The Wilcoxon rank sum test.
|
Value |
Medical |
Surgical |
Total |
Range of ranks |
Average rank |
|---|---|---|---|---|---|
|
3 |
0 |
1 |
1 |
1 |
1 |
|
4 |
1 |
1 |
2 |
2--3 |
2.5 |
|
5 |
3 |
1 |
4 |
4--7 |
5.5 |
|
6 |
0 |
4 |
4 |
8--11 |
9.5 |
|
7 |
3 |
0 |
3 |
12--14 |
13 |
|
8 |
2 |
0 |
2 |
15--16 |
15.5 |
|
9 |
0 |
1 |
1 |
17 |
17 |
|
10 |
0 |
1 |
1 |
18 |
18 |
|
11 |
0 |
3 |
3 |
19--21 |
20 |
|
12 |
0 |
2 |
2 |
22--23 |
22.5 |
|
14 |
0 |
2 |
2 |
24--25 |
24.5 |
|
Total |
9 |
16 |
25 |
|
|
R1=89, from Table 12 it follows that p>0.05 and there is no significant difference between the median white count of these two groups of patients.
9.16 A nonparametric test would be useful because the distribution of duration of effusion is very skewed and the assumptions about normality of the underlying distribution are unlikely to hold.
9.17 The Wilcoxon signed rank test should be used here because the breast and bottle fed babies are matched on age, sex, socioeconomic status, and type of mediations and thus form two paired samples.
9.18 Apply the signed rank test to these data. First compute the difference in duration of effusion between the breast and bottle fed babies in the matched paris.
Second, order the differences by absolute value.
Third, count the number of people with the same absolute value and assign an average rank to each absolute value.
Show that you can use the normal approximation
Compute the ranks sum:
R1=215, E(R1)=138, Var(R1)=1079, sd(R1)=32.85
Tstatistic=2.329
pvalue=0.01
Breast fed babies have significantly shorter effusion s than bottle fed babies do
9.19 The paired t-test
9.20 t statistic = 3.72 with 16 degrees of freedom
9.21 The Wilcoxon signed rank test
9.22 Tstatistic=2.87 (use normal approximation method). p value =0.004
9.23 A significant reduction in blood pressure after adopting the diet was found using both the paired t test in 9.20 and the signed test in 9.22. The distribution of blood pressure difference scores is probably close enough to being normal to allow for a valid use of the t test in this instance.
9.24 The Wilcoxon signed rank test should be used since we are comparing two paired samples.
9.25 First subtract the random zero readings from the standard cuff readings and order the differences by absolute value.
Then compute R1=33.5, E(R1)=68, Var(R1)=372, Tstatistic=1.76. There is no significant difference in mean blood pressure between the two types of machines, although there is a trend for the random zero readings to be lower.
9.26 Use Wilcoxon signed rank test.
9.38 Use Wilcoxon signed rank test.
9.39 First we calculate the itching score for the active eye minus the itching score for the placebo eye for each patient. Rank all patients with non-zero difference scores by the absolute value of the difference scores.
R1=3. (Be careful b/c we are concerned with Active-Placebo and need to refer to table 7.6 in conjunction with table 7.7!) Since there are <16 non-zero difference scores, we use the small sample test. Based on Table 11, using 5% signficance, the critical values for n=8 are 3 and 33. Since R1=3 <= 3 (less than or equal to the lower cut-off), it follows that p<.05. Therefore there is a significant difference between the itching scores for active and placebo treated eyes.