Question 1.
The data set lowbwt contains information for the sample of 100 low birth weight infants born in Boston, Massachusetts [3] (Appendix B, Table B.7).
The variable grmhem is a dichotomous random variable indicating whether an infant experienced a germinal matrix hemorrhage. The value 1 indicates that a hemorrhage occurred and 0 that it did not. The infants' five-minute apgar scores are saved under the name apgar5, and indicators of toxemia-where 1 represents a diagnosis of toxemia during pregnancy for the child's mother and 0 no such diagnosis-under the variable name tox.
a) Using germinal matrix hemorrhage as the response, fit a logistic regression model of the form ln[p^/(1-p^)] = α^ + β^1x1 Where X1 is five-minute apgar score. Interpret β1 the estimated coefficient of Apgar score.
b) If a particular child has a five-minute apgar score of 3, what is the predicted probability that this child will experience a brain hemorrhage? What is the probability if the child's score is 7?
c) At the 0.05 level of significance, test the null hypothesis that the population parameter PI is equal to 0. What do you conclude?
d) Now fit the regression model.
ln[p^/(1-p^)] = α^ + β^1x1 Where x2 represents toxemia status.
Interpret β2 the estimated coefficient of toxemia.
e) For a child whose mother was diagnosed with toxemia during pregnancy, what is the predicted probability of experiencing a germinal matrix hemorrhage? What is the probability for a child whose mother was not diagnosed with toxemia?
f) What are the estimated odds of suffering a germinal matrix hemorrhage for children whose mothers were diagnosed with toxemia relative to children whose mothers were not?
g) Construct a 95% confidence interval for the population odds ratio. Does this interval contain the value 1? What does this tell you?
TABLE- Data set lowbwt;
Variables sbp, sex, tox, grmhem, gestage, and apgar5
|
43
|
Male
|
No
|
No
|
29
|
7
|
62
|
Female
|
No
|
No
|
27
|
7
|
67
|
Female
|
No
|
No
|
27
|
8
|
|
51
|
Male
|
No
|
No
|
31
|
8
|
59
|
Female
|
No
|
No
|
27
|
8
|
40
|
Female
|
No
|
Yes
|
31
|
8
|
|
42
|
Female
|
No
|
No
|
33
|
0
|
36
|
Male
|
No
|
No
|
27
|
9
|
48
|
Female
|
No
|
No
|
26
|
8
|
|
39
|
Female
|
No
|
No
|
31
|
8
|
47
|
Female
|
No
|
No
|
32
|
8
|
36
|
Male
|
No
|
No
|
27
|
5
|
|
48
|
Female
|
Yes
|
No
|
30
|
7
|
45
|
Male
|
No
|
Yes
|
31
|
2
|
44
|
Male
|
No
|
No
|
27
|
6
|
|
31
|
Male
|
No
|
Yes
|
25
|
0
|
62
|
Female
|
No
|
Yes
|
28
|
5
|
53
|
Female
|
Yes
|
No
|
35
|
9
|
|
31
|
Male
|
Yes
|
No
|
27
|
7
|
75
|
Male
|
Yes
|
No
|
30
|
7
|
45
|
Female
|
Yes
|
No
|
28
|
6
|
|
40
|
Female
|
No
|
No
|
29
|
9
|
44
|
Male
|
No
|
No
|
29
|
0
|
54
|
Male
|
No
|
No
|
30
|
8
|
|
57
|
Female
|
No
|
No
|
28
|
6
|
39
|
Male
|
No
|
No
|
28
|
8
|
44
|
Male
|
Yes
|
No
|
31
|
2
|
|
64
|
Female
|
Yes
|
No
|
29
|
9
|
48
|
Female
|
No
|
Yes
|
31
|
7
|
42
|
Male
|
No
|
No
|
30
|
5
|
|
46
|
Female
|
No
|
No
|
26
|
7
|
43
|
Female
|
Yes
|
No
|
27
|
6
|
50
|
Female
|
No
|
No
|
27
|
0
|
|
47
|
Female
|
No
|
Yes
|
30
|
6
|
19
|
Female
|
No
|
Yes
|
25
|
4
|
48
|
Female
|
No
|
No
|
25
|
5
|
|
63
|
Female
|
No
|
No
|
29
|
8
|
63
|
Male
|
No
|
No
|
30
|
7
|
29
|
Female
|
No
|
Yes
|
25
|
5
|
|
56
|
Female
|
No
|
No
|
29
|
1
|
42
|
Male
|
No
|
No
|
28
|
6
|
30
|
Female
|
No
|
Yes
|
26
|
2
|
|
49
|
Male
|
No
|
No
|
29
|
8
|
44
|
Female
|
No
|
No
|
28
|
9
|
36
|
Female
|
No
|
No
|
29
|
0
|
|
87
|
Male
|
No
|
No
|
29
|
7
|
25
|
Female
|
No
|
No
|
25
|
8
|
44
|
Female
|
No
|
No
|
29
|
0
|
|
46
|
Female
|
No
|
No
|
29
|
8
|
26
|
Female
|
No
|
No
|
23
|
8
|
46
|
Female
|
Yes
|
No
|
34
|
9
|
|
66
|
Female
|
No
|
No
|
33
|
8
|
27
|
Male
|
No
|
No
|
27
|
9
|
51
|
Male
|
Yes
|
No
|
30
|
4
|
|
42
|
Female
|
Yes
|
No
|
33
|
8
|
35
|
Male
|
No
|
No
|
28
|
8
|
51
|
Male
|
No
|
No
|
29
|
5
|
|
52
|
Female
|
No
|
No
|
29
|
7
|
40
|
Male
|
No
|
No
|
27
|
7
|
43
|
Male
|
Yes
|
No
|
33
|
7
|
|
51
|
Male
|
No
|
No
|
28
|
7
|
44
|
Female
|
No
|
No
|
27
|
6
|
48
|
Male
|
No
|
No
|
30
|
5
|
|
47
|
Female
|
No
|
No
|
30
|
9
|
66
|
Male
|
No
|
No
|
26
|
8
|
52
|
Male
|
No
|
No
|
29
|
8
|
|
54
|
Male
|
No
|
No
|
27
|
4
|
59
|
Female
|
No
|
No
|
25
|
3
|
43
|
Male
|
No
|
No
|
24
|
6
|
|
64
|
Male
|
No
|
No
|
33
|
9
|
24
|
Female
|
No
|
No
|
23
|
7
|
42
|
Male
|
Yes
|
No
|
33
|
8
|
|
37
|
Female
|
No
|
No
|
32
|
7
|
40
|
Female
|
No
|
Yes
|
26
|
3
|
48
|
Male
|
No
|
Yes
|
25
|
5
|
|
36
|
Male
|
Yes
|
No
|
28
|
3
|
49
|
Female
|
No
|
No
|
24
|
5
|
49
|
Female
|
Yes
|
No
|
32
|
8
|
|
45
|
Female
|
No
|
Yes
|
29
|
7
|
53
|
Male
|
Yes
|
No
|
29
|
9
|
62
|
Male
|
Yes
|
No
|
31
|
7
|
|
39
|
Male
|
No
|
No
|
28
|
7
|
45
|
Female
|
No
|
No
|
29
|
9
|
45
|
Male
|
No
|
No
|
31
|
9
|
|
29
|
Female
|
No
|
No
|
29
|
4
|
50
|
Male
|
No
|
Yes
|
27
|
8
|
51
|
Female
|
Yes
|
Yes
|
31
|
6
|
|
61
|
Female
|
No
|
No
|
30
|
3
|
64
|
Male
|
No
|
No
|
30
|
7
|
52
|
Male
|
No
|
No
|
29
|
8
|
|
53
|
Male
|
No
|
No
|
31
|
7
|
48
|
Female
|
No
|
No
|
30
|
6
|
47
|
Male
|
Yes
|
No
|
32
|
5
|
|
64
|
Female
|
No
|
No
|
30
|
7
|
48
|
Female
|
No
|
Yes
|
32
|
4
|
40
|
Female
|
Yes
|
No
|
33
|
8
|
|
35
|
Female
|
No
|
No
|
31
|
6
|
58
|
Female
|
Yes
|
No
|
33
|
7
|
50
|
Female
|
No
|
No
|
28
|
7
|
|
34
|
Male
|
No
|
No
|
29
|
9
|
|
|
|
|
|
|
|
|
|
|
|
|
Question 2. Measurements of length and weight for a sample of 20 low birth weight infants are contained in the data set twenty (Appendix B, Table B.23).
The length measurements are saved under the variable name length, and the corresponding birth weights underweight.
(a) Construct a two-way scatter plot of birth weight versus length for the 20 infants in the sample. Without doing any calculations, sketch your best guess for the least-squares regression line directly on the scatter plot.
(b) Now compute the true least-squares regression line. Draw this line on the scatter plot. Does the actual least-squares line concur with your guess? Based on the two-way scatter plot, it is clear that one point lies outside the range of the remainder of the data. This point corresponds to the ninth infant in the sample of size 20. To illustrate the effect that the outlier has on the model, remove this point from the data set.
(c) Compute the new least-squares regression line based on the sample of size 19, and sketch this line on the original scatter plot. How does the least-squares line change? In particular, comment on the values of the slope and the intercept.
(d) Compare the coefficients of determination (R2) and the standard deviations from regression (sylx) for the two least-squares regression lines. Explain how these values changed when you removed the outlier from the original data set. Why did they change?
|
Length
|
weight
|
|
41
|
1360
|
|
40
|
1490
|
|
38
|
1490
|
|
38
|
1180
|
|
38
|
1200
|
|
32
|
680
|
|
33
|
620
|
|
38
|
1060
|
|
30
|
1320
|
|
34
|
830
|
|
32
|
880
|
|
39
|
1130
|
|
38
|
1140
|
|
39
|
1350
|
|
37
|
950
|
|
39
|
1220
|
|
38
|
980
|
|
42
|
1480
|
|
39
|
1250
|
|
38
|
1250
|