A normal-probit-binomial model for the analysis of repeated sum scores from multi-item questionnaires

Sum scores from questionnaire data are frequently analyzed under the normality assumption which is not always tenable due to for instance the skewness of the distribution, or the discreteness of the sum scores for small number of items. Alternatively, an ordinal regression analysis can be applied, but this approach seems appropriate only for very small number of items. The current state-of-the-art approach is to apply a beta-binomial model, though the beta-binomial model does not simply connect to item-response theory models that may describe the underlying items of the questionnaire, and it is more cumbersome to fit to multi domain sum scores or repeated longitudinal single sum scores. This paper proposes a normal-probit-binomial model for repeated sum scores that has an item level interpretation under certain assumptions without having to analyze the individual items. All parameters, the temporal correlation coefficients, and the mean difficulty parameter were almost unbiasedly estimated, irrespective of the set of item difficulty parameters. The coverage probability of the 95% confidence intervals were close to the nominal level, except for the correlation coefficients that were slightly liberal. Our model performed slightly better than the beta-binomial model on cross-sectional data and it handles missing items easily.