II. Special Case – Simple
Analysis
X 1 ¼E¼exposure (0, 1)
D¼disease (0, 1)
E¼ 1 E¼ 0
D¼ 1 ab
D¼ 0 cd
PðXÞ¼
1
1 þeðÞaþb^1 E,
whereE¼(0, 1) variable.
Note: Other coding schemes
(1,1), (1, 2), (2, 1)
logit P(X)¼aþb 1 E
P(X)¼Pr(D¼1|E)
E¼1: R 1 ¼Pr(D¼1|E¼1)
E¼0: R 0 ¼Pr(D¼1|E¼0)
We begin with the simple situation involving
one dichotomous independent variable, which
we will refer to as anexposurevariable and will
denote it asX 1 =E. Because the disease variable,
D, considered by a logistic model is dichoto-
mous, we can use a two-way table with four
cells to characterize this analysis situation,
which is often referred to as asimple analysis.For convenience, we define the exposure vari-
able as a (0, 1) variable and place its values in
the two columns of the table. We also define the
disease variable as a (0, 1) variable and place its
values in the rows of the table. The cell frequen-
cies within the fourfold table are denoted asa, b,
c,andd, as is typically presented for such a table.A logistic model for this simple analysis situa-
tion can be defined by the expression P(X)
equals 1 over 1 plus e to minus the quantitya
plusb 1 timesE, whereEtakes on the value 1
for exposed persons and 0 for unexposed per-
sons. Note that other coding schemes forEare
also possible, such as (1,1), (1, 2), or even
(2, 1). However, we defer discussing such alter-
natives until Chap. 3.The logit form of the logistic model we have
just defined is of the form logit P(X) equals the
simple linear sumaplusb 1 timesE. As stated
earlier in our review, this logit form is an alter-
native way to write the statement of the model
we are using.The term P(X) for the simple analysis model
denotes the probability that the disease vari-
ableDtakes on the value 1, given whatever the
value is for the exposure variableE. In epidemi-
ologic terms, this probability denotes therisk
for developing the disease, given exposure sta-
tus. When the value of the exposure variable
equals 1, we call this riskR 1 , which is the con-
ditional probability thatDequals 1 given thatE
equals 1. WhenEequals 0, we denote the risk
byR 0 , which is the conditional probability that
Dequals 1 given thatEequals 0.46 2. Important Special Cases of the Logistic Model