Bayesian Analysis of Individual Choice Behavior With Aggregate Data

The statistical analysis of individual choice behavior can be hampered by aggregate information. When a choice response variable is fully observed, individual choice behavior is typically studied with generalized linear models with the logit or probit link function. The individual choice response variable is, however, often available in an aggregate form because of data confidentiality, privacy protection, and a limited database quota. Then the generalized linear models are no longer directly applicable to such aggregate data. In this article, we confront such an aggregate data problem by using the method of data augmentation, where latent individual choice responses are augmented while satisfying a constraint on their aggregate sum. To do so, we devise the novel choice-wise sampling algorithm for a generalized linear model with aggregate binary responses. The proposed algorithm is applied to the 2006 Pennsylvania gubernatorial election data, where only aggregate votes cast for each candidate are available because of the privacy protection of voters. This article has supplementary material online.

[1]  M. Gail,et al.  Likelihood calculations for matched case-control studies and survival studies with tied death times , 1981 .

[2]  Aviv Nevo.  Measuring Market Power in the Ready-to-Eat Cereal Industry , 1998 .

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

[4]  Jun S. Liu,et al.  Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes , 1994 .

[5]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[6]  A A Schuessler,et al.  Ecological inference. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Xiao-Li Meng,et al.  The Art of Data Augmentation , 2001 .

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

[9]  Jun S. Liu,et al.  STATISTICAL APPLICATIONS OF THE POISSON-BINOMIAL AND CONDITIONAL BERNOULLI DISTRIBUTIONS , 1997 .

[10]  Steven T. Berry,et al.  Automobile Prices in Market Equilibrium , 1995 .

[11]  D. V. van Dyk,et al.  Partially Collapsed Gibbs Samplers: Illustrations and Applications , 2009 .

[12]  J. Albert.  Bayesian Estimation of Normal Ogive Item Response Curves Using Gibbs Sampling , 1992 .

[13]  Sha Yang,et al.  Estimating Disaggregate Models using Aggregate Data through Augmentation of Individual Choice , 2007 .

[14]  Jun S. Liu,et al.  Weighted finite population sampling to maximize entropy , 1994 .

[15]  Xiao-Li Meng,et al.  Fitting Full-Information Item Factor Models and an Empirical Investigation of Bridge Sampling , 1996 .

[16]  M. Schervish,et al.  A Bayesian Approach to a Logistic Regression Model with Incomplete Information , 2008, Biometrics.

[17]  A Gibbs sampler for mixed logit analysis of differentiated product markets using aggregate data , 2007 .

[18]  Eric T. Bradlow,et al.  Bayesian Estimation of Random-Coefficients Choice Models Using Aggregate Data , 2006 .