1. Wilcoxon Signed Rank Test
EPI 809 / Spring 2008 9 47

2. Signed Rank Test Example
You work in the finance department. Is the new financial package faster (.05 level)? You collect the following data entry times:
User Current New
Donna
Santosha
Sam
Tamika
Brian
Jorge
EPI 809 / Spring 2008
© T/Maker Co.

3. Wilcoxon Signed Rank Test
1. Tests Probability Distributions of 2 Related Populations
2. Corresponds to t-test for Dependent (Paired) Means
3. Assumptions
Random samples; Both populations are continuous; paired samples.
Can Use Normal Approximation
if #(non-zero differences)  16
n’ is the number of non-zero difference scores.
EPI 809 / Spring 2008

4. Signed Rank Test Procedure
Obtain Difference Scores, Di = X1i - X2i
Take Absolute Value |Di| and rank them (Do not count Di = 0)
Assign Ranks, Ri, with Smallest = 1
Calculate range and mean rank for |Di|
Sum ‘+’ Ranks (T+) & ‘-’ Ranks (T-)
Statistic (T+). mean = n(n+1)/4,
var = n(n+1)(2n+1)/24 if no ties.
EPI 809 / Spring 2008

5. Signed Rank Test Procedure
Table method for (n < 16).
Use Table 11 that provides the critical values at significant level.
2. If n ≥ 16, use normal approximation.
Let T0 = n(n+1)/4, If T+ =T0, then T = 0
Otherwise, if no ties,
If ties, ti =Ri
of ties.
EPI 809 / Spring 2008

6. Signed Rank Test Procedure (continued)
If T > Z1-α/2 then reject H0. Otherwise accept H0.
P – value:
p = 2 [1 - Φ(T)].
Condition: number of nonzero differences > 15.
EPI 809 / Spring 2008

7. Signed Rank Test Example Skin care treatments A vs B
Each subject received two treatments A and B randomly applied to left and right arms.
Measure redness after sunlight exposure of 10 min.
Data:
subject 1: X1A, X1B
subject 2: X2A, X2B .…
EPI 809 / Spring 2008

8. Signed Rank Test Computation Illustration
|di|
di (-)
fi (-)
di(+)
fi(+) k
(# subj)
Range (Rank)
Mean Rank 8 -8 1 40 7 -7 3
37-39 38 6 -6 2
35-36
35.5 5 -5
33-34
33.5 4 -4 32 -3
25-31 28 -2 10
15-24
19.5 -1 14
1-14
7.5
EPI 809 / Spring 2008

9. Testing with positive statistic T+
E(T+ ) = 40*(40+1)/4 = 410
Var = 40*(40+1)*(80+1)/24 – [(143-14) +
(103-10) + (73-7) +…+ (13-1)]/48 =
T = {|248-10|-.5}/√ = 2.19
P = 2*[1- Φ(2.19)] =
EPI 809 / Spring 2008

10. Signed Rank Test Example
You work in the finance department. Is the new financial package faster (.05 level)? You collect the following data entry times:
User Current New
Donna
Santosha
Sam
Tamika
Brian
Jorge
EPI 809 / Spring 2008
© T/Maker Co.

11. Signed Rank Test Computation Table
EPI 809 / Spring 2008

12. Signed Rank Test Solution
H0: Identical Distrib.
Ha: Current Shifted Right
 = .05 (one-sided)
n’ = 5 (not 6; 1 elim.)
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Look up n’ = 5 in table; not 6!
Rejection region does not include cut-off.
Do Not Reject
Reject TL Tu
EPI 809 / Spring 2008

13. Wilcoxon Signed Rank Table (Portion)
EPI 809 / Spring 2008

14. Signed Rank Test Solution
H0: Identical Distrib.
Ha: Current Shifted Right
 = .05 (one-sided)
n’ = 5 (not 6; 1 elim.)
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Look up n’ = 5 in table; not 6!
Rejection region does not include cut-off.
Do Not Reject 15
EPI 809 / Spring 2008

15. Signed Rank Test Solution
H0: Identical Distrib.
Ha: Current Shifted Right
 = .05 (one-sided)
n’ = 5 (not 6; 1 elim.)
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
Since one-sided Test & Current Shifted Right, Use T+
Look up n’ = 5 in table; not 6!
Rejection region does not include cut-off.
Do Not Reject 15
EPI 809 / Spring 2008

16. Signed Rank Test Solution
H0: Identical Distrib.
Ha: Current Shifted Right
 = .05 (one-sided)
n’ = 5 (not 6; 1 elim.)
Critical Value(s):
Test Statistic:
Conclusion:
Since One-sided Test & Current Shifted Right, Use T+ T+ = 15
Look up n’ = 5 in table; not 6!
Rejection region does not include cut-off.
Do Not Reject
Border line at  = .05 15
EPI 809 / Spring 2008

17. Kruskal-Wallis test
A k-sample non-parametric test on the means (k > 2).
Pool observations together N = ∑ni and assign ranks to individuals.
Compute the rank sum Ri for each sample.
Test statistic Chi-squares with df = (k-1)
one tail prob. Compare with χ2k-1, 1-α
Should only be used if smallest ni ≥ 5.
As a result of this class, you will be able to ...
EPI 809 / Spring 2008

18. Kruskal-Wallis test statistic
If no ties.
If ties. Denote ti, g as before.
As a result of this class, you will be able to ...
EPI 809 / Spring 2008

19. Conclusion 1. Distinguished Parametric & Nonparametric Test Procedures
2. Explained a Variety of Nonparametric Test Procedures
3. Solved Hypothesis Testing Problems Using Nonparametric Tests
As a result of this class, you will be able to ...
EPI 809 / Spring 2008

20. Summary Nonparametric Parametric Sign Rank test One sample t-test
Wilcoxon Rank – Sum test
(Mann-Whitney U test)
Two sample t-test
Wilcoxon Signed-Rank test
Two paired sample t-test
Kruskal-Wallis test
Multiple sample test.
As a result of this class, you will be able to ...
EPI 809 / Spring 2008