Summarises circular triads in the observed paired comparisons. A triad is circular when A is preferred to B, B to C, and C to A. For a complete tournament, the function reports Kendall's coefficient of consistency (Kendall and Babington Smith 1940). Judge-specific summaries are returned when judges are available.
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.
Details
A circular-triad rate of one quarter is the benchmark for a random tournament. It is not the expected rate under a fitted BTL model with unequal object locations, so this function is a descriptive consistency measure rather than a calibrated goodness-of-fit test. Comparisons involving an undefeated or winless object remain part of this observed-data summary, although that object is set aside from finite maximum-likelihood estimation.
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; random-tournament benchmark 25%) -> consistency 1.00
#> Kendall coefficient of consistency (complete design): 1.000
#> Note: the 0.25 chance rate is a random-tournament benchmark, not the fitted BTL expected circular rate; transitivity is descriptive