strans {eba} | R Documentation |
Checks the weak, moderate, and strong stochastic transitivity.
strans(M)
M |
a square matrix or a data frame consisting of absolute choice frequencies |
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).
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 |
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.
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