---
title: "Generalized Plackett-Luce Likelihoods"
author: "Robin K. S. Hankin"
bibliography: hyper2.bib
output: html_vignette
vignette: |
  %\VignetteIndexEntry{hyper3}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
---

```{r setup, include=FALSE}
library("hyper2",quietly=TRUE)
set.seed(0)
knitr::opts_chunk$set(echo = TRUE)
```

```{r label="hexsticker",out.width='20%', out.extra='style="float:right; padding:10px"',echo=FALSE}
knitr::include_graphics(system.file("help/figures/hyper2.png", package = "hyper2"))
```

To cite the `hyper2` package in publications, please use
@hankin2017_rmd; to cite `hyper3` functionality, use
@hankin2024_hyper3.

The `hyper2` package provides functionality to work with extensions of
the Bradley-Terry probability model such as Plackett-Luce likelihood
including team strengths and reified entities (monsters).  The package
allows one to use relatively natural R idiom to manipulate such
likelihood functions.  Here, I present a generalization of `hyper2` in
which multiple entities are constrained to have identical
Bradley-Terry strengths.  A new `S3` class `hyper3`, along with
associated methods, is motivated and introduced.  Three datasets are
analysed, each analysis furnishing new insight, and each highlighting
different capabilities of the package.


# Introduction

The `hyper2` package [@hankin2010,@hankin2017_rmd] furnishes
computational support for generalized Plackett-Luce [@plackett1975]
likelihood functions.  The preferred interpretation is a race (as in
track and field athletics): given six competitors $1-6$, we ascribe to
them nonnegative strengths $p_1,\ldots, p_6$; the probability that $i$
beats $j$ is $p_i/(p_i+p_j)$.  It is conventional to normalise so that
the total strength is unity, and to identify a competitor with his
strength.  Given an order statistic, say $p_1\succ p_2\succ p_3\succ
p_4$, the Plackett-Luce likelihood function would be

\begin{equation}\label{PL_like}
  \frac{p_1}{p_1+p_2+p_3+p_4}\cdot
  \frac{p_2}{    p_2+p_3+p_4}\cdot
  \frac{p_3}{        p_3+p_4}\cdot
  \frac{p_4}{            p_4}
  \qquad\mbox{Plackett-Luce}
\end{equation}


@mollica2014 call this a forward ranking process on the grounds
that the best (preferred; fastest; chosen) entities are identified in
the same sequence as their rank.  

Computational support for Bradley-Terry likelihood functions is
available in a range of languages.  @hunter2004, for example, presents
results in `MATLAB` [although he works with a nonlinear extension to
account for ties]; @allison1994 present related work for ranking
statistics in `SAS` and @maystre2022 has released a `python` package
for Luce-type choice datasets.

However, the majority of software is written in the R computer
language [@rcore2023], which includes extensive functionality for
working with such likelihood functions: @turner2020 discuss several
implementations from a computational perspective.  The `BradleyTerry`
package [@firth2005] considers pairwise comparisons using
`glm` but cannot deal with ties or player-specific
predictors; the `BradleyTerry2` package [@turner2012] implements a
flexible user interface and wider range of models to be fitted to
pairwise comparison datasets, specifically simple random effects.  The
`PlackettLuce` package [@turner2020] generalizes this to
likelihood functions of the form of the Plackett-Luce equation above,
and applies the Poisson transformation of @baker1994 to express the
problem as a log-linear model.  The `hyper2` package, in contrast,
gives a consistent language interface to create and manipulate
likelihood functions over the simplex
${S}_n=\left\lbrace\left(p_1,\ldots,p_n\right)\left|p_i\geq 0,\sum
p_i=1\right.\right\rbrace$.  A further extension in the package
generalizes this likelihood function to functions of ${\mathbf
p}=(p_1,\ldots,p_n)$ with

\begin{equation}\label{hyper2likelihood}
\mathcal{L}\left(\mathbf{p}\right)=
\prod_{s\in \mathcal{O}}\left({\sum_{i\in s}}p_i\right)^{n_s}\qquad\mbox{Hyperdirichlet likelihood}
\end{equation}

where $\mathcal{O}$ is a set of observations and $s$ a subset of
$\left\{1,2,\ldots,n\right\}$; numbers $n_s$ are integers which may be
positive or negative and we usually require $\sum_s n_s=0$.  The
`hyper2` package has the ability to evaluate such likelihood functions
at any point in $S_n$, thereby admitting a wide range of non-standard
nulls such as order statistics on the $p_i$ [@hankin2017_rmd].  It
becomes possible to analyse a wider range of likelihood functions than
standard Plackett-Luce [@turner2020].  For example, results involving
incomplete order statistics or teams are tractable.  Further, the
introduction of reified entities (monsters) allows one to consider
draws [@hankin2010], noncompetitive tactics such as collusion
[@hankin2020], and the phenomenon of team cohesion wherein the team
becomes stronger than the sum of its parts [@hankin2010].  Recent
versions of the package include experimental functionality
`cheering()` to investigate the relaxing of the assumption of
conditional independence of the forward-ranking process.

Here I present a different generalization.  Consider a race in which
there are six runners 1-6 but we happen to know that three of the
runners (1,2,3) are clones of strength $p_a$, two of the runners (4,5)
have strength $p_b$, and the final runner (6) is of strength $p_c$.
We normalise so $p_a+p_b+p_c=1$.  The runners race and the finishing
order is:

$$a\succ c\succ b\succ a\succ a \succ b$$

Thus the winner was $a$, second place for $c$, third for $b$, and so
on.  Alternatively we might say that $a$ came first, fourth, and
fifth; $b$ came third and sixth, and $c$ came second.  The
Plackett-Luce likelihood function for $p_a,p_b,p_c$ would be

\begin{equation}\label{plackettluce}
\frac{p_a}{3p_a+2p_b+p_c}\cdot
\frac{p_c}{2p_a+2p_b+p_c}\cdot
\frac{p_b}{2p_a+2p_b    }\cdot
\frac{p_a}{2p_a+ p_b    }\cdot
\frac{p_a}{ p_a+ p_b\vphantom{x_{x_{x_{x_{x_{x_{x_{x_{x_{x_{x_{x_{x_{x_{x_{x_{x_{x}}}}}}}}}}}}}}}}}}
                        }\cdot
\frac{p_b}{      p_b    },\qquad p_a+p_b+p_c=1\\
\mbox{generalized plackett-Luce likelihood}
\end{equation}

Here I consider such generalized Plackett-Luce likelihood functions,
and give an exact analysis of several simple cases.  I then show how
this class of likelihood functions may be applied to a range of
inference problems involving order statistics.  Illustrative examples,
drawn from Formula 1 motor racing, and track-and-field athletics, are
given.

## Computational methodology for generalized Plackett-Luce likelihood functions

Existing `hyper2` formalism as described by @hankin2017_rmd cannot
represent the Plackett-Luce likelihood equation, because the
hyperdirichlet likelihood equation uses sets as the indexing elements,
and in this case we need multisets\footnote{Note that the version of
`hyper2` presented by @hankin2017_rmd and reviewed by [@turner2020]
used integer-valued sets together with a print method that used a
complicated mapping system from integers to competitor names.  Current
methodology [following commit `51a8b46`] is to use sets of character
strings which represent the competitors directly; this allows for
easier combination of observations including different competitors.}.
The declarations

```
typedef map<string, long double> weightedplayervector;
typedef map<weightedplayervector, long double> hyper3;
```

show how the `map` class of the Standard Template Library is
used with `weightedplayervector` objects mapping strings to
long doubles (specifically, mapping player names to their
multiplicities), and objects of class `hyper3` are maps from a
`weightedplayervector` object to long doubles.  One advantage
of this is efficiency: search, removal, and insertion operations have
logarithmic complexity.  As an example, the following `C++`
pseudo code would create a log-likelihood function for the first term
in the Plackett-Luce likelihood equation above:

```
weightedplayervector n,d;
n["a"] = 1; 
d["a"] = 3; 
d["b"] = 2;
d["c"] = 1;

hyper3 L;
L[n] = 1;
L[d] = -1;
```

Above, we understand `n` and `d` to represent numerator and
denominator respectively.  Object `L` is an object of class
`hyper3`; it may be evaluated at points in probability space
[that is, a vector `[a,b,c]` of nonnegative values with unit sum]
using standard R idiom wrapping C++ back end.

## Package implementation

The package includes an `S3` class `hyper3` for this type of object;
extraction and replacement methods use `disordR` discipline
[@hankin2022].  Package idiom for creating such objects uses named
vectors:

```{r showLL}
LL <- hyper3()
LL[c(a = 1)] <- 1
LL[c(a = 3, b = 2, c = 1)] <- -1
LL
```

Above, we see object `LL` is a log-likelihood function of the
players' strengths, which may be evaluated at specified points in
probability space.  A typical use-case would be to assess $H_1\colon
p_a=0.9,p_b=0.05,p_c=0.05$ and $H_2\colon p_a=0.01,p_b=0.01,p_c=0.98$,
and we may evaluate these hypotheses using generic function
`loglik()`:

```{r loglikabc}
loglik(c(a = 0.01, b = 0.01, c = 0.98), LL)
loglik(c(a = 0.90, b = 0.05, c = 0.05), LL)
```

Thus we prefer $H_1$ over $H_2$ with about 3.5 units of support,
satisfying the standard two units of support criterion [@edwards1972],
and we conclude that our observation [in this case, that one of the
three clones of player $a$ beat the $b$ twins and the singleton $c$]
furnishes strong support against $H_2$ in favour of $H_1$.

The package includes many helper functions to work with order
statistics of this type.  Function `ordervec2supp3()`, for
example, can be used to generate a Plackett-Luce log-likelihood
function:

```{r orderacbaab}
(H <- ordervec2supp3(c("a", "c", "b", "a", "a", "b")))
```

(the package gives extensive documentation at `ordervec2supp.Rd`).  We
may find a maximum likelihood estimate for the players' strengths,
using generic function `maxp()`, dispatching to a specialist
`hyper3` method:

```{r mhmaxp}
(mH <- maxp(H))
```

(function `maxp()` uses standard optimization techniques to
locate the evaluate; it has access to first derivatives of the
log-likelihood and as such has rapid convergence, if its objective
function is reasonably smooth).

The package provides a number of statistical tests on likelihood
functions, modified from @hankin2017_rmd to work with `hyper3`
objects.  For example, we may assess the hypothesis that all three
players are of equal strength [viz $H_0\colon
p_a=p_b=p_c=\frac{1}{3}$]:

```{r testequality}
equalp.test(H)
```

showing, perhaps unsurprisingly, that this small dataset is consistent
with $H_0$. 

## Package helper functions

Arithmetic operations are implemented for `hyper3` objects in much the
same way as for `hyper2` objects: independent observations may be
combined using the overloaded `+` operator; an example is given below.

The original motivation for `hyper3` was the analysis of Formula 1
motor racing, and the package accordingly includes wrappers for
`ordervec2supp()` such as `ordertable2supp3()` and
`attemptstable2supp3()` which facilitate the analysis of commonly
encountered result formats.  Package documentation for order tables is
given at `ordertable.Rd` and an example is given below.

# Exact analytical solution for some simple generalized Plackett-Luce likelihood functions

Here I consider some order statistics with nontrivial maximum
likelihood Bradley-Terry strengths.  The simplest nontrivial case
would be three competitors with strengths $a,a,b$ and finishing order
$a\succ b\succ a$.  The Plackett-Luce likelihood function would be

\begin{equation}\label{aba}
\frac{a}{2a+b}\cdot\frac{b}{a+b}
\end{equation}

and in this case we know that $a+b=1$ so this is equal to
$\mathcal{L}=\mathcal{L}(a)=\frac{a(1-a)}{1+a}$.  The score would be
given by

\begin{equation}\label{aba_mle}
\frac{d\mathcal{L}}{da}=\frac{(1+a)(1-2a)-a(1-a)}{(1+a)^2}=
\frac{1-2a-a^2}{(1+a)^2}
\end{equation}

and this will be zero at $\sqrt{2}-1$; we also note that
$d^2\mathcal{L}/da^2=-4(1+a)^{-3}$, manifestly strictly negative for
$0\leq a\leq 1$: the root is a maximum.


```{r maxpaba}
maxp(ordervec2supp3(c("a", "b", "a")))
```

Above, we see close agreement with the theoretical value of
$(\sqrt{2}-1,2-\sqrt{2})\simeq (0.414,0.586)$.  Observe that the
maximum likelihood estimate for $a$ is strictly less than 0.5, even
though the finishing order is symmetric.  Using
$\mathcal{L}(a)=\frac{a(1-a)}{1+a}$, we can show that
$\log\mathcal{L}(\hat{a})=\log\left(3-2\sqrt{2}\right)\simeq -1.76$,
where $\hat{a}=\sqrt{2}-1$ is the maximum likelihood estimate for $a$.
Defining $\mathcal{S}=\log\mathcal{L}$ as the support [log-likelihood]
we have

\begin{equation}
\mathcal{S}=\mathcal{S}(a)=\log\left(\frac{a(1-a)}{1+a}\right)-\log\left(3-2\sqrt{2}\right)
\end{equation}

as a standard support function which has a maximum value of zero when
evaluated at $\hat{a}=\sqrt{2}-1$.  For example, we can test the null
that $a=b=\frac{1}{2}$, the statement that the competitors have equal
Bradley-Terry strengths:


```{r ahalf}
a <- 1/2 # null
(S_delta <- log(a * (1 - a)/(1 + a)) - log(3 - 2 * sqrt(2)))
```

Thus the additional support gained in moving from $a=\frac{1}{2}$ to
the evaluate of $a=\sqrt{2}-1$ is 0.029, rather small [as might be
expected given that we have only one rather uninformative observation,
and also given that the maximum likelihood estimate ($\simeq 0.41$) is
quite close to the null of $0.5$].  Nevertheless we can follow
[@edwards1972} and apply Wilks's theorem for a $p$ value:

```{r usechisq}
pchisq(-2 * S_delta, df = 1, lower.tail = FALSE)
```

The $p$-value is about 0.81, exceeding 0.05; thus we have no strong
evidence to reject the null of $a=\frac{1}{2}$.  The observation is
informative, in the sense that we can find a credible interval for
$a$.  With an $n$-units of support criterion the analytical solution
to $\mathcal{S}(p)=-n$ is given by defining $X=\log(3-2\sqrt{2})-n$
and solving $p(1-p)/(1+p)=X$, or
$p_\pm=\left(1-X\pm\sqrt{1+4X+X^2}\right)/2$, the two roots being the
lower and upper limits of the credible interval; see the figure below.


```{r, fig.width=7, fig.height=7, fig.cap="A support function for a>b>a"}
a <- seq(from = 0, by = 0.005, to = 1)
S <- function(a){log(a * (1 - a) / ((1 + a) * (3 - 2 * sqrt(2))))}
plot(a, S(a), type = 'b',xlab=expression(p[a]),ylab="support")
abline(h = c(0, -2))
abline(v = c(0.02438102, 0.9524271), col = 'red')
abline(v = sqrt(2) - 1)
```

## Fisher information

If we have two clones of $a$ and a singleton $b$, then there are three
possible rank statistics: (i), $a\succ a\succ b$ with probability
$\frac{2a^2}{1+a}$; (ii), $a\succ b\succ a$, with
$\frac{2a(1-a)}{(1+a)}$, (iii), $b\succ a\succ a$ at
$\frac{1-a}{1+a}$.  Likelihood functions for these order statistics
are given in the figure below.  It can be shown that the
Fisher information for such observations is

\begin{equation}
\mathcal{I}(a)=2\frac{1+a+a^2}{a(1-a)(1+a)^2}
\end{equation}

which has a minimum of about $6.21$ at at about $a=0.522$.  We can
compare this with the Fisher information matrix ${\mathcal I}$, for
the case of three distinct runners $a,b,c$, evaluated at its minimum
of $p_a=p_b=p_c=\frac{1}{3}$.  If we observe the complete order
statistic, $\left|{\mathcal I}\right| =\frac{1323}{16}\simeq 82.7$; if
we observe just the winner, $\left|{\mathcal I}\right|=27$, and if we
observe just the loser we have $\left|{\mathcal
I}\right|=\frac{16875}{256}\simeq 65.9$.  A brief discussion is given
at `inst/fisher_inf_PL3.Rmd`.

```{r, fig.width=7, fig.height=7, fig.cap="Support functions for observations a>a>b, a>b>a and b>a>a.  Horizontal dotted line represents two units of support"}
f_aab <- function(a){a^2 / (1 + a)}
f_aba <- function(a){a * (1 - a) / (1 + a)}
f_baa <- function(a){(1 - a) / (1 + a)}
p <- function(f, ...){
  a <- seq(from = 0, by = 0.005, to = 1)
  points(a, f(a) / max(f(a)), ...)
  }
plot(0:1, 0:1, xlab = expression(p[a]), ylab = "Likelihood", type = "n")
p(f_aab, type = "l", col = "black")
p(f_aba, type = "l", col = "red")
p(f_baa, type = "l", col = "blue")
text(0.8,0.8,"AAB")
text(0.8,0.4,"ABA",col="red")
text(0.8,0.05,"BAA",col="blue")
abline(h = exp(-2), lty = 2)
```

## Nonfinishers

If we allow non-finishers---that is, a subset of competitors who are
beaten by all the ranked competitors ([@turner2020] call this a _top
$n$ ranking_), there is another nontrivial order statistic, viz
$a\succ b\succ\left\lbrace a,b\right\rbrace$ [thus one of the two
$a$'s won, one of the $b$'s came second, and one of each of $a$ and
$b$ failed to finish].  Now

\begin{equation}
\mathcal{L}(a)=
\frac{a}{2a+2b}\cdot
\frac{b}{ a+2b}\propto\frac{a(1-a)}{2-a}
\end{equation}

(see how the likelihood function is actually simpler than for the
complete order statistic).  The evaluate would be $2-\sqrt{2}\simeq
0.586$:

```{r maxoab}
maxp(ordervec2supp3(c("a", "b"), nonfinishers=c("a", "b")))
```

The Fisher information for such observations has a minimum of
$\frac{68}{9}\simeq 7.56$ at $a=\frac{1}{2}$.  An inference problem
for a dataset including nonfinishers will be given below.

# An alternative to the Mann-Whitney test using generalized Plackett-Luce likelihood

The ideas presented above can easily be extended to arbitrarily large
numbers of competitors, although the analytical expressions tend to be
intractable and numerical methods must be used.  All results and
datasets presented here are maintained under version control and
available at `https://github.com/RobinHankin/hyper2`.  Given an
order statistic of the type considered above, the
Mann-Whitney-Wilcoxon test [@mann1947,wilcoxon1945] assesses a
null of identity of underlying distributions [@ahmad1996].
Consider the chorioamnion dataset [@hollander2013], used in
`wilcox.test.Rd`:

```{r define_xy_wilcox}
x <- c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46)
y <- c(1.15, 0.88, 0.90, 0.74, 1.21)
```

Here we see a measure of permeability of the human placenta at term
(`x`) and between 3 and 6 months' gestational age (`y`).  The order
statistic is straightforward to calculate:

```{r hyper3osdef}
names(x) <- rep("x", length(x))
names(y) <- rep("y", length(y))
(os <- names(sort(c(x, y))))
```

Then object `os` is converted to a `hyper3` object, again
with `ordervec2supp3()`, which may be assessed using the Method
of Support:

```{r hyper3xytest}
Hxy <- ordervec2supp3(os)
equalp.test(Hxy)
```

Above, we use generic function `equalp.test()` to test the null
that the permeability of the two groups both have Bradley-Terry
strength of $0.5$.  We see a $p$ value of about 0.09; compare 0.25 from
`wilcox.test()`.  However, observe that the `hyper3`
likelihood approach gives more information than Wilcoxon's analysis:
Firstly, we see that the maximum likelihood estimate for the
Bradley-Terry strength of `x` is about 0.24, considerably less
than the null of 0.5; further, we may plot a support curve for this
dataset, given in the figure below.

```{r, fig.width=7, fig.height=7, fig.cap="A support function for the Bradley-Terry strength of permeability at term.  The evaluate of 0.24 is shown; and the two-units-of support credible interval, which does not exclude the null of strength 0.5 (dotted line), is also shown"}
a <- seq(from = 0.02, to = 0.8, len = 40)
L <- sapply(a, function(p){loglik(p, Hxy)})
plot(a, L - max(L), type = 'b',xlab=expression(p[a]),ylab="likelihood")
abline(h = c(0, -2))
abline(v = c(0.24))
abline(v=c(0.5), lty=2)
```

## A generalization of the Mann-Whitney test using generalized Plackett-Luce likelihood

The ideas presented above may be extended to more than two types of
competitors.  Consider the following table, drawn from the men's
javelin, 2020 Olympics:

```{r javtab}
javelin_table
```

Thus Chopra threw 87.03m on his first throw, 87.58m on his second, and
so on.  No-throws, ignored here, are indicated with an `X`.  We
may convert this to a named vector with elements being the throw
distances, and names being the competitors, using
`attemptstable2supp3()`:

```{r converttosupp3}
javelin_vector <- attemptstable2supp3(javelin_table,
       decreasing = TRUE, give.supp = FALSE)
options(width = 60)
javelin_vector
```

Above we see that Chopra threw the longest and second-longest throws
of 87.58m and 87.03 respectively; Vadlejch threw the third-longest
throw of 86.67m, and so on (`NA` entries correspond to
no-throws.)  The attempts table may be converted to a `hyper3`
object, again using function `attemptstable2supp3()` but this
time passing `give.supp=TRUE`:

```{r javorder}
javelin <- ordervec2supp3(v = names(javelin_vector)[!is.na(javelin_vector)])
```

Above, object `javelin` is a `hyper3` likelihood function, so one has
access to the standard likelihood-based methods, such as finding and
displaying the maximum likelihood estimate, shown in the figure below.


```{r setoptsdig3,echo=FALSE}
options(digits = 3)
```

```{r, fig.width=7, fig.height=7, fig.cap="Maximum  likelihood estimate for javelin throwers' Bradley-Terry strengths"}
(mj <- maxp(javelin))
dotchart(mj, pch = 16,xlab="Estimated Bradley-Terry strength")
```

From this, we see that Vadlejch has the highest estimated
Bradley-Terry strength, but further analysis with `equalp.test()`
reveals that there is no strong evidence in the dataset to reject the
hypothesis of equal competitive strength ($p=0.26$), or that Vadlejch
has a strength higher than the null value of $\frac{1}{8}$ ($p=0.1$).

A particularly attractive feature of this analysis is that the
Bradley-Terry strengths have direct operational significance: If two
competitors, say Vadlejch and Vesely, were to throw a javelin, then we
would estimate the probability that Vadlejch would throw further than
Vesely at $\displaystyle p_{\mbox{Vad}}/\left(p_{\mbox{Vad}} +
p_{\mbox{Ves}}\right)\simeq 0.74$.  Indeed, from a training or
selection perspective we might follow @hankin2017_rmd and observe that
log-contrasts [@ohagan2004] appear to have approximately Gaussian
likelihood functions for observations of the type considered here.
Profile log-likelihood curves for log-contrasts are easily produced by
the package, below.

```{r, echo=FALSE, fig.width=7, fig.height=7, fig.cap="Profile likelihood for log-contrast as per the text. Null indicated with a dotted line, and two-units-of-support limit indicated with horizontal lines; thus the null is not rejected"}
M <-  structure(c(3.13462808462437, -3.49547133821576, 2.31958387909791, 
 -1.453478109001, 1.85404664879129, -0.591727993171389, 1.47038547592281, 
 -0.172268427320532, 1.17564017651567, -0.00849666385090586, 0.927885079194978, 
 0, 0.706501034186756, -0.0998759637247275, 0.536848604955139, 
 -0.281432246390324, 0.33818759910768, -0.521602929477154, 0.167613563612845, 
 -0.816172716636203, 0.0827382775966942, -1.20676576256501, -0.138220714577699, 
 -1.5361019338357, -0.273432638006251, -1.95101772253715, -0.409370691327909, 
 -2.40043826949947, -0.560827492984493, -2.88462938724336, -0.666618197447362, 
 -3.39664587496794), dim = c(2L, 16L), dimnames = list(c("logcontrast", 
 "support"), NULL))
plot(t(M), type = "b")
abline(h = c(0, -2))
abline(v = 0, lty = 2)
abline(v = log(0.32062833 / 0.11402735)) # these from mp
```

We see that the credible range for $\log\left(p_{\mbox{Vad}}/
p_{\mbox{Ves}}\right)$ includes zero and we have no strong evidence
for these athletes having different (Bradley-Terry) strengths.


# Formula 1 motor racing: The Constructors' Championship

Formula 1 motor racing is an important and prestigious motor sport
[@codling2017,jenkins2010].  In Formula 1 Grand Prix, the
constructors' championship takes place between _manufacturers_ of
racing cars (compare the drivers' championship, which is between
drivers).  In this analysis, the constructor is the object of
inference.  Each constructor typically fields two cars, each of which
separately accumulates ranking-based points at each venue.  Here we
use a generalized Plackett-Luce model to assess the constructors'
performance.  The following table, included in the `hyper2`
package as a dataset, shows rankings for the first 9 venues of the
2021 season:

```{r cons2021}
constructor_2021_table[, 1:9]
```

Above, we see that Mercedes ("`Merc`") came first and third at Bahrain
(`BHR`); and came second and retired at Emilia Romagna (`EMI`); full
details of the notation and conventions are given in the package at
`constructor.Rd`.  The identity of the driver is viewed as
inadmissible information and indeed may change during a season.
Alternatively, we may regard the driver and the constructor as a joint
entity, with the constructor's ability to attract and retain a skilled
driver being part of the object of inference.  The associated
generalized Plackett-Luce `hyper3` object is easily constructed using
package idiom, in this case `ordertable2supp3()`, and we may use this
to assess the Plackett-Luce strengths of the constructors:

```{r cheatmaxp2, echo=TRUE, eval=FALSE}
const2020 <- ordertable2supp3(constructor_2020_table)
const2021 <- ordertable2supp3(constructor_2021_table)
options(digits=4)
```

```
maxp(const2020)
##    ARRF     ATH Ferrari      HF    Merc      MR       R 
## 0.04530 0.06807 0.06063 0.02623 0.37783 0.10026 0.09767 
##    RBRH  RPBWTM      WM 
## 0.12072 0.08055 0.02273
```

```
maxp(const2021)
##     AMM      AR    ARRF     ATH Ferrari      HF    Merc 
## 0.05942 0.07543 0.06238 0.05611 0.16939 0.02023 0.19395 
##      MM    RBRH      WM 
## 0.14126 0.18334 0.03848
```


Above, we see the strength of Mercedes falling from about 0.38 in 2020
to less than 0.20 in 2021 and it is natural to wonder whether this can
be ascribed to random variation.  Observe that testing such a
hypothesis is complicated by the fact that constructors field multiple
cars, and also that constructors come and go, with two 2020 teams
dropping out between years and two joining.  We may test this
statistically by defining a combined likelihood function for both
years, keeping track of the year:

```{r defcomblike}
H <- (
      psubs(constructor_2020, "Merc", "Merc2020") +
      psubs(constructor_2021, "Merc", "Merc2021")
     )
```

Above, we use generic function `psubs()` to change the name of
Mercedes from `Merc` to `Merc2020` and `Merc2021`
respectively.  Note the use of `+` to represent addition of
log-likelihoods, corresponding to the assumption of conditional
independence of results.  The null would be simply that the strengths
of `Merc2020` and of `Merc2021` are identical.  Package
idiom would be to use generic function `samep.test()`:

```{r usesameptest,eval=FALSE}
options(digits = 4)
samep.test(H, c("Merc2020", "Merc2021"))
```

```
## 
##  Constrained support maximization
## 
## data:  H
## null hypothesis: Merc2020 = Merc2021
## null estimate:
##      AMM       AR     ARRF      ATH  Ferrari       HF 
##  0.04239  0.05413  0.04677  0.04374  0.07568  0.02323 
## Merc2020 Merc2021       MM       MR        R     RBRH 
##  0.13903  0.13903  0.09016  0.07944  0.07475  0.10024 
##   RPBWTM       WM 
##  0.06235  0.02905 
## (argmax, constrained optimization)
## Support for null:  -1189 + K
## 
## alternative hypothesis:  sum p_i=1 
## alternative estimate:
##      AMM       AR     ARRF      ATH  Ferrari       HF 
##  0.03766  0.04824  0.04333  0.04060  0.07036  0.02132 
## Merc2020 Merc2021       MM       MR        R     RBRH 
##  0.23135  0.09216  0.07893  0.07973  0.07455  0.09322 
##   RPBWTM       WM 
##  0.06177  0.02679 
## (argmax, free optimization)
## Support for alternative:  -1184 + K
## 
## degrees of freedom: 1
## support difference = 4.722
## p-value: 0.002119
```

Above, we see strong evidence for a real decrease in the strength of
the Mercedes team from 2020 to 2021, with $p=0.002$.

# Conclusions and further work

Plackett-Luce likelihood functions for rank datasets have been
generalized to impose identity of Bradley-Terry strengths for certain
groups; the preferred interpretation is a running race in which the
competitors are split into equivalence classes of clones.
Implementing this in `R` is accomplished via a `C++` back-end making
use of the `STL` "map" class which offers efficiency advantages,
especially for large objects.

New likelihood functions for simple cases with three or four
competitors were presented, and extending to larger numbers furnishes
a generalization of the Mann-Whitney-Wilcoxon test that offers a
specific alternative (Bradley-Terry strength) with a clear operational
definition.  Further generalizations allow the analysis of more than
two groups, here applied to Olympic javelin throw distances.
Generalized Plackett-Luce likelihood functions were used to assess the
Grand Prix constructors' championship and a reasonable null.
Specifically, the hypothesis that the strength of the Mercedes team
remained unchanged between 2020 and 2021 was tested and rejected.

Draws are not considered in the present work but in principle may be
accommodated, either using likelihoods comprising sums of
Plackett-Luce probabilities [@hankin2017_rmd]; or the introduction of
a reified draw entity [@hankin2010].

Further work might include a systematic comparison between
`hyper3` approach and the Mann-Whitney-Wilcoxon test, including
the characterisation of the power function of both tests.  The package
could easily be extended to allow non-integer multiplicities, which
might prove useful in the context of reified Bradley Terry
techniques [@hankin2020].


# References