Cultural Mixture Modeling: Identifying Cultural Consensus (and Disagreement) using Finite Mixture Modeling Shane T. Mueller (smueller@ara.com) and Elizabeth S. Veinott (bveinott@ara.com) Cognitive Science Group, Klein Associates Division, ARA Inc., Fairborn OH 45324 USA Abstract In this paper, we describe a new technique for identifying cultural consensus called Cultural Mixture Modeling (CMM). This technique adopts finite mixture modeling, and introduces a new probabilistic formulation of agreement, which we call the strong consensus model. We use this technique to examine the cultural belief data from Weller (1983; 1984) and social network data from Krackhardt (1987). We show that CMM can go beyond classic models of consensus and identify sit- uations in which multiple distinct but disagreeing beliefs ex- ist between subgroups of individuals. By identifying groups of shared belief, CMM offers a practical and useful technique for understanding and characterizing how socio-cultural fac- tors influence our beliefs and attitudes. Keywords: culture, mental models, consensus theory Background Understanding the underlying beliefs, attitudes, and mental models of individuals is an important goal in a number of domains of cognitive science. This is important for applied problems (in which these mental models might be elicited in order to develop training, design system interfaces, or under- stand a target population), as well as basic research problems (e.g., identifying a concept’s conceptual coherence; determin- ing typical associations from verbal stimuli). We have found it is especially useful when studying how those beliefs and mental models are affected by social or cultural factors, and identifying how different beliefs lead to different behaviors. For example, one sociological view of culture (cf. Atran, Medin & Ross, 2005; Sieck, Smith & McHugh, 2007) holds that culture is comprised primarily of the shared beliefs and practices of a group, rather than just the demographic and linguistic characteristics commonly equated with culture. A pernicious problem faced when eliciting such knowledge is in knowing whether variation among respondents simply repre- sents random noise, or whether that variation represents some more fundamental differences in what a group of individuals believe. One method that has been developed to understand whether a group of people share a set of common beliefs is called Cultural Consensus Theory (CCT; Romney, Batchelder, & Weller, 1986). CCT is a set of statistical tools designed to assess agreement in belief or knowledge among a set of respondents. Perhaps CCT’s most profound insight is that culturally-correct responses can be determined “without the answer key”: the culturally-correct beliefs are the ones that most members of that culture consistently agree with. CCT uses a matrix-algebra procedure known as eigenfactor de- composition to determine whether or not a consensus exists. This procedure starts by forming a dissimilarity matrix across respondents, and then decomposes the matrix into its princi- ple components, thus determining whether a consensus exists among the respondents. In essence, CCT is similar to factor analysis performed on the responses of a survey, but instead of determining sets of questions for which respondents give similar responses (i.e., the columns), it determines sets of re- spondents who share similar beliefs (i.e., the rows). If the respondents are well described by a single factor, then a con- sensus is deemed to exist. If a consensus does exist, one can estimate the extent to which each respondent agrees with the dominant belief set. Romney, Batchelder and Weller (1986) refer to this agree- ment as cultural competence. Cultural competence has of- ten been found to be related to demographic factors such as age (for example, older and more experienced individuals are more likely to believe the culturally correct answer). Thus, by determining the culturally-correct answers, CCT allows each individual to be given a score showing how well they know those answers. Limitations of Cultural Consensus Theory Although CCT has proven useful in understanding whether respondents in a survey or interview share common beliefs, it is not without its limitations. The most obvious limitation is that the model only determines whether or not an overall con- sensus exists, but not whether there are multiple subcultures who believe different things. If a consensus does not exist, there are several plausible explanations that CCT cannot dis- tinguish between. One possibility is that there is no consensus because each respondent is essentially unique. Another pos- sibility is that there are several subsets of consistent beliefs. As an illustration (expanded in Demonstration 1), consider using this method to understand the positions of U.S. politi- cians. Across a political body (such as the U.S. Senate), a consensus would be unlikely. However, lack of consensus does not mean that each Senator’s response patterns are com- pletely unique: we would likely find a handful of coherent be- liefs aligned with political party membership and geographic region. CCT can determine whether members agree, but if they do not agree, it is incapable of providing much insight without placing a priori beliefs about what the groups should be (e.g., political affiliation). But in that case, CCT may not be necessary; we can simply compare the range of responses for each pre-defined sub-group and determine whether they differ. The insights of CCT have proven useful, but some of its restrictions are difficult to surmount in principled ways. To address some of these problems, we have adapted a statisti- cal technique called finite mixture modeling (FMM; Leisch,