circular {eba} R Documentation

### 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

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


[Package eba version 1.10-0 Index]