Signature
dx_b = -b_a / b_b
| Inputs | Definition | Unit |
|---|---|---|
b_a | Part-worth of the change being valued, such as a local clinic against hospital | rating points or utility |
b_b | Part-worth of a one-unit change in b, such as one extra week of waiting; not zero | rating points or utility per unit of b |
dx_b | Change in attribute b that leaves the profile's value unchanged after a one-unit change in attribute a | units of attribute b |
|---|
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.
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Implementations
Excel
Offsetting change from named part-worths
With PartWorthA and PartWorthB named, the formula returns the offsetting change in b, held in OffsetChange. A per-unit part-worth from two levels is =PartWorthTwoLevel/(HighLevel-LowLevel).
=-PartWorthA/PartWorthB
Assumptions
Conjoint attribute b continuous and linear over the range used
The part-worth of b is the same for every unit; a two-level attribute gives a per-unit value only by assuming linearity between its levels, such as minus 3.00 over 8 weeks.
Both conjoint part-worths from the same model and scale
b_a and b_b come from one fitted model, so the scale cancels in the ratio.
Worked examples
Extra weeks of waiting accepted for a local clinic
A per-week part-worth of minus 3.00 / 8, or minus 0.375, and a local part-worth of 1.50 give an offsetting wait of 4.0 weeks, as in the article.
b_a = 1.5; b_b = -0.375; dx_b = 4
Extra weeks of waiting accepted to keep a consultant lead
Keeping a consultant lead is worth 1.00, so the offsetting wait is 1.00 / 0.375, about 2.6667 weeks (2.7 in the article).
b_a = 1; b_b = -0.375; dx_b = 2.6667
Orthodontic first appointment at a local clinic
In Ryan and Farrar's choice study a local first appointment was worth 0.77 and each month of waiting minus 0.59, so respondents would accept about 1.3051 extra months of waiting, reported as 1.3 months.
b_a = 0.77; b_b = -0.59; dx_b = 1.3051
Common errors
Reading a ratio from ratings data as willingness to pay
The cost ratio is a welfare measure only for choice data under random utility theory; in rating-based studies it is used in an ad hoc way, because the ratio does not follow from the statistical specification fitted to ratings.
Inverting the part-worth ratio
Dividing minus 0.375 by 1.50 gives 0.25, the value of a week of waiting in units of the local clinic, not 4.0 weeks of waiting.
Extrapolating a per-week part-worth beyond the waits studied
The 4.0 weeks rests on linearity between 4 and 12 weeks; a trade-off reaching beyond 12 weeks applies the line where no ratings were given.
Sources
Ratio of part-worths as the waiting time individuals would give up for a local clinic
Ryan M, Farrar S. Using conjoint analysis to elicit preferences for health care. BMJ. 2000;320(7248):1530-1533. doi:10.1136/bmj.320.7248.1530. Stage 5 and Results: in the orthodontic study the ratio of the location coefficient to the waiting time coefficient indicates how much waiting time individuals are willing to give up to have their first appointment at a local clinic; moving from a hospital to a local clinic increases benefit by 0.77 for the first appointment, each unit increase in waiting time reduces it by 0.59, and individuals are willing to wait an extra 1.3 months (0.77 / 0.59).
Willingness-to-pay ratios used ad hoc in conjoint studies
Louviere JJ, Flynn TN, Carson RT. Discrete choice experiments are not conjoint analysis. Journal of Choice Modelling. 2010;3(3):57-72. doi:10.1016/S1755-5345(13)70014-9. Section 5: for marginal willingness to pay the scale factor drops out in some simple specifications where the ratio of two parameters is the appropriate measure, a circumstance often used in an ad hoc manner in conjoint analysis studies even though it does not follow from the statistical specification fitted; traditional conjoint methods do not naturally yield willingness to pay.
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