#### Notes Data Applets Examples ## Statistical Tests for Ordinal Data

Traditional statistical tests, such as T-Tests and Analysis of Variance, require the computation of means and standard deviations.

• These computations only make sense when the data are numeric.

In this topic we present statistical tests that only use the ordinal information in the data.

• These tests do not involve adding, subtracting, dividing or multiplying the numbers in the data.
• They only involve using the order of the numbers.
We discuss two tests:
Mann-Whitney U-Test
This test is used when we obtain ordinal data in the independent groups situation.
Wilcoxon Signed-Ranks Test
This test is used when we obtain ordinal data in the paired samples situation.

## Mann-Whitney U-Test

The Mann-Whitney U-test evaluates the difference between two treatments (two populations) using data from the independent measures design.

### Conceptual Basis:

• A real difference between two treatments should make the scores in one group generally larger than those in the other.
• If the treatment had an effect, then when we combine the two samples together and rank order all the combined scores, the observations for one sample should be concentrated at one end of the scale, and the other sample's observations should be at the other end.
• If there is no effect, then large and small scores should be mixed together.

### Calculations

1. Observations are obtained for two groups, Group A and Group B.

2. All observations from the two groups are combined and rank ordered.

3. The Mann-Whitney U is calculated.
• Each observation in Group A gets a point for every observation in Group B that it is larger than. The total number of points for Group A is calculated.
• Each observation in Group B gets a point for every observation in Group A that it is larger than. The total number of points for Group B is calculated.
• The smaller of these two sums is the Mann-Whitney U.

1. The hypothesis test for the Mann-Whitney U is performed.
Null Hypothesis:
The two groups are identically distributed.
Alternative Hypothesis:
The two groups are not identically distributed.
• Note that when the treatment had such a large effect that all scores in one group are larger than those in the other, then U=0. 
• This test is called a non-parametric test because the hypotheses do not refer to a population parameter.
• The value of U is looked up in a table to determine its significance, or the computer calculates the significance.

### Example:

 These data are about the manual dexterity of three-year-old children. A sample of 13 children was obtained, 4 boys and 8 girls. They were asked to place a set of blocks into a specified pattern. The time (in seconds) required by each child to arrange the blocks was recorded. These data are from Gravetter and Walnau, (Ed. 4), p 607, who use them to demonstrate Mann-Whitney U-test. These data were analyzed with the Univariate Analysis module of ViSta. The report is shown below. The Mann-Whitney suggests that the difference between the two groups is border-line significant.  ## Wilcoxon Signed-Ranks Test

The Wilcoxon Signed-ranks test evaluates the difference between two treatments, using data from a paired-samples (repeated-measures) design.

### Conceptual Basis:

Since we have paired samples, we calculate the difference scores.
• If the treatment had an effect, the scores in one treatment would be consistently larger than those in the other, producing difference scores that are consistently positive or consistently negative.
• If there is no effect, then we would expect positive and negative differences to be intermixed evenly.
• The Wilcoxon test uses the signs and ranks of the differences to decide on the significance of the differences.

### Calculations

1. Rank the absolute values of the difference scores.

2. Separate the ranks into those associated with positive differences and those associated with negative differences.

3. Sum the ranks for the positive differences and sum the ranks for the negative differences.

4. The smaller sum is the Wilcoxon signed ranks test statistic, identified as the Wilcoxon T (not to be confused with the regular T-Test T value).

5. The hypothesis test for the Wilcoxon T is performed.
Null Hypothesis:
The two groups are identically distributed.
Alternative Hypothesis:
The two groups are not identically distributed.
• Note that when the treatment had such a large effect that all scores in one group are larger than those in the other, then T=0. 
• This test is called a non-parametric test because the hypotheses do not refer to a population parameter.
• The value of T is looked up in a table to determine its significance, or the computer calculates the significance.

### Example:

 These data are about an intensive campaign conducted by the Red Cross to increase blood donations. The campaign concentrated on 10 businesses. In each company the goal was to increase the percentage of employees who participated. The data are the percentage participation. These data are from Gravetter and Wallnau (4th Ed.) p. 614, who use them to demonstrate Wilcoxon Signed Rank test. These data were analyzed with the Univariate Analysis module of ViSta. The report is shown below. The Wilcoxon test suggests that the difference between the two groups is significant.  Back | Up | Next