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Ordinally Scale Variables

Greg C Elvers

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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

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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

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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

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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

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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

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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 conditionThe 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

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Steps in the Mann-Whitney U Test

Calculate the U (or U’) statistic:

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2221

111

21

R2

1NNNNU

R2

1NNNNU

N1 is the number of scores in the 1st conditionN2 is the number of scores in the 2nd conditionR1 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

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Steps in the Mann-Whitney U Test

Consult a table to find the critical U and U’ valuesThe 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

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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

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Mann-Whitney Example

An instructor taught two sections of PSY 216One 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

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Mann-Whitney U Example

Are the mean ratings of liking different?Write the hypotheses:

H0: SPSS = Hand

H1: SPSS Hand

Determine the tailsIt is a two-tailed, non-directional test

Specify the level = .05

Calculate the Mann-Whitney U

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Mann-Whitney Example

SPSS Hand

86 4853 2479 8291 6383 8176 7267 8094 9075 7152 5933 4987 3058 6847 5566 57

44

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Mann-Whitney Example

SPSS HandScore Rank Score Rank

86 27 48 653 9 24 179 22 82 2591 30 63 1483 26 81 2476 21 72 1967 16 80 2394 31 90 2975 20 71 1852 8 59 1333 3 49 787 28 30 258 12 68 1747 5 55 1066 15 57 11

44 4

R1 = 273 R2 = 223

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Mann-Whitney U Example

Calculate the statistic:

87

2732

115151615

R2

1NNNNU 1

1121

153

2232

116161615

R2

1NNNNU 2

2221

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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

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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

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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

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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:

U

E1

s

UUz

WhereU1 = sum of ranks of group 1

UE = sum expected under H0

su = standard error of the U statistics

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Special Considerations

2

1NNNU 211

E

12

1NNNNs 2121

u

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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

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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

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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

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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

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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

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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

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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

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Sign Test Example

An instructor taught two sections of PSY 216One section used SPSS for calculationsThe other section performed calculations by handThe 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

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Sign Test Example

Are the mean ratings of liking different?Write the hypotheses:

H0: P = .5

H1: P .5

Determine the tailsIt is a two-tailed, non-directional test

Specify the level = .05

Calculate the sign test

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Sign Test Example

SPSS Hand

86 4853 2479 8291 6383 8176 7267 8094 9075 7152 5933 4987 3058 6847 5566 57

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Sign Test Example

SPSS Hand Sign of Difference

86 48 +53 24 +79 82 -91 63 +83 81 +76 72 +67 80 -94 90 +75 71 +52 59 -33 49 -87 30 +58 68 -47 55 -66 57 +

# + Signs = 9

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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

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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

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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

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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

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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

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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

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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

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Wilcoxon Matched-Pairs Signed-Rank Test Example

We will use the same example as the sign-test

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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 tailsIt is a two-tailed, non-directional test

Specify the level = .05

Calculate the Wilcoxon test

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Wilcoxon Matched-Pairs Signed-Rank Test Example

SPSS Hand Difference Rank of Difference Ranks With - Value

86 48 38 1453 24 29 1379 82 -3 (-)2 (-)291 63 28 1283 81 2 176 72 4 467 80 -13 (-)10 (-)1094 90 4 475 71 4 452 59 -7 (-)6 (-)633 49 -16 (-)11 (-)1187 30 57 1558 68 -10 (-)9 (-)947 55 -8 (-)7 (-)766 57 9 8

T = -45

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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

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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