R Assignment 6
Nicole Hayek
2022-11-05
Problem 6
In this exercise, you will further analyze the Wage data set considered throughout this chapter.
- Perform polynomial regression to predict wage using age . Use cross-validation to select the optimal degree d for the polynomial. What degree was chosen, and how does this compare to the results of hypothesis testing using ANOVA? Make a plot of the resulting polynomial fit to the data.
library(ISLR)
library(boot)
set.seed(1)
degree <- 10
cv.errs <- rep(NA, degree)
for (i in 1:degree) {
fit <- glm(wage ~ poly(age, i), data = Wage)}The test MSE of degree 4 is small enough although the minimum is at degree 9. The ANOVA comparison allows degree 4 to be enough.
plot(wage ~ age, data = Wage, col = "darkgrey")
age.range <- range(Wage$age)
age.grid <- seq(from = age.range[1], to = age.range[2])
fit <- lm(wage ~ poly(age, 3), data = Wage)
preds <- predict(fit, newdata = list(age = age.grid))
lines(age.grid, preds, col = "red", lwd = 2)
- Fit a step function to predict wage using age , and perform cross-validation to choose the optimal number of cuts. Make a plot of the fit obtained.
cv.errs <- rep(NA, degree)
for (i in 2:degree)
Wage$age.cut <- cut(Wage$age, i)
fit <- glm(wage ~ age.cut, data = Wage)
cv.errs[i] <- cv.glm(Wage, fit)$delta[1]This Model shows that 8 is the minimum MSE.
plot(wage ~ age, data = Wage, col = "darkgrey")
fit <- glm(wage ~ cut(age, 8), data = Wage)
preds <- predict(fit, list(age = age.grid))
lines(age.grid, preds, col = "red", lwd = 2)
Problem 10
This question relates to the College data set.
- Split the data into a training set and a test set. Using out-of-state tuition as the response and the other variables as the predictors, perform forward stepwise selection on the training set in order to identify a satisfactory model that uses just a subset of the predictors.
library(ISLR)
library(leaps)
train <- sample(1: nrow(College), nrow(College)/2)
test <- -train
fit <- regsubsets(Outstate ~ ., data = College, subset = train, method = 'forward')
fit.summary <- summary(fit)
fit.summary## Subset selection object
## Call: regsubsets.formula(Outstate ~ ., data = College, subset = train,
## method = "forward")
## 17 Variables (and intercept)
## Forced in Forced out
## PrivateYes FALSE FALSE
## Apps FALSE FALSE
## Accept FALSE FALSE
## Enroll FALSE FALSE
## Top10perc FALSE FALSE
## Top25perc FALSE FALSE
## F.Undergrad FALSE FALSE
## P.Undergrad FALSE FALSE
## Room.Board FALSE FALSE
## Books FALSE FALSE
## Personal FALSE FALSE
## PhD FALSE FALSE
## Terminal FALSE FALSE
## S.F.Ratio FALSE FALSE
## perc.alumni FALSE FALSE
## Expend FALSE FALSE
## Grad.Rate FALSE FALSE
## 1 subsets of each size up to 8
## Selection Algorithm: forward
## PrivateYes Apps Accept Enroll Top10perc Top25perc F.Undergrad
## 1 ( 1 ) " " " " " " " " " " " " " "
## 2 ( 1 ) "*" " " " " " " " " " " " "
## 3 ( 1 ) "*" " " " " " " " " " " " "
## 4 ( 1 ) "*" " " " " " " " " " " " "
## 5 ( 1 ) "*" " " " " " " " " " " " "
## 6 ( 1 ) "*" " " " " " " " " " " " "
## 7 ( 1 ) "*" " " "*" " " " " " " " "
## 8 ( 1 ) "*" " " "*" "*" " " " " " "
## P.Undergrad Room.Board Books Personal PhD Terminal S.F.Ratio
## 1 ( 1 ) " " " " " " " " " " " " " "
## 2 ( 1 ) " " " " " " " " " " " " " "
## 3 ( 1 ) " " " " " " " " " " " " " "
## 4 ( 1 ) " " " " " " " " " " "*" " "
## 5 ( 1 ) " " "*" " " " " " " "*" " "
## 6 ( 1 ) " " "*" " " " " " " "*" " "
## 7 ( 1 ) " " "*" " " " " " " "*" " "
## 8 ( 1 ) " " "*" " " " " " " "*" " "
## perc.alumni Expend Grad.Rate
## 1 ( 1 ) " " "*" " "
## 2 ( 1 ) " " "*" " "
## 3 ( 1 ) " " "*" "*"
## 4 ( 1 ) " " "*" "*"
## 5 ( 1 ) " " "*" "*"
## 6 ( 1 ) "*" "*" "*"
## 7 ( 1 ) "*" "*" "*"
## 8 ( 1 ) "*" "*" "*"coef(fit, id = 6)## (Intercept) PrivateYes Room.Board Terminal perc.alumni
## -4205.9594359 2881.8919492 0.6364738 44.5484531 34.8779635
## Expend Grad.Rate
## 0.2967855 40.4176035- Fit a GAM on the training data, using out-of-state tuition as the response and the features selected in the previous step as the predictors. Plot the results, and explain your findings.
library(gam)## Loading required package: splines## Loading required package: foreach## Loaded gam 1.22gam.mod <- gam(Outstate ~ Private + s(Room.Board, 5) + s(Terminal, 5) + s(perc.alumni, 5) + s(Expend, 5) + s(Grad.Rate, 5), data = College, subset = train)
par(mfrow = c(2,3))
plot(gam.mod, se = TRUE)
Looking at the curves we see that Expend and Grad.Rate are strong
non-linear with out of state.
- Evaluate the model obtained on the test set, and explain the results obtained.
preds <- predict(gam.mod, College[test, ])
RSS <- sum((College[test, ]$Outstate - preds)^2) # based on equation (3.16)
TSS <- sum((College[test, ]$Outstate - mean(College[test, ]$Outstate)) ^ 2)
1 - (RSS / TSS)## [1] 0.7566204The R-squared is 0.785.
- For which variables, if any, is there evidence of a non-linear relationship with the response?
summary(gam.mod)##
## Call: gam(formula = Outstate ~ Private + s(Room.Board, 5) + s(Terminal,
## 5) + s(perc.alumni, 5) + s(Expend, 5) + s(Grad.Rate, 5),
## data = College, subset = train)
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -7090.88 -1070.48 -27.63 1140.73 4710.59
##
## (Dispersion Parameter for gaussian family taken to be 3137799)
##
## Null Deviance: 6312291527 on 387 degrees of freedom
## Residual Deviance: 1132742854 on 360.9992 degrees of freedom
## AIC: 6933.216
##
## Number of Local Scoring Iterations: NA
##
## Anova for Parametric Effects
## Df Sum Sq Mean Sq F value Pr(>F)
## Private 1 1965386166 1965386166 626.358 < 2.2e-16 ***
## s(Room.Board, 5) 1 1380289614 1380289614 439.891 < 2.2e-16 ***
## s(Terminal, 5) 1 530388128 530388128 169.032 < 2.2e-16 ***
## s(perc.alumni, 5) 1 265962578 265962578 84.761 < 2.2e-16 ***
## s(Expend, 5) 1 555598895 555598895 177.066 < 2.2e-16 ***
## s(Grad.Rate, 5) 1 143084955 143084955 45.600 5.811e-11 ***
## Residuals 361 1132742854 3137799
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Anova for Nonparametric Effects
## Npar Df Npar F Pr(F)
## (Intercept)
## Private
## s(Room.Board, 5) 4 2.2898 0.05938 .
## s(Terminal, 5) 4 0.9933 0.41112
## s(perc.alumni, 5) 4 0.7520 0.55722
## s(Expend, 5) 4 9.1565 4.687e-07 ***
## s(Grad.Rate, 5) 4 3.2102 0.01310 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Anova for Nonparametric Effects shows Expend has strong non-linear relationshop with the Outstate. Grad.Rate and PhD have moderate non-linear relationship with the Outstate. This matched what we saw in part b