Modelling health scores with the skew-normal distribution

Health care interventions which use quality of life or health scores often provide data which are skewed and bounded. The scores are typically formed by adding up responses to a number of questions. Different questions might have different weights, but the scores will be bounded, and are often scaled to the range 0 to 100. If improvement in health over time is measured, scores will tend to cluster near the ’healthy’ or ’good’ boundary as time progresses, leading to a skew distribution. Further, some patients will drop out as time progresses, so the scores reflect a selected population. We fit models based on the skew-normal distribution to data from a randomised controlled trial of treatments for sprained ankles, in which scores were recorded at baseline and 1, 3 and 9 months. We consider the extent to which skewness in the data can be explained by the clustering at the boundary via a comparison between a censored normal and a censored skew-normal model. As this analysis is based on the complete data only, a formula for the distortion of the treatment effects due to informative drop-out is given. This allows us to assess under which conditions the conclusions drawn on the complete data may be either reinforced or reversed, when the informative drop-out process is taken into account.