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A History of Conjoint
Paul Green—University of Pennsylvania
Joel Huber—Duke University
Rich Johnson—Sawtooth Software
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A History of Conjoint
• The psychometric roots of conjoint
• The development of ACA
• The development of choice models
• The application of conjoint
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Psychometric Dream
• To be able to build an axiomatic system of preferences akin to those in the physical sciences
• Requires interval scales over which mathematical operations are meaningful
• People have difficulty making numerically meaningful estimates
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Psychometric solution
• People can give preference orderings for compound or conjoint objects
• If you prefer a trip to Victoria for $1000 over a trip to Philadelphia for $500 implies that Victoria is worth at least $500 more than Philadelphia
• A number of such statements can produce asymptotically interval utility scales for cities and money
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Typical Early Conjoint Measurement
• Individuals rank order profiles
• Profiles developed from full factorials
• Test consistency with axioms: additivity, cancellation
• If test is passed, use monotone regression or LINMAP to estimate partworth utilities
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Early conjoint results
• People regularly violated the assumptions
• There was little correspondence between predictive accuracy and order violations
• The rank order task was more difficult but no more effective than a rating task
• Despite theoretical failure the derived utility functions predicted well
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Paul Green’s Orientation
• He knew the psychometricians and was instrumental in developments in multidimensional scaling as well as conjoint
• He came from Dupont and was concerned with managerial problems.
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Paul Green’s Paradigm Shift
• Full factorial Orthogonal arrays
• Ordinal estimation Linear estimation
• Focus on tests Focus on simulations
• Conjoint measurement Conjoint analysis
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Our debt to Psychometricians
• A focus on individual preferences
• The use of full profile stimuli
• Simple main-effects models
• Psychometricians tried to axiomatize behavior, we tried to predict it
• Their task largely failed, but with their help ours has been surprisingly successful
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A Tradeoff Matrix
Weight
Price 3 lbs. 4 lbs. 6 lbs.
$1,000 1
$2,000
$3,000 9
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A Respondent’s Preferences
Weight
Price 3 lbs. 4 lbs. 6 lbs.
$1,000 1 2 5
$2,000 3 4 6
$3,000 7 8 9
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A Tradeoff Matrix
Weight
Price 3 lbs. 4 lbs. 6 lbs.
$1,000 a b c
$2,000 d e f
$3,000 g h i
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The Evolution of Choice-Based Conjoint
• Why choices are better than ratings
• Problems with early linear choice models
• McFadden’s development of logit
• Louviere’s adoption of logit for experimental choice sets
• Hierarchical Bayes as the best way to account for heterogeneity
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Why choices over ratings?
• Choice reflects what people do in the marketplace
• Choice defines the competitive context
• Managers can immediately use the implications of a choice model
• People will answer choices about almost anything
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What is wrong with choices?
• Little information in each choice
• Analysis requires aggregation across respondents
• Linear model does not work
• Simple logit does not account for heterogeneity
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What’s wrong with linear probability model?
• Violates homoskediasticity assumptions
• Produces predictions greater than zero of less than one
• Assumes the marginal impact of a market action is the same regardless of initial share
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Which brand benefits most from a promotion or shelf tag?
1. A soft drink with 5% share of its market
2. A soft drink with 50% of its market
3. A soft drink with 95% of its market
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Typical sigmoid curve showing impact of effort on share
Typical Sigmoid Curve
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8
Choice Probability
Marketing effort
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Marginal impact of effort depends on share
0
0.5
1
1.5
2
2.5
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3.5
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4.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Marginal value of incremental effort
Original probability of choice
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Aggregate Logit
• Has the correct marginal properties
• But becomes undefined for choice probabilities of zero or one
• Ln (p/(1-p) is undefined where p=0 or 1
• Worse, it become very large for probabilities close to one and very small for probabilities close to zero
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McFadden’s 1976 breakthrough
• Builds choice from a random utility framework—errors are independent Gumbel
• MLE criterion—maximize probability actual choices occur given parameters—has no problem with zero’s or ones
• Critical statistics are defined and asymptotically consistent
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Louviere and Woodworth (1983) choice-based experimental
designs• Applied to experimental design (stated
choices) as opposed to actual choices
• Permitted predictions to alternatives that did not exist and teased out otherwise correlated characteristics in the marketplace
• Orthogonal arrays were adapted to create choice designs
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The red bus, blue bus problem
• Suppose people choose 50% red bus and 50% cars
• What happens to share if you add a blue bus that has is the same as the other bus?
• Logit says 33% for each• Logic says 50% cars, 50% red and blue bus• Logit assumes proportionality, but similar
items need to take share from similar ones
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Modeling heterogeneity resolves differential substitution
• People choose car or bus, then choose bus color
• Generally, businesses need to estimate shares for items that strongly violate proportionality– Demand for a new or revised offering– Estimate impact of revised offering on own and
competitors
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Ways to modify logit to accept differential substitution
• Include customer parameters in the aggregate utility function
• Car use is correlated with income, include income as a cross term
• Problem 1: there can be many cross terms
• Problem 2: demographics are poor at predicting choices
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Latent class
• Heterogeneity is reflected in mass points where responses are assumed to be consistently logit within those points
• Latent class produces the partworth values and the weights for each class
• Neat idea—used in Sawtooth’s ICE program
• Did not work as well as HB
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Random Parameter Logit
• Assumes that logit parameters are distributed over the population
• Sample enumeration over the population produces share estimates that are strongly non-proportional
• Works well, but sensitive to the assumption of the aggregate distribution
• Requires a new analysis or cross terms for subset analysis
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Hierarchical Bayes
• Estimates both aggregate distribution and individual distributions
• Individual means serve well in choice simulators, just like those from choice-based conjoint
• Very efficient, need only as many choices per person as you have parameters
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Why HB works
• It is robust against overfitting
• It is also less affected by assumptions about the aggregate distribution
• It’s magic has little to do with Bayesian philosophy
• Random parameter logit plus estimate at the individual level results in identical solution
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Lessons
• HB permits choice-based conjoint to be as user friendly as ratings-based conjoint
• Choices are not always the best input, but where they are, we can now accommodate them
• We naturally tend to use models with which we are most familiar, but progress is marked with unfamiliar victors