The single-dimension analogue for paired comparisons of the unidimensionality question. A Bradley-Terry-Luce scale implies that preferences stack into one consistent order: if A beats B and B beats C then A should beat C. A circular triad (A beats B, B beats C, C beats A) is a local contradiction, like rock-paper-scissors. A few are sampling noise; many, systematically, mean the comparisons are not being driven by a single attribute. The rate of circular triads is compared with the value expected from pure guessing (one quarter of triples), and, when every pair has been compared, Kendall's coefficient of consistency is reported (Kendall & Babington Smith 1940). With judges, each judge's own consistency is reported too, flagging judges whose choices approach chance.
Arguments
- fit
A paired-comparison fit from
btl.- min_triples
A judge is reported only if this many complete triples (all three pairs judged) are available.
Value
A list of class "rasch_btl_transitivity": summary (one
row: objects, pairs compared, complete triples, circular triads, the
circular rate, the chance rate 0.25, the consistency index
1 - rate/0.25, and Kendall's zeta when the design is a
complete round-robin with no exactly-tied pair – NA otherwise);
objects (each object's circular-triad
involvement); judges (per-judge consistency, when judges exist);
and notes.
References
Kendall, M. G., & Babington Smith, B. (1940). On the method of paired comparisons. Biometrika, 31(3/4), 324-345.
Examples
set.seed(1); objs <- LETTERS[1:6]; beta <- setNames(seq(-1.5, 1.5, len = 6), objs)
pr <- t(utils::combn(objs, 2))
d <- data.frame(a = rep(pr[, 1], each = 20), b = rep(pr[, 2], each = 20))
d$win <- ifelse(runif(nrow(d)) < plogis(beta[d$a] - beta[d$b]), d$a, d$b)
btl_transitivity(btl(d, "a", "b", "win"))
#> Paired-comparison transitivity: 6 objects, 20 complete triples
#> Circular triads: 0 (0.0% of triples; chance 25%) -> consistency 1.00
#> Kendall coefficient of consistency (complete design): 1.000