Analysis of Sales Promotion Effects on Household Purchase Behavior

Pr(event)
90
=
1+
1
z
e −
Here Z is the linear combination
Z = B + B X + B X + .......... +
0
1
1
.
2
2
B p X p
<strong>of</strong> constants B0,…,Bp and independent variables X1,…,Xp.
The probability <strong>of</strong> the event not occurring is estimated as
P( no event)
= 1 − P(
event)
.
.
In logistic regression the parameters are estimated using the maximum-likelihood method.
That is, the coefficients that make our observed results most likely are selected. Since the
logistic regression model is nonlinear, an iterative algorithm is needed for parameter
estimation. To understand the interpretation <strong>of</strong> the logistic coefficients, consider a
rearrangement <strong>of</strong> the equation <strong>of</strong> the logistic model. The logistic model can be rewritten in
terms <strong>of</strong> the odds <strong>of</strong> an event occurring (the odds <strong>of</strong> an event occurring are defined as the
ratio <strong>of</strong> the probability that it will occur to the probability that it will not). The log <strong>of</strong> the
odds, also known as logit, can be written as
Pr(event)
1 ...
Pr(noevent)
B X B B + + + =
⎛
⎞
⎜
⎟
⎝
⎠
log 0 1 p p X
.
The logistic coefficient can be interpreted as the change in log odds associated with a oneunit
change in the independent variable. Since it is easier to think <strong>of</strong> odds, the logistic
equation can be written in terms <strong>of</strong> odds as
P(event) B0
+ B X + ... + B X B B X B X B
B
0 B1
X1
p X p
P(noevent)
1 1 p p 0 1 1 p p
= e = e e ... e = e
( e
Then e raised to the power Bi is the factor by which the odds change when the ith
independent variable increases by one unit. If Bi is positive this factor will be greater than
1, which means that the odds are increased; if Bi is negative the factor will be less than 1,
which means that the odds are decreased. When Bi is zero the factor equals 1, which leaves
the odds unchanged.
)
... ( e
)
.