circular.RdNumber of circular triads and coefficient of consistency.
circular(mat, alternative = c("two.sided", "less", "greater"), exact = NULL, correct = TRUE, simulate.p.value = FALSE, nsim = 2000)
| 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 |
Kendall's coefficient of consistency,
$$zeta = 1 - T/T_{max},$$
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
\(T_{max} = n(n^2 - 4)/24\), if \(n\) is even, and
\(T_{max} = n(n^2 - 1)/24\) else, and \(n\) 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 \(n\), 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 \(n < 8\) and is only available for \(n > 4\).
number of circular triads
maximum possible number of circular triads
expected number of circular triads \(E(T)\) when choices are totally random
Kendall's coefficient of consistency
the chi-square statistic and degrees of freedom for the approximate test, and whether continuity correction has been applied
the p-value for the test (see Details)
whether the p-value is based on simulations, number of simulation runs
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
# 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#> #> Circular triads (intransitive cycles) #> #> T = 5, max(T) = 8, E(T) = 5, zeta = 0.375, p-value = 0.5093 #> alternative hypothesis: T is smaller than expected by chance #>#> triple perm p12 p23 p13 wst mst sst #> 7 2 meat.biscuit.apple 1 1 0 FALSE FALSE FALSE #> 25 5 meat.chocolate.apple 1 1 0 FALSE FALSE FALSE #> 44 8 meat.pear.apple 1 1 0 FALSE FALSE FALSE #> 50 9 meat.cheese.apple 1 1 0 FALSE FALSE FALSE #> 91 16 biscuit.pear.cheese 1 1 0 FALSE FALSE FALSE