trineq.RdChecks if binary choice probabilities fulfill the trinary inequality.
trineq(M, A = 1:I)For any triple of stimuli \(x, y, z\), the trinary inequality states that, if \(P(x, y) > 1/2\) and \((xy)z\), then $$R(x, y, z) > 1,$$ where \(R(x, y, z) = R(x, y) R(y, z) R(z, x)\), \(R(x, y) = P(x, y)/P(y, x)\), and \((xy)z\) denotes that \(x\) and \(y\) share at least one aspect that \(z\) does not have (Tversky and Sattath, 1979, p. 554).
inclusion.rule checks if a family of aspect sets is
representable by a tree.
Results checking the trinary inequality.
number of tests of the trinary inequality
proportion of triples confirming the trinary inequality
quantiles of \(R(x, y, z)\)
number of transitivity tests performed
data frame reporting \(R(x, y, z)\) for each triple where \(P(x, y) > 1/2\) and \((xy)z\)
Tversky, A., & Sattath, S. (1979). Preference trees. Psychological Review, 86, 542–573. doi:10.1037/0033-295X.86.6.542
data(celebrities) # absolute choice frequencies
A <- list(c(1,10), c(2,10), c(3,10),
c(4,11), c(5,11), c(6,11),
c(7,12), c(8,12), c(9,12)) # the structure of aspects
trineq(celebrities, A) # check trinary inequality for tree A
#>
#> Trinary Inequality
#>
#> Number of Tests: 54
#> % triples confirming trinary inequality: 0.87
#> Quantiles of R(xyz):
#> 25% 50% 75%
#> 1.14 1.39 1.69
#>
trineq(celebrities, A)$chkdf # trinary inequality for each triple
#> triple Pxy Rxyz trineq
#> 1 (LBJ,HW)JU 0.6794872 1.6745404 TRUE
#> 2 (LBJ,HW)CY 0.6794872 1.6390481 TRUE
#> 3 (LBJ,HW)AJF 0.6794872 1.4084252 TRUE
#> 4 (LBJ,HW)BB 0.6794872 1.4950289 TRUE
#> 5 (LBJ,HW)ET 0.6794872 1.0680446 TRUE
#> 6 (LBJ,HW)SL 0.6794872 1.4961569 TRUE
#> 7 (LBJ,CDG)JU 0.6965812 1.2610178 TRUE
#> 8 (LBJ,CDG)CY 0.6965812 1.3045396 TRUE
#> 9 (LBJ,CDG)AJF 0.6965812 1.0140206 TRUE
#> 10 (LBJ,CDG)BB 0.6965812 1.2056276 TRUE
#> 11 (LBJ,CDG)ET 0.6965812 1.1565990 TRUE
#> 12 (LBJ,CDG)SL 0.6965812 1.5656877 TRUE
#> 13 (HW,CDG)JU 0.5897436 0.9996317 FALSE
#> 14 (HW,CDG)CY 0.5897436 1.0565256 TRUE
#> 15 (HW,CDG)AJF 0.5897436 0.9557129 FALSE
#> 16 (HW,CDG)BB 0.5897436 1.0704787 TRUE
#> 17 (HW,CDG)ET 0.5897436 1.4375000 TRUE
#> 18 (HW,CDG)SL 0.5897436 1.3891286 TRUE
#> 19 (JU,CY)LBJ 0.7521368 2.5083596 TRUE
#> 20 (JU,CY)HW 0.7521368 2.5626761 TRUE
#> 21 (JU,CY)CDG 0.7521368 2.4246762 TRUE
#> 22 (JU,CY)BB 0.7521368 1.8397740 TRUE
#> 23 (JU,CY)ET 0.7521368 2.3209481 TRUE
#> 24 (JU,CY)SL 0.7521368 3.0344828 TRUE
#> 25 (AJF,JU)LBJ 0.5085470 1.1354119 TRUE
#> 26 (AJF,JU)HW 0.5085470 0.9549741 FALSE
#> 27 (AJF,JU)CDG 0.5085470 0.9130172 FALSE
#> 28 (AJF,JU)BB 0.5085470 0.8705091 FALSE
#> 29 (AJF,JU)ET 0.5085470 0.9280821 FALSE
#> 30 (AJF,JU)SL 0.5085470 0.8376491 FALSE
#> 31 (AJF,CY)LBJ 0.6709402 1.8493468 TRUE
#> 32 (AJF,CY)HW 0.6709402 1.5891337 TRUE
#> 33 (AJF,CY)CDG 0.6709402 1.4375000 TRUE
#> 34 (AJF,CY)BB 0.6709402 1.0399511 TRUE
#> 35 (AJF,CY)ET 0.6709402 1.3987076 TRUE
#> 36 (AJF,CY)SL 0.6709402 1.6505243 TRUE
#> 37 (ET,BB)LBJ 0.7136752 1.9002628 TRUE
#> 38 (ET,BB)HW 0.7136752 1.3575426 TRUE
#> 39 (ET,BB)CDG 0.7136752 1.8229858 TRUE
#> 40 (ET,BB)JU 0.7136752 1.2848411 TRUE
#> 41 (ET,BB)CY 0.7136752 1.6208781 TRUE
#> 42 (ET,BB)AJF 0.7136752 1.2051368 TRUE
#> 43 (SL,BB)LBJ 0.7948718 2.1088998 TRUE
#> 44 (SL,BB)HW 0.7948718 2.1104911 TRUE
#> 45 (SL,BB)CDG 0.7948718 2.7387218 TRUE
#> 46 (SL,BB)JU 0.7948718 1.2120665 TRUE
#> 47 (SL,BB)CY 0.7948718 1.9991558 TRUE
#> 48 (SL,BB)AJF 0.7948718 1.2596145 TRUE
#> 49 (SL,ET)LBJ 0.6282051 1.2061750 TRUE
#> 50 (SL,ET)HW 0.6282051 1.6896552 TRUE
#> 51 (SL,ET)CDG 0.6282051 1.6327988 TRUE
#> 52 (SL,ET)JU 0.6282051 1.0252861 TRUE
#> 53 (SL,ET)CY 0.6282051 1.3404923 TRUE
#> 54 (SL,ET)AJF 0.6282051 1.1359765 TRUE