---
title: "handover"
output: bookdown::html_document2
---

```{r setup, include=FALSE}
set.seed(0)
knitr::opts_chunk$set(echo = TRUE)
library("hyper2")
library("magrittr")
options("digits" = 5)
```

<p style="text-align: right;">
![](`r system.file("help/figures/hyper2.png", package = "hyper2")`){width=10%}
</p>


This short document is based on Altham and Hankin 2010, which
discussed an interesting dataset originally presented by Lin
et. al. (2009).  It uses more sophisticated idiom furnished by the
`hyper2` package, which is an improved and efficient upgrade of the
primitive `hyperdirichlet` package.  The dataset arose from 69 medical
malpractice claims and refers to the two surgeon–reviewers' answers to
the question "was there a communication breakdown in the hand-off
between physicians caring for the patient?".  The rows of Table 1
correspond to the answers that were given by reviewer 1, and the
columns to the answers given by reviewer 2.

```{r readcurlingtable}
handover_table <- as.matrix(read.table("handover.txt"))
names(dimnames(handover_table)) <- c("reviewer1","reviewer2")
handover_table[3,3] <- NA # structural zero
handover_table
```

Thus we see that in 26 cases, both reviewers answered "yes", and in 18
both said "no".  The table also records partial responses that include
observations missed by one or both reviewers.  Responses missed by
both reviewers are uninformative and automatically rejected:
structural zeros, in `aylmer` terminology.  We first apply the Aylmer
test to assess whether the two reviewers give the same proportion of
"yes" responses:

```
> aylmer.test(a,alternative=function(a){a[1,2]-a[2,1]})

	Aylmer functional test for count data

data:  a
p-value = 0.169
alternative hypothesis: test function exceeds observed

> 
```


(Compare the McNemar test which gives an exact $p$-value of
$7/64\simeq 0.1094$).  We follow the previous workers and assume that
cases are missing at random and seek a likelihood function on
$\theta_{11},\theta_{10},\theta_{01},\theta_{00}$ subject to
$\theta_{ij}\geq 0,\sum_{i,j \in\left\lbrace
0,1\right\rbrace}\theta_{ij}=1$.  Here "1" means "yes" and "0"
means "no", so $\theta_{01}$ is the probability of reviewer 1
responding "no" and reviewer 2 responding "yes".

```{r}
handover <- hyper2()
handover["t11"] %<>% inc(handover_table[1,1])
handover["t10"] %<>% inc(handover_table[1,2])
handover["t01"] %<>% inc(handover_table[2,1])
handover["t00"] %<>% inc(handover_table[2,2])

handover[c("t11","t10")] %<>% inc(handover_table[1,3])  # yes-yes plus yes-no  = 2 
handover[c("t01","t00")] %<>% inc(handover_table[2,3])  #  no-yes plus  no-no  = 9 
handover[c("t11","t01")] %<>% inc(handover_table[3,1])  # yes-yes plus  no-yes = 4 
handover[c("t10","t00")] %<>% inc(handover_table[3,2])  # yes-no  plus  no-no  = 4 
handover
handover_maxp <- maxp(handover)
handover_maxp
```


We may simply test the hypothesis that $\theta_{01} = \theta_{10}$:

```{r}
samep.test(handover,c("t01","t10"))
```

Alternatively we might use the Bayesian paradigm and calculate the
probability of $\theta_{01} > \theta_{10}$:

```{r}
probability(handover,disallowed=function(p){p[2]<p[3]},tol=0.05)
```

the result differing slightly from the previous value, probably due to
numerical issues.


We can now consider the distribution of
$\psi=\theta_{10}/(\theta_{10}+\theta_{01})$ under the posterior
distribution induced by the complete cases:

```{r,samplepsi,message=FALSE,cache=TRUE}
jj <- rp(300000,H=handover)
head(jj)
psifun <- function(x){x[3]/(x[2]+x[3])}
logoddsfun <- function(x){log(x[2]/x[3])}
psi <- apply(jj,1,psifun)
log_odds <- apply(jj,1,logoddsfun)
psi_complete <- rbeta(length(psi),2,6)
```

We can now reproduce the four graphics given by Altham and Hankin:

Sample of $\psi=\theta_{10}/(\theta_{10}+\theta_{01})$ under the
posterior induced by the whole dataset (histogram), together with the
posterior distribution of $\psi$ induced by the complete cases only
(a beta distribution with parameters 2 and 6).

```{r histpsi,echo=FALSE,fig.cap="Sample of $\\psi=\\theta_{10}/(\\theta_{10}+\\theta_{01})$ under the posterior induced  by the whole dataset (histogram), together with the posterior distribution of $\\psi$ induced by the complete cases only (a beta distribution with parameters 2 and 6)"}
hist(psi,nclass=70,probability=TRUE,main=expression(paste("PDF of ",psi)))
p <- seq(from=0,to=1,len=100)
points(p,dbeta(p,2,6),type="l")
abline(v=psifun(handover_maxp),col="red",lwd=5)
segments(x0=0.5,y0=0,y1=1.5,col="blue",lwd=5)
legend("topright",lty=1,lwd=c(1,5,5),col=c("black","red","blue"),
       legend=c(
           "complete cases",
           expression(hat(psi) * " (complete cases)"),
           expression(psi * " = 0.5")
           ))
```

```{r qqplotpsi,echo=FALSE,fig.cap="Quantile-quantile plot of $\\psi=\\theta_{10}/(\\theta_{10}+\\theta_{01})$ under the posterior induced  by the whole dataset against the posterior distribution of $\\psi$ induced by the complete cases only (a beta distribution with parameters 2 and 6)",cache=TRUE}
par(pty='s')
qqplot(psi,psi_complete,asp=1,xlim=0:1,ylim=0:1)
abline(0,1)
```

```{r ecdfpsi,echo=FALSE,fig.cap="Empirical cumulative distribution function of $\\psi=\\theta_{10}/(\\theta_{10}+\\theta_{01})$ under the posterior induced  by the whole dataset, and that of $\\psi$ induced by the complete cases only (a beta distribution with parameters 2 and 6)",cache=TRUE}
plot(psi,seq_along(psi)/length(psi),type="l",col="black",xlab=expression(psi),ylab="quantile")
points(psi_complete,seq_along(psi_complete)/length(psi_complete),type="l",col="red")
legend("topleft",lty=1,col=c("black","red"),legend=c("whole dataset","complete cases"))
```

```{r logoddsqqplot,echo=FALSE,fig.cap="QQ plot of $\\phi=\\log(\\theta_{01}/\\theta_{10})$",cache=TRUE}
qqnorm(log_odds,main="QQ plot of log-odds")
abline(mean(log_odds),sd(log_odds))
```

Figures \@ref(fig:histpsi) to  \@ref(fig:logoddsqqplot) show different aspects of the dataset.

### References {-}

* PME Altham and RKS Hankin 2010.  "Using recently developed software on
a 2x2 table of matched pairs with incompletely classified data".
_Journal of the Royal Statistical Society_, 59:2):377-379
* L West and RKS Hankin 2008. "Exact tests for two-way contingency
tables with structural zeros".  _Journal of Statistical Software_
28(11)
* RKS Hankin 2010. "A generalization of the Dirichlet distribution". 
 _Journal of Statistical Software_ 33(11)
* RKS Hankin 2017. "Partial rank data with the hyper2 package:
  likelihood functions for generalized Bradley-Terry models".  _The R
  Journal_ 9(2)


### Package dataset {-}

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

```{r,label=savecurlingdataset}
save(handover_table,handover_maxp,handover,file="handover.rda")
```
