Journal of Mathematical Psychology 61 (2014) 1–13 Contents lists available at ScienceDirect Journal of Mathematical Psychology journal homepage: www.elsevier.com/locate/jmp Cultural consensus theory for continuous responses: A latent appraisal model for information pooling R. Anders ∗ , Z. Oravecz, W.H. Batchelder Department of Cognitive Sciences, University of California, 3151 Social Science Plaza, Irvine, CA 92697-5100, United States highlights • A new Cultural Consensus Theory (CCT) model for continuous data. • Model-based clustering and detection of multiple cultures in the data. • Derivation of the mathematical and statistical properties of the model. • Hierarchical Bayesian inference for the model on real and simulated data. • User-friendly software that facilitates application of the model. article info Article history: Received 17 January 2014 Received in revised form 10 May 2014 Keywords: Cultural consensus theory Latent class models Latent trait models Continuous data Signal detection theory Mixture modeling Clustering abstract A Cultural Consensus Theory approach for continuous responses is developed, leading to a new model called the Continuous Response Model (CRM). It is a cognitive psychometric model that is applicable to consensus data, in which respondents (informants) have answered questions (items) regarding a shared knowledge or belief domain, and where a consensus (a latent set of ‘true’ answers applicable to the group) may exist. The model estimates the consensus answers to items, item difficulty, informant knowledge and response biases. The model can handle subcultures that have different consensuses from one another in the data, and can both detect and cluster respondents into these subcultures; it thus provides one of the first forms of model-based clustering for multicultural consensus data of the continuous response type. The model is demonstrated on both simulated and real multi-cultural data, using the hierarchical Bayesian framework for inference; two posterior predictive checks are developed to verify the central assumptions of the model; and software is provided to facilitate the application of the model by other researchers. Published by Elsevier Inc. 1. Introduction The purpose of the present paper is to introduce a Cultural Con- sensus Theory (CCT) model for continuous responses, and in tan- dem, supply user-friendly software that facilitates application of the model by others. CCT is a methodology conceived of in the mid 1980s (Batchelder & Romney, 1986, 1988; Romney, Weller, & Batchelder, 1986) that is applicable to consensus data: defined as data in which respondents (informants) have answered ques- tions (items) regarding a shared knowledge or belief domain, and where a consensus (a set of ‘true’ answers applicable to the group, or culture) may exist. Exemplary forms of consensus data may con- sist of eyewitness testimony, probability forecasting, political polls, ∗ Corresponding author. E-mail addresses: andersroyce@gmail.com, andersr@uci.edu (R. Anders), zoravecz@uci.edu (Z. Oravecz), whbatche@uci.edu (W.H. Batchelder). cultural beliefs, subjective assessment, or ideological beliefs. In or- der to estimate the consensus answers to items, as well as item response effects (e.g. knowledge level, response biases, item diffi- culty, and cultural membership) for these forms of data, CCT con- sists of a number of cognitive psychometric models. The first CCT model was developed for binary data (e.g. true/false data); it is called the General Condorcet Model (GCM), and makes the assumption that the consensus truth of each item is also a bi- nary value. This model has been widely applied in the social and behavioral sciences, especially cultural anthropology (e.g. Weller, 2007). The detection of multiple cultural truths (subcultures with differing consensuses), and cultural membership for the GCM, was developed by Anders and Batchelder (2012). An alternate assump- tion to that of the GCM, that continuous (fuzzy) truths in (0, 1) in- stead underlie binary data, was explored by Batchelder and Anders (2012). They introduced a new model for binary data that used a beta appraisal distribution to estimate these values in (0, 1); the Latent Truth Model (LTM), and these values could represent such http://dx.doi.org/10.1016/j.jmp.2014.06.001 0022-2496/Published by Elsevier Inc.