Signature
R_hat = b_0 + b_W * x_W + b_L * x_L + b_N * x_N
| Inputs | Definition | Unit |
|---|---|---|
b_0 | Predicted rating of the reference profile: a 4-week wait, in hospital, led by a consultant | rating points |
b_W | Part-worth of a 12-week wait against a 4-week wait | rating points |
x_W | 1 if the profile has a 12-week wait, 0 for a 4-week wait | none |
b_L | Part-worth of a local clinic against hospital | rating points |
x_L | 1 if the clinic is local, 0 if in hospital | none |
b_N | Part-worth of a specialist nurse lead against a consultant lead | rating points |
x_N | 1 if the clinic is nurse led, 0 if consultant led | none |
R_hat | Rating the additive model predicts for the profile | rating points |
|---|
Function
Decomposition of whole-profile ratings into attribute part-worths in conjoint analysis
Maps a respondent's ratings of profiles, each described by levels of several attributes, to a part-worth for each level under an additive model, so that a profile's value is the intercept plus the part-worths of its levels. One level of each attribute is the reference, with a part-worth of zero. Relative importance and trade-offs between attributes follow from the part-worths. The notation follows the Conjoint Analysis article, whose outpatient clinic example is used throughout.
Try this function
Implementations
Excel
Predicted clinic rating from named part-worths and codes
With Intercept, PartWorthWait, PartWorthLocal, PartWorthNurse and the profile's codes WaitCode, LocalCode and NurseCode named, the formula returns the prediction, held in PredictedRating.
=Intercept+PartWorthWait*WaitCode+PartWorthLocal*LocalCode+PartWorthNurse*NurseCode
Assumptions
Additive model holds across the profiles predicted
Predictions add independent part-worths, so they are valid only if preferences for one attribute do not depend on the levels of another.
Predicted profiles use only levels included in the study
Each code is 0 or 1 for a level that was rated; values between or beyond the studied levels need an assumption about the shape of the part-worth function.
Worked examples
Local nurse-led clinic with a 4-week wait
6.00 plus 1.50 minus 1.00 gives a predicted rating of 6.5, above the reference hospital consultant clinic at 6.0, as in the article.
b_0 = 6; b_W = -3; x_W = 0; b_L = 1.5; x_L = 1; b_N = -1; x_N = 1; R_hat = 6.5
Local nurse-led clinic with a 12-week wait
Adding the 12-week part-worth of minus 3.00 lowers the prediction to 3.5, below the reference clinic, as in the article.
b_0 = 6; b_W = -3; x_W = 1; b_L = 1.5; x_L = 1; b_N = -1; x_N = 1; R_hat = 3.5
Local consultant-led clinic with a 4-week wait
The prediction is 7.5 against an observed rating of 8 for profile 3, a residual of 0.5, one of the four misses that give the residual sum of squares of 1.0 in the article.
b_0 = 6; b_W = -3; x_W = 0; b_L = 1.5; x_L = 1; b_N = -1; x_N = 0; R_hat = 7.5
Common errors
Predicting for levels outside those rated
Entering a code of 2.75 for a 26-week wait assumes value falls linearly beyond 12 weeks, which the two observed waits cannot test.
Reading predicted ratings as uptake or market shares
A rating of 6.5 against 6.0 ranks the two clinics for one respondent; it does not say what share of patients would choose either, which needs a choice model.
Sources
Additive part-worths and preferential independence
Hauser JR, Rao VR. Conjoint analysis, related modeling, and applications. In: Wind Y, Green PE, eds. Marketing Research and Modeling: Progress and Prospects. Boston, MA: Springer; 2004:141-168. doi:10.1007/978-0-387-28692-1_7 (read as the MIT open-access pre-publication version). Section Decomposing the Product or Service: two features are preferentially independent of the remaining features if trade-offs between them do not depend on the remaining features, and if each set of features is preferentially independent of its complement the conjoint function can be represented by an additive decomposition; section Conjoint Analysis is a Journey not a Destination: Green and Wind assumed overall preference was an additive sum of the part-worths of the features, each represented by a series of dummy variables.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0