E¼ 1 E¼ 0
D¼ 1 ab
D¼ (^0) cd
RORd ¼e^b¼ad=bc
Simple analysis: does not need
computer
Other special cases: require computer
III. Assessing
Multiplicative
Interaction
X 1 ¼A¼(0, 1) variable
X 2 ¼B¼(0, 1) variable
Interaction: equation involving
RORs for combinations ofAandB
RAB¼risk givenA,B
¼PrðD¼ 1 jA,BÞ
B¼ 1 B¼ 0
A¼ 1 R 11 R 10
A¼ 0 R 01 R 00
The reader should not be surprised to find out
that an alternative formula for the estimated
odds ratio for the simple analysis model is the
familiaratimesdoverbtimesc, wherea, b, c,
anddare the cell frequencies in the fourfold
table for simple analysis. That is, e to the^b 1
obtained from fitting a logistic model for sim-
ple analysis can alternatively be computed as
addivided bybcfrom the cell frequencies of the
fourfold table.Thus, in the simple analysis case, we need
not go to the trouble of fitting a logistic model
to get an odds ratio estimate as the typical
formula can be computed without a computer
program. We have presented the logistic model
version of simple analysis to show that the
logistic model incorporates simple analysis as
a special case. More complicated special cases,
involving more than one independent variable,
require a computer program to compute the
odds ratio.We will now consider how the logistic model
allows the assessment of interaction between
two independent variables.Consider, for example, two (0, 1)Xvariables,
X 1 andX 2 , which for convenience we rename as
AandB, respectively. We first describe what we
mean conceptually by interaction between
these two variables. This involves an equation
involving risk odds ratios corresponding to dif-
ferent combinations of A and B. The odds
ratios are defined in terms of risks, which we
now describe.LetRABdenote the risk for developing the dis-
ease, given specified values forAand B;in
other words,RABequals the conditional proba-
bility thatDequals 1, givenAandB.BecauseAandBare dichotomous, there are
four possible values forRAB, which are shown
in the cells of a two-way table. WhenAequals 1
andBequals 1, the riskRABbecomesR 11. Sim-
ilarly, whenAequals 1 and B equals 0, the risk
becomesR 10. WhenAequals 0 and B equals 1,
the risk isR 01 , and finally, whenAequals 0 and
B equals 0, the risk isR 00.Presentation: III. Assessing Multiplicative Interaction 49