PART I
Question 1
install.packages("smss")
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C:\Users\erink\AppData\Local\Temp\RtmpUtOWzQ\downloaded_packageslibrary("smss")
data("house.selling.price.2")
install.packages('plyr', repos = "http://cran.us.r-project.org")
package 'plyr' successfully unpacked and MD5 sums checked
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C:\Users\erink\AppData\Local\Temp\RtmpUtOWzQ\downloaded_packagesFor the house.selling.price.2 data the tables show a correlation matrix and a model fit using four predictors of selling price.
house<- house.selling.price.2
head(house)
P S Be Ba New
1 48.5 1.10 3 1 0
2 55.0 1.01 3 2 0
3 68.0 1.45 3 2 0
4 137.0 2.40 3 3 0
5 309.4 3.30 4 3 1
6 17.5 0.40 1 1 0summary(house)
P S Be Ba
Min. : 17.50 Min. :0.40 Min. :1.000 Min. :1.000
1st Qu.: 72.90 1st Qu.:1.33 1st Qu.:3.000 1st Qu.:2.000
Median : 96.00 Median :1.57 Median :3.000 Median :2.000
Mean : 99.53 Mean :1.65 Mean :3.183 Mean :1.957
3rd Qu.:115.00 3rd Qu.:1.98 3rd Qu.:4.000 3rd Qu.:2.000
Max. :309.40 Max. :3.85 Max. :5.000 Max. :3.000
New
Min. :0.0000
1st Qu.:0.0000
Median :0.0000
Mean :0.3011
3rd Qu.:1.0000
Max. :1.0000 lm(P~ S +Be + Ba + New, data = house)
Call:
lm(formula = P ~ S + Be + Ba + New, data = house)
Coefficients:
(Intercept) S Be Ba New
-41.795 64.761 -2.766 19.203 18.984 cor<- cor(x=house)
cor
P S Be Ba New
P 1.0000000 0.8988136 0.5902675 0.7136960 0.3565540
S 0.8988136 1.0000000 0.6691137 0.6624828 0.1762879
Be 0.5902675 0.6691137 1.0000000 0.3337966 0.2672091
Ba 0.7136960 0.6624828 0.3337966 1.0000000 0.1820651
New 0.3565540 0.1762879 0.2672091 0.1820651 1.0000000#correlate(house)
With these four predictors,
A. For backward elimination, which variable would be deleted first? Why?
Call:
lm(formula = P ~ ., data = house)
Residuals:
Min 1Q Median 3Q Max
-36.212 -9.546 1.277 9.406 71.953
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -41.795 12.104 -3.453 0.000855 ***
S 64.761 5.630 11.504 < 2e-16 ***
Be -2.766 3.960 -0.698 0.486763
Ba 19.203 5.650 3.399 0.001019 **
New 18.984 3.873 4.902 4.3e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 16.36 on 88 degrees of freedom
Multiple R-squared: 0.8689, Adjusted R-squared: 0.8629
F-statistic: 145.8 on 4 and 88 DF, p-value: < 2.2e-16We would delete Bedrooms (Be) because it has the highest P Value.
B. For forward selection, which variable would be added first? Why?
S, size would be added first because that is the variable that is the most significant (has the smallest P Value)
C. Why do you think that BEDS has such a large P-value in the multiple regression model, even though it has a substantial correlation with PRICE?
I think that there are too many other factors that might affect the price of a house with just one bedroom or many bedrooms, this variable alone isn’t significant.
D. Using software with these four predictors, find the model that would be selected using each criterion:
R2
Adjusted R2
Call:
lm(formula = P ~ ., data = house)
Residuals:
Min 1Q Median 3Q Max
-36.212 -9.546 1.277 9.406 71.953
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -41.795 12.104 -3.453 0.000855 ***
S 64.761 5.630 11.504 < 2e-16 ***
Be -2.766 3.960 -0.698 0.486763
Ba 19.203 5.650 3.399 0.001019 **
New 18.984 3.873 4.902 4.3e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 16.36 on 88 degrees of freedom
Multiple R-squared: 0.8689, Adjusted R-squared: 0.8629
F-statistic: 145.8 on 4 and 88 DF, p-value: < 2.2e-16lm() model includes R2 and Adjusted R-Squared
- PRESS
PRESS <- function(linear.model) {
pr <- residuals(linear.model)/(1 - lm.influence(linear.model)$hat)
sum(pr^2)
}
fit <- lm( P ~ Be + Ba + S + New , data = house)
PRESS(fit)
[1] 28390.22Used the linear.model function to find PRESS (thank you Piazza.)
- AIC
- BIC
E. Explain which model you prefer and why. I prefer the multiple regression model, which seems to provide the most information.
Question 2
(Data file: trees from base R) From the documentation: “This data set provides measurements of the diameter, height and volume of timber in 31 felled black cherry trees. Note that the diameter (in inches) is erroneously labelled Girth in the data. It is measured at 4 ft 6 in above the ground.”
Tree volume estimation is a big deal, especially in the lumber industry. Use the trees data to build a basic model of tree volume prediction. In particular,
A. fit a multiple regression model with the Volume as the outcome and Girth and Height as the explanatory variables
Girth Height Volume
Min. : 8.30 Min. :63 Min. :10.20
1st Qu.:11.05 1st Qu.:72 1st Qu.:19.40
Median :12.90 Median :76 Median :24.20
Mean :13.25 Mean :76 Mean :30.17
3rd Qu.:15.25 3rd Qu.:80 3rd Qu.:37.30
Max. :20.60 Max. :87 Max. :77.00 head(trees)
Girth Height Volume
1 8.3 70 10.3
2 8.6 65 10.3
3 8.8 63 10.2
4 10.5 72 16.4
5 10.7 81 18.8
6 10.8 83 19.7fit<- lm(Volume~ Girth + Height, data = trees)
fit
Call:
lm(formula = Volume ~ Girth + Height, data = trees)
Coefficients:
(Intercept) Girth Height
-57.9877 4.7082 0.3393 B. Run regression diagnostic plots on the model. Based on the plots, do you think any of the regression assumptions is violated?
Residuals vs Fitted: I don’t see a horizontal line with equally spread residuals. Assumptions are violated
Normal Q-Q: Appears that residuals are normally distributed.
Scale-Location: I do not see a horizontal line with equally spread points, suggesting Heteroscedasticity?
Question 3
(inspired by ALR 9.16)
(Data file: florida in alr R package)
In the 2000 election for U.S. president, the counting of votes in Florida was controversial. In Palm Beach County in south Florida, for example, voters used a so-called butterfly ballot. Some believe that the layout of the ballot caused some voters to cast votes for Buchanan when their intended choice was Gore.
The data has variables for the number of votes for each candidate—Gore, Bush, and Buchanan. Run a simple linear regression model where the Buchanan vote is the outcome and the Bush vote is the explanatory variable. Produce the regression diagnostic plots. Is Palm Beach County an outlier based on the diagnostic plots? Why or why not?
install.packages("alr4")
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C:\Users\erink\AppData\Local\Temp\RtmpUtOWzQ\downloaded_packages Gore Bush Buchanan
ALACHUA 47300 34062 262
BAKER 2392 5610 73
BAY 18850 38637 248
BRADFORD 3072 5413 65
BREVARD 97318 115185 570
BROWARD 386518 177279 789
CALHOUN 2155 2873 90
CHARLOTTE 29641 35419 182
CITRUS 25501 29744 270
CLAY 14630 41745 186
COLLIER 29905 60426 122
COLUMBIA 7047 10964 89
DADE 328702 289456 561
DE SOTO 3322 4256 36
DIXIE 1825 2698 29
DUVAL 107680 152082 650
ESCAMBIA 40958 73029 504
FLAGLER 13891 12608 83
FRANKLIN 2042 2448 33
GADSDEN 9565 4750 39
GILCHRIST 1910 3300 29
GLADES 1420 1840 9
GULF 2389 3546 71
HAMILTON 1718 2153 24
HARDEE 2341 3764 30
HENDRY 3239 4743 22
HERNANDO 32644 30646 242
HIGHLANDS 14152 20196 99
HILLSBOROUGH 166581 176967 836
HOLMES 2154 4985 76
INDIAN RIVER 19769 28627 105
JACKSON 6868 9138 102
JEFFERSON 3038 2481 29
LAFAYETTE 788 1669 10
LAKE 36555 49963 289
LEE 73560 106141 305
LEON 61425 39053 282
LEVY 5403 6860 67
LIBERTY 1011 1316 39
MADISON 3011 3038 29
MANATEE 49169 57948 272
MARION 44648 55135 563
MARTIN 26619 33864 108
MONROE 16483 16059 47
NASSAU 6952 16404 90
OKALOOSA 16924 52043 267
OKEECHOBEE 4588 5058 43
ORANGE 140115 134476 446
OSCEOLA 28177 26216 145
PALM BEACH 268945 152846 3407
PASCO 69550 68581 570
PINELLAS 199660 184312 1010
POLK 74977 90101 538
PUTNAM 12091 13439 147
ST. JOHNS 19482 39497 229
ST. LUCIE 41559 34705 124
SANTA ROSA 12795 36248 311
SARASOTA 72854 83100 305
SEMINOLE 58888 75293 194
SUMTER 9634 12126 114
SUWANNEE 4084 8014 108
TAYLOR 2647 4051 27
UNION 1399 2326 26
VOLUSIA 97063 82214 396
WAKULLA 3835 4511 46
WALTON 5637 12176 120
WASHINGTON 2796 4983 88head(florida)
Gore Bush Buchanan
ALACHUA 47300 34062 262
BAKER 2392 5610 73
BAY 18850 38637 248
BRADFORD 3072 5413 65
BREVARD 97318 115185 570
BROWARD 386518 177279 789summary(florida)
Gore Bush Buchanan
Min. : 788 Min. : 1316 Min. : 9.0
1st Qu.: 3055 1st Qu.: 4746 1st Qu.: 46.5
Median : 14152 Median : 20196 Median : 114.0
Mean : 43341 Mean : 43356 Mean : 258.5
3rd Qu.: 45974 3rd Qu.: 56542 3rd Qu.: 285.5
Max. :386518 Max. :289456 Max. :3407.0
Call:
lm(formula = Buchanan ~ ., data = florida)
Residuals:
Min 1Q Median 3Q Max
-913.70 -65.68 -36.96 21.41 2204.20
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 82.3532929 52.1819372 1.578 0.11945
Gore 0.0043013 0.0013309 3.232 0.00194 **
Bush -0.0002379 0.0017476 -0.136 0.89216
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 330.7 on 64 degrees of freedom
Multiple R-squared: 0.4747, Adjusted R-squared: 0.4582
F-statistic: 28.91 on 2 and 64 DF, p-value: 1.132e-09installed.packages("ggplot2")
Package LibPath Version Priority Depends Imports LinkingTo
Suggests Enhances License License_is_FOSS License_restricts_use
OS_type Archs MD5sum NeedsCompilation Builtlibrary(ggplot2)
ggplot(data = florida) +
geom_point(mapping = aes(x = Buchanan, y = Bush))
Yes, I can visually see from the plots that there is an outlier (above 3000 for Buchanan), and when I check that plot in the data, I can see that it is Palm Beach.
See separate File for PART II