Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

IV. Covariance and
Correlation

Covariance and correlation are
measures of relationships between
variables.

Covariance

Population:

covðX;YÞ¼E½ðXmxÞðYmyފ

Sample:

covdðX;YÞ

¼

1

ðn 1 Þ

~

n

i¼ 1

ðXiXÞðYiYÞ

Correlation

Population:rxy¼

covðX;YÞ

sxsy

Sample:rxy¼

covdðX;YÞ

sxsy

Correlation:

 Standardized covariance

 Scale free

X 1 ¼height
(in feet)

cov (X 2 ,Y)

¼12 cov(X 1 ,Y)

X 2 ¼height
(in inches)

BUT

Y¼weight rx 2 y¼rx 1 y

In the sections that follow, we provide an over-

view of the mathematical foundation of the

GEE approach. We begin by developing some

of the ideas that underlie correlated analyses,

including covariance and correlation.

Covariance and correlation are measures that

express relationships between two variables.

ThecovarianceofXandYin a population is

defined as the expected value, or average, of the

product ofXminus its mean (mx) andYminus

its mean (my). With sample data, the covariance

is estimated using the formula on the left,

whereXandYare sample means in a sample

of sizen.

ThecorrelationofXandY in a population,

often denoted by the Greek letter rho (r), is

defined as the covariance ofXandYdivided

by the product of the standard deviation ofX

(i.e.,sx) and the standard deviation ofY(i.e.,

sy). The corresponding sample correlation,

usually denoted asrxy, is calculated by dividing

the sample covariance by the product of the

sample standard deviations (i.e.,sxandsy).

The correlation is a standardized measure of

covariance in which the units ofXandYare the

standard deviations ofXandY, respectively.

The actual units used for the value of variables

affect measures of covariance but not mea-

sures of correlation, which are scale-free. For

example, the covariance between height and

weight will increase by a factor of 12 if the

measure of height is converted from feet to

inches, but the correlation between height

and weight will remain unchanged.

500 14. Logistic Regression for Correlated Data: GEE