</a>Polyhedral methods of conjoint analysis

Polyhedral Empowerment of Networks through Symmetry: psycho-social implications for organization and global governance (Part #9)

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Conjoint analysis, also called multi-attribute compositional models or stated preference analysis, is a statistical technique that originated in mathematical psychology. Today it is used in many of the social sciences and applied sciences including marketing, product management, and operations research.

Olivier Toubia (New Approaches to Idea Generation and Consumer Input in the Product Development Process, 2001) describes a method, dependent on computer support, for an adaptive question design method that attempts to reduce respondent burden while simultaneously improving accuracy. For each respondent the question design method dynamically adapts the design of the next question using that respondent's answers to previous questions. The adaptive method interprets question de-sign as a mathematical program and estimates the solution to the program using recent develop-ments based on the interior points of polyhedra.

Toubia begins with a conceptual description that highlights the geometry of the conjoint-analysis parameter space, permitting analyses of decision challenges involving many "dimensions" -- understood as distinct features of a product (3 to 100, say) which respondents are called upon to evaluate. The polyhedral method is designed to "shrink" the feasible set of features -- reducing its dimensionality -- determining the key features of the product design. The respondent's answers to the first q questions define a (p-q)-dimensional hy-perplane which intersects the initial p-dimensional polyhedron to give a (p-q)-dimensional polyhedron, namely one of lower dimensionality. The challenge is to select questions that reduce the dimensionality of the polyhedron as fast as possible.

Conjoint analysis is one of a set of techniques of multivariate analysis in which, for example, potential customers are asked to compare pairs of products and make judgements about their similarity (or their dissimilarity in the case of ordination statistics). By contrast, whereas techniques such as factor analysis, discriminant analysis, and conjoint analysis obtain underlying dimensions from responses to product attributes identified by the researcher, multidimensional scaling obtains the underlying dimensions from respondents' judgements about the similarity of products rather than being dependent on researchers' judgments (in furnishing a list of attributes to be shown to the respondents). The underlying dimensions then emerge from respondents' judgements about pairs of products making it the most common approach to perceptual mapping.

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