strans: Stochastic Transitivity

Description Usage Arguments Details Value References See Also Examples

View source: R/strans.R

Description

Checks the weak, moderate, and strong stochastic transitivity.

Usage

1
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} ≥ 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).

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

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

Example output

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

eba documentation built on Jan. 13, 2021, 10:12 a.m.

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