circular: Circular Triads (Intransitive Cycles)

In eba: Elimination-by-Aspects Models

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 "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

See Also

eba, strans, kendall.u.

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

Example output

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

eba documentation built on Jan. 13, 2021, 10:12 a.m.