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)

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 "two.sided" (default), "less" or "greater"
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, $$zeta = 1 - T/T_{max},$$ 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 $n > 4$.

Value

T: number of circular triads
T.max: maximum possible number of circular triads
T.exp: expected number of circular triads $E(T)$ when choices are totally random
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

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
#> 
#> 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
#> 
subset(strans(dog)$violdf, !wst)   # list circular triads
#>    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