Intercepts and interactions
Choice Modeling for Marketing in R
Elea McDonnell Feit
Assistant Professor of Marketing, Drexel University
Why no intercepts?
v1 <- alpha * seat1 + beta * price1
v2 <- alpha * seat2 + beta * price2
v3 <- alpha * seat3 + beta * price3
p1 <- exp(v1) / ( exp(v1) + exp(v2) + exp(v3) )
p2 <- exp(v2) / ( exp(v1) + exp(v2) + exp(v3) )
p3 <- exp(v3) / ( exp(v1) + exp(v2) + exp(v3) )
Example with an intercept
intercept <- 5
u1 <- intercept + 1.5
u2 <- intercept + 0
u3 <- intercept + -1.5
p1 <- exp(u1) / ( exp(u1) + exp(u2) + exp(u3) )
p2 <- exp(u2) / ( exp(u1) + exp(u2) + exp(u3) )
p3 <- exp(u3) / ( exp(u1) + exp(u2) + exp(u3) )
c(p1, p2, p3)
[1] 0.78559703 0.17529039 0.03911257
Example with an intercept
intercept <- 0
u1 <- intercept + 1.5
u2 <- intercept + 0
u3 <- intercept + -1.5
p1 <- exp(u1) / ( exp(u1) + exp(u2) + exp(u3) )
p2 <- exp(u2) / ( exp(u1) + exp(u2) + exp(u3) )
p3 <- exp(u3) / ( exp(u1) + exp(u2) + exp(u3) )
c(p1, p2, p3)
0.78559703 0.17529039 0.03911257
Interactions
u1 <- alpha * seat1 + beta * price1 + gamma * seat1 * price1
u2 <- alpha * seat2 + beta * price2 + gamma * seat2 * price2
u3 <- alpha * seat3 + beta * price3 + gamma * seat3 * price3
Putting interactions into formulas
m4 <- mlogit(choice ~ 0 + seat + trans + convert + price + trans:price,
data = sportscar)
m4 <- mlogit(choice ~ 0 + seat + convert + trans * price,
data = sportscar)
Interpreting interactions
summary(m4)
...
Coefficients :
Estimate Std. Error z-value Pr(>|z|)
seat4 -0.0202794 0.0759464 -0.2670 0.789452
seat5 0.4255033 0.0752777 5.6524 1.582e-08 ***
transmanual -1.1437766 0.0926133 -12.3500 < 2.2e-16 ***
convertyes 0.2621652 0.0821798 3.1901 0.001422 **
price -0.1908102 0.0086784 -21.9869 < 2.2e-16 ***
transmanual:convertyes -0.1444570 0.1265649 -1.1414 0.253717
...
summary(sportscar$segment)
basic fun racer
3840 1530 630
m5 <- mlogit(choice ~ 0 + seat + convert + trans + price:segment,
data=sportscar)
summary(m5)
...
Coefficients :
Estimate Std. Error z-value Pr(>|z|)
seat4 -0.016206 0.076170 -0.2128 0.831511
seat5 0.426851 0.075682 5.6401 1.700e-08 ***
convertyes 0.200792 0.062343 3.2207 0.001279 **
transmanual -1.228724 0.066893 -18.3686 < 2.2e-16 ***
price:segmentbasic -0.228245 0.011483 -19.8771 < 2.2e-16 ***
price:segmentfun -0.133885 0.015677 -8.5405 < 2.2e-16 ***
price:segmentracer -0.132417 0.023398 -5.6594 1.519e-08 ***
...
Let's try some interactions in the chocolate data.
Choice Modeling for Marketing in R
