Application of negative binomial modeling for discrete outcomes: a case study in aging research.

We present a case study using the negative binomial regression model for discrete outcome data arising from a clinical trial designed to evaluate the effectiveness of a prehabilitation program in preventing functional decline among physically frail, community-living older persons. The primary outcome was a measure of disability at 7 months that had a range from 0 to 16 with a mean of 2.8 (variance of 16.4) and a median of 1. The data were right skewed with clumping at zero (i.e., 40% of subjects had no disability at 7 months). Because the variance was nearly 6 times greater than the mean, the negative binomial model provided an improved fit to the data and accounted better for overdispersion than the Poisson regression model, which assumes that the mean and variance are the same. Although correcting the variance and corresponding test statistics for overdispersion is a standard procedure in the Poisson model, the estimates of the regression parameters are inefficient because they have more sampling variability than is necessary. The negative binomial model provides an alternative approach for the analysis of discrete data where overdispersion is a problem, provided that the model is correctly specified and adequately fits the data.