Code Challenge 4

\[\newcommand{\E}{\mathbb{E}}\]

For this problem, we will use the Caschool data which you can load with the following command (you may need to install the Ecdat package first if you have not previously done so)

data(Caschool, package="Ecdat")

I would like for you to run the following two regressions:

1. A regression of mathscr on the following variables: str, avginc.

2. A regression of mathscr on the following variables: str, avginc, grspan, enrltot, teachers, calwpct, mealpct, computer, compstu, expnstu, elpct.

For each of the two models, compute \(R^2\), adjusted \(R^2\), \(AIC\), and \(BIC\).

To win

Email me, brantly.callaway@uga.edu all 8 of the numbers that you calculated AND which model you would choose to make predictions with according to each criteria.

Solution below…

data(Caschool, package="Ecdat")
reg1 <- lm(mathscr ~ str + avginc, data=Caschool)
reg2 <- lm(mathscr ~ str + avginc + grspan + enrltot + teachers + calwpct + mealpct + computer + compstu + expnstu + elpct, data=Caschool)

r_squared1 <- summary(reg1)$r.squared
r_squared2 <- summary(reg2)$r.squared
r_squared1

[1] 0.4903198

r_squared2

[1] 0.7319691

n <- nrow(Caschool)
k1 <- 3
k2 <- length(coef(reg2))


SSR1 <- sum(resid(reg1)^2)
SSR2 <- sum(resid(reg2)^2)
barY <- mean(Caschool$mathscr)
TSS <- sum( (Caschool$mathscr - barY)^2)
adj_r_squared1 <- 1 - (n-1)/(n-k1) * SSR1/TSS
adj_r_squared2 <- 1 - (n-1)/(n-k2) * SSR2/TSS
adj_r_squared1

[1] 0.4878753

adj_r_squared2

[1] 0.7247428

aic1 <- 2*k1 + n*log(SSR1)
aic2 <- 2*k2 + n*log(SSR2)
aic1

[1] 4721.229

aic2

[1] 4469.303

bic1 <- k1*log(n) + n*log(SSR1)
bic2 <- k2*log(n) + n*log(SSR2)
bic1

[1] 4733.349

bic2

[1] 4517.786

You can see that all four model selection criteria favor the second model. The \(R^2\) is higher, the adjusted \(R^2\) is higher, the AIC is lower, and the BIC is lower.