% -*- mode: noweb; noweb-default-code-mode: R-mode; -*-
\documentclass[article]{jss}

\usepackage{orcidlink,thumbpdf,lmodern}
\graphicspath{{Figures/}}

\author{Robin K. S. Hankin~\orcidlink{0000-0001-5982-0415}\\
 University of Stirling}
\title{Generalized Plackett-Luce Likelihoods}
\Plainauthor{Robin K. S. Hankin} %% comma-separated
\Plaintitle{Generalized Plackett-Luce Likelihoods}

%% an abstract and keywords
\Abstract{
  The \pkg{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 \proglang{R}~idiom to manipulate such likelihood functions.  Here, I
  present a generalization of \code{hyper2} in which multiple entities
  are constrained to have identical Bradley-Terry strengths.  A new
  \proglang{S3} class `\code{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.  }

\Keywords{Plackett-Luce, Bradley-Terry, Mann-Whitney}
\Plainkeywords{Plackett-Luce, Bradley-Terry, Mann-Whitney}

%% publication information
%% NOTE: Typically, this can be left commented and will be filled out by the technical editor
%% \Volume{50}
%% \Issue{9}
%% \Month{June}
%% \Year{2012}
%% \Submitdate{2012-06-04}
%% \Acceptdate{2012-06-04}

%% The address of (at least) one author should be given
%% in the following format:
\Address{
  Robin K. S. Hankin\\
  University of Stirling\\
  Scotland\\
  E-mail: \email{hankin.robin@gmail.com}\\
  URL: \url{https://www.stir.ac.uk/people/1966824}
}

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\begin{document}


\section{Introduction}

The \pkg{hyper2} package \citep{hankin2009,hankin2017} furnishes
computational support for generalized
Plackett-Luce \citep{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 normalize 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}.
\end{equation}
%
\citet{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.  \cite{hunter2004}, for example,
presents results in \proglang{MATLAB} (although he works with a
nonlinear extension to account for ties); \cite{allison1994} present
related work for ranking statistics in \proglang{SAS} and
\cite{maystre2022} has released a \proglang{Python} package for
Luce-type choice datasets.

However, the majority of software is written in the \proglang{R}~computer language \citep{rcore2023}, which includes extensive
functionality for working with such likelihood functions:
\cite{turner2020} discuss several implementations from a computational
perspective.  The \pkg{BradleyTerry} package \citep{firth2005}
considers pairwise comparisons using \code{glm} but cannot deal
with ties or player-specific predictors; the \pkg{BradleyTerry2}
package \citep{turner2012} implements a flexible user interface and
wider range of models to be fitted to pairwise comparison datasets,
specifically simple random effects.  The \pkg{PlackettLuce} package
\citep{turner2020} generalizes this to likelihood functions of the
form of Equation~\ref{PL_like} and applies the Poisson transformation
of~\cite{baker1994} to express the problem as a log-linear model.  The
\pkg{hyper2} package, in contrast, gives a consistent language
interface to create and manipulate likelihood functions over the
simplex ${S}_n=\left\lbrace\left(p_1,\ldots,p_n\right)\mid p_i\geq
0,\sum p_i=1\right\rbrace$.  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}
\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.  The \pkg{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 $p_i$ \citep{hankin2017}.  It becomes possible to
analyse a wider range of likelihood functions than standard
Plackett-Luce \citep{turner2020}.  For example, results involving
incomplete order statistics or teams are tractable.  Further, the
introduction of reified entities (monsters) allows one to consider
draws \citep{hankin2009}, noncompetitive tactics such as
collusion \citep{hankin2020}, and the phenomenon of team cohesion
wherein the team becomes stronger than the sum of its
parts \citep{hankin2010}.  Recent versions of the package include
experimental functionality (\code{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 normalize 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    }\cdot
\frac{p_b}{      p_b    },\qquad p_a+p_b+p_c=1.
\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.

\subsection{Computational methodology}

Existing \code{hyper2} formalism as described by \citet{hankin2017}
cannot represent Equation~\ref{plackettluce}, because
Equation~\ref{hyper2likelihood} uses sets as the indexing elements,
and in this case we need multisets\footnote{Note that the version of
\code{hyper2} presented by \citet{hankin2017} and reviewed
by~\cite{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
  \code{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
%
\begin{verbatim}
typedef map<string, long double> weightedplayervector;
typedef map<weightedplayervector, long double> hyper3;
\end{verbatim}
%
show how the `\code{map}' class of the Standard Template Library is
used with `\code{weightedplayervector}' objects mapping strings to
long doubles (specifically, mapping player names to their
multiplicities), and objects of class `\code{hyper3}' are maps from
a `\code{weightedplayervector}' object to long doubles.  One
advantage of this is efficiency: Search, removal, and insertion
operations have logarithmic complexity.  As an example, the following
\proglang{C++} pseudo code would create a log-likelihood function for
the first term in Equation~\ref{plackettluce}:
%
\begin{verbatim}
weightedplayervector n,d;
n["a"] = 1; 
d["a"] = 3; 
d["b"] = 2;
d["c"] = 1;

hyper3 L;
L[n] = 1;
L[d] = -1;
\end{verbatim}
%
Above, we understand \code{n} and \code{d} to represent numerator and
denominator respectively.  Object \code{L} is an object of class
`\code{hyper3}'; it may be evaluated at points in probability space
(that is, a vector \code{[a,b,c]} of nonnegative values with unit sum)
using standard \proglang{R} idiom wrapping \proglang{C++} back end.

\subsection{Package implementation}

Functionality for working with generalized Plackett-Luce likelihoods
is provided in the package \pkg{hyper2}~\citep{hankin2017,hankin2009},
available from the Comprehensive \proglang{R} Archive Network (CRAN)
at \url{https://CRAN.R-project.org/hyper2}.  The package includes an
\proglang{S}3 class `\code{hyper3}' for this type of object;
extraction and replacement methods use \pkg{disordR} discipline
\citep{hankin2022}.  Package idiom for creating such objects uses
named vectors:
%
\begin{Schunk}
\begin{Sinput}
R> LL <- hyper3()
R> LL[c(a = 1)] <- 1
R> LL[c(a = 3, b = 2, c = 1)] <- -1
R> LL
\end{Sinput}
\begin{Soutput}
log( (a=1)^1 * (a=3, b=2, c=1)^-1)
\end{Soutput}
\end{Schunk}
%
Above, we see object \code{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
\code{loglik()}:
%
\begin{Schunk}
\begin{Sinput}
R> loglik(c(a = 0.01, b = 0.01, c = 0.98), LL)
\end{Sinput}
\begin{Soutput}
[1] -4.634729
\end{Soutput}
\begin{Sinput}
R> loglik(c(a = 0.90, b = 0.05, c = 0.05), LL)
\end{Sinput}
\begin{Soutput}
[1] -1.15268
\end{Soutput}
\end{Schunk}
%
Thus we prefer $H_1$ over $H_2$ with about 3.5 units of support,
satisfying the standard two units of support
criterion \citep{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
favor of $H_1$.

The package includes many helper functions to work with order
statistics of this type.  Function \code{ordervec2supp3()}, for
example, can be used to generate a log-likelihood function for
Equation~\ref{PL_like}:
%
\begin{Schunk}
\begin{Sinput}
R> (H <- ordervec2supp3(c("a", "c", "b", "a", "a", "b")))
\end{Sinput}
\begin{Soutput}
log( (a=1)^3 * (a=1, b=1)^-1 * (a=2, b=1)^-1 * (a=2, b=2)^-1 * (a=2,
b=2, c=1)^-1 * (a=3, b=2, c=1)^-1 * (b=1)^1 * (c=1)^1)
\end{Soutput}
\end{Schunk}
%
(the package gives extensive documentation at
\code{ordervec2supp.Rd}).  We may find a maximum likelihood estimate
for the players' strengths, using generic function \code{maxp()},
dispatching to a specialist \code{hyper3} method:
%
\begin{Schunk}
\begin{Sinput}
R> (mH <- maxp(H))
\end{Sinput}
\begin{Soutput}
         a          b          c 
0.21324090 0.08724824 0.69951086 
\end{Soutput}
\end{Schunk}
%
(function \code{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~\citet{hankin2017} to work with `\code{hyper3}'
objects.  For example, we may assess the hypothesis that all three
players are of equal strength (viz $H_0\colon
p_a=p_b=p_c=\frac{1}{3}$):
%
\begin{Schunk}
\begin{Sinput}
R> equalp.test(H)
\end{Sinput}
\begin{Soutput}
	Constrained support maximization

data:  H
null hypothesis: a = b = c
null estimate:
        a         b         c 
0.3333333 0.3333333 0.3333333 
(argmax, constrained optimization)
Support for null:  -6.579251 + K

alternative hypothesis:  sum p_i=1 
alternative estimate:
         a          b          c 
0.21324090 0.08724824 0.69951086 
(argmax, free optimization)
Support for alternative:  -5.73209 + K

degrees of freedom: 2
support difference = 0.8471613
p-value: 0.42863 
\end{Soutput}
\end{Schunk}
%
showing, perhaps unsurprisingly, that this small dataset is consistent
with $H_0$. 

\subsection{Package helper functions}

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

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

\section{Exact analytical solutions}

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.
%
\begin{Schunk}
\begin{Sinput}
R> maxp(ordervec2supp3(c("a", "b", "a")))
\end{Sinput}
\begin{Soutput}
        a         b 
0.4142108 0.5857892 
\end{Soutput}
\end{Schunk}
%
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:
%
\begin{Schunk}
\begin{Sinput}
R> a <- 1/2
R> (S_delta <- log(a * (1 - a)/(1 + a)) - log(3 - 2 * sqrt(2)))
\end{Sinput}
\begin{Soutput}
[1] -0.0290123
\end{Soutput}
\end{Schunk}
%
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
\cite{edwards1972} and apply Wilks's theorem for a $p$~value:
%
\begin{Schunk}
\begin{Sinput}
R> pchisq(-2 * S_delta, df = 1, lower.tail = FALSE)
\end{Sinput}
\begin{Soutput}
[1] 0.8096458
\end{Soutput}
\end{Schunk}
%
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;
Figure~\ref{ABA_likelihood}.
%
\begin{Schunk}
\begin{Sinput}
R> a <- seq(from = 0, by = 0.005, to = 1)
R> S <- function(a){log(a * (1 - a) / ((1 + a) * (3 - 2 * sqrt(2))))}
R> plot(a, S(a), type = "b", xlab = expression(p[a]), ylab = "support")
R> abline(h = c(0, -2))
R> abline(v = c(0.02438102, 0.9524271), col = "red")
R> abline(v = sqrt(2) - 1)
\end{Sinput}
\end{Schunk}
%
\begin{figure}[t!]
\centering
\includegraphics[trim = 0 5 0 20,clip]{jss_hyper3-figaba}
\caption{A\label{ABA_likelihood} support function for $p_a$ with observation $a\succ b\succ a$.}
\end{figure}

\subsubsection{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 Figure~\ref{ABAetc_likelihood}.  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 \code{inst/fisher\_inf\_PL3.Rmd}.

\begin{figure}[t!]
\centering
\includegraphics[trim = 0 5 0 20,clip]{jss_hyper3-figabbabb}
\caption{Likelihood \label{ABAetc_likelihood} functions for
observations $a\succ a\succ b$, $a\succ b\succ a$, $b\succ a\succ a$.  Horizontal dotted
line represents two units of support}
\end{figure}

\subsection{Nonfinishers}

If we allow non-finishers---that is, a subset of competitors who are
beaten by all the ranked competitors (\cite{turner2020} call this a
{\em 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 $2-\sqrt{2}\simeq
0.586$:
%
\begin{Schunk}
\begin{Sinput}
R> maxp(ordervec2supp3(c("a", "b"), nonfinishers = c("a", "b")))
\end{Sinput}
\begin{Soutput}
        a         b 
0.5857892 0.4142108 
\end{Soutput}
\end{Schunk}
%
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 in
Section~\ref{f1}.

\section{An alternative to the Mann-Whitney test}

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 \url{https://github.com/RobinHankin/hyper2}.  Given an
order statistic of the type considered above, the
Mann-Whitney-Wilcoxon test \citep{mann1947,wilcoxon1945} assesses a
null of identity of underlying distributions \citep{ahmad1996}.
Consider the chorioamnion dataset \citep{hollander2013}, used in
\code{wilcox.test.Rd}:
%
\begin{Schunk}
\begin{Sinput}
R> x <- c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46)
R> y <- c(1.15, 0.88, 0.90, 0.74, 1.21)
\end{Sinput}
\end{Schunk}
%
Here we see a measure of permeability of the human placenta at term
(\code{x}) and between 3 and 6 months' gestational age (\code{y}).
The order statistic is straightforward to calculate:
%
\begin{Schunk}
\begin{Sinput}
R> names(x) <- rep("x", length(x))
R> names(y) <- rep("y", length(y))
R> (os <- names(sort(c(x, y))))
\end{Sinput}
\begin{Soutput}
 [1] "x" "y" "x" "x" "y" "y" "x" "y" "y" "x" "x" "x" "x" "x" "x"
\end{Soutput}
\end{Schunk}
%
Then object \code{os} is converted to a `\code{hyper3}' object, again
with \code{ordervec2supp3()}, which may be assessed using the method
of support:
%
\begin{Schunk}
\begin{Sinput}
R> Hxy <- ordervec2supp3(os)
R> equalp.test(Hxy)
\end{Sinput}
\begin{Soutput}
	Constrained support maximization

data:  Hxy
null hypothesis: x = y
null estimate:
  x   y 
0.5 0.5 
(argmax, constrained optimization)
Support for null:  -27.89927 + K

alternative hypothesis:  sum p_i=1 
alternative estimate:
        x         y 
0.2401539 0.7598461 
(argmax, free optimization)
Support for alternative:  -26.48443 + K

degrees of freedom: 1
support difference = 1.414837
p-value: 0.09253716 
\end{Soutput}
\end{Schunk}
%
Above, we use generic function \code{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
\code{wilcox.test()}.  However, observe that the \code{hyper3}
likelihood approach gives more information than Wilcoxon's analysis:
Firstly, we see that the maximum likelihood estimate for the
Bradley-Terry strength of \code{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 Figure~\ref{wilcox_likelihood}.

\begin{figure}[t!]
\centering
\includegraphics[trim = 0 5 0 40,clip]{jss_hyper3-plotwilcoxlike}
\caption{A\label{wilcox_likelihood} support function for the Bradley-Terry
strength $p_a$ of permeability at term.  The evaluate of 0.24 is
shown; and the two-units-of support credible interval, which does not
exclude $H_0\colon p_a=0.5$ (dotted line), is also shown.}
\end{figure}

\subsection{A generalization of the Mann-Whitney test}

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:
%
\begin{Schunk}
\begin{Sinput}
R> javelin_table
\end{Sinput}
\begin{Soutput}
          throw1 throw2 throw3 throw4 throw5 throw6
Chopra     87.03  87.58  76.79      X      X  84.24
Vadlejch   83.98      X      X  82.86  86.67      X
Vesely     79.73  80.30  85.44      X  84.98      X
Weber      85.30  77.90  78.00  83.10  85.15  75.72
Nadeem     82.40      X  84.62  82.91  81.98      X
Katkavets  82.49  81.03  83.71  79.24      X      X
Mardare    81.16  81.73  82.84  81.90  83.30  81.09
Etelatalo  78.43  76.59  83.28  79.20  79.99  83.05
\end{Soutput}
\end{Schunk}
%
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 \code{X}.  We
may convert this to a named vector with elements being the throw
distances, and names being the competitors, using
\code{attemptstable2supp3()}:
%
\begin{Schunk}
\begin{Sinput}
R> javelin_vector <- attemptstable2supp3(javelin_table,
+    decreasing = TRUE, give.supp = FALSE)
R> options(width = 60)
R> javelin_vector
\end{Sinput}
\begin{Soutput}
   Chopra    Chopra  Vadlejch    Vesely     Weber     Weber 
    87.58     87.03     86.67     85.44     85.30     85.15 
   Vesely    Nadeem    Chopra  Vadlejch Katkavets   Mardare 
    84.98     84.62     84.24     83.98     83.71     83.30 
Etelatalo     Weber Etelatalo    Nadeem  Vadlejch   Mardare 
    83.28     83.10     83.05     82.91     82.86     82.84 
Katkavets    Nadeem    Nadeem   Mardare   Mardare   Mardare 
    82.49     82.40     81.98     81.90     81.73     81.16 
  Mardare Katkavets    Vesely Etelatalo    Vesely Katkavets 
    81.09     81.03     80.30     79.99     79.73     79.24 
Etelatalo Etelatalo     Weber     Weber    Chopra Etelatalo 
    79.20     78.43     78.00     77.90     76.79     76.59 
    Weber  Vadlejch    Nadeem  Vadlejch    Chopra    Vesely 
    75.72        NA        NA        NA        NA        NA 
   Chopra Katkavets  Vadlejch    Vesely    Nadeem Katkavets 
       NA        NA        NA        NA        NA        NA 
\end{Soutput}
\end{Schunk}
%
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 (\code{NA} entries correspond to
no-throws.)  The attempts table may be converted to a `\code{hyper3}'
object, again using function \code{attemptstable2supp3()} but this
time passing \code{give.supp = TRUE}:
%
\begin{Schunk}
\begin{Sinput}
R> javelin <- ordervec2supp3(v = names(javelin_vector)[!is.na(javelin_vector)])
\end{Sinput}
\end{Schunk}
%
Above, object \code{javelin} is a \code{hyper3} likelihood function,
so one has access to the standard likelihood-based methods, such as
finding and displaying the maximum likelihood estimate, shown in
Figure~\ref{javelinbradleyterry}.  From this, we see that Vadlejch has
the highest estimated Bradley-Terry strength, but further analysis
with \code{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 $\frac{1}{8}$ ($p=0.1$).
%
\begin{Schunk}
\begin{Sinput}
R> (mj <- maxp(javelin))
\end{Sinput}
\begin{Soutput}
   Chopra Etelatalo Katkavets   Mardare    Nadeem  Vadlejch 
   0.0930    0.0482    0.0929    0.1173    0.1730    0.3206 
   Vesely     Weber 
   0.1140    0.0409 
\end{Soutput}
\begin{Sinput}
R> dotchart(mj, pch = 16, xlab = "Estimated Bradley-Terry strength")
\end{Sinput}
\end{Schunk}
%
\begin{figure}[t!]
\centering
\includegraphics[trim = 0 5 0 40,clip]{jss_hyper3-testthejav}
\caption{Maximum \label{javelinbradleyterry} likelihood estimate for javelin throwers' Bradley-Terry strengths.}
\end{figure}

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 $\displaystyle p_{\mbox{\tiny Vad}}/\left(p_{\mbox{\tiny
Vad}} + p_{\mbox{\tiny Ves}}\right)\simeq 0.74$.  Indeed, from a
training or selection perspective we might follow \cite{hankin2017}
and observe that log-contrasts \citep{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, Figure~\ref{profliklogcont}.  We
see that the credible range for~$\log\left(p_{\mbox{\tiny Vad}}/
p_{\mbox{\tiny Ves}}\right)$ includes zero and we have no strong
evidence for these athletes having different (Bradley-Terry)
strengths.

\begin{figure}[t!]
\centering
\includegraphics[trim = 0 5 0 20,clip]{jss_hyper3-plottheloglikcont}
\caption{Profile \label{profliklogcont} likelihood for log-contrast
$\log\left(p_{\mbox{\tiny Vad}}/ p_{\mbox{\tiny Ves}}\right)$.  Null
of $p_{\mbox{\tiny Vad}}=p_{\mbox{\tiny Ves}}$ indicated with a dotted
line, and two-units-of-support limit indicated with horizontal lines;
thus the null is not rejected.}
\end{figure}

\section{Formula 1 motor racing}\label{f1}

Formula 1 motor racing is an important and prestigious motor sport
\citep{codling2017,jenkins2010}.  In Formula 1 Grand Prix, the
constructors' championship takes place between {\em 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 \pkg{hyper2}
package as a dataset, shows rankings for the first 9 venues of the
2021 season:
%
\begin{Schunk}
\begin{Sinput}
R> constructor_2021_table[, 1:9]
\end{Sinput}
\begin{Soutput}
   Constructor BHR EMI POR ESP  MON AZE FRA STY
1         Merc   1   2   1   1    7  12   2   2
2         Merc   3 Ret   3   3  Ret  15   4   3
3         RBRH   2   1   2   2    1   1   1   1
4         RBRH   5  11   4   5    4  18   3   4
5      Ferrari   6   4   6   4    2   4  11   6
6      Ferrari   8   5  11   7 DNSP   8  16   7
7           MM   4   3   5   6    3   5   5   5
8           MM   7   6   9   8   12   9   6  13
9           AR  13   9   7   9    9   6   8   9
10          AR Ret  10   8  17   13 Ret  14  14
11         ATH   9   7  10  10    6   3   7  10
12         ATH  17  12  15 Ret   16   7  13 Ret
13         AMM  10   8  13  11    5   2   9   8
14         AMM  15  15  14  13    8 Ret  10  12
15          WM  14 Ret  16  14   14  16  12  17
16          WM  18 Ret  18  16   15  17  18 Ret
17        ARRF  11  13  12  12   10  10  15  11
18        ARRF  12  14 Ret  15   11  11  17  15
19          HF  16  16  17  18   17  13  19  16
20          HF Ret  17  19  19   18  14  20  18
\end{Soutput}
\end{Schunk}
%
Above, we see that Mercedes (``\code{Merc}'') came first and third at
Bahrain (\code{BHR}); and came second and retired at Emilia Romagna
(\code{EMI}); full details of the notation and conventions are given
in the package at \code{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 `\code{hyper3}' object is easily
constructed using package idiom, in this case
\code{ordertable2supp3()}, and we may use this to assess the
Plackett-Luce strengths of the constructors:
%
\begin{Schunk}
\begin{Sinput}
R> const2020 <- ordertable2supp3(constructor_2020_table)
R> const2021 <- ordertable2supp3(constructor_2021_table)
R> options(digits = 4)
R> maxp(const2020, n = 1)
\end{Sinput}
\begin{Soutput}
   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 
\end{Soutput}
\begin{Sinput}
R> maxp(const2021, n = 1)
\end{Sinput}
\begin{Soutput}
    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 
\end{Soutput}
\end{Schunk}
%
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:
%
\begin{Schunk}
\begin{Sinput}
R> H <- (psubs(constructor_2020, "Merc", "Merc2020") +
+      psubs(constructor_2021, "Merc", "Merc2021")
+    )
\end{Sinput}
\end{Schunk}
%
Above, we use generic function \code{psubs()} to change the name of
Mercedes from \code{Merc} to \code{Merc2020} and \code{Merc2021}
respectively.  Note the use of ``\code{+}'' to represent addition of
log-likelihoods, corresponding to the assumption of conditional
independence of results.  The null would be simply that the strengths
of \code{Merc2020} and of \code{Merc2021} are identical.  Package
idiom would be to use generic function \code{samep.test()}:
%
\begin{Schunk}
\begin{Sinput}
R> options(digits = 4)
R> samep.test(H, c("Merc2020", "Merc2021"))
\end{Sinput}
\begin{Soutput}
	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 
\end{Soutput}
\end{Schunk}
%
Above, we see strong evidence for a real decrease in the strength of
the Mercedes team from 2020 to 2021, with $p=0.002$.

\section{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 \proglang{R} is accomplished via a \proglang{C++}
back-end making use of the \proglang{STL} `\code{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 \citep{hankin2017}; or the introduction of
a reified draw entity \citep{hankin2010}.

Further work might include a systematic comparison between
\code{hyper3} approach and the Mann-Whitney-Wilcoxon test, including
the characterization 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 \citep{hankin2020}.

\bibliography{ref}
\end{document}

