WILCOXON MATCHED PAIRS SIGN- RANK TEST

Learning Outcomes

At the end of the topic, students should be able to:• Making inferences two related sample by using Wilcoxon

matched-pairs signed- rank test.

Definition • Need for case of two related sample when the

measurement scale allow to determine not only whether the member of a pair of observation differ, but also the magnitude of any difference.

Assumption The data for analysis consist of n values of the

differences Di = Yi- Xi. Each pairs of measurement (Xi, Yi) is taken on the same subject or non subject that have paired with respect to one or more variables. The sample of (Xi, Yi) pairs is random.

The differences represent observation on a continuous random variable.

The distribution of the population of differences is symmetric about their median, MD.

The differences are independent.The differences are measured on at least an interval

scale.

Hypothesis

Test statistic Obtain each of the signed differences

Di = Yi– Xi

Rank the absolute values of these differences from smallest to largest

| Di | = | Yi – Xi|

Assign to each of the resulting ranks the sign of the difference whose absolute valued yielded that rank.

Compute (depend on H1):

T+ = sum of the rank with positive signs

T- = sum of the rank with negative signs

Ties

There are two types:

When Yi=Xi for a given pair. Then we having Di = Yi– Xi = 0 and reduce n accordingly.

When two or more values of | Di | are equal

Decision Rule

Example Dickie et. al* studied hemodynamic changes in patients

with acute pulmonary thromboembolism. Table 4.6 shows the mean pulmonary artery pressure of 9 of these patients before and 24 hours after urokinase therapy. We wish to know whether these data provide sufficient evidence to indicate that urokinase theraphy lowers pulmonary artery pressure. Let α = 0.05

* Source:Kenneth J.Dickie,William J.de Groot, Robert N.Cooley,Ted P.Bond, and M.Mason Guest, “

Hemodynamic Effects of Bolus Infusion of Urokinase in Pulmonary

Thromboembolism,”Amer.Rev.Respir.Dis.,109(1974),48-56.

Mean Pulmonary Artery pressure , millimeters and mercury

SOLUTIONStep 1:

Patient 1 2 3 4 5 6 7 8 9

0 hours (X)

33 17 30 25 36 25 31 20 18

24 hours (Y)

21 17 22 13 33 20 19 13 9

Step 2: Test Statisticthe calculation of the test statistic is shown in table below:

Step 3: Since n= 8 , T+=0

From the table A.3 the probability of observing a value of T+=0 when H0 is true is 0.0039.P-value = 0.0039 < 0.05 Table A.3

Step 4: Decision Reject H0

Step 5: Conclusion Enough evidence to support the claim that urokinase therapy

lowers pulmonary artery pressure.

Exercise for small sample EXERCISE 4.4

Piggot et. al* paired 10 psychotic and 10 normal children on the age and gender. They then compared subjects for different in respiratory sinus arrhythmia under conditions of spontaneous and 5-, 10- and 15- second interval breathing. They recorded cardiac rate and respiratory changes simultaneously. Table 4.8 shows the differences in duration of the cardiac respiratory phase following the beginning of aspiratory (psychotics compared to the controls for the respiration). Do these data provide sufficient evidence to indicate a difference between psychotic and normal children? Let What is the P-value for this test?

*Source: Leonard R. Piggot, Albert F.Ax, Jaccqueline L.Bamford and Joanne M.Fetzer, “Respiration Sinus Arrhythmia in Psychotic Children, “Psychophysiology, 10(1973), 401-414; copyright 1973, The society for psycho physiological Research; reprinted with permission of the publisher.

Pair 1 2 3 4 5 6 7 8 9 10

Psychotic

(x)

1.74 1.44 2.12 1.80 2.00 2.70 1.96 1.46 1.82 1.40

Control (y) 2.46 1.88 2.38 1.94 2.14 1.60 1.96 1.82 1.80 1.84

Duration of cardiac acceleration, seconds, timed respiration means for 15-second interval breathing Table 4.8

Answer

Case 1 2 3 4 5 6 7 8

Before treatment

(x)

109 57 53 57 68 72 51 65

After treatment

( y)

56 44 55 40 62 46 49 41

Answer

Large Sample Approximation

When n larger than 30.

Exercise for large sampleExercise 4.16

Heiman et al* randomly placed one member of each of thirty two matched pairs of students reading below grade level in a supplementary reading program. Subjects were matched nearly as possible on the discrepancy between their reading level and their current grade level. The supplementary program consisted of a point system to reward attention to and identification of letter and word combinations. Table 4.25 derived from author’s results, shows the differences between the scores made by the subjects on a reading test after and before the program. Use the procedure based on the Wilcoxon test to construct a 95% confidence interval for the median difference.*Source: Julia R. Heiman, Mark J. Fisher, and Alan O. Ross, “ A Supplementary Behavioral Program to Improve Deficient reading Performance,” J. Abnormal Child Psychol. 1(1973), 390-399; published by Plenum Publishing Corporation New York.

Difference in reading test score made by thirty two matched pairs of subject, one member of which was assigned to an experimental program and the other to a control group ( before score subtracted from the score)

Pair 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Experimental (X) 0.5 1.0 0.6 0.1 1.3 0.1 1.0 0.2 0.4 0.6 0.7 0.9 1.2 0.8 0.5 1.4

Control (Y) 0.8 1.1 -0.1 0.3 0.2 1.5 1.3 0.6 1.2 0.3 0.8 1.0 0.5 1.6 0.1 0.5

Pair 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Experimental (X) 0.2 1.3 0.6 0.4 0.3 0.7 1.0 1.1 1.3 1.2 0.2 0.4 0.7 0.8 0.1 1.0

Control (Y) 1.2 0.2 0.8 1.1 1.1 1.6 0.4 1.16 0.5 0.5 0.8 0.9 1.1 1.1 0.3 0.9

Answer

http://youtu.be/r0oxMfmmrOE