Part VI: NonParamteric Analogues
Tests for Matched Groups
| The Wilcoxon Test | The Sign Test |
 
 
The Wilcoxon test is the most powerful of the non-parametric techniques. This occurs primarily because the Wilcoxon is done very much like a Matched Pairs t-test, where participants are measured twice or are matched on some third variable.

In order to calculate a Wlcoxon statistic, follow steps below:

  • Find the difference score (D) for all pairs. This is accomplished by simply subtracting the scores associated with each group or testing session.
  • Rank the absolute value for D, regardless of group, with the smallest D having a rank of 1. If any particular D = 0, exclude that pair from the calculation (it doesn't make sense to test for differences when you know there is no difference!).
  • Calculate T+. This is found by adding the ranks associated with positive difference scores.
  • Calculate T-. This is found by adding the ranks associated with negative difference scores.
  • Compare the smaller of T+ and T- to the critical value from a table in your statistics text. The Wilcoxon is opposite of all other tests; you will reject the null hypothesis only when you obtained value (T+ or T-) is smaller than the critical value!
  • For large n (>8), T approaches a normal distribution, and a z test can be used:

Picture (175x106, 2Kb)
 
 
Just as the Mann-Whitney had a less powerful partner (the Median test), the Wilcoxon has its partner. The Sign test is also a matched-pairs test, but is much less powerful than the Wilcoxon. The Sign test is essentially a Goodness of Fit test.

Say, for example, that judges rate which essay of 2 matched individuals is superior. In the chart below, a "+" indicates that a judge selected that person's essay over the other's.

  Judge A Judge B Judge C Judge D Judge E Judge F
Person 1 + +   +   +
Person 2     +   +  

Under the null hypothesis, we would expect that an equal number of judges would pick person 1 and Person 2. Thus, given the above data, we can do a Chi Square Goodness of Fit test with 1 df. Of course, since it is 1 df , you would use Yates' correction in the actual calculation.