Signature
P_seq = exp(V_b) / S_pos * exp(-V_w) / (S_neg - exp(-V_b))
| Inputs | Definition | Unit |
|---|---|---|
V_b | Systematic utility of item b | utility |
S_pos | Sum of exp(V_j) over all items in the set | none |
V_w | Systematic utility of item w, a different item from b | utility |
S_neg | Sum of exp(minus V_k) over all items in the set, including b | none |
P_seq | Probability that item b is chosen best and then item w worst from the remaining items | probability |
|---|
Function
Best-worst scaling design, scoring and choice probability function
Maps a planned series of choice sets, each answered with a best and a worst choice, to a scale of preference or priority. The design is usually a balanced incomplete block design, the simplest analysis counts best and worst choices and standardises their difference, and choice models place items on a latent utility scale through the probability of each best and worst pair. The notation follows the Best-Worst Scaling article.
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Implementations
Excel
Sequential best-then-worst probability from a range of set utilities
With the set's utilities in SetUtils and the chosen items' utilities in UtilBest and UtilWorst, the formula returns the probability, held in SeqProb.
=EXP(UtilBest)/SUMPRODUCT(EXP(SetUtils))*EXP(-UtilWorst)/(SUMPRODUCT(EXP(-SetUtils))-EXP(-UtilBest))
Assumptions
Worst chosen from the items left after the best
The respondent first picks the best item and then the worst from the rest, as in rank-ordered or exploded logit analysis; the joint probability is the product of the two choices.
Same utility scale for best and worst choices
The worst choice uses minus the same utilities and the same error scale as the best choice. Worst choices may follow a different mental process, in which case the scale can differ.
Worked examples
Article's set 1 under the sequential model
With the same utilities, A is best with probability 2.718 / 5.974, about 0.455, and G is then worst among C, E and G with probability 1.649 / 3.256, about 0.506, a joint probability of about 0.2305, as in the article.
V_b = 1; V_w = -0.5; S_pos = 5.973534; S_neg = 3.623131; P_seq = 0.2305
Equal utilities in a set of four under the sequential model
With four equal utilities the best item has probability 1/4 and the worst then 1/3, a joint probability of 1/12, the same as the maxdiff model in this case (computed here for illustration).
V_b = 0; V_w = 0; S_pos = 4; S_neg = 4; P_seq = 0.0833
Common errors
Keeping the best item in the worst-choice set
Dividing exp(minus V_w) by S_neg instead of S_neg minus exp(minus V_b) gives 0.455 x 0.455, about 0.2071 instead of 0.2305 in the article's set (computed here for illustration).
Appending worst choices without reversing the explanatory variables
When worst data are stacked under best data for a conditional logit, every explanatory variable in the worst rows is multiplied by minus one. Without that step the worst choices pull utilities the wrong way.
Sources
Sequential model for best and worst choices
Mühlbacher AC, Zweifel P, Kaczynski A, Johnson FR. Experimental measurement of preferences in health care using best-worst scaling (BWS): theoretical and statistical issues. Health Economics Review. 2016;6:5. Equation 18: the probability of choosing w as worst is based on the alternatives remaining after b is removed, and the joint probability is exp(V_b) over the sum of exp(V_j) times exp(minus V_w) over the sum of exp(minus V_k) across the set without b.
Canonical Identity
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