Incorporating the Luce–Krantz threshold model into cultural consensus theory for ordinal categorical data

Abstract

Cultural consensus theory (CCT), developed by Batchelder and colleagues in the mid-1980s, is a cognitively driven methodology to assess informants’ consensus in which the culturally correct answers are unknown to researchers a priori. The primary goal of CCT is to uncover the cultural knowledge, preferences or beliefs shared by group members. One of the models within CCT, the general Condorcet model (GCM), is designed for dichotomous response data (e.g., true/false) collected from a group of informants who share a common body of cultural knowledge. In many applied settings, however, responses are not limited to binary judgments; instead, informants often convey varying degrees of confidence, which are naturally represented using ordinal response formats such as Likert-type scales. To address this, we propose an extension of the GCM, termed the general Condorcet–Luce–Krantz (GCLK) model, which integrates the GCM with Luce–Krantz threshold theory to accommodate ordinal categorical responses. The model is intended for settings in which the latent cultural answer remains dichotomous while the observed response process is ordinal. In addition to finding out the consensus truth to the items, the GCLK also estimates other response characteristics, including the item-difficulty levels, informants’ competency levels, and guessing biases. We introduce the multicultural version of the GCLK that can help researchers detect the number of cultures for a given dataset. We use the hierarchical Bayesian modeling approach and the Markov chain Monte Carlo sampling method for estimation. A posterior predictive check is established to test the central assumptions of the model. Through a series of simulation studies, we provide a proof of concept by evaluating the applicability of the GCLK and demonstrate that the model achieves good parameter recovery.