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 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

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

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

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

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

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

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

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

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

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

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

Mann-Whitney Example

Mann-Whitney Example

Mann-Whitney U Example Calculate the statistic:

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

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

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

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

Special Considerations

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

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

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

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

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

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

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

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

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

Sign Test Example

Sign Test Example

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

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

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

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

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

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

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

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

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

Wilcoxon Matched-Pairs Signed-Rank Test Example

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

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