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

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570 14 Models for Tables

Roberts, E., Dawson, W. M., and Madden, M. (1939). Observed and theoretical ratios in

Mendelian inheritance. Biometrika, 31, 56–66.

Robins, J., Breslow, N., and Greenland, S. (1986). Estimators of the Mantel-Haenszel vari-

ance consistent in both sparse data and large-strata limiting models. Biometrics, 42,

311–323.

Rosenﬁeld, R. N., Bielefeldt, J., and Anderson, R. K. (1988). Effectiveness of broadcast calls

for detecting Cooper’s hawks. Wildl. Soc. Bull., 16, 210–212.

Schoener, T. W. (1968). The Anolis lizards of Bimini: Resource partitioning in a complex

fauna. Ecology, 48, 704–726.

Sleigh, A., Hoff, R., Mott, K., Barreto, M., de Paiva, T., Pedrosa, J., and Sherlock, I. (1982).

Comparison of ﬁltration staining (Bell) and thick smear (Kato) for the detection and

quantitation of Schistosoma mansoni eggs in faeces. Trop. Med. Hyg., 76, 3, 403–406.

Speckmann, J. D. (1965). Marriage and kinship among the Indians in Surinam. Assen: Van-

Gorgum, 32–33.

Spiegelhalter, D. J., Abrams, K. R., and Myles, J. P. (2004). Bayesian Approaches to Clinical

Trials and Health Care Evaluation. Wiley, New York.

Srole, L., Langner, T., Michael, S., Opler, M., and Rennie, T. (1962). Mental Health in the

Metropolis: The Midtown Manhattan Study, vol. 1. McGraw-Hill, New York.

Streissguth, A. P., Martin, D. C., Barr, H. M., Sandman, B. M., Kirchner, G. L., and Darby,

B. L. (1984). Intrauterine alcohol and nicotine exposure: attention and reaction time

in 4-year-old children. Dev. Psychol., 20, 533–541.

Williams, J. W. (1921). A critical analysis of twenty-one years’ experience with cesarean

section. Johns Hopkins Hosp. Bull., 32, 173–184.

Worchester, J. (1971). The relative odds in 2

contingency table. Am. J. Epidemiol., 93, 145–

149.

Chapter 15

Correlation

The invalid assumption that correlation implies cause is probably among the two or

three most serious and common errors of human reasoning.

– Stephen Jay Gould

WHAT IS COVERED IN THIS CHAPTER

• Calculating the Pearson Coefﬁcient of Correlation. Conditional Cor-

relation

• Testing the Null Hypothesis of No Correlation

• General Test for Correlation and Conﬁdence Intervals. Fisher z-

Transformation

• Inference for Two Correlation Coefﬁcients

• Nonparametric Correlation Measures: Spearman’s and Kendall’s

Correlation Coefﬁcient

15.1 Introduction

Roget’s New Thesaurus (3rd Edition, 1995) deﬁnes correlation as a logical or

natural association between two or more things, with synonyms connection,

interconnection, interdependence, interrelationship, link, linkage, relation, re-

lationship, and tie-in.

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

Springer Texts in Statistics, DOI 10.1007/978-1-4614-0394-4_15,

571

572 15 Correlation

Statistically, correlation is a measure of the particular afﬁnity between two

sets of comparable measurements. It is often incorrectly believed that the no-

tions of correlation and statistical dependence and causality coincide. Accord-

ing to an anecdote, it could be true that the number of drownings at a partic-

ular large beach in one season is positively and signiﬁcantly correlated with

the number of ice-creams sold at the beach during the same period of time.

However, nobody would argue that the relationship is causal. Purchasing an

ice-cream at the beach does not increase the risk of drowning – the positive

correlation is caused by a latent (lurking) variable, the number of visitors at

the beach.

Correlation is, in informal terms, a constrained dependence. For exam-

ple, the most common (Pearson) coefﬁcient of correlation measures the

strength and direction of the linear relationship between two variables.

If measurements are correlated, then they are dependent, but not neces-

sarily vice versa. A simple example involves points on a unit circle. One selects

n angles

, i =1,..., n, uniformly from [0,2π]. The angles determine n points

on the unit circle, (x

, y

) = (cosϕ

,sinϕ

), i =1, ..., n. Although x =(x

,... , x

)

and y

= (y

,... , y

) are functionally dependent (since x

+ y

= 1), their coefﬁ-

cient of correlation is 0 or very close to 0. See also Exercise 15.1. Only for a

normal distribution do the notions of correlation and independence coincide:

uncorrelated normally distributed measurements are independent.

15.2 The Pearson Coefﬁcient of Correlation

In Chap. 2, when discussing the summaries of multidimensional data we dis-

cussed correlations between the variables and connected the coefﬁcient of cor-

relation with scatterplots and a descriptive statistical methodology. In this

chapter we will take an inferential point of view on correlations. For develop-

ing the tests and conﬁdence intervals we will assume that the data come from

normal distributions for which the population coefﬁcient of correlation is

ρ.

Suppose that pairs (X

),(X

),... , (X

) are observed, and assume

that X s and Y s come from normal

N (0,σ

) and N (0,σ

) distributions and

that the correlation coefﬁcient between X and Y is

X Y

, or simply ρ. We are

interested in making an inference about

ρ when n pairs (X

) are observed.

From the observations the following sums of squares are formed: S

− X )

, S

−Y )

, and S

− X )(Y

−Y ). Then, an

estimator of

ρ is

15.2 The Pearson Coefﬁcient of Correlation 573

−4 −3 −2 −1 0 1 2 3

−1

−0.5

0.5

1.5

2.5

3.5

r = 0.04

−4 −3 −2 −1 0 1 2 3

−0.5

0.5

1.5

2.5

3.5

r = 0.54

(a) (b)

−4 −3 −2 −1 0 1 2 3

1.5

2.5

3.5

4.5

5.5

r = −0.77

−4 −3 −2 −1 0 1 2 3

−3

−2

−1

r = 0.96

Fig. 15.1 A simulated example showing scatterplots with correlated components (a) r

0.04, (b) r =0.54, (c) r =−0.77, and (d) r = 0.96.

An alternative expression for r is

−nX Y

(

−n(X )

)(

−n(Y )

)

This estimator is an MLE for

ρ, and it is asymptotically unbiased, that is,

lim

n→∞

=ρ.

Sample correlation coefﬁcient r is always in [

−1,1]. To show this, consider

the Cauchy–Schwartz inequality that states:

Substitutions u

= X

− X and v

= Y

−Y prove the inequality r

≤ 1. Figure

15.1 shows simulated correlated data (green dots) with coefﬁcients (a) r

=0.04,

(b) r

=0.54, (c) r =−0.77, and (d) r =0.96.

Partial Correlation. Let r

, r

, and r

be correlations between the pairs

of variables X , Y , and Z. The partial correlation between X and Y if Z is

accounted for (or excluding the inﬂuence of Z) is

xy.z

−r

1 −r

For any two vectors (u

, u

,... , u

) and (v

, v

,... , v

), it holds that (

)

≤

574 15 Correlation

When two variables Z and W are excluded from the correlation of X and Y ,

the partial coefﬁcient is

xy.zw

xy.w

−r

xz.w

yz.w

1 −r

xz.w

1 −r

yz.w

or, equivalently,

xy.zw

xy.z

−r

xw.z

yw.z

1 −r

xw.z

1 −r

yw.z

depending on the order of exclusion of W and Z.

15.2.1 Inference About ρ

To test H

: ρ = 0 against one of the alternatives ρ >,6=,< 0, the t-statistic is

used:

= r

n −2

1 −r

which has a t-distribution with d f

= n −2 degrees of freedom.

Alternative α-level rejection region p-value

: ρ >0 [t

d f ,1−α

,∞) 1-tcdf(t, df)

: ρ 6=0 (−∞, t

d f ,α/2

] ∪[t

d f ,1−α/2

,∞) 2

tcdf(-abs(t), df)

: ρ <0 (−∞, t

d f ,α

] tcdf(t, df)

Example 15.1. The table below gives the number of times X rats ran through

a maze and the time

it took them to run through the maze on their last trial.

Rat Trials (X ) Time (Y )

1 8 10.9

2 9 8.6

3 6 11.4

4 5 13.6

5 3 10.3

6 6 11.7

7 3 10.7

8 2 14.8

(a) Find r.

15.2 The Pearson Coefﬁcient of Correlation 575

(b) Is the maze learning signiﬁcant? Test the hypothesis that the popula-

tion correlation coefﬁcient

ρ is 0, versus the alternative that ρ is negative.

= 42,

= 264,

= 92,

= 1084.2,

= 463.8, X =

5.25, Y = 11.5.

463.8 −8 ·5.25 ·11.5

(264 −8 ·5.25

)(1084.2 −8 ·11.5

)

−

19.2

43.5 ·26.2

=−0.5687.

For testing H

: ρ =0 versus H

: ρ <0 we ﬁnd

= r

n −2

1 −r

=−1.69.

For

α =0.05 the critical value is t

6,0.05

=−1.943 and the null hypothesis is not

rejected.

Remark: Sample size is critical for our decision for the matching observed

values of r. If for the same r, the sample size n were 30, then t

=−3.6588, and

would be rejected, since t < t

28,0.05

=−1.7.

In MATLAB:

X = [8 9 6 5 3 6 3 2];

Y = [10.9 8.6 11.4 13.6 10.3 11.7 10.7 14.8];

n=8;

cxy=cov(X,Y)

% cxy =

% 6.2143 -2.7429

% -2.7429 3.7429

rxy=cxy(1,2)/sqrt(cxy(1,1)

cxy(2,2))

%rxy = -0.5687

tstat = rxy

sqrt(n-2)/sqrt(1-rxy^2)

%tstat = -1.6937

pval = tcdf(tstat, n-2)

%pval = 0.0706

%n=8, p-val = 7% > 5% do not reject H

%============

n=30; rxy=-0.5687;

tstat = rxy

sqrt(n-2)/sqrt(1-rxy^2)

%tstat =-3.6585

pval = tcdf(tstat, n-2)

% pval = 5.2070e-004

When partial correlations are of interest, the test is based on the statistic

= r

n−q−2

1−r

, which has d f = n−q−2 degrees of freedom. Here q is the number

of other variables accounted for.

Example 15.2. On the basis of n

= 28 records it was found that the education

level of the mother (X ) was negatively correlated with infant mortality (Y )

576 15 Correlation

as r

= −0.6. Socioeconomic status (Z) is a variable that was not taken into

account. Z is correlated with X and Y as r

=0.65 and r

=−0.7. Find r

xy.z

and test for the signiﬁcance of the population counterpart ρ

xy.z

rxy=-0.6; rxz=0.65; ryz=-0.7;

rxy

z = (rxy - rxz

ryz)/sqrt( (1-rxz^2)

(1-ryz^2))

%rxy

z = -0.2672

n=28;

tcdf( rxy

sqrt(n-2)/sqrt(1-rxy^2), n-2)

%p-value for testing H

0: rho

xy = 0, against

%the two sided alternative

%ans = 7.3810e-004

tcdf( rxy

sqrt(n-1-2)/sqrt(1-rxy

z^2), n-1-2)

%p-value for h

0: rho

xy.z = 0, against

%the two sided alternative

%ans = 0.1779

Thus, the signiﬁcant correlation between X and Y (p-value of 0.000738)

becomes insigniﬁcant (p-value of 0.1779) when variable Z is taken into ac-

count.

15.2.1.1 Conﬁdence Intervals for ρ.

Conﬁdence intervals for the population coefﬁcient of correlation

ρ are obtained

not by using direct measurements, but in a transformed domain. This trans-

formation is known as Fisher’s z-transformation and is introduced next.

Let (X

, X

),... , (X

, X

) be a sample from a bivariate normal distribu-

tion

(µ

,µ

,σ

,ρ), and X

i j

, i =1, 2.

The Pearson coefﬁcient of linear correlation

−X

)(X

−X

)

−X

)

−X

)

1/2

has a complicated distribution involving special functions, e.g., Anderson

(1984), p. 113. However, it is well known that the asymptotic distribution

for r is normal,

N (ρ,

(1−ρ

)

). Since the variance is a function of the mean

[

(ρ) =

(1−ρ

)

, see equation (6.4)], the transformation deﬁned as

15.2 The Pearson Coefﬁcient of Correlation 577

ϑ(ρ) =

σ(ρ)

1 −ρ

dρ

1 −ρ

1 +ρ

dρ

log

1 +ρ

1 −ρ

stabilizes the variance. It is known as Fisher’s z-transformation for the corre-

lation coefﬁcient (usually for c

=1/

n and k =0).

Assume that r and ρ are mapped to w and ζ as

log

1 +r

1 −r

arctanh r, ζ =

log

1 +ρ

1 −ρ

arctanh ρ.

The distribution of w is approximately normal,

N (ζ,

n−3

), and this ap-

proximation is quite accurate when

is small and n is as low as 20.

The inverse z-transformation is

=tanh(w) =

exp{2w}−1

exp{2w}+1

The use of Fisher’s z-transformation is illustrated on ﬁnding the conﬁdence

intervals for

ρ and testing hypotheses about ρ.

To exemplify the above-stated transformations, we generated n

=30 pairs

of normally distributed random samples with theoretical correlation

2/2.

This was done by generating two i.i.d. normal samples a and b of length 30

and taking the transformation x

= a + b, x

= b. The sample correlation co-

efﬁcient r was found. This was repeated M

= 10,000 times. A histogram of

10,000 sample correlation coefﬁcients is shown in Fig. 15.2a. A histogram of

the z-transformed rs is shown in Fig. 15.2b with a superimposed normal ap-

proximation

N (arctanh(

2/2),1/(30 −3)).

The sampling distribution for w is approximately normal with mean

ζ =

log

1+ρ

1−ρ

and variance 1/(n −3). This approximation is satisfactory when the

number of pairs exceeds 20. Thus, the (1

−α)100% conﬁdence interval for ζ =

log

1+ρ

1−ρ

578 15 Correlation

0.2 0.4 0.6 0.8 1

0 0.5 1 1.5 2

0.5

1.5

2.5

(a) (b)

Fig. 15.2 (a) Simulational run of 10,000 rs from a bivariate population having a theoretical

ρ =

2/2. (b) The same rs transformed into ws with the normal approximation superim-

posed.

, w

] =

w −

1−α/2

n −3

, w

1−α/2

n −3

where w

log

1+r

1−r

Since r

−1

, an approximate (1 −α)100% conﬁdence interval for ρ

, r

] =

−1

Remark. More accurate approximation for the sampling distribution of w has

the mean

ζ +ρ/(2n −2). The correction ρ/(2n −2) is often used in testing and

conﬁdence intervals when n is not large. In the above interval the corrected w

would be

log

1+r

1−r

−

2(n−1)

Example 15.3. If r

= −0.5687 and n = 8, then w = −0.6456. The bounds for

the 95% conﬁdence interval for

ζ =

log

1+ρ

1−ρ

are w

= −0.6456 −1.96/

5 =

−

1.522 and w

= −0.6456 +1.96/

5 = 0.2309. The conﬁdence interval for ρ

is obtained by back-transforming w

and w

using r =

−1

. The result is

[

−0.9091,0.2269].

15.2 The Pearson Coefﬁcient of Correlation 579

15.2.1.2 Test for ρ =ρ

We saw that when ρ

= 0, this test is a t-test with a statistic given as t =

n−2

1−r

. When ρ

6=0, then the test of H

: ρ = ρ

versus H

: ρ >,6=,<ρ

does

not have a simple generalization. An approximate test is based on a normal

approximation. Under H

: ρ =ρ

the test statistic

z =

n −3

log

1 +r

1 −r

−

log

1 +ρ

1 −ρ

¶¸

has an approximately standard normal distribution. This implies the following

testing rules:

Alternative α-level rejection region p-value

: ρ >ρ

1−α

,∞) 1-normcdf(z)

: ρ 6=ρ

(−∞, z

α/2

] ∪[z

1−α/2

,∞) 2

normcdf(-abs(z))

: ρ <ρ

(−∞, z

] normcdf(z)

Example 15.4. Interparticular Spacing in Nanoprisms. There is much

interest in knowing and understanding how nanoparticles interact optically.

One reason is the use of nanoparticles as plasmon rulers. Plasmon rulers are

beneﬁcial to the ﬁeld of biomedical engineering as they allow researchers to

measure changes and differences in DNA or cells at the nano level. This is

promising for diagnostics, especially with respect to genetic disorders, which

could be potentially identiﬁed by data from a plasmon ruler. In order to cre-

ate a basis for this idea, research must be done to determine the effect of

different interparticle spacing on the maximum wavelength of absorbance of

the particles. While a linear correlation between measured

separation and

wavelength is not strong, researchers have found that the correlation be-

tween the reciprocal of separation,

recsep=1/separation, and the logarithm

of a wavelength,

logwl=log(wavelength), is strong. The data from the lab of Dr.

Mostafa El-Sayed, Georgia Tech, are given in the table below, as well as in

nanoprism.dat.