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

Simple Analysis

E¼ 1 E¼ 0

D¼ 1 ab

D¼ 0 cd

Risk: only in follow-up
OR: case-control or cross-sectional

ORc¼ad=bc

Case-control and cross-sectional
studies:

¼

^PðÞE¼ 1 jD¼ 1.

^PðÞE¼ 0 jD¼ 1

^PðÞE¼ 1 jD¼ 0.

^PðÞE¼ 0 jD¼ 0

P(E|D) (general form)

P(E=1 | D=1)

P(E=1 | D=0)

ˆ

ˆ

Risk:PðÞDjE

RRc¼

P^ðÞD¼ 1 jE¼ 1

P^ðÞD¼ 1 jE¼ 0

The fact that only odds ratios, not individual

risks, can be estimated from logistic modeling

in case-control or cross-sectional studies is not

surprising. This phenomenon is a carryover of

a principle applied to simpler data analysis

situations, in particular, to the simple analysis

of a 22 table, as shown here.

For a 22 table,risk estimatescan be usedonly

if the data derive from a follow-up study,

whereas only odds ratios are appropriate if

the data derive from a casecontrol or cross-

sectional study.

To explain this further, recall that for 2 2

tables, the odds ratio is calculated asORcequals

atimesdoverbtimesc, wherea,b,c, anddare

the cell frequencies inside the table.

In case-control and cross-sectional studies, this

OR formula can alternatively be written, as

shown here, as a ratio involving probabilities

for exposure status conditional on disease status.

In this formula, for example, the term

^PðE¼ 1 jD¼ 1 Þis the estimated probability of

being exposed, given that you are diseased. Sim-

ilarly, the expression^PðE¼ 1 jD¼ 0 Þis the esti-

mated probability of being exposed given that

you are not diseased. All the probabilities in this

expression are of the general form P(E|D).

In contrast, in follow-up studies, formulae for

risk estimates are of the form P(D|E), in which

the exposure and disease variables have been

switched to the opposite side of the “given” sign.

For example, the risk ratio formula for follow-

up studies is shown here. Both the numerator

and denominator in this expression are of the

form P(D|E).

Presentation: V. Study Design Issues 13