Home Work 5

Author

Megha Joseph

Published

December 11, 2022

Code

# Setup
library(tidyverse)

── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
✔ ggplot2 3.3.6      ✔ purrr   0.3.4 
✔ tibble  3.1.8      ✔ dplyr   1.0.10
✔ tidyr   1.2.1      ✔ stringr 1.4.1 
✔ readr   2.1.3      ✔ forcats 0.5.2 
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()

Code

library(plyr)

Warning: package 'plyr' was built under R version 4.2.2

------------------------------------------------------------------------------
You have loaded plyr after dplyr - this is likely to cause problems.
If you need functions from both plyr and dplyr, please load plyr first, then dplyr:
library(plyr); library(dplyr)
------------------------------------------------------------------------------

Attaching package: 'plyr'

The following objects are masked from 'package:dplyr':

    arrange, count, desc, failwith, id, mutate, rename, summarise,
    summarize

The following object is masked from 'package:purrr':

    compact

Code

library(alr4)

Warning: package 'alr4' was built under R version 4.2.2

Loading required package: car

Warning: package 'car' was built under R version 4.2.2

Loading required package: carData

Warning: package 'carData' was built under R version 4.2.2


Attaching package: 'car'

The following object is masked from 'package:dplyr':

    recode

The following object is masked from 'package:purrr':

    some

Loading required package: effects

Warning: package 'effects' was built under R version 4.2.2

lattice theme set by effectsTheme()
See ?effectsTheme for details.

Code

library(smss)

Warning: package 'smss' was built under R version 4.2.2

Question 1

Part A

The first variable to be deleted would be beds because it has the largest p-value.

Part B

The first variable added in forward selection would be size because it has the smallest p-value.

Part C

Beds has such a large p-value despite its correlation with price because it also has strong correlations with other variables. This may cause multicollinearity.

Part D

Code

#test regression models
library(smss)
data(house.selling.price.2)
full <- lm(P ~ ., data = house.selling.price.2)
forw1 <- lm(P ~ S, data = house.selling.price.2)
forw2 <- lm(P ~ S + New, data = house.selling.price.2)
forw3 <- lm(P ~ S + New + Ba, data = house.selling.price.2)

a.

I used the forward selection method to fit models, adding variables, one at a time, based on t-values (highest to lowest).

\(R^{2}\) is highest for the full model with all variables.

b.

Adjusted \(R^{2}\) is highest for the model of Price as a function of Size, Baths, and New.

c.

Code

#calculate PRESS statistics
PRESS <- function(linear.model) {
  pr <- residuals(linear.model)/(1-lm.influence(linear.model)$hat) 
  PRESS <- sum(pr^2) 
  return(PRESS)
}
PRESS(full)

[1] 28390.22

Code

PRESS(forw1)

[1] 38203.29

Code

PRESS(forw2)

[1] 31066

Code

PRESS(forw3)

[1] 27860.05

The model with Price as a function of Size, Baths, and New has the lowest PRESS calculation.

d.

Code

#calculate AIC values
AIC(full, k=2)

[1] 790.6225

Code

AIC(forw1, k=2)

[1] 820.1439

Code

AIC(forw2, k=2)

[1] 800.1262

Code

AIC(forw3, k=2)

[1] 789.1366

The model with Price as a function of Size, Baths, and New has the lowest AIC calculation.

e.

Code

#calculate BIC values
BIC(full)

[1] 805.8181

Code

BIC(forw1)

[1] 827.7417

Code

BIC(forw2)

[1] 810.2566

Code

BIC(forw3)

[1] 801.7996

The model with Price as a function of Size, Baths, and New has the lowest BIC calculation.

Part E

As stated before, Model 1 has better results in four out of the five criteria (Adjusted R-Squared, PRESS, AIC, and BIC). Thus, the model I would prefer overall is Model 1, which omitted the Bed variable. The Bed variable also had an extremely high p-value compared to the other variables, so it would make sense to construct a model without it.

Question 2

Part A

Code

model <- lm(Volume ~ Girth + Height, data = trees)
summary(model)


Call:
lm(formula = Volume ~ Girth + Height, data = trees)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.4065 -2.6493 -0.2876  2.2003  8.4847 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -57.9877     8.6382  -6.713 2.75e-07 ***
Girth         4.7082     0.2643  17.816  < 2e-16 ***
Height        0.3393     0.1302   2.607   0.0145 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.882 on 28 degrees of freedom
Multiple R-squared:  0.948, Adjusted R-squared:  0.9442 
F-statistic:   255 on 2 and 28 DF,  p-value: < 2.2e-16

Part B

Code

plot(model)

There are some regression assumptions that are violated because of the data present in the plots. In the Residuals vs. Fitted plot, the line is not linear, indicating that the variances of the error terms are not equal and there may not be a linear relationship. The Scale-Location plot also shows a non-linear line, indicating that the assumption of constant variance is violated. The other two plots, Normal Q-Q and Residuals vs. Leverage, are normal.

Question 3

Part A

Code

model <- lm(Buchanan ~ Bush, data = florida)
plot(model)

Palm Beach is an outlier based on the diagnostic plots for the model because, while all the other data points are fairly close together, the Palm Beach data point is extremely far in each plot. Additionally, in the Residuals vs. Leverage plot, the Palm Beach point is outside of Cook’s distance, meaning it is an outlier with extreme influence on the data.

Part B

Code

model <- lm(log(Buchanan) ~ log(Bush), data = florida)
plot(model)

The findings do change somewhat because in the new model using logs, the Palm Beach data point is now inside Cook’s distance, meaning it has less influence over the data and is less of an outlier. The distance between Palm Beach and the other data points has been reduced, but it still seems to remain somewhat of an outlier but much less than in the previous model.