circular {eba} | R Documentation |

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)
```

`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`

.

`T` |
number of circular triads |

`T.max` |
maximum possible number of circular triads |

`T.exp` |
expected number of circular triads |

`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 |

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

[Package *eba* version 1.10-0 Index]