Answers to selected problems in chapters 18, 19

Simple and Multiple Regression
Chapter 18 Simple Linear Regression
(Data file lowbwt.sav (SPSS format) is in the CD that came with the book)
9.
a)
90
80
70
RESPONSE OF SYSTOLIC BLOOD
PRESSURE VERSUS GESTATIONAL AGE
systolic blood pressure
60
50
40
30
20
10
22
24
26
28
30
32
34
36
gestational age
b) If using SPSS the steps would be Analyze -> Regression -> Linear and select the sbp
(systolic blood pressure) as the dependent variable and gestage (gestational age) as the
independent variable. Click on save button and check on save the unstandardized predicted
values and unstandardized residuals, and also check the prediction intervals for mean and
individual, and click Continue and click OK. You should see a list of output tables in the
SPSS output window.
Coefficients a
Model
1
(Constant)
gestational age
Unstandardized
Coefficients
B
Std. Error
Standardiz
ed
Coefficient
s
Beta
10.552 12.651 .834 .406
1.264 .436 .281 2.898 .005
t
Sig.
a. Dependent Variable: systolic blood pressure
Slope: 1.264
y-intercept of line: 10.552
sbp = 10.552 + 1.264*gestage
The rate of increasing in systolic blood pressure is 1.264 for per unit (week) increases in
gestational age.
1

Simple and Multiple Regression
c) Ho: β = 0
H1: β ≠ 0
p-value is equal to .005, which is greater than .05 significance level, we reject the null
hypothesis. There is significant linear relation between the response (sbp) and the explanatory
variables (gestational age).
d) sbp = 10.552 + 1.264*gestage,
gestage = 31 => sbp = 10.552 + 1.264*31 = 49.736
e) The 95% prediction interval for the mean systolic blood pressure is:
(46.90159, 52.59409) This can be found in the data editor and in the case that has
gestational age of 31.
f) The predicted systolic blood pressure for the child is =10.552 + 1.264*31= 49.736.
g) The 95% prediction interval for the new value of systolic blood pressure is:
(27.73488, 71.76080) This can be found in the data editor and in the case that has
gestational age of 31.
h) The coefficient of determination is .079 that means about 8 percent of variability in the
variation of systolic blood pressure can be explained by gestational age. The variability of
error seems pretty uniformly spread throughout the range of the predicted values. Therefore
the regression model seems to fit the data.
There is no evidence that the assumption of homoscedasticity has been violated or that a
transformation of either the response or the explanatory variable is necessary.
Model Summary b
Model
1
R
R Square
Adjusted R
Square
Std. Error of
the Estimate
.281 a .079 .070 11.00
a. Predictors: (Constant), gestational age
b. Dependent Variable: systolic blood pressure
50
40
30
20
10
Unstandardized Residual
0
-10
-20
-30
38
40
42
44
46
48
50
52
54
56
Unstandardized Predicted Value
2

Simple and Multiple Regression
Chapter 19
Multiple Regression
8.
a)
90
80
response of systolic blood
pressure versus five-minute apgar
70
systolic blood pressure
60
50
40
30
20
10
-2
0
2
4
6
8
10
apgar score at five minutes
It seems like there is linear but weak relationship between these two variables.
b) apgar5 in SPSS file represents apgar score at five minutes. Do the same thing SPSS as the
exercise in chapter 18, except to include one more variable apgar5 in the independent variable
list.
Model
1
(Constant)
gestational age
apgar score at five minutes
a. Dependent Variable: systolic blood pressure
Sbp =9.803+1.185*gestage + .448*apgar5
Coefficients a
Unstandardized
Coefficients
B
Std. Error
Standardiz
ed
Coefficient
s
Beta
9.803 12.663 .774 .441
1.185 .442 .263 2.678 .009
.488 .461 .104 1.057 .293
The 1.185 means that rate of increasing in systolic blood pressure is 1.185 for per unit (week)
increases in gestational age.
The .448 means that rate of increasing in systolic blood pressure is .448 for per unit increases
in apgar score.
t
Sig.
3

Simple and Multiple Regression
c) Estimated mean systolic blood pressure is
= 9.803+1.185*31 + .448*7
= 49.674
d) The 95% prediction interval for the systolic blood pressure at gestage = 31 and apgar5 = 7 is:
(47.07655, 52.81469) See data editor for a case that gestage = 31 and apgar5 = 7.
e) H0: β
2
= 0
H1: β2
≠ 0
Refer to the SPSS table in question b, p-value is equal to .293 which is greater than the
significance level .05, again we fail to reject the null hypothesis. This means the apgar score
is not a significant factor in predicting the systolic blood pressure.
f)
Model
1
R
Model Summary b
R Square
Adjusted R
Square
Std. Error of
the Estimate
.299 a .089 .071 10.99
a. Predictors: (Constant), apgar score at five minutes,
gestational age
b. Dependent Variable: systolic blood pressure
R square is increased from .079 (model without gestage - table in f) to .089(model with
gestage -table in d)
g) The graph shows that there is no evidence for to say that the assumption of homoscedasticity
has been violated or that a transformation of either the response or the explanatory variable is
necessary.
40
30
20
10
Unstandardized Residual
0
-10
-20
-30
30
40
50
60
Unstandardized Predicted Value
4