strans.RdChecks the weak, moderate, and strong stochastic transitivity.
strans(M)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).
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.
number of violations of WST
number of violations of MST
number of violations of SST
number of transitivity tests performed
a vector containing \(0.5 - P_{ik}\) for all triples that violate WST
a vector containing \(\min(P_{ij}, P_{jk}) - P_{ik}\) for all triples that violate MST
a vector containing \(\max(P_{ij}, P_{jk}) - P_{ik}\) for all triples that violate SST
the permuted square matrix of relative choice frequencies
the ranking of the objects, which corresponds to the colnames of pcm
data frame reporting the choice proportions for each triple in each permutation
data frame reporting for each triple which type of transitivity holds or does not hold
likelihood ratio test of WST (see Details)
restricted matrix that satisfies WST
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
data(celebrities) # absolute choice frequencies
strans(celebrities) # WST and MST hold, but not SST
#>
#> Stochastic Transitivity
#>
#> Violations ErrorRatio MeanDev MaxDev Deviance Df Pr(>Chi)
#> Weak 0 0.000 0.0000 0.000 0 0 1
#> Moderate 0 0.000 0.0000 0.000 NA NA NA
#> Strong 34 0.405 0.0566 0.197 NA NA NA
#> ---
#> Number of Tests: 84
#>
strans(celebrities)$pcm # reordered relative frequencies
#> <
#> > LBJ HW CDG SL ET AJF JU
#> LBJ 0.0000000 0.6794872 0.6965812 0.6068376 0.6837607 0.7649573 0.7478632
#> HW 0.3205128 0.0000000 0.5897436 0.5213675 0.5213675 0.6837607 0.7008547
#> CDG 0.3034188 0.4102564 0.0000000 0.5128205 0.5213675 0.5897436 0.6196581
#> SL 0.3931624 0.4786325 0.4871795 0.0000000 0.6282051 0.6965812 0.7393162
#> ET 0.3162393 0.4786325 0.4786325 0.3717949 0.0000000 0.6068376 0.6324786
#> AJF 0.2350427 0.3162393 0.4102564 0.3034188 0.3931624 0.0000000 0.5085470
#> JU 0.2521368 0.2991453 0.3803419 0.2606838 0.3675214 0.4914530 0.0000000
#> BB 0.2606838 0.3333333 0.4017094 0.2051282 0.2863248 0.4273504 0.4700855
#> CY 0.2179487 0.2649573 0.3290598 0.2606838 0.3076923 0.3290598 0.2478632
#> <
#> > BB CY
#> LBJ 0.7393162 0.7820513
#> HW 0.6666667 0.7350427
#> CDG 0.5982906 0.6709402
#> SL 0.7948718 0.7393162
#> ET 0.7136752 0.6923077
#> AJF 0.5726496 0.6709402
#> JU 0.5299145 0.7521368
#> BB 0.0000000 0.5940171
#> CY 0.4059829 0.0000000
strans(celebrities)$violdf # transitivity violations
#> triple perm p12 p23 p13 wst mst sst
#> 1 1 LBJ.HW.CDG 0.6794872 0.5897436 0.6965812 TRUE TRUE TRUE
#> 7 2 LBJ.HW.JU 0.6794872 0.7008547 0.7478632 TRUE TRUE TRUE
#> 13 3 LBJ.HW.CY 0.6794872 0.7350427 0.7820513 TRUE TRUE TRUE
#> 19 4 LBJ.HW.AJF 0.6794872 0.6837607 0.7649573 TRUE TRUE TRUE
#> 25 5 LBJ.HW.BB 0.6794872 0.6666667 0.7393162 TRUE TRUE TRUE
#> 31 6 LBJ.HW.ET 0.6794872 0.5213675 0.6837607 TRUE TRUE TRUE
#> 37 7 LBJ.HW.SL 0.6794872 0.5213675 0.6068376 TRUE TRUE FALSE
#> 43 8 LBJ.CDG.JU 0.6965812 0.6196581 0.7478632 TRUE TRUE TRUE
#> 49 9 LBJ.CDG.CY 0.6965812 0.6709402 0.7820513 TRUE TRUE TRUE
#> 55 10 LBJ.CDG.AJF 0.6965812 0.5897436 0.7649573 TRUE TRUE TRUE
#> 61 11 LBJ.CDG.BB 0.6965812 0.5982906 0.7393162 TRUE TRUE TRUE
#> 67 12 LBJ.CDG.ET 0.6965812 0.5213675 0.6837607 TRUE TRUE FALSE
#> 73 13 LBJ.CDG.SL 0.6965812 0.5128205 0.6068376 TRUE TRUE FALSE
#> 79 14 LBJ.JU.CY 0.7478632 0.7521368 0.7820513 TRUE TRUE TRUE
#> 86 15 LBJ.AJF.JU 0.7649573 0.5085470 0.7478632 TRUE TRUE FALSE
#> 91 16 LBJ.JU.BB 0.7478632 0.5299145 0.7393162 TRUE TRUE FALSE
#> 98 17 LBJ.ET.JU 0.6837607 0.6324786 0.7478632 TRUE TRUE TRUE
#> 104 18 LBJ.SL.JU 0.6068376 0.7393162 0.7478632 TRUE TRUE TRUE
#> 110 19 LBJ.AJF.CY 0.7649573 0.6709402 0.7820513 TRUE TRUE TRUE
#> 116 20 LBJ.BB.CY 0.7393162 0.5940171 0.7820513 TRUE TRUE TRUE
#> 122 21 LBJ.ET.CY 0.6837607 0.6923077 0.7820513 TRUE TRUE TRUE
#> 128 22 LBJ.SL.CY 0.6068376 0.7393162 0.7820513 TRUE TRUE TRUE
#> 133 23 LBJ.AJF.BB 0.7649573 0.5726496 0.7393162 TRUE TRUE FALSE
#> 140 24 LBJ.ET.AJF 0.6837607 0.6068376 0.7649573 TRUE TRUE TRUE
#> 146 25 LBJ.SL.AJF 0.6068376 0.6965812 0.7649573 TRUE TRUE TRUE
#> 152 26 LBJ.ET.BB 0.6837607 0.7136752 0.7393162 TRUE TRUE TRUE
#> 158 27 LBJ.SL.BB 0.6068376 0.7948718 0.7393162 TRUE TRUE FALSE
#> 164 28 LBJ.SL.ET 0.6068376 0.6282051 0.6837607 TRUE TRUE TRUE
#> 169 29 HW.CDG.JU 0.5897436 0.6196581 0.7008547 TRUE TRUE TRUE
#> 175 30 HW.CDG.CY 0.5897436 0.6709402 0.7350427 TRUE TRUE TRUE
#> 181 31 HW.CDG.AJF 0.5897436 0.5897436 0.6837607 TRUE TRUE TRUE
#> 187 32 HW.CDG.BB 0.5897436 0.5982906 0.6666667 TRUE TRUE TRUE
#> 193 33 HW.CDG.ET 0.5897436 0.5213675 0.5213675 TRUE TRUE FALSE
#> 199 34 HW.CDG.SL 0.5897436 0.5128205 0.5213675 TRUE TRUE FALSE
#> 205 35 HW.JU.CY 0.7008547 0.7521368 0.7350427 TRUE TRUE FALSE
#> 212 36 HW.AJF.JU 0.6837607 0.5085470 0.7008547 TRUE TRUE TRUE
#> 217 37 HW.JU.BB 0.7008547 0.5299145 0.6666667 TRUE TRUE FALSE
#> 224 38 HW.ET.JU 0.5213675 0.6324786 0.7008547 TRUE TRUE TRUE
#> 230 39 HW.SL.JU 0.5213675 0.7393162 0.7008547 TRUE TRUE FALSE
#> 236 40 HW.AJF.CY 0.6837607 0.6709402 0.7350427 TRUE TRUE TRUE
#> 242 41 HW.BB.CY 0.6666667 0.5940171 0.7350427 TRUE TRUE TRUE
#> 248 42 HW.ET.CY 0.5213675 0.6923077 0.7350427 TRUE TRUE TRUE
#> 254 43 HW.SL.CY 0.5213675 0.7393162 0.7350427 TRUE TRUE FALSE
#> 259 44 HW.AJF.BB 0.6837607 0.5726496 0.6666667 TRUE TRUE FALSE
#> 266 45 HW.ET.AJF 0.5213675 0.6068376 0.6837607 TRUE TRUE TRUE
#> 272 46 HW.SL.AJF 0.5213675 0.6965812 0.6837607 TRUE TRUE FALSE
#> 278 47 HW.ET.BB 0.5213675 0.7136752 0.6666667 TRUE TRUE FALSE
#> 284 48 HW.SL.BB 0.5213675 0.7948718 0.6666667 TRUE TRUE FALSE
#> 290 49 HW.SL.ET 0.5213675 0.6282051 0.5213675 TRUE TRUE FALSE
#> 295 50 CDG.JU.CY 0.6196581 0.7521368 0.6709402 TRUE TRUE FALSE
#> 302 51 CDG.AJF.JU 0.5897436 0.5085470 0.6196581 TRUE TRUE TRUE
#> 307 52 CDG.JU.BB 0.6196581 0.5299145 0.5982906 TRUE TRUE FALSE
#> 314 53 CDG.ET.JU 0.5213675 0.6324786 0.6196581 TRUE TRUE FALSE
#> 320 54 CDG.SL.JU 0.5128205 0.7393162 0.6196581 TRUE TRUE FALSE
#> 326 55 CDG.AJF.CY 0.5897436 0.6709402 0.6709402 TRUE TRUE TRUE
#> 332 56 CDG.BB.CY 0.5982906 0.5940171 0.6709402 TRUE TRUE TRUE
#> 338 57 CDG.ET.CY 0.5213675 0.6923077 0.6709402 TRUE TRUE FALSE
#> 344 58 CDG.SL.CY 0.5128205 0.7393162 0.6709402 TRUE TRUE FALSE
#> 349 59 CDG.AJF.BB 0.5897436 0.5726496 0.5982906 TRUE TRUE TRUE
#> 356 60 CDG.ET.AJF 0.5213675 0.6068376 0.5897436 TRUE TRUE FALSE
#> 362 61 CDG.SL.AJF 0.5128205 0.6965812 0.5897436 TRUE TRUE FALSE
#> 368 62 CDG.ET.BB 0.5213675 0.7136752 0.5982906 TRUE TRUE FALSE
#> 374 63 CDG.SL.BB 0.5128205 0.7948718 0.5982906 TRUE TRUE FALSE
#> 380 64 CDG.SL.ET 0.5128205 0.6282051 0.5213675 TRUE TRUE FALSE
#> 389 65 AJF.JU.CY 0.5085470 0.7521368 0.6709402 TRUE TRUE FALSE
#> 392 66 JU.BB.CY 0.5299145 0.5940171 0.7521368 TRUE TRUE TRUE
#> 401 67 ET.JU.CY 0.6324786 0.7521368 0.6923077 TRUE TRUE FALSE
#> 407 68 SL.JU.CY 0.7393162 0.7521368 0.7393162 TRUE TRUE FALSE
#> 411 69 AJF.JU.BB 0.5085470 0.5299145 0.5726496 TRUE TRUE TRUE
#> 420 70 ET.AJF.JU 0.6068376 0.5085470 0.6324786 TRUE TRUE TRUE
#> 426 71 SL.AJF.JU 0.6965812 0.5085470 0.7393162 TRUE TRUE TRUE
#> 431 72 ET.JU.BB 0.6324786 0.5299145 0.7136752 TRUE TRUE TRUE
#> 437 73 SL.JU.BB 0.7393162 0.5299145 0.7948718 TRUE TRUE TRUE
#> 444 74 SL.ET.JU 0.6282051 0.6324786 0.7393162 TRUE TRUE TRUE
#> 448 75 AJF.BB.CY 0.5726496 0.5940171 0.6709402 TRUE TRUE TRUE
#> 456 76 ET.AJF.CY 0.6068376 0.6709402 0.6923077 TRUE TRUE TRUE
#> 462 77 SL.AJF.CY 0.6965812 0.6709402 0.7393162 TRUE TRUE TRUE
#> 468 78 ET.BB.CY 0.7136752 0.5940171 0.6923077 TRUE TRUE FALSE
#> 474 79 SL.BB.CY 0.7948718 0.5940171 0.7393162 TRUE TRUE FALSE
#> 480 80 SL.ET.CY 0.6282051 0.6923077 0.7393162 TRUE TRUE TRUE
#> 485 81 ET.AJF.BB 0.6068376 0.5726496 0.7136752 TRUE TRUE TRUE
#> 491 82 SL.AJF.BB 0.6965812 0.5726496 0.7948718 TRUE TRUE TRUE
#> 498 83 SL.ET.AJF 0.6282051 0.6068376 0.6965812 TRUE TRUE TRUE
#> 504 84 SL.ET.BB 0.6282051 0.7136752 0.7948718 TRUE TRUE TRUE