# Statistics 5601 (Geyer, Spring 2006) Examples: Wilcoxon Rank Sum Test

## General Instructions

To do each example, just click the "Submit" button. You do not have to type in any R instructions or specify a dataset. That's already done for you.

## The Wilcoxon Rank Sum Test

### Example 4.1 in Hollander and Wolfe.

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### Summary

• Lower-tailed rank sum test
• Test statistic: W = 30 (Wilcoxon form)
• Test statistic: U = 15 (Mann-Whitney form)
• Sample sizes: nx = 10, ny = 5
• P-value: P = 0.1272061

• Line one assigns the value of the parameter (population median difference) assumed under the null hypothesis. Usually zero.
• The reason for the `NA` removal in lines two and three is that Rweb insists on reading variables of the same length, so whichever of `x` or `y` is shorter must be padded with `NA` (not applicable) values.
• The test statistic `w` is the Wilcoxon form defined in equation (4.3) in Hollander and Wolfe.
• The test statistic `u` is the Mann-Whitney form defined in equation (4.15) in Hollander and Wolfe.
• In the last line we see that the R function giving the probability distribution of the test statistic under the null hypothesis uses the Mann-Whitney form. So we have to use it too, although Hollander and Wolfe use the other for most of their discussion.
• For an upper-tailed test the last line would be replaced by any of the following, which all do the same thing.
```1 - pwilcox(u - 1, nx, ny)
pwilcox(u - 1, nx, ny, lower.tail=FALSE)
pwilcox(nx * ny - u, nx, ny)
```
• For a two-tailed test do both the lower-tailed and the upper-tailed test and double the P-value of the smaller of the two results. (Two tails is twice one tail because of the symmetry of the null distribution of the test statistic.)
• For handling zeros and tied ranks, see Hollander and Wolfe, the class discussion, and the the section on the `wilcox.exact` function below.

## The Associated Point Estimate (Median of the Pairwise Differences)

The Hodges-Lehmann estimator associated with the rank sum test is the median of the pairwise differences, which are the nx ny differences

Yj - Xi,     for all i and j

### Example 4.3 in Hollander and Wolfe.

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### Summary

• Point Estimate (sample median of pairwise differences): -0.305

## The Associated Confidence Interval

Very similar to the confidence intervals associated with the sign test and signed rank test, the confidence interval has the form

(D(k), D(m + 1 - k))

where m = nx ny is the number of pairwise differences, the Di are the pairwise differences, and, as always, parentheses on subscripts indicates order statistics. That is, one counts in k from each end in the list of sorted pairwise differences to find the confidence interval.

### Example 4.4 in Hollander and Wolfe.

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### Summary

• Achieved confidence level: 96.004%
• Confidence interval for the population median difference: (-0.76, 0.15)

• Some experimentation may be needed to achieve the confidence level you want. The possible confidence levels are shown by
```1 - 2 * pwilcox(k - 1, nx, ny)
```
for different values of `k`. The vectorwise operation of R functions can give them all at once
```k <- seq(1, 100)
conf <- 1 - 2 * pwilcox(k - 1, nx, ny)
conf[conf > 1 / 2]
```
If one adds these lines to the form above, one sees that the choice is fairly restricted. There are nine possible achieved levels between 0.99 and 0.80 are
0.9873, 0.9807, 0.9720, 0.9600, 0.9447, 0.9247, 0.9008, 0.8708, 0.8355
• Alternatively, you can just assign `k` to be any integer between one and `n / 2` just before the second to last line in the form (`cat ...`). A confidence interval with some achieved confidence level will be produced.
• For a one-tailed confidence interval (called upper and lower bounds by Hollander and Wolfe) just use `alpha` rather than `alpha / 2` in the fifth line of the form. Then make either the lower limit minus infinity or the upper limit plus infinity, as desired.

## The R Function `wilcox.test`

All of the above can be done in one shot with the R function `wilcox.test` (on-line help).

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Only one complaint. It does not report the actual achieved confidence level (here 96.0%) but rather the confidence level asked for (here 95%, the default). If you want to know the actual achieved confidence level, you'll have to use the code in the confidence interval section above. But you can use `wilcox.test` as a convenient check (the intervals should agree).

### Warning About Ties and Zeros

Do not use the `wilcox.test` function when there are ties or zeros in the data. See the following section.

## The R Function `wilcox.exact`

There is an R function `wilcox.exact` (on-line help) that does do hypothesis tests correctly in the presence of ties.

It does not do confidence intervals or point estimates correctly in the presence of ties. Use the code in the confidence interval section or the point estimate section above.

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In order to see what's going on, let's copy some of the code from the beginning of the calculation without the function. This shows the ranks so we see the tied ranks and shows the calculation of the test statistic `u` so we can see that it agrees with the test statistic calculated by `wilcox.exact`.

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## Fuzzy Procedures

These are analogous to the fuzzy procedures for the sign test explained on the sign test and related procedures page and on the fuzzy confidence intervals and P-values page.

Since they are so similar, we won't belabor the issues and interpretations. The only difference is that for fuzzy confidence intervals the jumps in the plot are at YX differences (no surprise) rather than at order statistics and for fuzzy P-values they are at numbers in the CDF table for the null distribution of the test statistic, which is now the Mann-Whitney distribution rather than the symmetric binomial distribution.

In short, the distributions have changed but everything else remains the same.

### Fuzzy P-Values

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The fuzzy P-value is not very uniformly distributed over the interval from 0.0041 to 0.0175. This is fairly strong evidence against the null hypothesis.

The only virtue this procedure has over the `wilcox.exact` procedure illustrated in the preceding section is that this procedure is exact at all significance levels, whereas the `wilcox.exact` only gives an exact procedure for the significance levels that appear in the CDF table of the null distribution of the test statistic (which is not tabulated anywhere, just calculated by `wilcox.exact`, since the presence of tied ranks changes the distribution, so it is not the distribution calculated by `pwilcox` or tabulated in the textbook).

Both procedures are exact in some sense (not exactly the same sense). Both say more or less the same thing. Certainly, fairly strong evidence against the null hypothesis is what both say.

You can use whichever you like. What you should not do is follow traditional procedures, described by the textbook and implemented in `wilcox.test`, when there are ties.