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A Bayesian Multinomial Probit Model for The Analysis of Panel Choice Data

Published online by Cambridge University Press:  01 January 2025

Duncan K. H. Fong ,
Sunghoon Kim ,
Zhe Chen  and
Wayne S. DeSarbo
Duncan K. H. Fong
Affiliation:
The Pennsylvania State University
Sunghoon Kim
Affiliation:
Arizona State University
Zhe Chen
Affiliation:
Google Inc.
Wayne S. DeSarbo*
Affiliation:
The Pennsylvania State University
*
Correspondence should be made to Wayne S. DeSarbo, The Pennsylvania State University, University Park, PA 16802 USA. Email: wsd6@psu.edu
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Abstract

A new Bayesian multinomial probit model is proposed for the analysis of panel choice data. Using a parameter expansion technique, we are able to devise a Markov Chain Monte Carlo algorithm to compute our Bayesian estimates efficiently. We also show that the proposed procedure enables the estimation of individual level coefficients for the single-period multinomial probit model even when the available prior information is vague. We apply our new procedure to consumer purchase data and reanalyze a well-known scanner panel dataset that reveals new substantive insights. In addition, we delineate a number of advantageous features of our proposed procedure over several benchmark models. Finally, through a simulation analysis employing a fractional factorial design, we demonstrate that the results from our proposed model are quite robust with respect to differing factors across various conditions.

Keywords

Bayesian analysisheterogeneitymultinomial probit modelpanel dataparameter expansionmarketingconsumer psychology
Type
Original paper
Information
Psychometrika , Volume 81 , Issue 1 , March 2016 , pp. 161 - 183
Copyright
Copyright © 2014 The Psychometric Society

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References

Addelman, S. (1962). Orthogonal main-effect plans for asymmetrical factorial experiments. Technometrics, 4, 21–46.CrossRefGoogle Scholar
Albert, J.H., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88(422), 669–679.CrossRefGoogle Scholar
Barnard, J., McCulloch, R., & Meng, X. (2000). Modeling covariance matrices in terms of standard deviations and correlations, with applications to shrinkage. Statistica Sinica, 10, 1281–1311.Google Scholar
Burgette, L.F., & Nordheim, E.V. (2012). The trace restriction: An alternative identification strategy for the Bayesian multinomial probit model. Journal of Business and Economic Statistics, 30(3), 404–410.CrossRefGoogle Scholar
Chib, S., & Greenberg, E. (1998). Analysis of multivariate probit models. Biometrika, 85(2), 347–361.CrossRefGoogle Scholar
Chib, S., Greenberg, E., Chen, Y. (1998). MCMC methods for fitting and comparing multinomial response models. Working Paper, Olin School of Business, Washington University.Google Scholar
Daganzo, C. (1980). Multinomial probit. New York: Academic Press.Google Scholar
Dawid, A. (1981). Some matrix-variate distribution theory: Notational considerations and a Bayesian application. Biometrika, 68(1), 265–274.CrossRefGoogle Scholar
DeSarbo, W.S. (1982). GENNCLUS: New models for general nonhierarchical clustering analysis. Psychometrika, 47(4), 449–475.CrossRefGoogle Scholar
DeSarbo, W.S., & Carroll, J.D. (1985). Three-way metric unfolding via alternating weighted least squares. Psychometrika, 50(3), 275–300.CrossRefGoogle Scholar
DeSarbo, W.S., & Cron, W.L. (1988). A maximum likelihood methodology for clusterwise linear regression. Journal of Classification, 5(2), 249–282.CrossRefGoogle Scholar
Dotson, J., Lenk, P., Brazell, J., Otter, T., MacEachern, S., Allenby G. M. (2010). A probit model with structured covariances for similarity effects and source of volume calculations. Working paper.CrossRefGoogle Scholar
Fiebig, D.G., Keane, M.P., Louviere, J., & Wasi, N. (2010). The generalized multinomial logit model: Accounting for scale and coefficient heterogeneity. Marketing Science, 29, 393–421.CrossRefGoogle Scholar
Fong, D.K.H., Ebbes, P., & DeSarbo, W.S. (2012). A heterogeneous Bayesian regression model for cross sectional data involving a single observation per response unit. Psychometrika, 77(2), 293–314.CrossRefGoogle Scholar
Gupta, A.K., & Nagar, D.K. (2000). Matrix variate distributions. Monographs and surveys in pure and applied mathematics (Vol. 104). London: Chapman & Hall/CRC.Google Scholar
Hausman, J., & Wise, D. (1978). A conditional probit model for qualitative choice: Discrete decisions recognizing interdependence and heterogeneous preferences. Econometrica, 45, 319–339.Google Scholar
Hobert, J.P., & Marchev, D. (2008). A theoretical comparison of the data augmentation, marginal augmentation and px-da algorithms. Annals of Statistics, 36(2), 532–554.CrossRefGoogle Scholar
Imai, K., & van Dyk, D.A. (2005). A Bayesian analysis of the multinomial probit model using marginal data augmentation. Journal of Econometrics, 124(2), 311–334.CrossRefGoogle Scholar
Jedidi, K., & DeSarbo, W.S. (1991). A stochastic multidimensional scaling procedure for the spatial representation of three-mode, three-way pick any/J data. Psychometrika, 56(3), 471–494.CrossRefGoogle Scholar
Liechty, J.C., Liechty, M.W., & Muller, P. (2004). Bayesian correlation estimation. Biometrika, 91(1), 1–14.CrossRefGoogle Scholar
Liu, C. (2001). Discussion on the art of data augmentation. Journal of Computational and Graphical Statistics, 10(1), 75–81.CrossRefGoogle Scholar
Liu, J.S., & Wu, Y.N. (1999). Parameter expansion for data augmentation. Journal of the American Statistical Association, 94(448), 1264–1274.CrossRefGoogle Scholar
Liu, X., & Daniels, M. (2006). A new efficient algorithm for sampling a correlation matrix based on parameter expansion and re-parameterization. Journal of Computational and Graphical Statistics, 15(4), 897–914.CrossRefGoogle Scholar
Maydeu-Olivares, A., & Hernández, A. (2007). Identification and small sample estimation of Thurstone’s unrestricted model for paired comparisons data. Multivariate Behavioral Research, 42(2), 323–347.CrossRefGoogle ScholarPubMed
McCulloch, R., & Rossi, P.E. (1994). An exact likelihood analysis of the multinomial probit model. Journal of Econometrics, 64, 207–240.CrossRefGoogle Scholar
McCulloch, R.E., Polson, N.G., & Rossi, P.E. (2000). A Bayesian analysis of the multinomial probit model with fully identified parameters. Journal of Econometrics, 99(1), 173–193.CrossRefGoogle Scholar
Nobile, A. (1998). A hybrid Markov chain for the Bayesian analysis of the multinomial probit model. Statistics and Computing, 8, 229–242.CrossRefGoogle Scholar
Nobile, A. (2000). Comment: Bayesian multinomial probit models with normalization constraint. Journal of Econometrics, 99(1), 335–345.CrossRefGoogle Scholar
Rossi, P.E., Allenby, G.M., & McCulloch, R.E. (2005). Bayesian statistics and marketing. Chichester: Wiley.CrossRefGoogle Scholar
Rossi, P.E., McCulloch, R.E., & Allenby, G.M. (1996). The value of purchase history data in target marketing. Marketing Science, 15(4), 321–340.CrossRefGoogle Scholar
Rousseeuw, P., & Molenberghs, G. (1994). The shape of correlation matrices. The American Statistician, 48, 276–279.CrossRefGoogle Scholar
Thurstone, L. (1927). A law of comparative judgment. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
Tsai, R. (2000). Remarks on the identifiability of Thurstonian ranking models: Case V, case III, or neither?. Psychometrika, 65(2), 233–240.CrossRefGoogle Scholar
Tsai, R. (2003). Remarks on the identifiability of Thurstonian paired comparison models under multiple judgment. Psychometrika, 68(1), 361–372.CrossRefGoogle Scholar