Wilcoxon Rank Sum Test

1
Wilcoxon Rank Sum Test

• 1. Wilcoxon with both n1 and n2 lt 10
• 2. Wilcoxon with both n1 and n2 10
• 3. Examples

2
Wilcoxon Rank Sum Test

• Recall from last week
• When we test a hypothesis about the difference
  between two independent population means, we do
  so using the difference between two sample means.
• When the two sample variances are tested and
  found 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
Population 1
Population 2
µ1
µ2
Sample1
Sample2
4
Wilcoxon Rank Sum Test

• The Z test and the t test are parametric tests
  that is, they answer a question about the
  difference between populations by comparing
  sample statistics (e.g., X1 and X2) and making an
  inference to the population parameters (µ1 and
  µ2).
• The Wilcoxon, in contrast, allows inferences
  about whole populations

5
Note that distribution B is shifted to the right
of distribution A
6
1b. Small samples, independent groups

• Wilcoxon Rank Sum Test
• 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
1b. Small samples, independent groups

• Wilcoxon One-tailed Hypotheses
• H0 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
1b. Small samples, independent groups

• Wilcoxon Two-tailed Hypotheses
• H0 Prob. distributions for 2 sampled populations
  are identical.
• HA Prob. distribution for Population A shifted
  to right or left of distribution for Population B.

9
1b. Small samples, independent groups

• Wilcoxon Rejection region
• (With Sample taken from Population A being
  smaller than sample for Population B) reject H0
  if
• TA TU or TA TL

10
1b. Small samples, independent groups

• Wilcoxon for n1 10 and n2 10
• Test statistic
• Z TA n1(n1 n2 1)
• 2
• n1n2(n1 n2 1)
• 12

11
Wilcoxon for n1 10 and n2 10

• Rejection region
• One-tailed Two-tailed
• Z gt Za Z gt Za/2
• Note use this only when n1 10 and n2 10

12
Example 1

• These are small samples, and they are independent
  (random samples of Cajun and Creole dishes).
  Therefore, we have to begin with the test of
  equality of variances.

13
Test of hypothesis of equal variances

• H0 ?12 ?22
• HA ?12 ? ?22
• Test statistic F S12
• S22
• Rej. region F gt Fa/2 F(6,6,.025) 5.82
• or F lt (1/5.82) .172

14
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 we do the
  Wilcoxon.

15
Example 1 Wilcoxon Rank Sum Test

• H0 Prob. distributions for Cajun and Creole
  populations are identical.
• HA Prob. distribution for Cajun is shifted to
  right of distribution for Creole.
• Statistical test T

16
Example 1 Wilcoxon Rank Sum Test

• Rejection region
• Reject H0 if TCajun gt 66 (or if TCreole lt 39)
• (Note We shall give lower heat values lower rank
  values)

17
Example 1 Wilcoxon Rank Sum Test

• Cajun Creole
• 3500 3100
• 4200 4700
• 4100 2700
• 4700 3500
• 4200 2000
• 3705 3100
• 4100 1550

6.5
13.5
3
13.5
6.5
2
8
1
S 70 35
18
Example 1 Wilcoxon Rank Sum Test

• Calculation check
• Sum of the ranks should (n) (n1)
• 2
• 70 35 105 (14)(15)
• 2

19
Example 1 Wilcoxon Rank Sum Test

• TCajun 70 gt 66 (and TCreole 35 lt 39)
• Therefore, reject H0 Cajun dishes are
  significantly hotter than Creole dishes.

20
Example 2 Wilcoxon Rank Sum Test

• H0 ?12 ?22
• HA ?12 ? ?22
• Test statistic F S12
• S22
• Rej. region F gt Fa/2 F(7,8,.025) 4.53
• or F lt (1/4.90) .204

21
Example 2 Wilcoxon Rank Sum Test

• Fobt 4.316 9.38
• .46
• Reject H0 do Wilcoxon

22
Example 2 Wilcoxon Rank Sum Test

• H0 Prob. distributions for females and males
  populations are identical.
• HA Prob. distribution for females is shifted to
  left of distribution for males.
• Statistical test T
• Rejection region T? gt TU 90
• (or T? lt TL 54)

23
Example 2 Wilcoxon Rank Sum Test

• 6.4 16 2.7 3
• 1.7 1 3.9 10
• 3.2 5 4.6 12
• 5.9 15 3.0 4
• 2.0 2 3.4 6.5
• 3.6 8 4.1 11
• 5.4 14 3.4 6.5
• 7.2 17 4.7 13
• 3.8 9
• S 78 75

24
Example 2 Wilcoxon Rank Sum Test

• T? 78 lt TU 90
• Therefore, do not reject H0 no evidence that
  mean distance in females is less than that in
  males.

25
Example 3 Wilcoxon Rank Sum Test

• H0 ?12 ?22
• HA ?12 ? ?22
• Test statistic F S12
• S22
• Rej. region F gt Fa/2 F(5,5,.025) 7.15
• or F lt (1/7.15) .140

26
Example 3 Wilcoxon Rank Sum Test

• Fobt (7.563)2 57.20
• (2.04)2 4.16
• 13.74
• Reject H0 do Wilcoxon

27
Example 3 Wilcoxon Rank Sum Test

• 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 gt 52 or lt 26

28
Example 3 Wilcoxon Rank Sum Test

• 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
• S 33 45

29
Example 3 Wilcoxon Rank Sum Test

• Check TH TM 78
• (12)(13) 78
• 2
• TH 33 gt 26 and lt 52
• Do not reject H0 no evidence for a significant
  difference between teams.