volleyball dataset

knitr::opts_chunk$set(echo = TRUE)
options("digits" = 5)

This short document discusses a dataset first presented by Hankin (2010), although here only the first 52 observations are used. A volleyball \dfn{set} is a Bernoulli trial between two disjoint subsets of the players. The two subsets are denoted (after the game) as the winners and the losers: these are denoted by 1 and 0 respectively.

volleyball_table <- as.matrix(read.table("volleyball.txt",header=TRUE))

Each row of volleyball_table is a set. Thus the first line shows a game between a team comprising p1, p4, and p8 against a team comprising p5 and p6. Player p9 did not play; team p1 p4 p8 won.

We may use function volley() to convert this to a likelihood function:

volleyball <- volley(volleyball_table)
(volleyball_maxp <- maxp(volleyball))

The original synthetic dataset was prepared using Zipf's law for the players' strengths, so we may test the hypothesis that this is the case; $H_0\colon p_i\propto i^{-1}$:

zipf <- function(n){jj <- 1/(1:n); jj/sum(jj)}
(null_support <- loglik(indep(zipf(9)),volleyball))
(alternative_support <- loglik(indep(volleyball_maxp),volleyball))
(Lambda <- 2*(alternative_support-null_support))

somewhat disappointingly rejecting the null with a $p$-value of about 4\% (and indeed--just---with a two units of support per degree of freedom criterion). However, it is not clear to me the extent to which Wilks's theorem is applicable here [Wilks is an asymptotic result; recall that we have only 52 observations here], nor whether the support criterion is appropriate with 8 degrees of freedom.

Package dataset {-}

Following lines create volleyball.rda, residing in the data/ directory of the package.


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hyper2 documentation built on March 4, 2021, 9:09 a.m.