A hierarchical latent variable model for ordinal data from a customer satisfaction survey with "no answer" responses

We propose an item response theory model for ordinal customer satisfaction data where the probability of each response is a function of latent person and question parameters and of cutoffs for the ordinal response categories. This structure was incorporated into a Bayesian hierarchical model by Albert and Chib. We extend this formulation by modeling item nonresponse, coded as "no answer" (NA), as due to either lack of a strong opinion or indifference to the entire question. Because the probability of an NA is related to the latent opinion, the missing-data model is nonignorable. In our hierarchical Bayesian framework, prior means for the person and item effects are related to observed covariates. This structure supports model inferences about satisfaction of individual customers and about associations between customer characteristics and satisfaction levels or propensity to respond. We contrast this with exploratory and standard regression analyses that do not fully support these inferences. Our motivating example, an analysis of a DuPont Corporation 1992 customer satisfaction survey, is described in detail. The nonconjugate likelihood and prior prevent closed-form posterior inference. We present a Markov chain Monte Carlo solution using data augmentation. We diagnose case influence and identify outliers by importance reweighting, and apply posterior predictive model checks. The methods illustrated have application in other situations in which categorical observations can be determined by several latent variables.