strans {eba}

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; row stimuli are chosen over column stimuli

Details

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

P_{ik} \ge 0.5 (WST),

P_{ik} \ge \min(P_{ij}, P_{jk}) (MST),

P_{ik} \ge \max(P_{ij}, P_{jk}) (SST).

See Suppes, Krantz, Luce, and 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 \ge 0.5. This rearranged 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); for a more exact test of WST see Iverson and 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. doi: 10.1016/0165-4896(85)90031-9

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. doi: 10.1037/h0026750

See Also

eba, circular, kendall.u, trineq.

Examples

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