A History of Conjoint

1. A History of Conjoint Paul Green—University of Pennsylvania Joel Huber—Duke University Rich Johnson—Sawtooth Software

2. A History of Conjoint • The psychometric roots of conjoint • The development of ACA • The development of choice models • The application of conjoint

3. 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

4. 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

5. 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

6. 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

7. 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.

8. Paul Green’s Paradigm Shift • Full factorial  Orthogonal arrays • Ordinal estimation Linear estimation • Focus on tests  Focus on simulations • Conjoint measurement  Conjoint analysis

9. 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

10. A Tradeoff Matrix

11. A Respondent’s Preferences

12. A Tradeoff Matrix

13. 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

14. 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

15. 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

16. 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

17. Which brand benefits most from a promotion or shelf tag? • A soft drink with 5% share of its market • A soft drink with 50% of its market • A soft drink with 95% of its market

18. Typical sigmoid curve showing impact of effort on share

19. Marginal impact of effort depends on share

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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