PROPHET StatGuide: Examining Wilcoxon paired signed rank test results to detect assumption violations

Results for sample values:

• Normality tests: detecting violation of normality assumption
• Histograms: detecting assumption violations graphically
• Boxplots: detecting assumption violations graphically
• Normal probability plot: detecting assumption violations graphically

Normality tests:

Although normality is not assumed for the paired signed rank test, departures from normality can suggest the presence of outliers in the data, or of skewness in the paired differences. Conversely, if the population from which the paired differences were drawn is in fact normally distributed, the paired two-sample t test may be a more powerful alternative to the paired signed rank test.

The normality test will give an indication of whether the population from which the paired differences were drawn appears to be normally distributed, but will not indicate the cause(s) of the nonnormality. The smaller the sample size, the less likely the normality test will be able to detect nonnormality.

Histograms:

The histogram for paired differences has a reference normal distribution curve for a normal distribution with the same mean and variance as the sample. This provides a reference for detecting gross nonnormality (such as severe skewness) when the sample sizes are large.

Boxplots:

Suspected outliers appear in a boxplot as individual points o or x outside the box. If these appear on both sides of the box, they also suggest the possibility of a heavy-tailed distribution. If they appear on only one side, they suggest the possibility of a skewed distribution. Skewness is also suggested if the mean (+) does not lie on or near the central line of the boxplot, or if the central line of the boxplot does not evenly divide the box. Examples of these plots will help illustrate the various situations.

Normal probability plot:

For values sampled from a normal distribution, the normal probability plot, (normal Q-Q plot) has the points all lying on or near the straight line drawn through the middle half of the points. Scattered points lying away from the line are suspected outliers. Examples of these plots will help illustrate the various situations.

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Last modified: March 11, 1997