Ordinal DataOrdinal Data

Ordinal TestsOrdinal Tests

Non-parametric testsNon-parametric tests No assumptions about the shape of No assumptions about the shape of

the distributionthe distribution Useful When: Useful When:

– Scores are ranksScores are ranks– Violated assumptionsViolated assumptions– There are outliersThere are outliers

Frequently Used Ordinal Frequently Used Ordinal TestsTests

1.1. Spearman’s Rank Correlation Spearman’s Rank Correlation Coefficient (Chapter 16)Coefficient (Chapter 16)

2.2. Mann-Whitney U-test Mann-Whitney U-test

3.3. Wilcoxon Signed Rank TestWilcoxon Signed Rank Test

4.4. Kruskal Wallis Kruskal Wallis HH-Test-Test

5.5. Friedman testFriedman test

Spearman’s Rank Spearman’s Rank Correlation Coefficient (rCorrelation Coefficient (rss))

Designed to measure the relationship Designed to measure the relationship between variables measured on an between variables measured on an ordinal scale of measurementordinal scale of measurement

Alternative to Pearson correlationAlternative to Pearson correlation– Treatment of ordinal dataTreatment of ordinal data– Good even if data is interval or ratioGood even if data is interval or ratio– Spearman can be used for nonlinear Spearman can be used for nonlinear

relationshipsrelationships

Spearman’s Rank Spearman’s Rank Correlation Coefficient (rs)Correlation Coefficient (rs)

yx

sSSSS

SPr

Alternative FormulaAlternative Formula

2

2

61

( 1)i

s

dr

n n

2

2

61

( 1)i

s

dr

n n

where: where: nn = number of items being ranked = number of items being ranked dd = = difference between the X rank and Y difference between the X rank and Y

rank for each individualrank for each individual

ExampleExample

X Y

7 19

2 4

11 34

15 28

32 104

Original Data

XRank

X YRank

Y

7 2 19 2

2 1 4 1

11 3 34 4

15 4 28 3

32 5 104 5

Mann-Whitney U-testMann-Whitney U-test When to use:When to use:

– Two independent samples in your Two independent samples in your experimentexperiment

Data have only ordinal properties (e.g. Data have only ordinal properties (e.g. rating scale data) rating scale data) OROR there is some other there is some other problem with the data problem with the data

–Non-normalityNon-normality

–Non-homogeneity of varianceNon-homogeneity of variance

Ranked DataRanked Data

The Test ProcedureThe Test Procedure

We compute two “U” values (UA and UB) based on the sum of the ranks for each sample

AAA

BAA Rnn

nnU

2

)1(

BBB

BAB Rnn

nnU

2

)1(

Where: nA = number in sample A nB = number in sample BRA = sum of ranks group ARB = sum of ranks group B

Worked Out ExampleWorked Out ExampleSpecies A Species B

7 24

9 19

14 21

20 26

16 21

18 29

10 13

22 28

25 32

13 17

DV: amount of food consumed

nA: 10

nB: 10

Calculate UCalculate UAA and U and UBB

AAA

BAA Rnn

nnU

2

)1(

Calculate UCalculate UAA and U and UBB

BBB

BAB Rnn

nnU

2

)1(

Wilcoxon Signed Ranks TestWilcoxon Signed Ranks Test

Each participant observed twice

Compute difference scores

Analogous to related samples t-test

Preliminary Steps of the TestPreliminary Steps of the Test

Rank difference scores

Compute sum of ranks of “+” and “-” difference scores separately

If tied differences, use tied ranks

Preliminary Steps of the TestPreliminary Steps of the Test

If difference is 0: – ignore and reduce n– do not discard

Compromise: if there’s only one difference Compromise: if there’s only one difference score of 0, then we discard it. score of 0, then we discard it. – If there’s more than one, we divide them evenly If there’s more than one, we divide them evenly

into positive and negative ranks. It doesn’t into positive and negative ranks. It doesn’t matter which is which, because they’re all 0. matter which is which, because they’re all 0.

– If you have an odd number of 0 differences, then If you have an odd number of 0 differences, then discard one, and divide the rest evenly into discard one, and divide the rest evenly into positive and negative ranks. positive and negative ranks.

The smaller sum is denoted as TThe smaller sum is denoted as T

T = smaller of TT = smaller of T+ + and Tand T--

If Ho true, sum of “+” and “-” ranks approx. equal

ExampleExample

Participant Before After

1 2.1 2.2

2 3.9 2.8

3 3.8 2.5

4 2.5 2.6

5 2.4 1.9

6 3.6 1.8

7 3.4 2

8 2.4 1.6

Is there enough evidence to conclude that there is a difference in headache hours before and after the new drug?

= 0.01

Kruskal-Wallis TestKruskal-Wallis Test

Used to test for differences between three or more Used to test for differences between three or more treatment conditions from an independent treatment conditions from an independent measures designmeasures design

Analogous to the one-way independent measures Analogous to the one-way independent measures ANOVA ANOVA EXCEPT EXCEPT data consist of ranksdata consist of ranks

Does not require the assumption of normally Does not require the assumption of normally distributed populationsdistributed populations

Ri, is the sum of ranks for each groupN is the total sample sizeni is the sample size of the particular group

Friedman TestFriedman Test A nonparametric test invented by Milton A nonparametric test invented by Milton

Friedman (the Nobel prize winning Friedman (the Nobel prize winning economist)economist)

Used to test for differences between three or Used to test for differences between three or more treatment conditions from an dependent more treatment conditions from an dependent measures designmeasures design

Analogous to the one-way repeated Analogous to the one-way repeated measures ANOVA measures ANOVA EXCEPT EXCEPT data consist of data consist of ranksranks

Follows the Follows the 22 distribution when we have at distribution when we have at least 10 scores in each of the 3 columns or at least 10 scores in each of the 3 columns or at least 5 scores in each of 4 columnsleast 5 scores in each of 4 columns

July 31, 1912 – November 16, 2006

Friedman TestFriedman Test

There are specific tables for Friedman’s test statistic for There are specific tables for Friedman’s test statistic for up to k=5 variablesup to k=5 variables

Otherwise use chi-square tables because Fr is Otherwise use chi-square tables because Fr is distributed approximately as chi-square with df= k-1distributed approximately as chi-square with df= k-1

If If 22FF>= the tabled value for df =k-1, then the result is >= the tabled value for df =k-1, then the result is

significant, and we can say the difference in total ranks significant, and we can say the difference in total ranks between the k conditions is not due to chance variationbetween the k conditions is not due to chance variation

)1(3)1(

12

1

22

kNRkNk

k

jjF

Summary Table: Parametric Tests & Summary Table: Parametric Tests & Their Non-Parametric CounterpartsTheir Non-Parametric Counterparts

Parametric Test Non-Parametric Test

Independent Samples t-test Mann-Whitney U Test

Related Samples

t-test

Wilcoxon Signed Rank Test

One Way Between Subjects ANOVA

Kruskal-Wallis Test

One Way Repeated Subjects ANOVA

Friedman Test

Pearson Correlation Spearman Rank-Order Correlation