Joint modelling of choice and rating data: Theory and examples

In many cases, ordinal data, for example rating objects on a scale from 1 to 5, is observed only for those objects that have been chosen from a set of discrete alternatives, with no ratings for unchosen objects. An example is customer ratings of goods sold by online retailers. The joint modelling of choice and rating is made difficult by the missing ratings for unchosen alternatives. A method of jointly modelling choice and rating data termed a choice-ordered logit (COL) model is presented. Two types of COL model are defined: two-step, which places a positive probability on the chosen alternative not having the highest rating, and one-step, where the highest rated alternative is always chosen. Three case studies exemplifying the use of COL models are given. One uses simulated data and two use data from discrete choice experiments. It is shown that COL models can produce robust estimates. Two-step models provided a better fit than one-step, and most participants seemed to use two-step decision-making. However, a sizeable minority used one-step decision-making in one case study. It is argued that COL models have benefits over standard approaches, in particular adding information on strength-of-preference to discrete choices.