Vidakovic B. Statistics for Bioengineering Sciences: With Matlab and WinBugs Support

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590 15 Correlation

and in the case of ties, use

τ =

−n

Example 15.9. Prevention of Vitreous Loss. Limbal incisions were made

in rabbit eyes to mirror the initial steps of lens extraction. The vitreous body

loses water when the eye is open and decreases in weight, as reported by

Galin et al. (1971). The results had implications in the context of cataract

surgery. The authors measured the vitreous body weight for each eye of 15

New Zealand albino rabbits. One eye has been open for 5 min. ( y), while the

other served as a control (x). The measurements of vitreous weight (in mg) are

provided next:

Rabbit # 1 2 3 4 5 6 7 8

Control eye (x)

1848 1532 1460 1947 1810 1718 1686 1617

Open eye (y)

1738 1440 1388 1756 1692 1629 1583 1499

Rabbit # 9 10 11 12 13 14 15

Control eye (x)

1724 1873 1928 2226 1708 1605 1822

Open eye (y)

1596 1794 1785 2044 1602 1491 1702

We will ﬁnd Kendall’s

τ and provide an approximate 95% conﬁdence inter-

val.

The sample variance of

τ when no ties are present is approximately

(

τ) =4

i=1

−2

i=1

−

(2n −3)

With the presence of ties the expression for sample variance is more com-

plicated, but the above expression can serve as an approximation. Then

−α)100% conﬁdence interval is

τ −

1−α/2

τ),

τ +

1−α/2

τ)

∩

[−1,1].

Using

rabbits.m we found no ties, n

= 100, n

= 5,

τ = 0.9048, and a

95% conﬁdence interval of [0.8091,1.0000]. Using the difference T

= n

−n

one can test for independence of the two components. The test has a p-value

=2P

Z ≥

3(|T|+1)

n(n −1)(2n −5)

where Z is standard normal. In our example a strong dependence between

control and open eye measurements is found

15.5 Cum hoc ergo propter hoc 591

T=nc-nd %95

p = 2

(1-normcdf(3

(abs(T)+1)

sqrt(2)/sqrt(n

(n-1)

n -5))))

%1.8965e-008

15.5 Cum hoc ergo propter hoc

We conclude this chapter with a discussion on the misuses of correlations. The

fallacy that correlation implies causation is summarized by Gould’s quote at

the chapter’s beginning (Latin Cum hoc ergo propter hoc meaning “With this,

therefore because of this”). We already mentioned the “link” between ice-cream

sold on the beach and the number of drowning accidents, but the fallacy causes

more serious damage to science. Spurious correlations are often misused in

medical and health science and attributed to causations. The number of pub-

lished studies with voodoo causations, often conﬂicted from study to study, is

stunning.

200 300 400 500 600

14.8

15.2

15.4

15.6

15.8

Import of fresh lemons (metric tons)

Highway fatality rate (per 100,000)

1996

1997

1998

1999

2000

Fig. 15.6 Fresh lemons imported to USA from Mexico (in metric tons; U.S. Department of

Agriculture) and total U.S. highway fatality rate (per 100,000; U.S. NHTSA, DOT HS 810

780).

As an extreme case of spurious correlation we give an example (popular

among bloggers on the Web) involving data on imports of fresh lemons from

Mexico (1996–2000) and U.S. highway fatality rates (1996–2000), Fig. 15.6.

The correlation is r

= −0.986 and is highly signiﬁcant (p < 0.0002) even with

sample size n

=5. Some bloggers provided “causal links” citing less expensive

592 15 Correlation

car air-fresheners that make drivers happy or slower trafﬁc caused by trucks

from Mexico transporting lemons.

There are two possible errors in correlation inference caused by grouping

data. The ﬁrst one is if two separate groups are combined. For each group there

may not be correlation, but when the groups are combined, the correlation

may be signiﬁcant and, of course, spurious. Figure 15.7 illustrated this point.

Observations represented by red circles (group 1, r

= 0.0643), as well as the

pairs represented by blue circles (group 2, r

= 0.0079), show no signiﬁcant

correlation. However, when the groups are combined, the correlation increases

to r

= 0.7031, and it is signiﬁcant with a p-value of 1.5 ×10

−5

. Details can be

found in

spur.m.

3 4 5 6 7 8

Group 1

Group 2

Fig. 15.7 Spurious correlation when two groups of uncorrelated pairs are combined.

The second error is more subtle. Often, repeated bivariate measurements

are considered as independent and an artiﬁcial correlation due to a blocking

factor is introduced. For example, if for 15 subjects one measures weight (X )

and skinfold thickness (Y ) before and after a diet and combines the measure-

ments, then due to the “increased sample size” a signiﬁcance of correlation

between X and Y is more likely.

15.6 Exercises

15.1. Correlation Between Uniforms and Their Squares. Generate 10,000

uniform random numbers between

−1 and 1 in the form of a vector x.

15.6 Exercises 593

Demonstrate that y=x.2 has a small correlation with x, regardless of their

perfect functional relationship.

15.2. Muscle Strength of “Ethanol Abusers.” It is estimated that 10% of Eu-

ropean and North American adults, and up to one-third of acute hospital ad-

missions, are alcoholics. Obviously, the high proportion of alcoholics in the

hospitalized population imposes severe ﬁnancial constraints on health au-

thorities and emphasizes the need for primary caretakers to focus on min-

imizing alcohol misuse. A staggering two-thirds of chronic ethanol abusers

have skeletal muscle myopathy (Martin et al., 1985; Worden, 1976).

Hickish et al. (1989) provide height, quadriceps muscle strength, and age

data in 41 male alcoholics, as in the table below. The data are available as

alcos.xls or alcos.ascii.

Height Quadriceps Age Height Quadriceps Age

(cm) muscle (years)

strength (N)

155 196 55 172 147 32

159 196 62

173 441 39

159 216 53

173 343 28

160 392 32

173 441 40

160 98 58

173 294 53

161 387 39

175 304 27

162 270 47

175 404 28

162 216 61

175 402 34

166 466 24

175 392 53

167 294 50

175 196 37

167 491 35

176 368 51

168 137 65

177 441 49

168 343 41

177 368 48

168 74 65

177 412 32

170 304 55

178 392 49

171 294 47

178 540 41

172 294 31

178 417 42

172 343 38

178 324 55

172 147 31

179 270 32

172 319 39

180 368 34

172 466 53

(a) Find the sample correlation between Height and Strength, r

. Test the

hypothesis that the population correlation coefﬁcient between the Height

and Strength

is signiﬁcantly positive at the level α =0.01.

Since an increase in Age is expected to decrease the Strength (negative

correlation), ﬁnd the correlation between Height and Strength when Age is

594 15 Correlation

accounted for, that is, ﬁnd r

HS.A

. Test the hypothesis that ρ

HS.A

is positive

at the level

α =0.01.

Find an approximate 95% conﬁdence interval for

15.3. Vending Machine and Pharmacy Errors. Mr. Joseph Bentley, the

owner of a pharmacy store, wants to remove the Coke vending machine

standing in front of his store because he believes the vending machine in-

ﬂuences the number of errors the store employees make. More precisely, as

more Coke is sold outside his store, more errors are made. He provided the

following data:

Errors made 5 3 10 9 5 7 8 4

Coke sold 112 100 220 250 100 200 160 100

Find the coefﬁcient of correlation. Comment on why this correlation is high.

Is there a causation – are Coke sales by themselves inﬂuencing the phar-

macy employees?

15.4. Vending Machine and Pharmacy Errors Revisited. Refer to Exercise

15.3. In addition to Errors and Coke, Mr. Bentley provided the count of

people that pass by his store (and the vending machine):

Errors made 5 3 10 9 5 7 8 4

Coke 112 100 220 250 100 200 160 100

People 10000 6000 17000 20000 9000 15000 14000 8000

Find the coefﬁcient of correlation between Errors and Coke sales while ac-

counting for the number of people. Comment.

15.5. Corn Yields and Rainfall. The following table published by Misner

(1928) has been analyzed by Ezekiel and Fox (1959). The variables are

years:

rain (X): rainfall measurements in inches, in the six states, from 1890 to

1927. Year 1 in the data below corresponds to 1890.

yield (Y): yearly corn yield in bushels per acre, in six Corn Belt states

(Iowa, Illinois, Nebraska, Missouri, Indiana, and Ohio).

year X Y year X Y

1 9.6 24.5 20 12.0 32.3

2 12.9 33.7 21 9.3 34.9

3 9.9 27.9 22 7.7 30.1

4 8.7 27.5 23 11.0 36.9

5 6.8 21.7 24 6.9 26.8

6 12.5 31.9 25 9.5 30.5

7 13.0 36.8 26 16.5 33.3

8 10.1 29.9 27 9.3 29.7

9 10.1 30.2 28 9.4 35.0

10 10.1 32.0 29 8.7 29.9

11 10.8 34.0 30 9.5 35.2

12 7.8 19.4 31 11.6 38.3

13 16.2 36.0 32 12.1 35.2

14 14.1 30.2 33 8.0 35.5

15 10.6 32.4 34 10.7 36.7

16 10.0 36.4 35 13.9 26.8

17 11.5 36.9 36 11.3 38.0

15.6 Exercises 595

18 13.6 31.5 37 11.6 31.7

19 12.1 30.5 38 10.4 32.6

Find the sample correlation coefﬁcient r and a 95% conﬁdence interval for

the population coefﬁcient

ρ.

15.6. Drosophilæ. Sokoloff (1966) reported the correlation between body weight

and wing length in Drosophila pseudoobscura as 0.52 in a sample of n

39 at the Grand Canyon, and as 0.67 in a sample of n

= 20 at Flagstaff,

Arizona. Do the correlations in these two populations differ signiﬁcantly?

Use

α =0.05.

15.7. Conﬁdence Interval for the Difference of Two Correlation Coefﬁ-

cients. Using the results on testing the equality of two correlation coefﬁ-

cients develop a (1

−α)100% conﬁdence interval for their difference.

15.8. Oxygen Intake. The human body takes in more oxygen when exercising

than when it is at rest, and to deliver the oxygen to the muscles, the heart

must beat faster. Heart rate is easy to measure, but the measurement of

oxygen uptake requires elaborate equipment. If oxygen uptake (VO2) is

strongly correlated with the heart rate (HR) under a particular set of ex-

ercise conditions, then its predicted, rather than measured, values could be

used for various research purposes.

HR VO2 HR VO2

94 0.473 108 1.403

96 0.753

110 1.499

95 0.929

113 1.529

95 0.939

113 1.599

94 0.832

118 1.749

95 0.983

115 1.746

94 1.049

121 1.897

104 1.178

127 2.040

104 1.176

135 2.231

106 1.292

Find the sample correlation r and calculate a 95% conﬁdence interval for

its population counterpart,

ρ.

15.9. Obesity and Pain. Khimich (1997) found that a pain threshold increases

in obese subjects and increases with age. Obesity is measured as the per-

centage over ideal weight (X ). The response to pain is measured by using

the threshold of the nociceptive ﬂexion reﬂex (Y ), which is a measure of

the pricking pain sensation in an individual. Measurements X and Y are

Data provided by Paul Waldsmith from experiments conducted in Don Corrigan’s lab at

Purdue University, West Lafayette, Indiana.

596 15 Correlation

considered to be normal. We are interested in an inference about the corre-

lation between X and Y . The following data were obtained:

X 89 90 75 30 51 75 62 45 90 20

Y 2 3 4 4.5 5.5 7 9 13 15 14

77,

=799.5 calculate the Pearson coefﬁcient of linear correlation, r.

(b) Test the hypothesis that the population coefﬁcient of correlation,

ρ, is 0,

against the alternative H

: ρ <0. Use α =0.05.

(in the corresponding order): 20 18 23 19 44 51 36 47 60 55. Find the partial

coefﬁcient of correlation r

xy.z

if r

=−0.2089 and r

=0.8627.

(d) Find a 95% conﬁdence interval for

ρ.

MATLAB AND WINBUGS FILES AND DATA SETS USED IN THIS CHAPTER

http://springer.bme.gatech.edu/Ch15.Corr/

corrs.m, errorscoke.m, fisherzsimu.m, histo.m, iriscorr.m, lemon.m,

nanoprism.m, ObesityPain.m, rabbits.m, spur.m, variouscorrs.m

corr.odc

nanoprism.dat

CHAPTER REFERENCES

Anderson, T. W., (1984). An Introduction to Multivariate Statistical Analysis, 2nd Ed. Wiley,

New York.

Arvin, D. V. and Spaeth, R. (1998). Trends in Indiana’s water use, 1986–1996. Indiana De-

partment of Natural Resources Special Report, No. 1.

Brower, L. P. (1959). Speciation in butterﬂies of the Papilio glaucus group. I: Morphological

relationships and hybridizations. Evolution, 13, 40–63.

Ezekiel, M. and Fox, K. A. (1959). Methods of Correlation and Regression Analysis. Wiley,

New York.

(a) Using results n =10,

=4461.5,

=627,

=45141,

Chapter References 597

Galin, M. A., Robbins, R., and Obstbaum, S. (1971). Prevention of vitreous loss. Brit. J.

Ophthal., 55, 533–537.

Hickish, T., Colston, K., Bland, J. M., and Maxwell J. D. (1989). Vitamin D deﬁciency and

muscle strength in male alcoholics. Clin. Sci., 77, 171–176.

Kendall, M. (1938). A new measure of rank correlation. Biometrika, 30, 1–2, 81–89.

Khimich, S. (1997). Level of sensitivity of pain in patients with obesity. Acta Chir. Hung., 36,

166–167.

Martin, R., Ward, K., Slavin, G., Levi, J., and Peters, T. J. (1985). Alcoholic skeletal myopa-

thy, a clinical and pathological study. Q. J. Med., 55, 233–251.

Misner, E. G. (1928). Studies of the relationship of weather to the production and price of

farm products. I: Corn, mimeographed publication, Cornell University, March 1928.

Sokoloff, A. (1966). Morphological variation in natural and experimental populations of

Drosophila pseudoobscura and Drosophila persimilis. Evolution, 20, 49–71.

Spearman, C. (1904). General intelligence: objectively determined and measured. Am. J.

Psychol., 15, 201–293.

Stichler, R. G., Richey, G. G., and Mandel, J. (1953). Measurement of treadware of commer-

cial tires. Rubber Age, 73, 2.

Worden R. E. (1976). Pattern of muscle and nerve pathology in alcoholism. N.Y. Acad. Sci.,

273, 351–359.

Chapter 16

Regression

The experiments showed further that the mean ﬁlial regression towards mediocrity

was directly proportional to the parental deviation from it.

– Francis Galton, F.R.S. & c. (1886)

WHAT IS COVERED IN THIS CHAPTER

• Ordinary Linear Regression.

• Conﬁdence Intervals and Hypothesis Tests for Parameters and Re-

sponses in Linear Regression.

• Multivariable Regression. Matrix Formulation

• Model Selection and Assessment in Multivariable Regression.

• Power Analysis in Regression.

• Regression Nonlinear in Predictors. Errors-in-Variables Regression.

• Analysis of Covariance (ANCOVA)

16.1 Introduction

The rather curious name regression was given to a statistical methodology

by British scientist Sir Francis Galton, who analyzed the heights of sons

and the average heights of their parents. From his observations, Galton

(Fig. 16.1a) concluded that sons of very tall (or short) parents were generally

B. Vidakovic, Statistics for Bioengineering Sciences: With MATLAB and WinBUGS Support,

Springer Texts in Statistics, DOI 10.1007/978-1-4614-0394-4_16,

599

600 16 Regression

taller (shorter) than average, but not as tall (short) as their parents. The re-

sults were published in 1886 under the title Regression Towards Mediocrity in

Hereditary Stature. In the course of time the word regression became synony-

mous with the statistical study of the functional relationship between two or

more variables. The data set illustrating Darwin’s ﬁnding and used by Pearson

is given in

pearson.dat. The scatterplot and regression ﬁts are analyzed in

galton.m and summarized in Fig. 16.1b. The circles correspond to pairs of

father–son heights, the black line is the line y

= x, the red line is the regres-

sion line, and the green line is the regression line constrained to pass through

the origin. Darwin’s ﬁndings can be summarized by the observation that the

slope of the regression (red) line was signiﬁcantly smaller than the slope of the

◦

-line.

(a)

60 65 70 75

Height of Fathers (in)

Height of Sons (in)

(father, son)

y=x line

regr line

regr line int=0

(b)

Fig. 16.1 (a) Sir Francis Galton (1822–1911). (b) Galton’s father–son height data (used by

Pearson). The circles correspond to pairs of father–son heights, the black line is the line

= x, the red line is the regression line, and the green line is the regression line constrained

to pass through the origin.

Usually the response variable y is “regressed” on several predictors or co-

variates, x

,... , x

, and this raises many interesting questions involving the

model choice and ﬁt, collinearity among the predictors, and others. When we

have a single predictor x and a linear relation between y and x, the regression

is called a simple linear regression.

16.2 Simple Linear Regression

Assume that we observed n pairs (x

, y

),... , (x

, y

), and each observation y

can be modeled as a linear function of x

, plus an error,

=β

+β

+²

, i =1, . .., n.