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Ordinally Scale Variables

Ordinally Scale Variables. Greg C Elvers. 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

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Ordinally Scale Variables

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  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  .5where 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

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