WilcoxonWilcos

8/22/2019 WilcoxonWilcos

1/29

1

Wilcoxon

Wilcoxon Rank Sum Test

1. Wilcoxon with both n1 and n2 < 10

2. Wilcoxon with both n1 and n2 10

3. Examples


2/29

2

Wilcoxon


Recall from last week:

When we test a hypothesis about the differencebetween two independent population means, wedo so using the difference between two samplemeans.

When the two sample variances are tested andfound not to be equal we cannot pool the sample variances thus we cannot use the t-test for independent samples.Instead, we use the Wilcoxon Rank Sum Test.


3/29

3

Wilcoxon

Population 1 Population 2

1 2

Sample1 Sample2X1 X2

tells us about the

population

The sample mean

tells us about


4/29

4

Wilcoxon


The Z test and the t test are parametrictests that is, they answer a questionabout the difference between populations by

comparing sample statistics (e.g., X1 and X2)

and making an inference to the population

parameters (1 and 2).

The Wilcoxon, in contrast, allows inferencesabout whole populations


5/29

5

Wilcoxon

X

X

Distribution B

Distribution A

Note that distribution B

is shifted to the right of

distribution A


6/29

6

Wilcoxon

1b. Small samples, independent groups


first, combine the two samples and rank order

all the observations.

smallest number has rank 1, largest number

has rank N (= sum of n1 and n2).

separate samples and add up the ranks for the

smaller sample. (If n1 = n2, choose either one.) test statistic : rank sum T for smaller sample.


7/29

7

Wilcoxon


Wilcoxon One-tailed HypothesesH0: Prob. distributions for 2 sampled

populations are identical.

HA: Prob. distribution for Population A

shifted to right of distribution for Population

B. (Note: could be to the left, but must be

one or the other, not both.)


8/29

8

Wilcoxon


Wilcoxon Two-tailed HypothesesH0: Prob. distributions for 2 sampled


HA: Prob. distribution for Population A

shifted to right or left of distribution for

Population B.


9/29

9

Wilcoxon


Wilcoxon Rejection region:

(With Sample taken from Population A beingsmaller than sample for Population B)reject H0 if

TA TU or TA TL


10/29

10

Wilcoxon


Wilcoxon for n1 10 and n2 10:

Test statistic:

Z = TA n1(n1 + n2 + 1)

2n1n2(n1 + n2 + 1)

12


11/29

11

Wilcoxon

Wilcoxon for n1 10 and n2 10

Rejection region:

One-tailed Two-tailed

Z > Z Z > Z/2

Note: use this only when n1 10 and n2 10


12/29

12

Wilcoxon

Example 1

These are small samples, and they are

independent (random samples of Cajun andCreole dishes). Therefore, we have to beginwith the test of equality of variances.


13/29

13

Wilcoxon

Test of hypothesis of equal variances

H0: 12 = 2

2

HA: 1222

Test statistic: F = S12

S22

Rej. region: F > F/2 = F(6,6,.025) = 5.82

or F < (1/5.82) = .172


14/29

14

Wilcoxon

Test of hypothesis of equal variances

S2Cajun = (385.27)2 = 148432.14

S2Creole = (1027.54)2 = 1055833.33

Fobt = 148432.14 = 7.11

1055833.33

Reject H0 variances are not equal, so wedo the Wilcoxon.


15/29

15

Wilcoxon

Example 1 Wilcoxon Rank Sum Test

H0: Prob. distributions for Cajun and Creole


HA

: Prob. distribution for Cajun is shifted to

right of distribution for Creole.

Statistical test: T


16/29

16

Wilcoxon


Rejection region:

Reject H0 if TCajun > 66 (or if TCreole 66 (and TCreole = 35 < 39)

Therefore, reject H0 Cajun dishes aresignificantly hotter than Creole dishes.


20/29

20

Wilcoxon


H0: 12 = 2

2

HA: 1222


S22

Rej. region: F > F/2 = F(7,8,.025) = 4.53

or F < (1/4.90) = .204


21/29

21

Wilcoxon


Fobt = 4.316 = 9.38

.46

Reject H0 do Wilcoxon


22/29

22

Wilcoxon


H0: Prob. distributions for females and males


HA

: Prob. distribution for females is shifted to

left of distribution for males.

Statistical test: TRejection region: T > TU = 90

(or T < TL = 54)


23/29

23

Wilcoxon


6.4 16 2.7 3

1.7 1 3.9 10

3.2 5 4.6 12

5.9 15 3.0 42.0 2 3.4 6.5

3.6 8 4.1 11

5.4 14 3.4 6.5

7.2 17 4.7 133.8 9

78 75


24/29

24

Wilcoxon


T = 78 < TU = 90

Therefore, do not reject H0 no evidencethat mean distance in females is less thanthat in males.


25/29

25

Wilcoxon


H0: 12 = 2

2

HA: 1222


S22

Rej. region: F > F/2 = F(5,5,.025) = 7.15

or F < (1/7.15) = .140


26/29

26

Wilcoxon


Fobt = (7.563)2 = 57.20

(2.04)2 4.16

= 13.74

Reject H0 do Wilcoxon


27/29

27

Wilcoxon


H0: Prob. distributions for Hoodoo and

Mukluk populations are identical.

HA

: Prob. distribution for Hoodoos is shifted

to right or left of distribution for Mukluks.

Statistical test: T

Rejection region: TH > 52 or < 26


28/29

28

Wilcoxon


Hoodoo Mukluk

2 1 6 5

6 5 8 9.5

4 2.5 7 7.5

23 12 10 11

7 7.5 8 9.5

6 5 4 2.5

33 45


29/29

29

Wilcoxon


Check: TH + TM = 78

(12)(13) = 78

2

TH = 33 > 26 and < 52

Do not reject H0 no evidence for asignificant difference between teams.

Documents

WilcoxonWilcos