To cite the hyper2 package in publications, please use Hankin (2017). This file presents some speculative experimental work which is under development. Do not confuse this file with icons.Rmd, which creates the icons objects in the package.

icons
## log(L^24 * (L + NB + OA + THC)^-20 * (L + NB + OA + WAIS)^-9 * (L + NB
## + THC + WAIS)^-15 * (L + OA + PB + THC)^-11 * (L + OA + PB + WAIS)^-18
## * (L + PB + THC + WAIS)^-16 * NB^32 * (NB + OA + PB + THC)^-18 * (NB +
## OA + PB + WAIS)^-8 * (NB + PB + THC + WAIS)^-18 * OA^14 * PB^30 *
## THC^24 * WAIS^9)
icons_maxp <- sort(icons_maxp,decreasing=TRUE)  # sort in decreasing order
icons_maxp
##       NB       PB        L      THC       OA     WAIS 
## 0.252304 0.224582 0.173644 0.170113 0.110686 0.068671

Recall samep.test():

samep.test(icons,c("NB","THC"))
## 
##  Constrained support maximization
## 
## data:  icons
## null hypothesis: NB = THC
## null estimate:
##       NB        L       PB      THC       OA     WAIS 
## 0.209060 0.175554 0.227870 0.209060 0.109874 0.068582 
## (argmax, constrained optimization)
## Support for null:  -175.98 + K
## 
## alternative hypothesis:  sum p_i=1 
## alternative estimate:
##       NB        L       PB      THC       OA     WAIS 
## 0.252304 0.173644 0.224582 0.170113 0.110686 0.068671 
## (argmax, free optimization)
## Support for alternative:  -175 + K
## 
## degrees of freedom: 1
## support difference = 0.98583
## p-value: 0.16027

I will calculate all pairwise distances between the icons, as measured by the support difference [penultimate line of the output above, named statistic in the object for consistency with other tests in R]

M <- matrix(0,6,6)
p <- names(icons_maxp)
rownames(M) <- p
colnames(M) <- p
for(i in seq_len(6)){
  for(j in seq_len(6)){
    if(i != j){M[i,j] <- samep.test(icons,p[c(i,j)])$statistic}
  }
}
M
##            NB       PB         L       THC      OA   WAIS
## NB   0.000000 0.084169 0.8130461 0.9858289 3.20674 6.4970
## PB   0.084169 0.000000 0.3847325 0.4603617 2.32110 5.4088
## L    0.813046 0.384732 0.0000000 0.0023084 0.86195 2.9809
## THC  0.985829 0.460362 0.0023084 0.0000000 0.75044 2.7563
## OA   3.206739 2.321100 0.8619462 0.7504358 0.00000 0.5954
## WAIS 6.496962 5.408845 2.9809307 2.7562598 0.59540 0.0000
(M <- round(M*100))
##       NB  PB   L THC  OA WAIS
## NB     0   8  81  99 321  650
## PB     8   0  38  46 232  541
## L     81  38   0   0  86  298
## THC   99  46   0   0  75  276
## OA   321 232  86  75   0   60
## WAIS 650 541 298 276  60    0

Observations: the triangle inequality is broken, consider NB-L (81), L-PB (38) and PB-NB (8)

plot(phangorn::upgma(as.dist(M)))

References

Hankin, R. K. S. 2017. “Partial Rank Data with the hyper2 Package: Likelihood Functions for Generalized Bradley-Terry Models.” The R Journal 9 (2): 429–39.