% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/btl-independence.R
\name{btl_transitivity}
\alias{btl_transitivity}
\title{Transitivity of paired comparisons}
\usage{
btl_transitivity(fit, min_triples = 5L)
}
\arguments{
\item{fit}{A paired-comparison fit from \code{\link{btl}}.}

\item{min_triples}{A judge is reported only if this many complete triples
(all three pairs judged) are available.}
}
\value{
A list of class \code{"rasch_btl_transitivity"}: \code{summary} (one
  row: objects, pairs compared, complete triples, circular triads, the
  circular rate, the chance rate 0.25, the consistency index
  \code{1 - rate/0.25}, and Kendall's \code{zeta} when the design is a
  complete round-robin with no exactly-tied pair -- \code{NA} otherwise);
  \code{objects} (each object's circular-triad
  involvement); \code{judges} (per-judge consistency, when judges exist);
  and \code{notes}.
}
\description{
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 \emph{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.
}
\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"))
}
\references{
Kendall, M. G., & Babington Smith, B. (1940). On the method of
  paired comparisons. \emph{Biometrika}, 31(3/4), 324-345.
}
