Allenby-Rossi (2006) Hierarchical Bayes Model

Summary

Allenby and Rossi provide a pedagogical introduction to hierarchical Bayes (HB) models for marketing data, covering the motivation (limited individual-level data), the HB model specification, Gibbs sampling mechanics, and a full empirical application to the same N=946 credit card conjoint study used in Allenby-Ginter (1995) and Allenby-Rossi-McCulloch (2005). The chapter's empirical contribution is a complete 14×14 posterior covariance matrix $V_\beta$ (Table 20.3) and a refined $\Gamma$ matrix (Table 20.2) showing demographic effects on part-worths. The final 25 pages form an annotated bibliography of ~75 Bayesian marketing papers through 2004. The chapter is Chapter 20 (pp. 418–440) of The Handbook of Marketing Research: Uses, Misuses, and Future Advances, eds. R. Grover and M. Vriens, Sage (2006).

Key Claims

• HB models are necessary because individual-level marketing data is extremely sparse (15–20 conjoint responses, fewer than 20 scanner panel observations per household for most categories); hierarchical pooling from the population distribution is the only viable identification strategy.
• The HB logit model: $\Pr(i)_h = \exp[x_i'\beta_h]/\sum_j \exp[x_j'\beta_h]$; $\beta_h = \Gamma z_h + \xi_h$, $\xi_h \sim MVN(0, V_\beta)$ (multivariate normal); four-level Gibbs chain (per-respondent $\beta_h$, $\Gamma$, $V_\beta$, plus hyperpriors).
• Full $V_\beta$ diagonal (Table 20.3): Out-of-State (OOS) Bank = 15.9, Low Annual Fee = 13.5, Med. Variable Interest = 10.2, High Rebate = 8.5, Low Fixed Interest = 8.8, Long Grace Period = 5.0, High Credit Line = 6.3, Bank B = 3.5.
• Key off-diagonals: cov(Low Annual Fee, OOS Bank) = 8.1, $\rho$ = 0.55; cov(Low Fixed Interest, OOS Bank) = 2.1, $\rho$ = 0.18; cov(Low Annual Fee, Long Grace Period) = 5.1, $\rho$ = 0.62.
• Posterior mean of $\Gamma$ (Table 20.2): Low Fixed Interest intercept = 4.883; Out-of-State Bank intercept = −3.758; Low Annual Fee intercept = 4.158; Long Grace Period = 3.399.
• Demographic effects: females → Low Annual Fee coefficient = 1.302; age → Low Fixed Interest = −0.025/year (−1.25 utiles over 50 years); income → High Rebate = +0.021/$1K.
• Targeting: 7.5% of respondents exceed utility threshold 10.0 under low-fee product vs. 4.5% under low-interest, despite low interest having mean utility 0.7 utiles higher — driven entirely by the off-diagonal $V_\beta$ structure.
• Chain convergence: 20,000 iterations, convergence after ~6,000.
• Across dozens of studies, heterogeneity is better represented by a continuous (normal) distribution than by finite mixture (discrete) segments; asserting a small number of homogeneous groups is a systematic error.
• Markov chain Monte Carlo (MCMC) is described as "a breakthrough in statistical science" enabling models with thousands of parameters.

Concepts Introduced or Extended

• Conjoint Analysis — provides full $V_\beta$ covariance matrix (Table 20.3) and tutorial Gibbs derivation
• Bayesian Hierarchical Model — pedagogical treatment: Bayes theorem via heart-attack example, MCMC mechanics
• Gibbs Sampler — four-block chain for hierarchical logit; convergence at ~6,000/20,000 iterations
• Discrete Choice Models — motivates HB extension over aggregate logit

Entities Mentioned

• Greg M. Allenby
• Peter E. Rossi

Quotes

"Bayesian methods are ideally suited for analysis with limited data, and have resulted in new developments in modeling individual-level decision making, new characterizations of preferences and sensitivities across respondents, and models that include the analysis of a firm's decision in response to consumer demands."

"Across dozens of studies, the distribution of heterogeneity has been shown to be better represented by a continuous, not a discrete distribution (e.g., from a finite mixture model) of heterogeneity. This has important implications for analysis connected with market segmentation, where researchers often incorrectly assert the existence of a small number of homogeneous groups."

"Even though the average utility for low interest is 0.7 units larger, the low fee distribution has almost twice the mass in the region of the distribution corresponding to respondents with strong preference."

My Take

This is a tutorial chapter, not a research paper — its contributions are pedagogical and bibliographic rather than theoretical. The empirical application replicates the credit card study from Allenby-Ginter (1995) and Allenby-Rossi-McCulloch (2005) with slightly refined estimates. The chapter's distinctive value for the wiki is the complete 14×14 $V_\beta$ posterior covariance matrix (Table 20.3) not reproduced elsewhere, and the annotated bibliography providing a comprehensive map of Bayesian marketing applications through 2004. The heart-attack Bayes theorem example (sensitivity=0.80, specificity=0.70, likelihood ratio (LR)=2.67) is a clean pedagogical device. The coefficient estimates differ slightly from Allenby-Ginter 1995 (female/Low Annual Fee = 1.302 here vs. 1.092 there) — presumably from re-estimation with the full model rather than the simplified 1995 specification.