Analyzing data using the chance-corrected beta-binomial model: Parameter estimates and their confidence regions
Data arising from replicated sensory discrimination test methods with a correct response are often modelled using the chance-corrected beta-binomial distribution. The model can provide maximum likelihood estimates of the mean proportion of correct responses in the population (and discriminal distances under Thurstonian assumptions) and of the assessor heterogeneity (overdispersion). Both of these parameters are estimated with uncertainty. Previously uncertainty associated with these parameters has been considered only one parameter at a time.
In this manuscript we give likelihood profile confidence intervals for the individual parameter estimates, which are more useful because they can be computed for (more extreme) data which is nearer to the boundaries of the solution space. Furthermore, we propose considering the mean proportion and the assessor heterogeneity as bivariate parameters, and investigating their joint uncertainty via confidence regions. Three approaches are proposed. First, likelihood confidence regions are obtained, with parameters corresponding to a 95% confidence level. Second, bootstrap procedures are used to obtain a scatter of parameter estimates from virtual panels over which either ellipses that enclose 95% of the bootstrap points, or 50% of points enclosed via the bagplot. The partial bootstrap is proposed for this purpose based on simulation studies involving three potential bootstrap procedures. Both likelihood and partial bootstrap confidence regions can be considered valid, with interpretation connecting to assumptions. Implications on statistical testing for the purposes of making conclusions related to sensory differences and sensory equivalencies are discussed.
Castura, J.C., Stachlewska, K.A., Brockhoff, P.B., & Christensen, R.H.B. (2019). Analyzing data using the chance-corrected beta-binomial model: Parameter estimates and their confidence regions. 13th Pangborn Sensory Science Symposium. 28 July-1 August. Edinburgh, UK. (Poster).