Hierarchical Bayes Conjoint Choice Models - Model Framework, Bayesian Inference, Model Selection, and Interpretation of Estimation Results
Authors: Goeken Nils , Kurz Peter , Steiner Winfried 
Published: December 2021

Choice-based conjoint (CBC) is nowadays the most widely used variant of conjoint analysis, a class of methods for measuring consumer preferences. The primary reason for the increasing dominance of the CBC approach over the last 35 years is that it closely mimics real choice behavior of consumers by asking respondents repeatedly to choose their preferred alternative from a set of several offered alternatives (choice sets). Within the framework of CBC analysis, the multinomial logit (MNL) model is the most frequently used discrete choice model due to the existence of closed form solutions for conditional choice probabilities. The popularity of CBC and the MNL model has grown even more since the introduction of hierarchical Bayesian (HB) estimation techniques that accommodate individual consumer heterogeneity in choice data, and which have now become state-of-the-art in marketing theory and practice. Still, researchers and practitioners have to make further decisions under this framework (CBC, MNL, HB estimation), such as how to represent preference heterogeneity. Here, using a normal distribution (and therefore a unimodal distribution) has become the standard approach in the marketing literature. However, the thin tails of the normal distribution suggest that the standard HB-MNL model should not be the “go-to” approach to approximate multimodal preference distributions, because individual preference patterns lying at the tails of the normal distribution (i.e., that do not fit well with the assumption of a unimodal distribution) tend to be shrunk to the population mean. This shrinkage, especially in multimodal data settings, could mask important information (e.g., new or different structures in the data). A mixture of normal distributions avoids this limited flexibility of the most simple continuous approach of assuming a unimodal prior heterogeneity distribution. There are currently two prominent HB-CBC modeling approaches embedding the mixture-of-normals (MoN) approach: the more widespread MoN-HB-MNL model, and the Dirichlet process mixture (DPM)-HB-MNL model. In this article, we review the prominent HB-MNL model (with its normal prior), the MoN-HB-MNL model, and the DPM-HB-MNL model and apply them to an empirical multi-country CBC data set. We compare the statistical performance of the three models in terms of goodness-of-fit and predictive accuracy, show how to include consumer background characteristics in the upper level of these models, and illustrate how to interpret the estimation results (with a special focus on cross-county heterogeneity). In sum, our article serves as a kind of user guide to the estimation and interpretation of Hierarchical Bayes Conjoint Choice Models. For our data, we observed that all three choice models (both with and without consumer background characteristics) resulted in a one-component solution. The DPM-HB-MNL model nevertheless yielded a higher cross-validated hit rate compared to the MoN-HB-MNL and the HB-MNL models due to its even more flexible prior assumptions. The two latter models tended to slightly overfit our empirical data, which was reflected by higher goodness-of-fit statistics but a lower predictive accuracy compared to the DPM-HB-MNL model. We showed that this result could be attributed to the weaker extent of Bayesian shrinkage of these two models. The DPM-HB-MNL model showed a stronger shrinkage effect and seems therefore somewhat more robust against overfitting. Including consumer background characteristics in terms of country of origin information for the respondents did not improve the statistical model performance (especially not the predictive performance). Still, using the country of origin information for respondents in a post-hoc segmentation analysis helped us to explain some differences in brand preferences between the five countries.