set.seed(0) knitr::opts_chunk$set(echo = TRUE) library("hyper2") library("magrittr") 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)) nrow(volleyball_table) head(volleyball_table)

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)} zipf(9) (null_support <- loglik(indep(zipf(9)),volleyball)) (alternative_support <- loglik(indep(volleyball_maxp),volleyball)) (Lambda <- 2*(alternative_support-null_support)) pchisq(Lambda,df=8,lower.tail=FALSE)

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.

Following lines create `volleyball.rda`

, residing in the `data/`

directory of the package.

save(volleyball_table,volleyball_maxp,volleyball,file="volleyball.rda")

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