Question

Suppose that you are interested in examining the effects of the transition from fetal to postnatal circulation among the premature infants. For each of the 14 healthy newborns, respiratory rate is measured at two different times - once when the infant is less than 15 days old, and again when he or she is more than 25 days old.

Respiratory Rate
Subject, Time 1, Time 2
1, 62, 46
2, 35, 42
3, 38, 40
4, 80, 42
5, 48, 36
6, 48, 46
7, 68, 45
8, 26, 40
9, 48, 42
10, 27, 40
11, 43, 46
12, 67, 31
13, 52, 44
14, 88, 48

a) What is the population under study?

b) Using the Wilcoxon signed-rank test, evaluate the null hypothesis that the median difference in respiratory rates for the two times is equal to 0

Answer 1

(a) The population under study is premature infants. The study is to examine the effect of transition from fatal to postnatal circulation and the same is tested by respiratory rates.  

(b) WILCOXON signed rank test is used as the sample size is small and also it is non-parametric test.

Step 1:

Ho: Median difference = 0

Ha: Median difference 0

The null hypothesis states that the median difference in respiratory rates for the two times is equal to 0.

Step 2: Create below Table

Find the difference between Time 1 and Time 2 and then take Absolute difference.

Subject	Time 1	Time 2	Difference	Abs. Difference	Sign
1	62	46	16	16	1
2	35	42	-7	7	-1
3	38	40	-2	2	-1
4	80	42	38	38	1
5	48	36	12	12	1
6	48	46	2	2	1
7	68	45	23	23	1
8	26	40	-14	14	-1
9	48	42	6	6	1
10	27	40	-13	13	-1
11	43	46	-3	3	-1
12	67	31	36	36	1
13	52	44	8	8	1
14	88	48	40	40	1

Arrange the absolute differences in ascending order and Rank the data.

Subject	Time 1	Time 2	Abs. Difference	Rank	Sign
3	38	40	2	1.5	-1
6	48	46	2	1.5	1
11	43	46	3	3	-1
9	48	42	6	4	1
2	35	42	7	5	-1
13	52	44	8	6	1
5	48	36	12	7	1
10	27	40	13	8	-1
8	26	40	14	9	-1
1	62	46	16	10	1
7	68	45	23	11	1
12	67	31	36	12	1
4	80	42	38	13	1
14	88	48	40	14	1

The sum of positive ranks is:

W+=1.5+4+6+7+10+11+12+13+14=78.5

and the sum of negative ranks is:

W−=1.5+3+5+8+9=26.5

Hence, the test statistic T or W Statistics is =min⁡{W+,W−} = min⁡{78.5,26.5} = 26.5

This is a two tailed test as median difference in the respiratory rates can be more or less than 0.

Sample size = n = 14

The value of W is 26.5.

Level of significance is not mentioned. Hence we will check the hypothesis at both level of significance 0.05 and 0.01

Level of significance = 0.01

The critical value for W at N = 14 (p < .01) is 12

So here also W stat = 26.5 is greater than W critical at = 0.01. We fail to reject the null hypothesis.

There is not enough evidence to claim that the population median of differences is different than 0, hence we fail to reject the Null hypothesis.

Level of significance = 0.05

The critical value for W at N = 14 (p < .05) is 21

So here W stat = 26.5 is greater than W critical at = 0.05. We fail to reject the null hypothesis.