The Will-Coxon Rank Sum Test Lecture 52

 The Wilcoxon Rank Sum Test

 Lecture 52

The Wilcoxon signed-rank test is applicable when the observations of two samples are dependent, meaning matched or paired. The Wilcoxon's rank sum test is an improvement over the Wilcoxon's signed rank test. The Wilcoxon rank-sum test is a nonparametric counterpart of the student’s t two independent samples test or non-matched paired. The Wilcoxon rank sum test is similar to the Mann-Whitney U test. Let’s review the student’s two-sample t-test assumptions for comparing two population means:

i. The observations of both samples are independent.

ii. The sample populations have identical variances.

iii. The sampled populations follow normal distribution.

The Wilcoxon's rank sum test is used when the above assumptions are not met. The Wilcoxon rank sum test can be used to test the null hypothesis that the medians of the two populations are identical or the two populations have the same distribution.

Procedure

Step 1: Combine and arrange the observations of both samples in ascending order of magnitude.

Step 2: Assign ranks to the arranged observations in step 1.

Step 3: Calculate the sum of ranks denoted by R and assign it to the smaller sample.

The test statistic is denoted by R, and reject H₀ if R ≤ lower table value OR R ≥ upper table value. The upper pair of the table values is used for a two-tailed test, and the lower pair of the table values is for a one-tailed test.

Step 4: If the sample sizes are large, R is approximately normally distributed with the mean and standard deviation given below:

The test statistic is given by;

Example 13.11: The two companies, A and B, manufacture tubeless tyres. Two independent random samples of about the length of the life of tubeless tyres are measured in 1000 kilometres. The lengths of the lives of two companies are given below:

Manufacture A	29	27	23	30
Manufacture B	24	37	35	19	40	32

Use Wilcoxon rank sum to test if there is any difference in the length of tubeless life of the two types of tyres.

Solution:

i. State null and alternative hypothesis

H0: M1 = M2 vs. H1: M1 ≠ M2 

ii. The significance level; α = 0.05

iii. The test statistic: The sample sizes are small; then R is used as the test statistic.

iv. Reject H0, if R ≤ 12 OR R ≥ 32

v. Computation:

Arranged the observations of both samples combined and assigned ranks. If sample 1 is smaller, then add the ranks of sample 1.

R = 2 + 4 + 5 + 6 = 17 

vi. Remarks: The R (sum of ranks assigned to a smaller sample) falls in the acceptance region; the sample data does not provide sufficient evidence to reject the null hypothesis. Thus, it is concluded that the medians of both types of tyres are identical.

Example 13.12: An agriculture researcher claims that the local farmers' collected potatoes have lower producing ability than the newly developed hybrid variety of potato. To check the claim, two independent random samples of sizes 13 and 16 of production are selected. The collected data of the two samples are given below:

Local	26	25	38	33	42	40	44	26	25	43	35	48	37
Hybrid	44	30	34	47	35	46	35	47	48	34	32	42	43	49	46	47

Test the null hypothesis that the population medians are equal against the alternative that M1 < M2.

Solution:

i. State null and alternative hypothesis

H0: M1 = M2 vs. H1: M1 < M2 

ii. The significance level; α = 0.05

iii. The test statistic: 

iv. Reject H0, When z < -1.645

v. Computation: Arrange both samples combined in ascending order of magnitude and assign ranks. Assigned average ranks to tied observations.

R1 = 1.5 + 1.5 + 3.5 + 3.5 + 7 + 10 + 13 + 14 + 15 + 16 + 18 + 20.5 + 27 = 152.5

vi. Remarks: The calculated z value falls in the rejection region; the sample data does not provide sufficient evidence to accept the null hypothesis of the equality of medians. Thus, it is concluded that the yield capabilities of the local are less than those of the hybrid.

Example 13.3: It is claimed that the working women spent more time on Saturday visiting than on Sunday in the shopping mall. To check the claim, a random sample of the time they spent in the shopping mall, nearest to minutes, is given below:

Sat Day	35	48	63	49
Sun Day	29	49	105	35	60	69

Test the null hypothesis that the population medians of the time spent in the shopping mall on Saturday are more than the time spent on Sunday.

Solution: Let M1 and M2 be the medians of time spent in the shopping mall on Saturday and Sunday, respectively.

i. State null and alternative hypothesis

H0: M1 = M2 vs. H1: M1 > M2 

ii. The significance level; α = 0.05

iii. The test statistic: As the sample sizes are small, the test statistic is R.

iv. Reject H0, if R ≤  13 OR R ≥ 31

v. Computation:

R = 2.5 + 4 + 5.5 + 8 = 20

vi. Remarks: The computed R value in the acceptance region; the sample does not provide sufficient evidence to accept H0. Thus, it is concluded that population medians of the time spent in the shopping mall on Saturday are equal to the time spent on Sunday.

• Read More: Median Test