Conjoint Analysis

Definition

Conjoint analysis is a survey-based technique for measuring how consumers trade off product attributes. Respondents evaluate or choose among product profiles that vary on multiple attribute levels; the inferred part-worths (utilities for each attribute level) reveal the relative importance of attributes in purchase decisions. Choice-based conjoint (CBC), in which respondents select a preferred alternative from a set, is the standard format in modern marketing research and pairs naturally with multinomial logit or probit likelihood specifications.

Key Ideas

Random utility model: $\Pr_h(i) = \exp(x_i'\beta_h)/\sum_j \exp(x_j'\beta_h)$ links attribute-level vectors $x_i$ to choice probabilities via household-specific part-worth vectors $\beta_h$ .
Individual-level estimation problem: fractional conjoint designs give each respondent far fewer observations than parameters to estimate; hierarchical pooling is essential for reliable respondent-level part-worths.
Hierarchical Bayes logit (HB conjoint): $\beta_h = \Gamma z_h + \xi_h$ , $\xi_h \sim \mathcal{N}(0, V_\beta)$ , where $z_h$ is a vector of respondent demographics and $\Gamma$ maps demographics to mean part-worths; individual posteriors $\pi(\beta_h|\text{data})$ emerge from the Gibbs chain.
Covariance matrix $V_\beta$ has managerial content: off-diagonal elements reveal which attribute preferences co-vary across respondents — positive covariance between attribute levels $A$ and $B$ means respondents who highly value $A$ also tend to highly value $B$ ; this determines which product configurations attract the strongest preference tails.
Focusing on extremes: target marketing is inherently a tail-area problem. The fraction of respondents above a profitability threshold depends critically on $V_\beta$ , not just on the mean utility $\Gamma z_h$ .

How It Works

Study Design

Attributes (e.g., interest rate, annual fee, bank brand) are crossed with attribute levels (high/medium/low) to form a full factorial; fractional profiles keep the questionnaire manageable. Each respondent evaluates 13–17 paired comparisons in a typical commercial study.

Four-Block Gibbs Sampler (Allenby-Rossi-McCulloch 2005, eqs. 16–19)

Under the hierarchical logit model:

$y_h | x, \beta_h \sim \text{Logit}(x, \beta_h) \qquad (16)$ $\beta_h | z_h, \Gamma, V_\beta \sim \mathcal{N}(\Gamma z_h, V_\beta) \qquad (17)$ $\Gamma | a, A \sim \text{prior} \qquad (18)$ $V_\beta | w, W \sim \text{Inv-Wishart} \qquad (19)$

Gibbs cycle:

Draw each $\beta_h$ one respondent at a time given $\{y_h, x\}$ and current hyper-parameters (Metropolis step for the logit likelihood)
Draw $\Gamma | \{\beta_h\}, V_\beta$ — Normal-Normal conjugate
Draw $V_\beta | \{\beta_h\}, \Gamma$ — Inverse-Wishart conjugate
Repeat

Conditional independence: given $\{\beta_h\}$ , blocks 2–3 do not depend on the raw choices — all information flows through $\{\beta_h\}$ .

Original Extremes Paper (Allenby-Ginter 1995)

The foundational application uses a continuous-normal (rather than logit) paired-comparison likelihood: $y_{i,h} = x_{i,h}'(\alpha + \beta_h) + \varepsilon_{i,h}$ , $\varepsilon_{i,h} \sim N(0, \sigma^2)$ , where $\alpha$ is a common fixed-effects vector and $\beta_h \sim \mathcal{N}(\Gamma z_h, D)$ (Multivariate Normal, MVN). The six-block Gibbs sampler cycles through $\alpha$ , each $\beta_h$ , $\Gamma$ , $D$ , $\sigma^2$ , and a rejection-sampling step for the paired-comparison latent utilities; priors: $D \sim \mathrm{IW}(15, 15I)$ (Inverse-Wishart), $\sigma^2 \sim \text{Inv-}\chi^2(1, 5)$ ; 2000 iterations, last 1000 used.

The extremes argument: Low annual fee and low interest rate have almost identical mean utility (~4.2 utiles each). Yet 24.2% of respondents exceed net utility > 5.0 for low fee vs. only 19.2% for low interest, because cov(low fee, out-of-state bank) = 5.34 produces a fatter right tail. This translates to 111 vs. 58 respondents with >70% segment membership probability. The insight: product design is a tail-area problem, and the right objective is to find the product that generates the largest extreme segment.

Credit Card Application (Allenby-Rossi-McCulloch 2005)

$N=946$ bank customers, 14,799 paired comparisons, 7 attributes (interest rate, rewards, annual fee, bank brand, rebate, credit line, grace period), plus 3 demographic covariates (age, income, gender). Key results from posterior mean of $\Gamma$ :

Out-of-state bank penalty: −3.614 utiles (average respondent)
Older respondents less sensitive to interest rates (−0.017 per year of age)
Richer respondents more sensitive to rebates (+0.021 per $1,000 income)
Females particularly responsive to low annual fees

Tail-area comparison: suppose the target market requires net utility > 10. Low annual fee puts 7.5% of respondents above threshold; low fixed interest puts only 4.5% — despite low interest having higher mean utility (+0.8 utiles). The difference is driven by $V_\beta$ : covariance(low fee, out-of-state) = 8.5 ( $\rho=0.80$ ) vs. covariance(low interest, out-of-state) = 1.9 ( $\rho=0.23$ ). The low-fee distribution has nearly twice the tail mass even though it has lower mean.

Full $V_\beta$ diagonal (from Allenby-Rossi HB Chapter, Table 20.3): out-of-state (OOS) Bank = 15.9, Low Annual Fee = 13.5, Med. Variable Interest = 10.2, Low Fixed Interest = 8.8, High Rebate = 8.5, High Credit Line = 6.3, Long Grace Period = 5.0, Bank B = 3.5, Med. Annual Fee = 4.1, Rewards programs = 1.2–3.0. Key off-diagonals: cov(Low Fee, OOS Bank) = 8.1 ( $\rho=0.55$ ); cov(Low Fixed Int., OOS Bank) = 2.1 ( $\rho=0.18$ ); cov(Low Fee, Long Grace) = 5.1 ( $\rho=0.62$ ). The $\Gamma$ matrix (Table 20.2): Low Fixed Interest intercept = 4.883, OOS Bank = −3.758; female → Low Annual Fee = 1.302, age → Low Fixed Interest = −0.025/year.

Point Estimates vs. Full Posterior

Using only posterior means of $\Gamma$ and $V_\beta$ (plug-in) systematically overestimates market potential by ignoring hyper-parameter uncertainty. The correct tail-area uses all posterior draws of $\{\beta_h\}$ :

$\text{Market potential} = \frac{1}{G}\sum_{g=1}^G \mathbf{1}[\text{net utility}^{(g)} > \text{threshold}]$

Why It Matters

HB conjoint produces individual posteriors that enable tail-area targeting, optimal product design, and coupon valuation; all require nonlinear functions of $\beta_h$ for which point estimates introduce systematic overconfidence.
Prior to HB, individual-level part-worths were unreliable (16 observations to estimate 16 part-worths in a typical full-profile design); hierarchical pooling resolves the identification problem by borrowing from the population distribution.
The covariance structure $V_\beta$ is the key output for product design decisions — it tells the firm which market segments have extreme preferences for specific configurations and which incentives will efficiently attract them.

Open Questions

Adaptive conjoint design: which profiles to present given respondent answers so far?
Non-normal heterogeneity: when preferences are genuinely multimodal (distinct segments), normal mixing understates structure; mixture-of-normals for $\pi(\beta_h)$ or Dirichlet process priors are alternatives.
Partial-profile designs: when respondents evaluate subsets of attributes, bridging across respondents requires additional modelling assumptions.
Scalability: with many attributes and many respondents, the Gibbs chain (especially the $\beta_h$ block) is computationally intensive; modern implementations use parallelized per-respondent draws.