Interaction Terms.R

# Cleaning the workspace
rm(list = ls())

# Loading the data
library(Matching)
data(lalonde)
View(lalonde)
str(lalonde)

# Let's look at two individuals. For the purposes of this tutorial, 
# they will be the youngest and oldest person in the data.
youngest <- which(lalonde$age == min(lalonde$age))[1]
oldest <- which(lalonde$age == max(lalonde$age))[1]

sprintf("The youngest person is %d years old, had %d years of schooling, and earned %0.f dollars in 1978.", 
        lalonde[youngest, ]$age, lalonde[youngest, ]$educ, lalonde[youngest, ]$re78)
sprintf("The oldest person is %d years old, had %d years of schooling, and earned %0.f dollars in 1978.", 
        lalonde[oldest, ]$age, lalonde[oldest, ]$educ, lalonde[oldest, ]$re78)

# Let's fit a linear model based on age and education.
lm.additive <- lm(re78 ~ age + educ, data = lalonde)
summary(lm.additive)

# Let's look at the prediction of our model for these two people.
# At this point, we're not concerned with accuracy.
sprintf("The youngest person is predicted to have earned %0.f dollars in 1978.", 
        predict(lm.additive, lalonde[youngest, ]))
sprintf("The oldest person is predicted to have earned %0.f dollars in 1978.", 
        predict(lm.additive, lalonde[oldest, ]))

# How would their predicted income change if they had
# had an additional year of schooling?
lalonde_mod <- lalonde
lalonde_mod[c(youngest, oldest), ]$educ <- lalonde_mod[c(youngest, oldest), ]$educ + 1

# Let's compare the two datasets
lalonde[c(youngest, oldest), ]
lalonde_mod[c(youngest, oldest), ]

# What's the updated prediction?
predict(lm.additive, lalonde_mod[youngest, ])
predict(lm.additive, lalonde_mod[oldest, ])

# What's the difference?
predict(lm.additive, lalonde_mod[youngest, ]) - predict(lm.additive, lalonde[youngest, ])
predict(lm.additive, lalonde_mod[oldest, ]) - predict(lm.additive, lalonde[oldest, ])
coef(lm.additive)

# What happens if we add an interaction term to our model?
lm.non_additive <- lm(re78 ~ age*educ, data = lalonde)
summary(lm.non_additive)

# Let's again check our predictions
# For the two people with their years of schooling unchanged
predict(lm.non_additive, lalonde[youngest, ])
predict(lm.non_additive, lalonde[oldest, ])

# Adding one year of schooling...
predict(lm.non_additive, lalonde_mod[youngest, ])
predict(lm.non_additive, lalonde_mod[oldest, ])

# What's the difference?
predict(lm.non_additive, lalonde_mod[youngest, ]) - predict(lm.non_additive, lalonde[youngest, ])
predict(lm.non_additive, lalonde_mod[oldest, ]) - predict(lm.non_additive, lalonde[oldest, ])

# Why is this the case?
## re78 = c1*age + c2*educ + c3*age*educ
## re78_mod = c1*age + c2*(educ+1) + c3*age*(educ+1)
## re78_mod = c1*age + c2*educ + c2 + c3*age*educ + c3*age
## re78_mod = re78 + c2 + c3*age

coef(lm.non_additive)[3] + lalonde[youngest, ]$age*coef(lm.non_additive)[4]
coef(lm.non_additive)[3] + lalonde[oldest, ]$age*coef(lm.non_additive)[4]