Conjoint Analysis Applications Research Paper

(a) independent variables or ‘attributes’ that explain preferences are identified and defined (e.g., bus travel times and fares; MHz and prices of PCs);

(b) attribute ‘levels’ relevant to the research are assigned (e.g., 30, 60 min travel time or $1, $3 fare for buses; 600, 733 MHz and $2,500, $2,850 prices for PCs);

(c) statistical design techniques are used to combine the attribute levels and create descriptions of options (e.g., 15 min travel time for $2 or 733 MHz for $2,850); (d) a sample of individuals evaluate all or a subset of the descriptions;

(e) individual- or group-level models are estimated from the evaluation data (e.g., transport modes or PC options); and

(f) the estimated models are used to measure subjective quantities (e.g., the subjective value of 600 vs. 733 MHz or implied willingness to pay to save 30 min travel time).

Despite commonalities, CA methods also differ in experimental designs, response tasks, model form(s), and measurement of subjective quantities. For example, initial CA applications were based on an axiomatic theory of the behavior of a ranking of all possible combinations of a discrete set of attribute levels (e.g., Krantz and Tversky 1971). All possible combinations of attribute levels are given by complete factorial enumeration, such that if A₁(I ), A₂(J ), …, A_N(N ) represent the number of levels of attributes 1,2, …, N, then all combinations are given by the expression I×J×···× N. Thus, in a full factorial experiment the total combinations equal the product of the levels of each attribute. For example, if the attributes (levels) are bus fare (3), travel time (3), and service frequency (3), then there are 3×3×3 (or 3²)=27 possible bus descriptions.

If a subject ranks all designed descriptions in order of preference (or another subjective dimension), and certain axiomatic conditions are satisfied, the ranking can be represented by an algebraic model defined by marginal and joint attribute effects. For example, for two or more factors, a ranking of a factorial set of options is additive in the attribute effects, if, and only if, the effects of all attributes on the ranking are independent of one another and there are no joint attribute effects (i.e., only marginal effects apply). That is, an additive process underlying a ranking of all combinations of, say, the ith, jth, and kth levels of attributes A_1i, A_2j, and A_3k, respectively, can be expressed as:

where V_ijk is the overall value of attribute level combination ijk, c is a constant that sets the response scale origin and v(A_1i), (A_2j), and (A_3k) are values of the ith, jth, and kth levels of each attribute.

Statistically, if the underlying process is additive, all attribute interactions will be nonsignificant. If the underlying process is multiplicative, the model is as follows:

where all terms were previously defined, except c*, a constant to set the scale origin. Equation (2) requires all attribute main effects and interactions to be significant. Early applications lacked error theories, hence used goodness-of-fit tests to evaluate and compare models.

Full factorial designs proved too large, so ways to reduce numbers of combinations were proposed, such as fractional or other partial factorial designs, ranking all pairs of attribute levels or using incomplete block designs to select attributes and levels (e.g., Green 1974). Most applications rely on non replicated fractional factorials that cannot be used to test model forms and produce few observations relative to parameters for individuals (Louviere 1988). However one can estimate individual models if response, design, and model assumptions are satisfied (e.g., equal interval response scales, insignificant interactions, independent and identically distributed normal errors).

A new paradigm emerged when McFadden (1974) developed computationally tractable extensions of RUT to the multiple-choice case. This provided ways to estimate and test choice models with a formal theory of errors based on RUT. RUT assumes an unobservable value, U_i, for each of i=1, 2, …, I options that is a sum of observable (explainable) and random (unexplainable) components:

where V_i and ε_i are, respectively, observable and random components. RUT assumes that individuals try to choose their preferred options, but the random component implies that choices are inherently stochastic from a researcher’s perspective: that is, the probability that a subject will choose an option from a set of options is predictable, not the exact choice. If P(i|C ) is the probability of choosing option i in set C, and Max = maximum operator, then

Different discrete choice models are derived by making assumptions about the distributions of the ε’s. For example, McFadden (1974) derived the multinomial logit model (MNL) by assuming that the ε’s were independently and identically distributed Gumbel random variates. MNL choice probabilities have a simple closed-form, but if the ε’s are non-independently and identically distributed normals, the nonclosed form multinomial probit (MNP) model results.

Louviere and Woodworth (1983) integrated RUT-based choice models with CA, and proposed ways to design choice experiments consistent with RUT-based choice models. This choice-based (CB) approach designs attribute level combinations (to estimate evaluation utility functions) and selects choice sets in which they appear (to satisfy properties of choice models), and the error theory is used to select and test model forms. The CB approach typically estimates models from samples of subjects because individuals provide too few observations to satisfy asymptotic theory and obtain reliable model estimates.

Choice experiments mimic real choice situations. For example, to model choice of travel mode to work, one designs experiments to offer commuters sets of mode choices; to model choice of health insurance plan, one designs experiments that offer subjects sets of health insurance options. Thus, the CB approach designs attribute level combinations to describe sets of options (say M of them). If M 1, subjects evaluate each option separately, making M binary choices (e.g., will will not choose); if M 1, subjects evaluate sets of multiple options (say S of them), making S multiple choices (Louviere and Woodworth 1983, Louviere et al. 2000).

Significant progress has been made in linking behavior in choice experiments and real environments. Specifically, in RUT models the magnitude (or ‘scale’) of the parameters are inversely proportional to choice variability. For example, if we denote the scale as λ, it can be defined as follows for MNL modvels (Ben-Akiva and Lerman 1985 p p. 104–5): λ²= π² /6σ_ε² (π is the natural constant; σ_ε² error variance). In empirical applications, analysts estimate the quantities λβ_k, not β_k ( βk is a k-element parameter vector associated with design matrix, X_ki, and the ith choice option).

Thus, λ is not identified in any one data source, and typically is set equal to one for convenience. Ben-Akiva and Morikawa (1990) showed that ratios of λ’s can be identified and estimated from two or more choice data sources; hence, if experimental and actual choices differ only in response variability, parameters estimated from each will be proportional. Empirical tests in several fields suggest that parameter proportionality often holds to a close first approximation (see Louviere et al. 1999).

This property of RUT-based choice models allows many sources of preference and choice data to be compared, including traditional CA data. For example, one can transform CA ranks into implied choice sets by ‘exploding’ subjects’ complete or partial orders into one or more choice sets (Louviere et al. 2000). That is, for a ranking of M options, the first is preferred to the other M–1, the second to the remaining M–2, etc; hence M rankings yields M–1 choice sets (if A, B, C, D are ranked 1, 2, 3, 4 the first set contains A, B, C, D with A chosen; the second set contains B, C, D with B chosen; etc). Such explosions transform preference responses into discrete choices that can be compared using RUT-based statistical models (e.g., Louviere et al. 2000). Recently Bayesian methods have been used to estimate individual-level parameters (Wedel et al. 1999), but the pros and cons of these methods in CA and CB applications have not been fully explored.

Statistical properties of choice experiments also remain elusive. In particular, humans interact with CA and choice experiments and particular designs can impact response variability. Statistically efficient designs may require many choices, which may increase choice variability; increases in choice variability may offset higher statistical design efficiency. Hence, re-search into behavioral impacts of designs and effects of non-design issues like task layouts, formats, response tasks, etc, on choice variability also are needed.

Bibliography:

1. Anderson N H 1970 Functional measurement and psycho-physical judgment. Psychological Review 77: 153–77
2. Ben-Akiva M E, Lerman S 1985 Discrete Choice Analysis: Theory and Application to Travel Demand. MIT Press, Cambridge, MA
3. Ben-Akiva M, Morikawa T 1990 Estimation of switching models from revealed preferences and stated intentions. Transportation Research A 24A: 485–95
4. Green P E 1974 On the design of choice experiments involving multifactor alternatives. Journal of Consumer Research 1: 61–8
5. Hensher D A, Louviere J J, Swait J 1999 Combining sources of preference data. Journal of Econometrics 89: 197–221
6. Krantz D H, Tversky A 1971 Conjoint measurement analysis of composition rules in psychology. Psychological Review 78: 151–69
7. Louviere J J 1988 Analyzing decision making: Metric conjoint analysis. Sage University Papers Series in Quantitative Applications in the Social Sciences, Newbury Park, CA, No. 67
8. Louviere J J, Woodworth G G 1983 Design and analysis of simulated consumer choice or allocation experiments: An approach based on aggregate data. Journal of Marketing Research 20: 350–67
9. Louviere J J, Hensher D A, Swait J 2000 Stated Choice Methods: Analysis and Applications in Marketing, Transportation and Environmental Valuation. Cambridge University Press, Cam-bridge, UK
10. Louviere J, Meyer R, Bunch D, Carson R, Delleart B, Hanemann M, Hensher D, Irwin J 1999 Combining sources of preference data for modelling complex decision processes. Marketing Letters 10(3): 205–18
11. Luce R D 1959 Individual Choice Behavior. Wiley, New York
12. McFadden D 1974 Conditional logit analysis of qualitative choice behaviour. In: Zarembka P (ed.) Frontiers in Econometrics. Academic Press, New York, pp. 105–42
13. Wedel W, Kamakura W, Arora N, Bemmaor A, Chiang J, Elrod T, Johnson R, Lenk P, Neslin S, Poulsen C S 1999 Discrete and continuous representation of unobserved heterogeneity in choice modelling. Marketing Letters 10(3): 219–32