ROR (no interaction):bs only
ROR (interaction):bs andds
The previous odds ratio formula that we
gave for several exposures but no interaction
involved onlybcoefficients for the exposure
variables. Because the model we are now
considering contains interaction terms, the
corresponding odds ratio will involve not only
theb coefficients, but alsodcoefficients for
all interaction terms involving one or more
exposure variables.The odds ratio formula for our example then
becomes e to the quantity (SMK*SMK**)
times b 1 plus (PAL*PAL**) timesb 2 plus
(SBP*SBP**) times b 3 plus the sum of
terms involving adcoefficient times the differ-
ence betweenE*andE**values of one of the
exposures times aWvariable.For example, the first of the interaction terms
isd 11 times the difference (SMK*SMK**)
times AGE, and the second of these terms is
d 12 times the difference (SMK*SMK**)
times SEX.When we substitute into the odds ratio formula
the values forE*andE**, we obtain the expres-
sion e to the quantity (01) timesb 1 plus
(2510) timesb 2 plus (160120) timesb 3
plus several terms involving interaction coeffi-
cients denoted asds.The first set of these terms involves inter-
actions of AGE and SEX with SMK. These
terms are d 11 times the difference (01)
times AGE plus d 12 times the difference
(01) times SEX. The next set of dterms
involves interactions of AGE and SEX with
PAL. The last set ofdterms involves interactions
of AGE and SEX with SBP.After subtraction, this expression reduces to
the expression shown here at the left.We can simplify this expression further by fac-
toring out AGE and SEX to obtain e to the
quantity minusb 1 plus 15 times b 2 plus 40
timesb 3 plus AGE times the quantity minus
d 11 plus 15 timesd 21 plus 40 timesd 31 plus
SEX times the quantity minus d 12 plus 15
timesd 22 plus 40 timesd 32.EXAMPLE (continued)
RORE*vs:E**¼exp½SMK*SMK**
b 1
þPAL*PAL**
b 2
þSBP*SBP**
b 3
þd 11 SMK*SMK**
AGE
þd 12 SMK*SMK**
SEX
þd 21 PAL*PAL**
AGE
þd 22 PAL*PAL**
SEX
þd 31 SBP*SBP**
AGE
þd 32 SBP*SBP**
SEXROR¼exp½ðÞ 0 1 b 1 þðÞ 25 10 b 2
þðÞ 160 120 b 3interaction with SMKinteraction with PALinteraction with SBP+ d 11 (0 – 1) AGE + d 12 (0 – 1) SEX+ d 21 (25 – 10) AGE + d 22 (25 – 10) SEX+ d 31 (160 – 120) AGE + d 32 (160 – 120) SEX¼expðb 1 þ 15 b 2 þ 40 b 3
d 11 AGEd 12 SEX
þ 15 d 21 AGEþ 15 d 22 SEX
þ 40 d 31 AGEþ 40 d 32 SEXÞ¼expðb 1 þ 15 b 2 þ 40 b 3
þAGEðÞd 11 þ 15 d 21 þ 40 d 31
þSEXðÞd 12 þ 15 d 22 þ 40 d 3288 3. Computing the Odds Ratio in Logistic Regression