circular {eba}R Documentation

Circular Triads (Intransitive Cycles)


Number 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)



a square matrix or a data frame consisting of (individual) binary choice data; row stimuli are chosen over column stimuli


a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "less" or "greater"


a logical indicating whether an exact p-value should be computed


a logical indicating whether to apply continuity correction in the chi-square approximation for the p-value


a logical indicating whether to compute p-values by Monte Carlo simulation


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

chi2, df, correct

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)

simulate.p.value, nsim

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

See Also

eba, strans, kendall.u.


# 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

[Package eba version 1.10-0 Index]