strans {eba}R Documentation

Stochastic Transitivity

Description

Checks the weak, moderate, and strong stochastic transitivity.

Usage

strans(M)

Arguments

M a square matrix or a data frame consisting of absolute choice frequencies

Details

The weak (WST), moderate (MST), and strong (SST) stochastic transitivity hold for a set of choice probabilities P, whenever if P_{ij} >= 0.5 and P_{jk} >= 0.5, then

P_{ik} >= 0.5 (WST),

P_{ik} >= min(P_{ij}, P_{jk}) (MST),

P_{ik} >= max(P_{ij}, P_{jk}) (SST).

See Suppes, Krantz, Luce, & Tversky (1989/2007, chap. 17) for an introduction to the representation of choice probabilities.

If WST holds, a permutation of the indices of the matrix exists such that the proportions in the upper triangular matrix are >= 0.5. This re-arranged matrix is stored in pcm. If WST does not hold, cells in the upper triangular matrix that are smaller than 0.5 are replaced by 0.5. The deviance resulting from this restriction is reported in wst.fit.

The approximate likelihood ratio test for significance of the WST violations is according to Tversky (1969); a more exact test of WST is suggested by Iverson & Falmagne (1985).

Value

A table displaying the number of violations of the weak, moderate, and strong stochastic transitivity, the number of tests, the error ratio (violations/tests), and the mean and maximum deviation from the minimum probability for which the corresponding transitivity would hold.

weak number of violations of WST
moderate number of violations of MST
strong number of violations of SST
n.tests number of transitivity tests performed
wst.violations a vector containing 0.5 - P_{ik} for all triples that violate WST
mst.violations a vector containing min(P_{ij}, P_{jk}) - P_{ik} for all triples that violate MST
sst.violations a vector containing max(P_{ij}, P_{jk}) - P_{ik} for all triples that violate SST
pcm the permuted square matrix of relative choice frequencies
ranking the ranking of the objects, which corresponds to the colnames of pcm
chkdf data.frame reporting the choice proportions for each triple in each permutation
violdf data.frame reporting for each triple which type of transitivity holds or does not hold
wst.fit likelihood ratio test of WST (see details)
wst.mat restricted matrix that satisfies WST

References

Iverson, G., & Falmagne, J.-C. (1985). Statistical issues in measurement. Mathematical Social Sciences, 10, 131–153.

Suppes, P., Krantz, D.H., Luce, R.D., & Tversky, A. (1989/2007). Foundations of measurement. Volume II. Mineola, N.Y.: Dover Publications.

Tversky, A. (1969). Intransitivity of preferences. Psychological Review, 76, 31–48.

See Also

eba, circular, kendall.u.

Examples

data(celebrities)           # absolute choice frequencies
strans(celebrities)         # WST and MST hold, but not SST
strans(celebrities)$pcm     # re-ordered relative frequencies
strans(celebrities)$violdf  # log transitivity violations

[Package eba version 1.5-6 Index]