Description Usage Arguments Details Value References See Also Examples
Checks the weak, moderate, and strong stochastic transitivity.
1 | strans(M)
|
M |
a square matrix or a data frame consisting of absolute choice frequencies; row stimuli are chosen over column stimuli |
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, 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 ≥ 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.
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. 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
eba
, circular
, kendall.u
,
trineq
.
1 2 3 4 | 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
|
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
<
> 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
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
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