`ordertrans()`

function in the `hyper2`

package```
## function (x, players)
## {
## if (missing(players)) {
## return(x[order(names(x))])
## }
## else {
## if (is.hyper2(players) | is.hyper3(players)) {
## players <- pnames(players)
## }
## }
## stopifnot(length(x) == length(players))
## stopifnot(all(sort(names(x)) == sort(players)))
## stopifnot(all(table(names(x)) == 1))
## x[apply(outer(players, names(x), `==`), 1, which)]
## }
```

To cite the `hyper2`

package in publications, please use
Hankin (2017). The `ordertrans()`

function can be difficult to
understand and this short document provides some sensible use-cases.
The manpage provides a very simple example, but here we are going to
use an even simpler example:

```
## d a b c
## 2 3 1 4
```

In the above, object `x`

is a named vector with elements
`seq_along(x)`

in some order. It means that competitor `d`

came
second, competitor `a`

came third, `b`

came first and `c`

came fourth.
Technically `x`

is an order vector because it answers the question
“where did a particular competitor come?” However, it might equally
be a rank vector because it answers the question “who came first? who
came second?” (see also the discussion at `rrank.Rd`

).

But note that `x`

is not helpfully structured to answer either of
these questions: you have to go searching through the names or the
ranks respectively—both of which may appear in any order—to find
the competitor or rank you are interested in. To find the rank vector
(that is, who came first, second etc), one would use `sort()`

:

```
## b d a c
## 1 2 3 4
```

But this is suboptimal to find the order vector [“where did a
particular competitor come?”]. For the order vector, we want to
rearrange the elements of `x`

so that the names are in alphabetical
order. That way we can see straightaway where competitor `a`

placed
(in this case, third). This nontrivial task is accomplished by
function `ordertrans()`

:

```
## a b c d
## 3 1 4 2
```

Observe that objects `x`

and `o`

are equal in the sense that they are
a rearrangement of one another:

`## [1] TRUE`

`## [1] TRUE`

One consequence of this is that the resulting Plackett-Luce support functions will be the same:

`## log( a * (a + b + c + d)^-1 * (a + c)^-1 * (a + c + d)^-1 * b * d)`

`## log( a * (a + b + c + d)^-1 * (a + c)^-1 * (a + c + d)^-1 * b * d)`

Note carefully that the two support functions above can be
mathematically identical but not formally `identical()`

because
neither the order of the brackets, nor the order of terms within a
bracket, is defined. They look the same on my system but YMMV. It is
possible that the two support functions might appear to be different
even though they are mathematically the same. We can verify that the
two are mathematically identical using package idiom `==`

:

`## [1] TRUE`

How about this:

```
## d a c b e
## 2 3 4 3 6
```

```
## e c a b d
## 3 2 4 5 1
```

```
## d a c b e
## 5 5 8 8 7
```

Above, observe that R is behaving as intended \(\ldots\) but is
potentially confusing. Take the first element. This has value
`2+5=7`

and name `d`

from the name of element 1 of `x`

. But it would
be reasonable to ask for the first element to be the sum of element
`d`

of `x`

and element `d`

of `y`

, which would be `2+1=3`

.
This can be done with `ordertrans()`

:

```
## a b c d e
## 7 8 6 3 9
```

See how `ordertrans()`

has rearranged both `x`

and `y`

so that the
names are in alphabetical order.

But this is not perfect. Consider:

```
## f g h a b
## 3 2 4 5 1
```

```
## a b c d e
## 8 4 7 4 10
```

In this case the result is arguably incorrect.

Let us consider the skating dataset and use `ordertrans()`

to study it
(note that the skating dataset is analysed in more depth in
`skating.Rmd`

).

```
## J1 J2 J3 J4 J5 J6 J7 J8 J9
## hughes 1 4 3 4 1 2 1 1 1
## slutskaya 3 1 1 1 4 1 2 3 2
## kwan 2 3 2 2 2 3 3 2 3
## cohen 5 2 4 3 3 4 4 4 4
## suguri 4 8 5 5 5 7 5 5 5
## butyrskaya 6 5 8 7 12 5 8 7 6
## robinson 7 7 7 9 6 8 10 6 7
## sebestyen 8 10 12 8 7 6 12 8 8
## kettunen 9 9 13 6 13 10 7 11 14
## volchkova 10 6 14 11 10 12 6 9 15
## maniachenko 13 12 11 12 16 11 11 10 9
## fontana 14 11 18 16 9 15 9 12 10
## liashenko 15 13 6 10 8 14 13 14 16
## onda 11 14 10 15 15 13 15 13 11
## hubert 12 17 17 13 11 16 14 15 13
## meier 16 16 9 14 14 9 16 16 12
## gusmeroli 17 15 15 17 17 18 17 17 17
## soldatova 19 18 22 20 21 17 18 18 19
## hegel 20 21 16 22 18 19 21 19 18
## giunchi 18 19 20 21 19 20 20 20 20
## babiakova 22 20 19 19 20 21 19 22 22
## kopac 21 22 23 18 22 22 22 21 21
## luca 23 23 21 23 23 23 23 23 23
```

We might ask how judges `J1`

and `J2`

compare to one another? We need to
create vectors like `x`

and `y`

above:

```
j1 <- skating_table[,1] # column 1 is judge number 1
names(j1) <- rownames(skating_table)
j2 <- skating_table[,2] # column 2 is judge number 2
names(j2) <- rownames(skating_table)
j1
```

```
## hughes slutskaya kwan cohen suguri butyrskaya
## 1 3 2 5 4 6
## robinson sebestyen kettunen volchkova maniachenko fontana
## 7 8 9 10 13 14
## liashenko onda hubert meier gusmeroli soldatova
## 15 11 12 16 17 19
## hegel giunchi babiakova kopac luca
## 20 18 22 21 23
```

```
## hughes slutskaya kwan cohen suguri butyrskaya
## 4 1 3 2 8 5
## robinson sebestyen kettunen volchkova maniachenko fontana
## 7 10 9 6 12 11
## liashenko onda hubert meier gusmeroli soldatova
## 13 14 17 16 15 18
## hegel giunchi babiakova kopac luca
## 21 19 20 22 23
```

```
## j1 j2
## hughes 1 4
## slutskaya 3 1
## kwan 2 3
## cohen 5 2
## suguri 4 8
## butyrskaya 6 5
## robinson 7 7
## sebestyen 8 10
## kettunen 9 9
## volchkova 10 6
## maniachenko 13 12
## fontana 14 11
## liashenko 15 13
## onda 11 14
## hubert 12 17
## meier 16 16
## gusmeroli 17 15
## soldatova 19 18
## hegel 20 21
## giunchi 18 19
## babiakova 22 20
## kopac 21 22
## luca 23 23
```

In the above, see how objects `j1`

and `j2`

have identical names, in
the same order. Observe that `hughes`

is ranked 1 (that is, first) by
`J1`

, and 4 (that is, 4th) by `J2`

. This makes it sensible to plot
`j1`

against `j2`

:

```
par(pty='s') # forces plot to be square
plot(j1,j2,asp=1,pty='s',xlim=c(0,25),ylim=c(0,25),pch=16,xlab='judge 1',ylab='judge 2')
abline(0,1) # diagonal line
for(i in seq_along(j1)){text(j1[i],j2[i],names(j1)[i],pos=4,col='gray',cex=0.7)}
```

In figure 1, we see general agreement but differences
in detail. For example, `hughes`

is ranked first by judge 1 and
fourth by judge 2. However, other problems are not so easy. Suppose
we wish to compare the ranks according to likelihood with the ranks
according to some points system.

```
## babiakova butyrskaya cohen fontana giunchi
## 5.04723518e-06 5.30446955e-03 9.02414906e-02 3.68328234e-04 1.13732552e-05
## gusmeroli hegel hubert hughes kettunen
## 8.31691418e-05 9.78563138e-06 2.68983087e-04 2.98316647e-01 1.37641229e-03
## kopac kwan liashenko luca maniachenko
## 2.99666669e-06 2.65505941e-01 5.18862556e-04 1.00000004e-06 8.06879165e-04
## meier onda robinson sebestyen slutskaya
## 2.92808931e-04 4.88018960e-04 5.30826157e-03 2.58830213e-03 3.08892999e-01
## soldatova suguri volchkova
## 1.26927514e-05 1.84571521e-02 1.13837922e-03
```

Note that in the above, the competitors’ names are in alphabetical order. We first need to convert strengths to ranks:

```
## babiakova butyrskaya cohen fontana giunchi gusmeroli
## 21 7 4 14 19 17
## hegel hubert hughes kettunen kopac kwan
## 20 16 2 9 22 3
## liashenko luca maniachenko meier onda robinson
## 12 23 11 15 13 6
## sebestyen slutskaya soldatova suguri volchkova
## 8 1 18 5 10
```

(note that the names are in the same order as before, alphabetical).
In the above we see that `slutskya`

ranks first, `hughes`

second, and
so on. Another way of ranking the skaters is to use a Borda-type
system: essentially the rowsums of the table (and, of course, the
*lowest* score wins):

```
## hughes slutskaya kwan cohen suguri butyrskaya
## 1 2 3 4 5 6
## robinson sebestyen kettunen volchkova maniachenko fontana
## 7 8 9 10 11 13
## liashenko onda hubert meier gusmeroli soldatova
## 12 14 16 15 17 18
## hegel giunchi babiakova kopac luca
## 19 20 21 22 23
```

It is not at all obvious how to compare `mP`

and `mL`

. For example,
we might be interested in `hegel`

. It takes some effort to find that
her likelihood rank is 20 and her Borda rank is 19. Function
`ordertrans()`

facilitates this:

```
## babiakova butyrskaya cohen fontana giunchi gusmeroli
## 21 6 4 13 20 17
## hegel hubert hughes kettunen kopac kwan
## 19 16 1 9 22 3
## liashenko luca maniachenko meier onda robinson
## 12 23 11 15 14 7
## sebestyen slutskaya soldatova suguri volchkova
## 8 2 18 5 10
```

See above how `ordertrans()`

shows the points-based ranks but in
alphabetical order, to facilitate comparison with `mL`

. We can now
plot these against one another:

However, figure 2 is a bit crude. Function
`ordertransplot()`

gives a more visually pleasing output, see figure
3.

So now we may compare judge 1 against likelihood:

In figure 4, looking at the lower-left corner, we see (reading horizontally) that the likelihood method placed Slutskya first, then Hughes second, then Kwan third; while (reading vertically) judge 1 placed Hughes first, then Kwan, then Slutskya.

Hankin, R. K. S. 2017. “Partial Rank Data with the *The R Journal* 9 (2): 429–39.

`hyper2`

Package: Likelihood Functions for Generalized Bradley-Terry Models.”