Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences

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of variance, which tests the significance of the regression mean square. The

strength of the relationship between two variables under the correlation model is

assessed by testing the null hypothesis that there is no correlation in the popula-

tion. If this hypothesis can be rejected we may conclude, at the chosen level of

signiﬁcance, that the two variables are correlated.

5. Use the equation. Once it has been determined that it is likely that the regression

equation provides a good description of the relationship between two variables, X

and Y, it may be used for one of two purposes:

a. To predict what value Y is likely to assume, given a particular value of X, or

b. To estimate the mean of the subpopulation of Y values for a particular value

of X.

This necessarily abridged treatment of simple linear regression and correlation may

have raised more questions than it has answered. It may have occurred to the reader, for

example, that a dependent variable can be more precisely predicted using two or more inde-

pendent variables rather than one. Or, perhaps, he or she may feel that knowledge of the

strength of the relationship among several variables might be of more interest than knowl-

edge of the relationship between only two variables. The exploration of these possibilities

is the subject of the next chapter, and the reader’s curiosity along these lines should be at

least partially relieved.

For those who would like to pursue further the topic of regression analysis a num-

ber of excellent references are available, including those by Dielman (7), Hocking (8),

Mendenhall and Sincich (9), and Neter et al. (10).

SUMMARY OF FORMULAS FOR CHAPTER 9

Formula Name Formula

Number

9.2.1 Assumption of

linearity

9.2.2 Simple linear

regression model

9.2.3 Error (residual) term

9.3.1 Algebraic

representation

of a straight line

9.3.2 Least square

estimate of the

slope of a

regression line

i = 1

- x21y

- y2

i = 1

- x2

y = a + bx

P=y - 1b

+ b

x2= y - m

y = b

+ b

x +P

= b

+ b

SUMMARY OF FORMULAS FOR CHAPTER 9 457

(Continued)

9.3.3 Least square estimate

of the intercept of a

regression line

9.4.1 Deviation equation

9.4.2 Sum-of-squares

equation

9.4.3 Estimated

population

coefﬁcient of

determination

9.4.4–9.4.7 Means and

variances of

point estimators

a and b

9.4.8 z statistic for testing

hypotheses

about

9.4.9 t statistic for testing

hypotheses

about

9.5.1 Prediction

interval for Y

for a given X

9.5.2 Conﬁdence

Interval for the

mean of Y for

a given X

9.7.1–9.7.2 Correlation coefﬁcient

y ; t

(1- a>2)

- x2

y ; t

(1- a>2)

y>x

1 +

- x2

t =

- 1b

z =

- 1b

= 1 -

>1n - 22

- y2

>1n - 12

- y2

- y2= 1

- y2+ 1y

= y -

458 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION

y>x

i = 1

1x - x2

= b

i = 1

- x2

= b

r =

SUMMARY OF FORMULAS FOR CHAPTER 9 459

9.7.3 t statistic for

correlation coefﬁcient

9.7.4 z statistic for

correlation coefﬁcient

9.7.5 Estimated standard

deviation for z

statistic

9.7.6 Z statistic for

correlation coefﬁcient

9.7.7 Z statistic for

correlation coefﬁcient

when n  25

9.7.8 Standard Deviation

for z*

9.7.9 Z* statistic

for correlation

coefﬁcient

9.7.10 Conﬁdence interval

for

Symbol Key •  regression intercept term

•  estimated regression intercept

•  probability of type I error or regression intercept

•  estimated regression slope

•  regression slope

•  error term

•  population mean of statistic/variable x

• n  sample size

•  population variance of statistic/variable x

•  population correlation coefﬁcient

• r  sample correlation coefﬁcient

• r

 sample coefﬁcient of determination

• t  t statistic

• x

 value of independent variable at i

•  sample mean of independent variable

• y

 value of dependent variable at i

•  sample mean of dependent variable

•  estimated y

• z  z statistic

= Z

1>1n - 3

1n - 3

z* = z

+ r

Z =

- z

1>1n - 3

1n - 3

1 + r

1 - r

t = r

n - 2

1 - r

j* = Z

13z

+ r2

Z* =

z* - j*

1>1n - 1

= 1z* - j*21n - 1, where

460 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION

REVIEW QUESTIONS AND EXERCISES

1. What are the assumptions underlying simple linear regression analysis when one of the objectives

is to make inferences about the population from which the sample data were drawn?

2. Why is the regression equation called the least-squares equation?

3. Explain the meaning of in the sample regression equation.

4. Explain the meaning of in the sample regression equation.

5. Explain the following terms:

(a) Total sum of squares

(b) Explained sum of squares

6. Explain the meaning of and the method of computing the coefﬁcient of determination.

7. What is the function of the analysis of variance in regression analysis?

8. Describe three ways in which one may test the null hypothesis that .

9. For what two purposes can a regression equation be used?

10. What are the assumptions underlying simple correlation analysis when inference is an objective?

11. What is meant by the unit of association in regression and correlation analysis?

12. What are the possible explanations for a signiﬁcant sample correlation coefﬁcient?

13. Explain why it is risky to use a sample regression equation to predict or to estimate outside the

range of values of the independent variable represented in the sample.

14. Describe a situation in your particular area of interest where simple regression analysis would be

useful. Use real or realistic data and do a complete regression analysis.

15. Describe a situation in your particular area of interest where simple correlation analysis would be

useful. Use real or realistic data and do a complete correlation analysis.

In each of the following exercises, carry out the required analysis and test hypotheses at the indi-

cated signiﬁcance levels. Compute the p value for each test.

16. A study by Scrogin et al. (A-11) was designed to assess the effects of concurrent manipulations of

dietary NaCl and calcium on blood pressure as well as blood pressure and catecholamine responses

to stress. Subjects were salt-sensitive, spontaneously hypertensive male rats. Among the analyses

performed by the investigators was a correlation between baseline blood pressure and plasma epi-

nephrine concentration (E). The following data on these two variables were collected. Let

BP PlasmaE BP PlasmaE

163.90 248.00 143.20 179.00

195.15 339.20 166.00 160.40

170.20 193.20 160.40 263.50

171.10 307.20 170.90 184.70

a = .01.

= 0

(Continued)

BP PlasmaE BP PlasmaE

148.60 80.80 150.90 227.50

195.70 550.00 159.60 92.35

151.00 70.00 141.60 139.35

166.20 66.00 160.10 173.80

177.80 120.00 166.40 224.80

165.10 281.60 162.00 183.60

174.70 296.70 214.20 441.60

164.30 217.30 179.70 612.80

152.50 88.00 178.10 401.60

202.30 268.00 198.30 132.00

171.70 265.50

Source: Karie E. Scrogin. Used with permission.

17. Dean Parmalee (A-12) wished to know if the year-end grades assigned to Wright State University

Medical School students are predictive of their second-year board scores. The following table

shows, for 89 students, the year-end score (AVG, in percent of 100) and the score on the second-

year medical board examination (BOARD).

AVG BOARD AVG BOARD AVG BOARD

95.73 257 85.91 208 82.01 196

94.03 256 85.81 210 81.86 179

91.51 242 85.35 212 81.70 207

91.49 223 85.30 225 81.65 202

91.13 241 85.27 203 81.51 230

90.88 234 85.05 214 81.07 200

90.83 226 84.58 176 80.95 200

90.60 236 84.51 196 80.92 160

90.30 250 84.51 207 80.84 205

90.29 226 84.42 207 80.77 194

89.93 233 84.34 211 80.72 196

89.83 241 84.34 202 80.69 171

89.65 234 84.13 229 80.58 201

89.47 231 84.13 202 80.57 177

88.87 228 84.09 184 80.10 192

88.80 229 83.98 206 79.38 187

88.66 235 83.93 202 78.75 161

88.55 216 83.92 176 78.32 172

88.43 207 83.73 204 78.17 163

88.34 224 83.47 208 77.39 166

87.95 237 83.27 211 76.30 170

87.79 213 83.13 196 75.85 159

87.01 215 83.05 203 75.60 154

86.86 187 83.02 188 75.16 169

86.85 204 82.82 169 74.85 159

86.84 219 82.78 205 74.66 167

REVIEW QUESTIONS AND EXERCISES 461

(Continued)

AVG BOARD AVG BOARD AVG BOARD

86.30 228 82.57 183 74.58 154

86.13 210 82.56 181 74.16 148

86.10 216 82.45 173 70.34 159

85.92 212 82.24 185

Source: Dean Parmalee, M.D. and the Wright State University Statistical

Consulting Center. Used with permission.

Perform a complete regression analysis with AVG as the independent variable. Let

for all tests.

18. Maria Mathias (A-13) conducted a study of hyperactive children. She measured the children’s atti-

tude, hyperactivity, and social behavior before and after treatment. The following table shows for

31 subjects the age and improvement scores from pre-treatment to post-treatment for attitude

(ATT), social behavior (SOC), and hyperactivity (HYP). A negative score for HYP indicates an

improvement in hyperactivity; a positive score in ATT or SOC indicates improvement. Perform an

analysis to determine if there is evidence to indicate that age (years) is correlated with any of the

three outcome variables. Let for all tests.

Subject

No. AGE ATT HYP SOC

191.2 1.2 0.0

2 9 0.0 0.0 1.0

3130.4 0.0 0.2

460.4 0.2 1.2

5 9 1.0 0.8 0.2

6 8 0.8 0.2 0.4

780.6 0.2 0.6

891.2 0.8 0.6

9 7 0.0 0.2 0.8

10 12 0.4 0.8 0.4

11 9 0.8 0.8 0.2

12 10 1.0 0.8 1.2

13 12 1.4 1.6 0.6

14 9 1.0 0.2 0.2

15 12 0.8 0.8 1.0

16 9 1.0 0.4 0.4

17 10 0.4 0.2 0.6

18 7 0.0 0.4 0.6

19 12 1.1 0.6 0.8

20 9 0.2 0.4 0.2

21 7 0.4 0.2 0.6

22 6 0.0 3.2 1.0

23 11 0.6 0.4 0.0

24 11 0.4 0.4 0.0

25 11 1.0 0.7 0.6

26 11 0.8 0.8 0.0

a = .05

462 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION

(Continued)

Subject

No. AGE ATT HYP SOC

27 11 1.2 0.6 1.0

28 11 0.2 0.0 0.2

29 11 0.8 1.2 0.3

30 8 0.0 0.0 0.4

31 9 0.4 0.2 0.2

19. A study by Triller et al. (A-14) examined the length of time required for home health-care nurses

to repackage a patient’s medications into various medication organizers (i.e., pill boxes). For the

19 patients in the study, researchers recorded the time required for repackaging of medications.

They also recorded the number of problems encountered in the repackaging session.

Repackaging Repackaging

Patient No. No. of Problems Time (Minutes) Patient No. No. of Problems Time (Minutes)

1 9 38 11 1 10

2 2 25 12 2 15

30513117

4 6 18 14 0 18

5 5 15 15 0 23

6 3 25 16 10 29

7 3 10 17 0 5

81518122

9 2 10 19 1 20

10 0 15

Source: Darren M. Triller, Pharm.D. Used with permission.

Perform a complete regression analysis of these data using the number of problems to predict the

time it took to complete a repackaging session. Let for all tests. What conclusions can

be drawn from your analysis? How might your results be used by health-care providers?

20. The following are the pulmonary blood ﬂow (PBF) and pulmonary blood volume (PBV) values

recorded for 16 infants and children with congenital heart disease:

PBV (ml/sqM) PBF (L/min/sqM)

168 4.31

280 3.40

391 6.20

420 17.30

303 12.30

429 13.99

605 8.73

522 8.90

224 5.87

291 5.00

a = .05

REVIEW QUESTIONS AND EXERCISES 463

Source: Maria Mathias, M.D.

and the Wright State University

Statistical Consulting Center.

Used with permission.

(Continued)

PBV (ml/sqM) PBF (L/min/sqM)

233 3.51

370 4.24

531 19.41

516 16.61

211 7.21

439 11.60

Find the regression equation describing the linear relationship between the two variables, compute

and test by both the F test and the t test. Let

21. Fifteen specimens of human sera were tested comparatively for tuberculin antibody by two methods.

The logarithms of the titers obtained by the two methods were as follows:

Method

A (X) B (Y)

3.31 4.09

2.41 3.84

2.72 3.65

2.41 3.20

2.11 2.97

2.11 3.22

3.01 3.96

2.13 2.76

2.41 3.42

2.10 3.38

2.41 3.28

2.09 2.93

3.00 3.54

2.08 3.14

2.11 2.76

Find the regression equation describing the relationship between the two variables, compute

and test by both the F test and the t test.

22. The following table shows the methyl mercury intake and whole blood mercury values in 12 sub-

jects exposed to methyl mercury through consumption of contaminated ﬁsh:

Methyl Mercury in

Mercury Intake Whole Blood

( g Hg/day) (ng/g)

180 90

200 120

230 125

410 290

: b

= 0

a = .05.H

: b

= 0r

464 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION

(Continued)

Methyl Mercury in

Mercury Intake Whole Blood

( g Hg/day) (ng/g)

600 310

550 290

275 170

580 375

105 70

250 105

460 205

650 480

Find the regression equation describing the linear relationship between the two variables, compute

and test by both the F and t tests.

23. The following are the weights (kg) and blood glucose levels (mg/100 ml) of 16 apparently healthy

adult males:

Weight (X) Glucose (Y)

64.0 108

75.3 109

73.0 104

82.1 102

76.2 105

95.7 121

59.4 79

93.4 107

82.1 101

78.9 85

76.7 99

82.1 100

83.9 108

73.0 104

64.4 102

77.6 87

Find the simple linear regression equation and test using both ANOVA and the t

test. Test and construct a 95 percent confidence interval for What is the predicted

glucose level for a man who weighs 95 kg? Construct the 95 percent prediction interval for his

weight. Let for all tests.

24. The following are the ages (years) and systolic blood pressures of 20 apparently healthy adults:

Age (X) BP (Y) Age (X) BP (Y)

20 120 46 128

43 128 53 136

a = .05

r.H

: r = 0

: b

= 0

: b

= 0r

REVIEW QUESTIONS AND EXERCISES 465

(Continued)

Age (X) BP (Y) Age (X) BP (Y)

63 141 70 146

26 126 20 124

53 134 63 143

31 128 43 130

58 136 26 124

46 132 19 121

58 140 31 126

70 144 23 123

Find the simple linear regression equation and test using both ANOVA and the t test.

Test and construct a 95 percent conﬁdence interval for Find the 95 percent prediction

interval for the systolic blood pressure of a person who is 25 years old. Let for all tests.

25. The following data were collected during an experiment in which laboratory animals were inocu-

lated with a pathogen. The variables are time in hours after inoculation and temperature in degrees

Celsius.

Time Temperature Time Temperature

24 38.8 44 41.1

28 39.5 48 41.4

32 40.3 52 41.6

36 40.7 56 41.8

40 41.0 60 41.9

Find the simple linear regression equation and test using both ANOVA and the t test.

Test and construct a 95 percent conﬁdence interval for Construct the 95 percent pre-

diction interval for the temperature at 50 hours after inoculation. Let for all tests.

For each of the studies described in Exercises 26 through 28, 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 independent variable?

(d) What are the appropriate null and alternative hypotheses?

(e) Do you think the null hypothesis was rejected? Explain why or why not.

(f) Which is the more relevant objective, prediction or estimation, or are the two equally

relevant?

(g) What is the sampled population?

(h) What is the target population?

(i) Are the variables directly or inversely related?

26. Lamarre-Cliche et al. (A-15) state, “The QT interval corrected for heart rate (QTc) is believed to

reﬂect sympathovagal balance. It has also been established that -blockers inﬂuence the autonomic

nervous system.” The researchers performed correlation analysis to measure the association

a = .05

r.H

: r = 0

: b

= 0

a = .05

r.H

: r = 0

: b

= 0

466 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION