---

layout: page

title: Rank tests

---

```{r options, echo=FALSE}

library(knitr)

opts_chunk$set(fig.path=paste0("figure/", sub("(.*).Rmd","\\1",basename(knitr:::knit_concord$get('infile'))), "-"))

```

## Wilcoxon Rank Sum Test

We learned how the sample mean and SD are susceptible to outliers. The

t-test is based on these measures and is susceptible as well. The

Wilcoxon rank test (equivalent to the Mann-Whitney test) provides an

alternative. In the code below, we perform a t-test on data for which

the null is true. However, we change one sum observation by mistake

in each sample and the values incorrectly entered are different. Here

we see that the t-test results in a small p-value, while the Wilcoxon

test does not:

```{r}

set.seed(779) ##779 picked for illustration purposes

N=25

x<- rnorm(N,0,1)

y<- rnorm(N,0,1)

```

Create outliers:

```{r}

x[1] <- 5

x[2] <- 7

cat("t-test pval:",t.test(x,y)$p.value)

cat("Wilcox test pval:",wilcox.test(x,y)$p.value)

```

The basic idea is to 1) combine all the data, 2) turn the values into ranks, 3) separate them back into their groups, and 4) compute the sum or average rank and perform a test.

```{r rank-test-illustration, fig.cap="Data from two populations with two outliers. The left plot shows the original data and the right plot shows their ranks. The numbers are the w values ",fig.width=10.5,fig.height=5.25}

library(rafalib)

mypar(1,2)

stripchart(list(x,y),vertical=TRUE,ylim=c(-7,7),ylab="Observations",pch=21,bg=1)

abline(h=0)

xrank<-rank(c(x,y))[seq(along=x)]

yrank<-rank(c(x,y))[-seq(along=x)]

stripchart(list(xrank,yrank),vertical=TRUE,ylab="Ranks",pch=21,bg=1,cex=1.25)

ws <- sapply(x,function(z) rank(c(z,y))[1]-1)

text( rep(1.05,length(ws)), xrank, ws, cex=0.8)

W <-sum(ws)

```

`W` is the sum of the ranks for the first group relative to the second

group. We can compute an exact p-value for $W$ based on

combinatorics. We can also use the CLT since

statistical theory tells us that this `W` is approximated by the

normal distribution. We can construct a z-score as follows:

```{r}

n1<-length(x);n2<-length(y)

Z <- (mean(ws)-n2/2)/ sqrt(n2*(n1+n2+1)/12/n1)

print(Z)

```

Here the `Z` is not large enough to give us a p-value less

than 0.05. These are part of the calculations performed by the R function

`wilcox.test`.