A History of Conjoint

Paul Green—University of Pennsylvania

Joel Huber—Duke University

Rich Johnson—Sawtooth Software

A History of Conjoint

• The psychometric roots of conjoint

• The development of ACA

• The development of choice models

• The application of conjoint

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

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

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

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

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.

Paul Green’s Paradigm Shift

• Full factorial Orthogonal arrays

• Ordinal estimation Linear estimation

• Focus on tests Focus on simulations

• Conjoint measurement Conjoint analysis

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

A Tradeoff Matrix

Weight

Price 3 lbs. 4 lbs. 6 lbs.

$1,000 1

$2,000

$3,000 9

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

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

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

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

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

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

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

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

Marginal impact of effort depends on share

0

0.5

1

1.5

2

2.5

3

3.5

4

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

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

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

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

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

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

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

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

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

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

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

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