circular {eba} | R Documentation |
Circular Triads (Intransitive Cycles)
Description
Number of circular triads and coefficient of consistency.
Usage
circular(mat, alternative = c("two.sided", "less", "greater"),
exact = NULL, correct = TRUE, simulate.p.value = FALSE,
nsim = 2000)
Arguments
mat |
a square matrix or a data frame consisting of (individual) binary choice data; row stimuli are chosen over column stimuli |
alternative |
a character string specifying the alternative hypothesis,
must be one of |
exact |
a logical indicating whether an exact p-value should be computed |
correct |
a logical indicating whether to apply continuity correction in the chi-square approximation for the p-value |
simulate.p.value |
a logical indicating whether to compute p-values by Monte Carlo simulation |
nsim |
an integer specifying the number of replicates used in the Monte Carlo test |
Details
Kendall's coefficient of consistency,
lies between one (perfect consistency) and zero,
where T
is the observed number of circular triads,
and the maximum possible number of circular triads is
, if
is even, and
else, and
is the
number of stimuli or objects being judged. For details see Kendall and
Babington Smith (1940) and David (1988).
Kendall (1962) discusses a test of the hypothesis that the number of
circular triads T
is different (smaller or greater) than expected
when choosing randomly. For small , an exact p-value is computed,
based on the exact distributions listed in Alway (1962) and in Kendall
(1962). Otherwise, an approximate chi-square test is computed. In this
test, the sampling distribution is measured from lower to higher values of
T
, so that the probability that T
will be exceeded is the
complement of the probability for chi2
. The chi-square approximation
may be incorrect if and is only available for
.
Value
T |
number of circular triads |
T.max |
maximum possible number of circular triads |
T.exp |
expected number of circular triads |
zeta |
Kendall's coefficient of consistency |
chi2 , df , correct |
the chi-square statistic and degrees of freedom for the approximate test, and whether continuity correction has been applied |
p.value |
the p-value for the test (see Details) |
simulate.p.value , nsim |
whether the p-value is based on simulations, number of simulation runs |
References
Alway, G.G. (1962). The distribution of the number of circular triads in paired comparisons. Biometrika, 49, 265–269. doi: 10.1093/biomet/49.1-2.265
David, H. (1988). The method of paired comparisons. London: Griffin.
Kendall, M.G. (1962). Rank correlation methods. London: Griffin.
Kendall, M.G., & Babington Smith, B. (1940). On the method of paired comparisons. Biometrika, 31, 324–345. doi: 10.1093/biomet/31.3-4.324
See Also
Examples
# A dog's preferences for six samples of food
# (Kendall and Babington Smith, 1940, p. 326)
dog <- matrix(c(0, 1, 1, 0, 1, 1,
0, 0, 0, 1, 1, 0,
0, 1, 0, 1, 1, 1,
1, 0, 0, 0, 0, 0,
0, 0, 0, 1, 0, 1,
0, 1, 0, 1, 0, 0), 6, 6, byrow=TRUE)
dimnames(dog) <- setNames(rep(list(c("meat", "biscuit", "chocolate",
"apple", "pear", "cheese")), 2),
c(">", "<"))
circular(dog, alternative="less") # moderate consistency
subset(strans(dog)$violdf, !wst) # list circular triads