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

Perfect linear relationship

Y¼b 0 þb 1 X, for a givenX

XandYindependent)r¼ 0

BUT

r¼ 0 )

XandYindependent

or

XandYhave nonlinear

relationship

8

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>>

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Correlations on same variable

ðY 1 ;Y 2 ;...;YnÞ

rY 1 Y 2 ;rY 1 Y 3 ;...;etc:

Correlations between dichotomous
variables may also be considered.

By a perfect linear relationship we mean that,

given a value ofX, the value ofYcan be exactly

ascertained from that linear relationship of

Xand Y (i.e.,Y¼b 0 þb 1 Xwhere b 0 is the

intercept andb 1 is the slope of the line). IfX

andYare independent, then their correlation

will be zero. The reverse does not necessarily

hold. A zero correlation may also result from a

nonlinear association betweenXandY.

We have been discussing correlation in terms

of two different variables such as height

and weight. We can also consider correlations

between repeated observations (Y 1 ,Y 2 ,...,Yn)

on the same variableY.

Consider a study in which each subject has sev-

eral systolic blood pressure measurements over

a period of time. We might expect a positive

correlation between pairs of blood pressure

measurements from the same individual (Yj,Yk).

The correlation might also depend on the time

period between measurements. Measurements

5 min apart on the same individual might be

more highly correlated than measurements

2 years apart.

This discussion can easily be extended from

continuous variables to dichotomous vari-

ables. Suppose a study is conducted examining

daily inhaler use by patients with asthma. The

dichotomous outcome is coded 1 for the event

(use) and 0 for no event (no use). We might

expect a positive correlation between pairs of

responses from the same subject (Yj,Yk).

EXAMPLE

Daily inhaler use (1¼yes, 0¼no) on

same individual over time

ExpectrYjYk>0 for same subject

EXAMPLE

Systolic blood pressure on same

individual over time

ExpectrYjYk>0 for somej, k

Also,

t 1 t 2

Y 1 Y 2 Y 3 Y 4

t 3 t 4

(time)

ExpectrY 1 Y 2 orrY 3 Y 4 >

rY 1 Y 3 ,rY 1 Y 4 ,rY 2 Y 3 ,rY 2 Y 4

502 14. Logistic Regression for Correlated Data: GEE