strans function

Stochastic Transitivity

Checks the weak, moderate, and strong stochastic transitivity.

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).

Returns

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. tools:::Rd_expr_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. tools:::Rd_expr_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