1

Statistics for the Behavioral Sciences (5th ed.) Gravetter & Wallnau

Chapter 19

Statistical Techniques for Ordinal Data: Mann-Whitney, Wilcoxon, and Kruskal-Wallis

Tests

University of GuelphPsychology 3320 — Dr. K. Hennig

Winter 2003 Term

2

Introduction Widespread use of ranking:

longest river in the world = Nile most popular magazine = Reader’s Digest your favourite ice-cream?

Such data are easy to obtain; no need to rate 31 flavours on a 100-point scale

easy but less sophisticated than interval or ratio scales

3

Table 19-1 (p. 634) The best cities for singles as ranked by Forbes.com (2002).

no distance datare. closeness of ranking - whatmeans and SDsprovide

Reasons for using: simplicity (who’s

taller?) where basic

assumptions areviolate(Normal distribution, homog’-of-var, infinite score)

4

Hypotheses tests with ordinal data

Mann-Whitney test for independent samples (alternative to a independent-measures t-test)

Wilcox test - alternative to repeated-measures t

Kruskall-Wallis test - alternative to sg-factor, independent measures ANOVA

5

In part (a), the scores from the two samples are clustered at opposite ends of the rank ordering. In this case, the data suggest a systematic difference between the two treatments. Part (b) shows the two samples intermixed evenly along the scale, indicating no consistent difference between treatments.

I. Mann-Whitney U-test

6

Compare two distributions (vs. Ms)

Ho: there is no difference between the two treatments; no tendency for the ranks in one treatment condition to be systematically higher (or lower) than the ranks in the other condition

H1: There is a difference, ranks in one condition are systematically higher (or lower) than the ranks in the other condition

7

Example 19.1 Sample A: 27, 2, 9, 48, 6, 15 Sample B:71, 63, 18, 68, 94, 8

Calculate U-valuefor Sample A & B, UA = 6+6+5+4+4+4 = 30

8

Formula

∑RA = 1+2+4+5+7+8=27,nA = n of sample A UA = 33, UB = 7; final U is the smaller of the two Reported as U = 7, p > .05

9

Assumptions not require normal distributions or

homogenity-of-variances, but does require independent observations

and assumes dependent variable is continuous

Note. if you find many ties (unlikely by chance) than question whether Mann-Whitney U is correct test.

10

II. Wilcox Signed-Rank test ∑R+ = 6+2+5+3 = 16 ∑R- = 1+4 = 5; select the smaller, so T = 5, p>.05

with ranks for increases totaling … decreases… Two companies with no change were discarded prior to analysis

11

Original data consisting of numerical scores are shown. The original scores are listed in order and assigned ranks using the standard procedure for ranking tied scores. The ranks are then substituted for the original scores to create the set of ordinal data shown in table (b).

III. Kruskal-Wallis test

compute mean of the tied scores (8+9+10)/3

12

An example of ordinal data including the notation used for a Kruskal-Wallis test - they include some tied measurements that appear as tied ranks.Evaluate with χ2 table.

HN N

Tn

N

1 21

3 12

( )( ( ) H =3.79 (2, N = 15), p

> .05df = #groups-1