Question

Neurofibromatosis Type 1 (NF1) is a human genetic disorder. As well as physical symptoms, affected children often suffer from impaired cognition and learning. A cognitive task that involves solving a puzzle is administered to a group of children. For each child the time taken (in seconds) to solve the task is recorded. It is not known whether there is any suitable parametric model for the times so we will investigate non-parametric methods.

Carry out the Wilcoxon signed-rank test on these data to test the null hypothesis that the mean time to solve the puzzle for children with NF1 is the same as for healthy controls.

d) Calculate the value of the test statistic and give the approximate normal distribution of the test statistic under the null hypothesis.

e) Calculate the p-value for the test assuming a two-sided alternative hypothesis.

f) What do you conclude about the time taken by children with NF1 to solve the puzzle compared to healthy controls?

g) Describe in a few sentences how you would calculate a 95% confidence interval for the mean time without assuming any particular parametric model for the data. You do not need to calculate the interval.

The data below are the observed times ?? taken by 57 children with NF1.

ID	yi
1	51
2	60
3	75
4	43
5	92
6	72
7	49
8	39
9	62
10	127
11	51
12	75
13	69
14	59
15	25
16	58
17	95
18	63
19	91
20	63
21	32
22	50
23	108
24	41
25	93
26	43
27	74
28	50
29	55
30	60
31	62
32	91
33	79
34	71
35	85
36	86
37	78
38	100
39	146
40	62
41	134
42	41
43	40
44	51
45	68
46	59
47	59
48	38
49	66
50	79
51	111
52	69
53	68
54	110
55	69
56	62
57	91

Answer 1

Solution :

d)

H0: The median difference is zero versus

H1: The median difference is not zero α=0.05

Wilcoxon's signed-rank test statistic is given by

W =  W+ -  W-

where,

W+ =  sum of the observations with positive ranks

W- =  sum of the observations with negative ranks

In the given data,

Median = 66

ID	yi	yi-Median
1	51	-15
2	60	-6
3	75	9
4	43	-23
5	92	26
6	72	6
7	49	-17
8	39	-27
9	62	-4
10	127	61
11	51	-15
12	75	9
13	69	3
14	59	-7
15	25	-41
16	58	-8
17	95	29
18	63	-3
19	91	25
20	63	-3
21	32	-34
22	50	-16
23	108	42
24	41	-25
25	93	27
26	43	-23
27	74	8
28	50	-16
29	55	-11
30	60	-6
31	62	-4
32	91	25
33	79	13
34	71	5
35	85	19
36	86	20
37	78	12
38	100	34
39	146	80
40	62	-4
41	134	68
42	41	-25
43	40	-26
44	51	-15
45	68	2
46	59	-7
47	59	-7
48	38	-28
49	66	0
50	79	13
51	111	45
52	69	3
53	68	2
54	110	44
55	69	3
56	62	-4
57	91	25

W+ = 2506

W- = 1428

W = 1078

The approximate normal distribution has an expected value of 0 and a variance of

.

= 63365

e)

p-value = P(|W| > 1078) = 1.848324e-05 ... from normal probability table.

The calculations are as shown below:

f)

As p-value < 0.05, we can reject H0. Hence, the time taken by children with NF1 to solve the puzzle compared to healthy controls is significantly different.

g)

If we have to use no parametric distribution, we can find the 2.5th and the 97.5th quantile to find a 95% confidence interval for the mean time.

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