Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences
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HIV DNA Blood HIV Co-Culture HIV DNA Rectal HIV RNA Rectal
(Y) Blood (X
1
) Tissue (X
2
) Tissue (X
3
)
51 .09 31 36
73 .06 158 5
47 .08 773 5
48 .12 545 67
16 .03 5 5
Source: Peter A. Anton, M.D. Used with permission.
(a) Compute the simple correlation coefficients between all possible pairs of variables.
(b) Compute the multiple correlation coefficient among the four variables. Test the overall corre-
lation for significance.
(c) Calculate the partial correlations between HIV DNA blood and each one of the other variables
while controlling for the other two. (These are called second-order partial correlation coefficients.)
(d) Calculate the partial correlation between HIV co-culture blood and HIV DNA, controlling for
the other two variables.
(e) Calculate the partial correlation between HIV co-culture blood and HIV RNA, controlling for
the other two variables.
(f) Calculate the partial correlations between HIV DNA and HIV RNA, controlling for the other
two variables.
10.6.2 The following data were obtained on 12 males between the ages of 12 and 18 years (all measure-
ments are in centimeters):
Height Radius Length Femur Length
(Y)(X
1
)(X
2
)
149.0 21.00 42.50
152.0 21.79 43.70
155.7 22.40 44.75
159.0 23.00 46.00
163.3 23.70 47.00
166.0 24.30 47.90
169.0 24.92 48.95
172.0 25.50 49.90
174.5 25.80 50.30
176.1 26.01 50.90
176.5 26.15 50.85
179.0 26.30 51.10
Total 1992.1 290.87 573.85
(a) Find the sample multiple correlation coefficient and test the null hypothesis that
(b) Find each of the partial correlation coefficients and test each for significance. Let for
all tests.
(c) Determine the p value for each test.
(d) State your conclusions.
a = .05
r
y.12
= 0.
EXERCISES 517
10.6.3 The following data were collected on 15 obese girls:
Weight in Lean Body Mean Daily
Kilograms Weight Caloric Intake
(Y)(X
1
)(X
2
)
79.2 54.3 2670
64.0 44.3 820
67.0 47.8 1210
78.4 53.9 2678
66.0 47.5 1205
63.0 43.0 815
65.9 47.1 1200
63.1 44.0 1180
73.2 44.1 1850
66.5 48.3 1260
61.9 43.5 1170
72.5 43.3 1852
101.1 66.4 1790
66.2 47.5 1250
99.9 66.1 1789
Total 1087.9 741.1 22739
(a) Find the multiple correlation coefficient and test it for significance.
(b) Find each of the partial correlation coefficients and test each for significance. Let for
all tests.
(c) Determine the p value for each test.
(d) State your conclusions.
10.6.4 A research project was conducted to study the relationships among intelligence, aphasia, and
apraxia. The subjects were patients with focal left hemisphere damage. Scores on the following
variables were obtained through the application of standard tests.
The results are shown in the following table. Find the multiple correlation coefficient and
test for significance. Let and find the p value.
Subject YX
1
X
2
X
3
X
4
1 66 7.6 7.4 2296.87 2
2 78 13.2 11.9 2975.82 8
3 79 13.0 12.4 2839.38 11
a = .05
X
4
= severity of aphasia
X
3
= lesion volume 1pixels2
X
2
= constructive apraxia
X
1
= ideomotor apraxia
Y = intelligence
a = .05
518 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION
(Continued)
Subject YX
1
X
2
X
3
X
4
4 84 14.2 13.3 3136.58 15
5 77 11.4 11.2 2470.50 5
6 82 14.4 13.1 3136.58 9
7 82 13.3 12.8 2799.55 8
8 75 12.4 11.9 2565.50 6
9 81 10.7 11.5 2429.49 11
10 71 7.6 7.8 2369.37 6
11 77 11.2 10.8 2644.62 7
12 74 9.7 9.7 2647.45 9
13 77 10.2 10.0 2672.92 7
14 74 10.1 9.7 2640.25 8
15 68 6.1 7.2 1926.60 5
10.7 SUMMARY
In this chapter we examine how the concepts and techniques of simple linear regression
and correlation analysis are extended to the multiple-variable case. The least-squares
method of obtaining the regression equation is presented and illustrated. This chapter
also is concerned with the calculation of descriptive measures, tests of significance, and
the uses to be made of the multiple regression equation. In addition, the methods and
concepts of correlation analysis, including partial correlation, are discussed.
When the assumptions underlying the methods of regression and correlation pre-
sented in this and the previous chapter are not met, the researcher must resort to alter-
native techniques such as those discussed in Chapter 13.
SUMMARY OF FORMULAS FOR CHAPTER 10 519
SUMMARY OF FORMULAS FOR CHAPTER 10
Formula
Number Name Formula
10.2.1 Representation of
the multiple
linear regression
equation
10.2.2 Representation
of the multiple
linear regression
equation with
two independent
variables
y
j
= b
0
+ b
1
x
1j
+ b
2
x
2j
+P
j
y
j
= b
0
+ b
1
x
1j
+ b
2
x
2j
+
Á
+ b
k
x
kj
+P
j
(Continued)
520 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION
10.2.3 Random deviation
of a point from a
plane when there
are two
independent
variables
10.3.1 Sum-of-squared
residuals
10.4.1 Sum-of-squares
equation
10.4.2 Coefficient of
multiple
determination
10.4.3 t statistic for
testing hypotheses
about
10.5.1 Estimation equation
for multiple
linear regression
10.5.2 Confidence interval
for the mean of Y
for a given X
10.5.3 Prediction interval
for Y for a given X
10.6.1 Multiple correlation
model
10.6.2 Multiple correlation
coefficient
10.6.3 F statistic for
testing the multiple
correlation
coefficient
10.6.4–10.6.6 Partial correlation
between two
variables (1 and 2)
after controlling
for a third (3)
r
12.3
= 1r
12
- r
13
r
23
2>211 - r
2
13
211 - r
2
23
2
F =
R
2
y.12...k
1 - R
2
y.12...k
#
n - k - 1
k
R
y.12...k
= 2R
y
2
.12...k
=
D
g1y
i
- y
j
N
2
2
g1y
i
- y2
2
=
A
SSR
SST
y
j
= b
0
+ b
1
x
1j
+ b
2
x
2j
+
Á
+ b
k
x
kj
+P
j
y
j
N
; t
11- a>22, n- k - 1
s¿
y
N
j
y
j
N
; t
11- a>22, n- k - 1
s
y
N
j
y
j
N
= b
0
N
+ b
1
N
x
1j
+ b
2
N
x
2j
+
Á
+ b
k
N
x
kj
b
i
t =
b
j
N
- b
jo
s
b
N
i
R
2
y.12...k
=
g1y
j
- y
j
N
2
2
g1y
j
- y2
2
=
SSR
SST
a
P
2
j
=
a
1y
j
- y
j
N
2
2
P
j
= y
j
- b
0
- b
1
x
1j
- b
2
x
2j
SST SSR SSE
a
1y
j
- y2
2
=
a
1y
j
N
- y
2
2
+
a
1y
j
- y
j
N
2
2
REVIEW QUESTIONS AND EXERCISES 521
10.6.7 t statistic for testing
hypotheses about
partial correlation
coefficients
Symbol Key • estimated regression/correlation coefficient x
• regression/correlation coefficient
• model error term
• k number of independent variables
• n sample size
•r
12.3
sample partial correlation coefficient between 1 and 2
after controlling for 3
• R sample correlation coefficient
• R
2
multiple coefficient of determination
• t t statistic
• x
i
value of independent variable at i
• sample mean of independent variable
• y
i
value of dependent variable at i
• sample mean of dependent variable
• estimated y
• z z statistic
y
N
y
x
P
b
x
b
x
N
t = r
y1.2...k
A
n - k - 1
1 - r
2
y1.2..k
REVIEW QUESTIONS AND EXERCISES
1. What are the assumptions underlying multiple regression analysis when one wishes to infer about
the population from which the sample data have been drawn?
2. What are the assumptions underlying the correlation model when inference is an objective?
3. Explain fully the following terms:
(a) Coefficient of multiple determination (b) Multiple correlation coefficient
(c) Simple correlation coefficient (d) Partial correlation coefficient
4. Describe a situation in your particular area of interest where multiple regression analysis would
be useful. Use real or realistic data and do a complete regression analysis.
5. Describe a situation in your particular area of interest where multiple correlation analysis would
be useful. Use real or realistic data and do a complete correlation analysis.
In Exercises 6 through 11 carry out the indicated analysis and test hypotheses at the indicated sig-
nificance levels. Compute the p value for each test.
6. We learned in Example 9.7.1 that the purpose of a study by Kwast-Rabben et al. (A-6) was to
analyze somatosensory evoked potentials (SEPs) and their interrelations following stimulation of
digits I, III, and V in the hand. Healthy volunteers were recruited for the study. Researchers applied
stimulation below-pain-level intensity to the fingers. Recordings of spinal responses were made
with electrodes fixed by adhesive electrode cream to the subject’s skin. Results are shown in the
522 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION
following table for 114 subjects. Use multiple regression to see how well you can predict the peak
spinal latency (Cv) of the SEP for digit I when age (years) and arm length (cm) are the predictor
variables. Evaluate the usefulness of your prediction equation.
Age Arm Length Cv Dig.I Age Arm Length Cv Dig.I Age Arm Length Cv Dig.I
35.07 76.5 13.50 32.00 82.0 16.30 42.08 94.0 17.70
35.07 76.5 13.50 32.00 82.0 15.40 40.09 94.0 17.70
21.01 77.0 13.00 38.09 86.5 16.60 40.09 94.0 17.40
21.01 77.0 13.60 38.09 86.5 16.00 42.09 92.5 18.40
47.06 75.5 14.30 58.07 85.0 17.00 20.08 95.0 19.00
47.06 75.5 14.90 58.07 85.0 16.40 50.08 94.5 19.10
26.00 80.0 15.40 54.02 88.0 17.60 50.08 94.5 19.20
26.00 80.0 14.70 48.10 92.0 16.80 47.11 97.5 17.80
53.04 82.0 15.70 48.10 92.0 17.00 47.11 97.5 19.30
53.04 82.0 15.80 54.02 88.0 17.60 26.05 96.0 17.50
43.07 79.0 15.20 45.03 91.5 17.30 26.05 96.0 18.00
39.08 83.5 16.50 45.03 91.5 16.80 43.02 98.0 18.00
39.08 83.5 17.00 35.11 94.0 17.00 43.02 98.0 18.80
43.07 79.0 14.70 26.04 88.0 15.60 32.06 98.5 18.30
29.06 81.0 16.00 51.07 87.0 16.80 32.06 98.5 18.60
29.06 81.0 15.80 51.07 87.0 17.40 33.09 97.0 18.80
50.02 86.0 15.10 26.04 88.0 16.50 33.09 97.0 19.20
25.07 81.5 14.60 35.11 94.0 16.60 35.02 100.0 18.50
25.07 81.5 15.60 52.00 88.5 18.00 35.02 100.0 18.50
25.10 82.5 14.60 44.02 90.0 17.40 26.05 96.0 19.00
47.04 86.0 17.00 44.02 90.0 17.30 26.05 96.0 18.50
47.04 86.0 16.30 24.05 91.0 16.40 25.08 100.5 19.80
37.00 83.0 16.00 24.00 87.0 16.10 25.06 100.0 18.80
37.00 83.0 16.00 24.00 87.0 16.10 25.06 100.0 18.40
34.10 84.0 16.30 24.00 87.0 16.00 25.08 100.5 19.00
47.01 87.5 17.40 24.00 87.0 16.00 30.05 101.0 18.00
47.01 87.5 17.00 53.05 90.0 17.50 30.05 101.0 18.20
30.04 81.0 14.10 53.05 90.0 17.50 36.07 104.5 18.90
23.06 81.5 14.20 52.06 90.0 18.00 36.07 104.5 19.20
23.06 81.5 14.70 52.06 90.0 17.90 35.09 102.0 21.00
30.04 81.0 13.90 53.04 93.0 18.40 35.09 102.0 19.20
78.00 81.0 17.20 22.04 90.0 16.40 21.01 101.5 18.60
41.02 83.5 16.70 22.04 90.0 15.80 21.01 101.5 18.60
41.02 83.5 16.50 46.07 95.5 18.80 40.00 95.5 20.00
28.07 78.0 14.80 46.07 95.5 18.60 42.09 92.5 18.40
28.07 78.0 15.00 47.00 93.5 18.00 42.08 94.0 18.50
36.05 88.0 17.30 47.00 93.5 17.90 35.04 86.0 16.00
35.04 86.0 15.30 39.05 94.5 17.40 36.05 88.0 16.60
Source: Olga Kwast-Rabben, Ph.D. Used with permission.
7. The following table shows the weight and total cholesterol and triglyceride levels in 15 patients
with primary type II hyperlipoproteinemia just prior to initiation of treatment:
X
1
X
2
Total Cholesterol Triglyceride
Y Weight (kg) (mg/100 ml) (mg/100 ml)
76 302 139
97 336 101
83 220 57
52 300 56
70 382 113
67 379 42
75 331 84
78 332 186
70 426 164
99 399 205
75 279 230
78 332 186
70 410 160
77 389 153
76 302 139
Compute the multiple correlation coefficient and test for significance at the .05 level.
8. In a study of the relationship between creatinine excretion, height, and weight, the data shown in
the following table were collected on 20 infant males:
Creatinine
Excretion
(mg/day) Weight (kg) Height (cm)
Infant YX
1
X
2
1 100 9 72
2 115 10 76
352 6 59
485 8 68
5 135 10 60
658 5 58
790 8 70
860 7 65
945 4 54
10 125 11 83
11 86 7 64
12 80 7 66
13 65 6 61
14 95 8 66
15 25 5 57
16 125 11 81
17 40 5 59
REVIEW QUESTIONS AND EXERCISES 523
(Continued)
Creatinine
Excretion
(mg/day) Weight (kg) Height (cm)
Infant YX
1
X
2
18 95 9 71
19 70 6 62
20 120 10 75
(a) Find the multiple regression equation describing the relationship among these variables.
(b) Compute and do an analysis of variance.
(c) Let and and find the predicted value of Y.
9. A study was conducted to examine those variables thought to be related to the job satisfaction
of nonprofessional hospital employees. A random sample of 15 employees gave the following
results:
Coded Index of
Score on Job Intelligence Personal
Satisfaction Score Adjustment
Test (Y )(X
1
)(X
2
)
54 15 8
37 13 1
30 15 1
48 15 7
37 10 4
37 14 2
31 8 3
49 12 7
43 1 9
12 3 1
30 15 1
37 14 2
61 14 10
31 9 1
31 4 5
(a) Find the multiple regression equation describing the relationship among these variables.
(b) Compute the coefficient of multiple determination and do an analysis of variance.
(c) Let and and find the predicted value of Y.
10. A medical research team obtained the index of adiposity, basal insulin, and basal glucose values
on 21 normal subjects. The results are shown in the following table. The researchers wished to
investigate the strength of the association among these variables.
X
2
= 5X
1
= 10
X
2
= 60X
1
= 10
R
2
524 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION
Index of Basal Insulin Basal Glucose
Adiposity ( U/ml) (mg/100 ml)
YX
1
X
2
90 12 98
112 10 103
127 14 101
137 11 102
103 10 90
140 38 108
105 9 100
92 6 101
92 8 92
96 6 91
114 9 95
108 9 95
160 41 117
91 7 101
115 9 86
167 40 106
108 9 84
156 43 117
167 17 99
165 40 104
168 22 85
Compute the multiple correlation coefficient and test for significance at the .05 level.
11. As part of a study to investigate the relationship between stress and certain other variables, the
following data were collected on a simple random sample of 15 corporate executives.
(a) Find the least-squares regression equation for these data.
(b) Construct the analysis of variance table and test the null hypothesis of no relationship among
the five variables.
(c) Test the null hypothesis that each slope in the regression model is equal to zero.
(d) Find the multiple coefficient of determination and the multiple correlation coefficient. Let
and find the p value for each test.
Annual
Number of Years Salary
Measure of Measure of in Present (1000)
Stress (Y) Firm Size ( X
1
) Position (X
2
)(X
3
) Age ( X
4
)
101 812 15 $30 38
60 334 8 20 52
10 377 5 20 27
27 303 10 54 36
a = .05
M
REVIEW QUESTIONS AND EXERCISES 525
(Continued)
Annual
Number of Years Salary
Measure of Measure of in Present (1000)
Stress (Y) Firm Size ( X
1
) Position (X
2
)(X
3
) Age ( X
4
)
89 505 13 52 34
60 401 4 27 45
16 177 6 26 50
184 598 9 52 60
34 412 16 34 44
17 127 2 28 39
78 601 8 42 41
141 297 11 84 58
11 205 4 31 51
104 603 5 38 63
76 484 8 41 30
For each of the studies described in Exercises 12 through 16, answer as many of the following
questions as possible:
(a) Which is more relevant, regression analysis or correlation analysis, or are both techniques
equally relevant?
(b) Which is the dependent variable?
(c) What are the independent variables?
(d) What are the appropriate null and alternative hypotheses?
(e) Which null hypotheses do you think were rejected? Why?
(f) Which is the more relevant objective, prediction or estimation, or are the two equally relevant?
Explain your answer.
(g) What is the sampled population?
(h) What is the target population?
(i) Which variables are related to which other variables? Are the relationships direct or inverse?
(j) Write out the regression equation using appropriate numbers for parameter estimates.
(k) What is the numerical value of the coefficient of multiple determination?
(l) Give numerical values for any correlation coefficients that you can.
12. Hashimoto et al. (A-7) developed a multiple regression model to predict the number of visits to emer-
gency rooms at Jikei University hospitals in Tokyo for children having an asthma attack. The
researchers found that the number of visits per night increased significantly when climate conditions
showed a rapid decrease from higher barometric pressure, from higher air temperature, and from higher
humidity, as well as lower wind speed. The final model demonstrated that 22 percent of the variation
in the number of visits was explained by variation in the predictor variables mentioned above with
eight other significant climate variables.
13. Correlation was one of many procedures discussed in a study reported by Stenvinkel et al. (A-8).
In a cohort of 204 subjects with end-stage renal disease, they found no significant correlations
between log plasma adiponectin levels and age and no significant correlation between log plasma
adiponectin and glomerular filtration rate.
526 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION