1. Ordinally Scale Variables
Greg C Elvers

2. Why Special Statistics for Ordinally Scaled Variables
The parametric tests (e.g. t, ANOVA) rely on estimates of variance which cannot be meaningfully obtained from ordinally scaled data
The non-parametric tests for nominally scaled variables (e.g. binomial, 2) do not use all the information that is present in ordinally scaled variables

3. Types of Statistics for Ordinally Scaled Variables
There are three main statistics that are used with ordinally scaled variables:
Mann-Whitney U
Sign test
Wilcoxon matched-pairs signed-rank test

4. Mann-Whitney U Test The Mann-Whitney U test can be used when:
the dependent variable is ordinally scaled (or above), and
the design is a two-sample design, and
the design is between subjects, and
the participants are not matched across conditions

5. Mann-Whitney U Test
The Mann-Whitney U test is a useful alternative to the t-test if
the dv is ordinally scaled, or
you do not meet the assumption of normality, or
you do not meet the assumption of homogeneity of variance

6. Steps in the Mann-Whitney U Test
Write the hypotheses:
H0: 1 = 2 or H0: 1  2
H1: 1  2 or H1: 1 > 2
Decide if the hypothesis is one- or two-tailed
Specify the  level
Calculate the Mann-Whitney U

7. Steps in the Mann-Whitney U Test
Rank order all of the data (from both control and experimental conditions) from lowest to highest
Lowest score has a rank of 1
Sum the ranks of the scores in the first condition
The sum of the ranks is called R1
Sum the ranks of the scores in the second condition
The sum of the ranks is called R2

8. Steps in the Mann-Whitney U Test
Calculate the U (or U’) statistic:
N1 is the number of scores in the 1st condition
N2 is the number of scores in the 2nd condition
R1 is the sum of the ranks of the scores in the 1st condition
R2 is the sum of the ranks of the scores in the 2nd condition

9. Steps in the Mann-Whitney U Test
Consult a table to find the critical U and U’ values
The tails and  level determine which table you will use
Find N1 across the top of the table and N2 down the left side of the table
The critical U and U’ values are given at the intersection of the N1 column and N2 row
Critical U is the smaller number in the pair
Critical U’ is the larger number in the pair

10. Steps in the Mann-Whitney U Test
Decide whether to reject H0 or not:
If the observed U is less than or equal to the critical U, reject H0
If the observed U’ is greater than or equal to the critical U’, reject H0

11. Mann-Whitney Example An instructor taught two sections of PSY 216
One section used SPSS for calculations
The other section performed calculations by hand
At the end of the course, the students rated how much they liked statistics
The questionnaire asked 20 questions on a 5 point scale

12. Mann-Whitney U Example
Are the mean ratings of liking different?
Write the hypotheses:
H0: SPSS = Hand
H1: SPSS  Hand
Determine the tails
It is a two-tailed, non-directional test
Specify the  level
 = .05
Calculate the Mann-Whitney U

13. Mann-Whitney Example

14. Mann-Whitney Example

15. Mann-Whitney U Example
Calculate the statistic:

16. Mann-Whitney U Example
Find the critical U and U’ values
Consult the table of critical U values with  = .05, two-tailed
Column = N1 = 15
Row = N2 = 16
Critical U = 70
Critical U’ = 170

17. Mann-Whitney U Example
Decide whether to reject H0:
If observed U (87) is  critical U (70), reject H0
If observed U’ (153) is  critical U’ (170), reject H0
Fail to reject H0
There is insufficient evidence to conclude that the mean ratings are different

18. Special Considerations
If a score in one condition is identical to a score in the other condition (i.e. the ranks are tied) then a special form of the Mann-Whitney U test should be used
Failure to use the special form increases the probability of a Type-II error

19. Special Considerations
When N1 and / or N2 exceed 20, then the sampling distributions are approximately normal (due to the central limit theorem) and the z test can be used:
Where
U1 = sum of ranks of group 1
UE = sum expected under H0
su = standard error of the U statistics

20. Special Considerations

21. Sign Test The sign test can be used when:
the dependent variable is ordinally scaled (or above), and
the design is a two-sample design, and
the participants are matched across conditions

22. Sign Test
The basic idea of the sign test is that we take the difference of each pair of matched scores
Then we see how many of the differences have a positive sign and how many have a negative sign

23. Sign Test
If the groups are equivalent (i.e. no effect of the treatment), then about half of the differences should be positive and about half should be negative
Because there are only two categories (+ and -), the sign test is no different from the binomial test

24. Steps in the Sign Test Write the hypotheses:
H0: P = .5
H1: P  .5 where P = probability of a positive sign in the difference
Decide if the hypothesis is one- or two-tailed
Specify the  level
Calculate the Sign Test

25. Calculate the Sign Test
For each pair of scores, take the difference of the scores
Count the number of differences that have a positive sign

26. Steps in the Sign Test
Determine the critical value from a table of binomial values
Find the column with the appropriate number of tails and  level
Find the row with the number of pairs of scores
The critical value is at the intersection of the row and column

27. Steps in the Sign Test Decide whether to reject H0 or not:
If the number of differences that are positive is greater than or equal to the critical value from the binomial table, reject H0

28. Sign Test Example An instructor taught two sections of PSY 216
One section used SPSS for calculations
The other section performed calculations by hand
The students in the sections were matched on their GPA
At the end of the course, the students rated how much they liked statistics
The questionnaire asked 20 questions on a 5 point scale

29. Sign Test Example Are the mean ratings of liking different?
Write the hypotheses:
H0: P = .5
H1: P  .5
Determine the tails
It is a two-tailed, non-directional test
Specify the  level
 = .05
Calculate the sign test

30. Sign Test Example

31. Sign Test Example

32. Sign Test Example Determine the critical value
=.05, two-tailed, N = 15
Critical value = 12
Decide whether to reject H0:
If observed value (9) is greater than or equal to critical (12), then reject H0
Fail to reject H0 - there is insufficient evidence to conclude that the groups are different

33. Wilcoxon Matched-Pairs Signed-Rank Test
The sign test considers only the direction of the difference, and not the magnitude of the difference
If the magnitude of the difference is also meaningful, then the Wilcoxon matched-pairs signed-rank test may be a more powerful alternative

34. Wilcoxon Matched-Pairs Signed-Rank Test
The Wilcoxon matched-pairs signed-rank test can be used when:
the dependent variable is ordinally scaled (or above), and
the design is a two-sample design, and
the participants are matched across conditions, and
the magnitude of the difference is meaningful

35. Steps in the Wilcoxon Matched-Pairs Signed-Rank Test
Write the hypotheses:
H0: 1 = 2
H1: 1  2
Decide if the hypothesis is one- or two-tailed
Specify the  level
Calculate the Wilcoxon matched-pairs signed-rank test

36. Steps in the Wilcoxon Matched-Pairs Signed-Rank Test
For each pair of scores, take the difference of the scores
Rank order the differences from lowest to highest, ignoring the sign of the difference
Sum the ranks that have a negative sign

37. Steps in the Wilcoxon Matched-Pairs Signed-Rank Test
Determine the critical value from a table of critical Wilcoxon values
Find the column with the appropriate number of tails and  level
Find the row with the number of pairs of scores
The critical value is at the intersection of the row and column

38. Steps in the Wilcoxon Matched-Pairs Signed-Rank Test
Decide whether to reject H0 or not:
If the absolute value of the sum of the negative ranks is less than the critical value from the table, reject H0

39. Wilcoxon Matched-Pairs Signed-Rank Test Example
We will use the same example as the sign-test

40. Wilcoxon Matched-Pairs Signed-Rank Test Example
Are the mean ratings of liking different?
Write the hypotheses:
H0: SPSS = Hand
H1: SPSS  Hand
Determine the tails
It is a two-tailed, non-directional test
Specify the  level
 = .05
Calculate the Wilcoxon test

41. Wilcoxon Matched-Pairs Signed-Rank Test Example

42. Wilcoxon Matched-Pairs Signed-Rank Test Example
Determine the critical value
=.05, two-tailed, N = 15
Critical value = 25
Decide whether to reject H0:
If the absolute value of the observed value (-45) is less than the critical (25), then reject H0
Fail to reject H0 - there is insufficient evidence to conclude that the groups are different

43. Special Considerations
The Wilcoxon test assumes that the differences are ordinally scaled
This assumption is often incorrect, and is hard to verify
If we cannot verify it, we should not use the Wilcoxon test