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 Interpreting results: Wilcoxon signed rank test

The nonparametric Wilcoxon signed rank test compares the median of a single column of numbers against a hypothetical median. Don't confuse it with the Wilcoxon matched pairs test which compares two paired or matched groups.

## Interpreting the confidence interval

The signed rank test compares the median of the values you entered with a hypothetical population median you entered. Prism reports the difference between these two values, and the confidence interval of the difference. Prism subtracts the median of the data from the hypothetical median, so when the hypothetical median is higher, the result will be positive. When the hypothetical median is lower, the result will be negative

Since the nonparametric test works with ranks, it is usually not possible to get a confidence interval with exactly 95% confidence. Prism finds a close confidence level, and reports what it is. So you might get a 96.2% confidence interval when you asked for a 95% interval.

## Interpreting the P value

The P value answers this question:

If the data were sampled from a population with a median equal to the hypothetical value you entered, what is the chance of randomly selecting N data points and finding a median as far (or further) from the hypothetical value as observed here?

If the P value is small, you can reject the idea that the difference is a due to chance and conclude instead that the population has a median distinct from the hypothetical value you entered.

If the P value is large, the data do not give you any reason to conclude that the population median differs from the hypothetical median. This is not the same as saying that the medians are the same. You just have no compelling evidence that they differ. If you have small samples, the Wilcoxon test has little power. In fact, if you have five or fewer values, the Wilcoxon test will always give a P value greater than 0.05, no matter how far the sample median is from the hypothetical median.

## Assumptions

The Wilcoxon signed rank test does not assume that the data are sampled from a Gaussian distribution. However it does assume that the data are distributed symmetrically around the median. If the distribution is asymmetrical, the P value will not tell you much about whether the median is different than the hypothetical value.

Like all statistical tests, the Wilcoxon signed rank test assumes that the errors are independent. The term “error” refers to the difference between each value and the group median. The results of a Wilcoxon test only make sense when the scatter is random – that any factor that causes a value to be too high or too low affects only that one value.

## How the Wilcoxon signed rank test works

1.Calculate how far each value is from the hypothetical median.

2.Ignore values that exactly equal the hypothetical value. Call the number of remaining values N.

3.Rank these distances, paying no attention to whether the values are higher or lower than the hypothetical value.

4.For each value that is lower than the hypothetical value, multiply the rank by negative 1.

5.Sum the positive ranks. Prism reports this value.

6.Sum the negative ranks. Prism also reports this value.

7.Add the two sums together. This is the sum of signed ranks, which Prism reports as W.

If the data really were sampled from a population with the hypothetical median, you would expect W to be near zero. If W (the sum of signed ranks) is far from zero, the P value will be small.

With fewer than 200 values, Prism computes an exact P value, using a method explained in Klotz(2). With 200 or more values, Prism uses a standard approximation that is quite accurate.

Prism calculates the confidence interval for the discrepancy between the observed median and the hypothetical median you entered  using the method explained on page 234-235 of Sheskin (1) and 302-303 of Klotz (2).

## How Prism deals with values that exactly equal the hypothetical median

What happens if a value is identical to the hypothetical median?

When Wilcoxon developed this test, he recommended that those data simply be ignored. Imagine there are ten values. Nine of the values are distinct from the hypothetical median you entered, but  the tenth is identical to that hypothetical median (to the precision recorded). Using Wilcoxon's original method, that tenth value would be ignored and the other nine values would be analyzed.This is how InStat and previous versions of Prism (up to version 5) handle the situation.

Pratt(3,4) proposed a different method that accounts for the tied values. Prism 6 offers the choice of using this method.

Which method should you choose? Obviously, if no value equals the hypothetical median, it doesn't matter. Nor does it matter much if there is, for example, one such value out of 200.

It makes intuitive sense that data should not be ignored, and so Pratt's method must be better.  However, Conover (5) has shown that the relative merits of the two methods depend on the underlying distribution of the data, which you don't know.

## Why results in Prism 6 can be different than from previous versions of Prism

Results from Prism 6 can differ from prior versions because Prism 6 does exact calculations in two situations where Prism 5 did approximate calculations. All versions of Prism report whether it uses an approximate or exact methods.

Prism 6 can perform the exact calculations much faster than did Prism 5, so does exact calculations with some sample sizes that earlier versions of Prism could only do approximate calculations.

If two values are the same, prior versions of Prism always used the approximate method. Prism 6 uses the exact method unless the sample is huge.

Another reason for different results between Prism 6 and prior versions is if a value exactly matches the hypothetical value you are comparing against. Prism 6 offers a new option (method of Pratt) which will give different results than prior versions did. See the previous section.

## References

1. D.J. Sheskin, Handbook of Parametric and Nonparametric Statistical Procedures, fourth edition.

2. JH Klotz, A computational approach to statistics, 2006, self-published book,  Chapter 15.2 The Wilcoxon Signed Rank Test.

3. Pratt JW (1959) Remarks on zeros and ties in the Wilcoxon signed rank procedures. Journal of the American Statistical Association, Vol. 54, No. 287 (Sep., 1959), pp. 655-667

4. Pratt, J.W. and Gibbons, J.D. (1981), Concepts of Nonparametric Theory, New York: Springer Verlag.

5. WJ Conover, On Methods of Handling Ties in the Wilcoxon Signed-Rank Test, Journal of the American Statistical Association, Vol. 68, No. 344 (Dec., 1973), pp. 985-988