error of the regression coefficients will be biased by the positive kurtosis and skew.
I was hoping there was a generalized linear model that I wasn't familiar with whose distribution fit my outcome variable; however, maybe that was naively optimistic...