Signature
r = b * k / v; lambda = r * (k - 1) / (v - 1)
| Inputs | Definition | Unit |
|---|---|---|
b | Number of choice sets (blocks) shown to each respondent or design version | count |
k | Number of objects shown in each set, fewer than v | count |
v | Number of objects to be scaled | count |
r | Number of sets in which each object appears | count |
|---|---|---|
lambda | Number of sets in which any given pair of objects appears together | count |
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
Best-worst design appearances and pair count from named cells
With the numbers of sets, objects per set and objects in Sets, SetSize and Objects, the formulas return the appearances, held in Appearances, and the pair count, held in PairCount.
=Sets*SetSize/Objects; =Appearances*(SetSize-1)/(Objects-1)
Assumptions
Equal set size and equal appearances in a balanced design
Every set has k objects and every object appears r times, so no object gains from the way sets were drawn. Orthogonal main effects designs give sets of unequal size and are now seen as inappropriate for the object case.
Whole-number parameters before constructing the design
The identities give r and lambda for chosen v, b and k; when either is not a whole number no balanced incomplete block design has those parameters, and another number of sets or set size is chosen. Designs are then constructed or taken from published tables.
Worked examples
Seven objects in seven sets of four
The seven-set design used by Hollin and colleagues shows seven objects in sets of four: each object appears four times and each pair twice, as in the article (7 x 4 = 28 appearances).
b = 7; k = 4; v = 7; r = 4; lambda = 2
Four objects in four sets of three
Four objects in four sets of three give r = 3 and lambda = 2, the first example in the Penn State course notes.
b = 4; k = 3; v = 4; r = 3; lambda = 2
Eight objects in eight sets of four with no balanced design
Eight objects in eight sets of four give r = 4 but lambda of about 1.7143, not a whole number, so no balanced incomplete block design has these parameters (computed here for illustration).
b = 8; k = 4; v = 8; r = 4; lambda = 1.7143
Common errors
Using an orthogonal plan with unequal set sizes for the object case
An orthogonal main effects plan gives sets of different sizes, so objects shown mostly in small sets are chosen best or worst more often by construction.
Checking appearances but not pairings
A design in which every object appears four times can still show some pairs together more often than others. The pair count, not only the appearance count, needs to be equal for the design to be balanced.
Sources
Balanced incomplete block design identities
Penn State Eberly College of Science. STAT 503: Design of Experiments. Lesson 4: Blocking, section 4.7 Incomplete Block Designs. With t treatments, block size k, b blocks and r replicates, the number of observations is N = t r = b k; lambda = r (k minus 1) / (t minus 1); for t = 7, b = 7, k = 4 and r = 4, lambda is 2.
Balanced incomplete block design in object case best-worst scaling
Hollin IL, Paskett J, Schuster ALR, Crossnohere NL, Bridges JFP. Best-worst scaling and the prioritization of objects in health: a systematic review. PharmacoEconomics. 2022;40(9):883-899. Introduction: seven objects compared in sets of four by a BIBD in seven tasks, each task the same size, each object shown four times and with each other object twice; Discussion: orthogonal designs now seen as inappropriate because of non-uniform set sizes.
Canonical Identity
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