Bayesian estimation of multivariate-normal models when dimensions are absent

Multivariate economic and business data frequently suffer from a missing data phenomenon that has not been sufficiently explored in the literature: both the independent and dependent variables for one or more dimensions are absent for some of the observational units. For example, in choice based conjoint studies, not all brands are available for consideration on every choice task. In this case, the analyst lacks information on both the response and predictor variables because the underlying stimuli, the excluded brands, are absent. This situation differs from the usual missing data problem where some of the independent variables or dependent variables are missing at random or by a known mechanism, and the “holes” in the data-set can be imputed from the joint distribution of the data. When dimensions are absent, data imputation may not be a well-poised question, especially in designed experiments. One consequence of absent dimensions is that the standard Bayesian analysis of the multi-dimensional covariances structure becomes difficult because of the absent dimensions. This paper proposes a simple error augmentation scheme that simplifies the analysis and facilitates the estimation of the full covariance structure. An application to a choice-based conjoint experiment illustrates the methodology and demonstrates that naive approaches to circumvent absent dimensions lead to substantially distorted and misleading inferences.

[1]  S. Chib,et al.  Bayesian analysis of binary and polychotomous response data , 1993 .

[2]  Peter E. Rossi,et al.  The Value of Purchase History Data in Target Marketing , 1996 .

[3]  Peter E. Rossi,et al.  Marketing models of consumer heterogeneity , 1998 .

[4]  Roderick J. A. Little,et al.  Statistical Analysis with Missing Data , 1988 .

[5]  Michael D. Gordon,et al.  A utility theoretic examination of the probability ranking principle in information retrieval , 1991, J. Am. Soc. Inf. Sci..

[6]  Peter E. Rossi,et al.  A Bayesian analysis of the multinomial probit model with fully identified parameters , 2000 .

[7]  Peter E. Rossi,et al.  Bayesian analysis of the multinomial probit model , 2000 .

[8]  Pradeep K. Chintagunta,et al.  Investigating Category Pricing Behavior at a Retail Chain , 2002 .

[9]  D. V. Dyk,et al.  A Bayesian analysis of the multinomial probit model using marginal data augmentation , 2005 .

[10]  Patricia J. Labaw.  Advanced questionnaire design , 1981 .

[11]  Pradeep K. Chintagunta,et al.  Estimating a Multinomial Probit Model of Brand Choice Using the Method of Simulated Moments , 1992 .

[12]  Wayne S. DeSarbo,et al.  Bayesian inference for finite mixtures of generalized linear models with random effects , 2000 .

[13]  Greg M. Allenby,et al.  Reassessing Brand Loyalty, Price Sensitivity, and Merchandising Effects on Consumer Brand Choice , 1995 .

[14]  Michel Wedel,et al.  The information content of response latencies in conjoint choice experiments , 2000 .

[15]  Michael D. Gordon,et al.  When is the probability ranking principle suboptimal , 1992 .

[16]  A. Zellner.  An Introduction to Bayesian Inference in Econometrics , 1971 .

[17]  Peter E. Rossi,et al.  An exact likelihood analysis of the multinomial probit model , 1994 .

[18]  W. Wong,et al.  The calculation of posterior distributions by data augmentation , 1987 .

[19]  P. Lenk,et al.  Hierarchical Bayes Conjoint Analysis: Recovery of Partworth Heterogeneity from Reduced Experimental Designs , 1996 .

[20]  WedelMarco VriensTom Wansbeek.  Utility Covariancesand Context Effectsin Conjoint MNP Models , 1998 .

[21]  Terry Elrod,et al.  A Factor-Analytic Probit Model for Representing the Market Structure in Panel Data , 1995 .

[22]  D. Wise,et al.  A CONDITIONAL PROBIT MODEL FOR QUALITATIVE CHOICE: DISCRETE DECISIONS RECOGNIZING INTERDEPENDENCE AND HETEROGENEOUS PREFERENCES' , 1978 .

[23]  Eric T. Bradlow,et al.  A Learning-Based Model for Imputing Missing Levels in Partial Conjoint Profiles , 2004 .

[24]  Erik Brynjolfsson,et al.  Consumer Decision-Making at an Internet Shopbot , 2001 .

[25]  Erik Brynjolfsson,et al.  Consumer Decision-Making at an Internet Shopbot , 2001 .

[26]  Roderick J. A. Little,et al.  Statistical Analysis with Missing Data: Little/Statistical Analysis with Missing Data , 2002 .

[27]  Michel Wedel,et al.  Response Latencies in the Analysis of Conjoint Choice Experiments , 2000 .

[28]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .

[29]  Michel Wedel,et al.  Conjoint Choice Experiments: General Characteristics and Alternative Model Specifications , 2007 .

[30]  Greg M. Allenby,et al.  Modeling Household Purchase Behavior with Logistic Normal Regression , 1994 .