## Some Nonparametric Statistics

ANOVA is a useful procedure, as is its simpler cousin the t-test. Both of these tests are “parametric” — a term that means they rely on assumptions about the parameters or characteristics of the underlying population from which the data were drawn. Tests exist that make no such assumptions. These tesrs are called “non-parametric.”

### Test of Exact Probability

Expt: 1, 2, 4 -- Sum = 7, Mean = 7/3

Ctrl: 3, 5, 6 -- Sum = 14, Mean = 14/3

Possible Data Group Sums 1 2 3 | 4 5 6 6 | 15

1 2 4 | 3 5 6 7 | 14

1 2 5 | 3 4 6 8 | 13

1 2 6 | 3 4 5 9 | 12

1 3 4 | 2 5 6

1 3 5 | 2 4 6 .

1 3 6 | 2 4 5

1 4 5 | 2 3 6 .

1 4 6 | 2 3 5

1 5 6 | 2 3 4 .

2 3 4 | 1 5 6

2 3 5 | 1 4 6 .

2 3 6 | 1 4 5

2 4 5 | 1 3 6 .

2 4 6 | 1 3 5

2 5 6 | 1 3 4 .

3 4 5 | 1 2 6

3 4 6 | 1 2 5 13 | 8

3 5 6 | 1 2 4 14 | 7

4 5 6 | 1 2 3 15 | 6

N k Outcomes

6 3 20

7 3 35

8 4 70

9 4 126

10 5 252

11 5 462

12 6 924

13 6 1716

14 7 3432

15 7 6435

16 8 12870

17 8 24310

18 9 48620

19 9 92378

20 10 184756

### Alternatives to Test of Exact Probability:

Person A B C D E F G

First 1 3 5 7 2 4 5

Second 6 6 8 8 8 2 8

Person A B C D E F G

First 6 3 8 8 2 4 5

Second 1 6 5 7 8 2 8

#### The Sign Test

Person A B C D E F G

First 1 3 5 7 2 4 5

Second 6 6 8 8 8 2 8

Diff -5 -3 -3 -1 -6 2 -3

Sign - - - - - + -

#### The Wilcoxon Signed Ranks Test

Person A B C D E F G

First 1 3 5 7 2 4 5

Second 6 6 8 8 8 2 8

Diff -5 -3 -3 -1 -6 2 -3

|Diff| 5 3 3 1 6 2 3

Rank 5 4 4 1 6 2 4

Signed Rank -5 -4 -4 -1 -6 2 -4

### Mann-Whitney U test

For independent groups, the Wilcoxon or Sign tests will not work, as there is no way to pair up the scores across the conditions. Mann-Whitney U test is the test to use as a nonparametric test of two independent samples.

Rank the scores across both groups, with the lowest score given the rank of 1, and the highest score the rank of (n1 + n2). Ties share a rank. Once this is done. calculate the value of U for each of the two groups, as follows:

where R1 and R2 are the sums of the ranks for each group. The test statistic U is the lowest of these two Us.

Consult a table of critical values. If your U is less than or equal to the critical U, then the p is < alpha.

Example: here are two groups:

Our U is the smaller of the two: 15

Critical U (from the table): 6. Our U is LARGER THAN the critical U, so our p is GREATER THAN .05. Retain the null hypothesis of no effect, and conclude that the groups do not differ.

[Some info and the formulas come from this source.]

### Summary

*Nonparametric statistics for the behavioral sciences*. New York : McGraw-Hill.