A Probit Model with Structured Covariance for Similarity Effects and Source of Volume Calculations

Distributional assumptions for random utility models play an important role in relating observed product attributes to choice probabilities. Choice probabilities derived with independent errors have the IIA property, which often does not match consumer behavior and leads to inaccurate source of volume predictions. Correlated errors, such as in the correlated probit model, give more realistic results. However, the source of the correlation among the utility functions for different alternatives is often not well studied. In practice, covariance matrices are frequently associated with presentation order or brands in conjoint studies. However, other structures allow richer specifications of substitution patterns. In this paper, we parameterize the covariance matrix for probit models so that similar brands in the preference space have higher correlation than dissimilar brands, resulting in higher rates of substitution. We investigate alternative measures of similarity in the context of a conjoint model, and compare the resulting substitution patterns to those of standard choice models. The proposed model fits the data better and results in more realistic measures of substitution for a product line extension. The structured covariance matrix approach allows marketing managers to predict the substitution pattern for product profiles not included in the conjoint analysis