getwd()[1] "/cloud/project"regression1 <- read.csv("incidents.csv",header=T, sep=",")
str(regression1)'data.frame': 16 obs. of 4 variables:
$ area : chr "Boulder" "California-lexington" "Huntsville" "Seattle" ...
$ zone : chr "west" "east" "east" "west" ...
$ population: chr "107,353" "326,534" "444,752" "750,000" ...
$ incidents : int 605 103 161 1703 1003 527 721 704 105 403 ...summary(regression1) area zone population incidents
Length:16 Length:16 Length:16 Min. : 103.0
Class :character Class :character Class :character 1st Qu.: 277.8
Mode :character Mode :character Mode :character Median : 654.0
Mean : 695.2
3rd Qu.: 853.0
Max. :2072.0 # make sure the packages for this chapter
# are installed, install if necessary
pkg <- c("ggplot2", "scales", "maptools",
"sp", "maps", "grid", "car" )
new.pkg <- pkg[!(pkg %in% installed.packages())]
if (length(new.pkg)) {
install.packages(new.pkg) }Warning in install.packages :
package ‘maptools’ is not available for this version of R
A version of this package for your version of R might be available elsewhere,
see the ideas at
https://cran.r-project.org/doc/manuals/r-patched/R-admin.html#Installing-packagestrying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/xts_0.14.1.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/TTR_0.24.4.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/quantmod_0.4.28.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/lmtest_0.9-40.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/tseries_0.10-58.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/tibble_3.3.0.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/tidyr_1.3.2.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/pillar_1.11.1.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/tidyselect_1.2.1.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/cowplot_1.2.0.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/forecast_8.24.0.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/modelr_0.1.11.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/broom_1.0.11.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/dplyr_1.1.4.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/doBy_4.7.1.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/reformulas_0.4.3.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/Rdpack_2.6.4.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/RcppEigen_0.3.4.0.2.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/gtable_0.3.6.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/isoband_0.3.0.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/S7_0.2.1.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/withr_3.0.2.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/farver_2.1.2.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/labeling_0.4.3.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/RColorBrewer_1.1-3.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/viridisLite_0.4.2.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/carData_3.0-5.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/abind_1.4-8.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/Formula_1.2-5.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/pbkrtest_0.5.5.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/quantreg_6.1.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/lme4_1.1-38.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/ggplot2_4.0.1.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/scales_1.4.0.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/sp_2.2-0.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/maps_3.4.3.tar.gz'
trying URL 'http://rspm/default/__linux__/focal/latest/src/contrib/car_3.1-3.tar.gz'Warning in install.packages :
installation of package ‘RcppEigen’ had non-zero exit status
Warning in install.packages :
installation of package ‘lme4’ had non-zero exit status
The downloaded source packages are in
‘/tmp/RtmpEKPcRW/downloaded_packages’regression1$population <- as.numeric(gsub(",","",regression1$population))
regression1$population [1] 107353 326534 444752 750000 64403 2744878 1600000 2333000 1572816 712091 6900000 2700000 4900000 4200000 5200000 7100000regression2<-regression1[,-1]head(regression2)reg.fit1<-lm(regression2$incidents ~ regression2$population)
summary(reg.fit1)
Call:
lm(formula = regression2$incidents ~ regression2$population)
Residuals:
Min 1Q Median 3Q Max
-684.5 -363.5 -156.2 133.9 1164.7
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.749e+02 2.018e+02 2.353 0.0337 *
regression2$population 8.462e-05 5.804e-05 1.458 0.1669
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 534.9 on 14 degrees of freedom
Multiple R-squared: 0.1318, Adjusted R-squared: 0.0698
F-statistic: 2.126 on 1 and 14 DF, p-value: 0.1669Based on the output obtained above, please answer the following question: Is Population significant at a 5% significance level? What is the adjusted-R squared of the model?
Ans: Population is not a statistically significant predictor of incidents at the 5% significance level (p = 0.1669). The model has a low adjusted R-squared of 0.0698, indicating limited explanatory power.
reg.fit2<-lm(incidents ~ zone+population, data = regression2)
summary(reg.fit2)
Call:
lm(formula = incidents ~ zone + population, data = regression2)
Residuals:
Min 1Q Median 3Q Max
-537.21 -273.14 -57.89 188.17 766.03
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.612e+02 1.675e+02 0.962 0.35363
zone 7.266e+02 1.938e+02 3.749 0.00243 **
population 6.557e-05 4.206e-05 1.559 0.14300
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 384.8 on 13 degrees of freedom
Multiple R-squared: 0.5828, Adjusted R-squared: 0.5186
F-statistic: 9.081 on 2 and 13 DF, p-value: 0.003404Based on the output obtained above, please answer the following question: Are Population and/or Zone significant at a 5% significance level? What is the adjusted-R squared of the model?
At the 5% significance level, Zone is a statistically significant predictor of incidents, while Population is not. The adjusted R-squared of the model is 0.5186, indicating that approximately 52% of the variation in incidents is explained by the model.
regression2$zone <- ifelse(regression2$zone == "west", 1, 0)interaction<-regression2$zone*regression2$populationreg.fit3<-lm(regression2$incidents~interaction+regression2$population+regression2$zone)
summary(reg.fit3)
Call:
lm(formula = regression2$incidents ~ interaction + regression2$population +
regression2$zone)
Residuals:
Min 1Q Median 3Q Max
-540.91 -270.93 -59.56 187.99 767.99
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.659e+02 2.313e+02 0.717 0.4869
interaction 2.974e-06 9.469e-05 0.031 0.9755
regression2$population 6.352e-05 7.868e-05 0.807 0.4352
regression2$zone 7.192e+02 3.108e+02 2.314 0.0392 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 400.5 on 12 degrees of freedom
Multiple R-squared: 0.5829, Adjusted R-squared: 0.4786
F-statistic: 5.589 on 3 and 12 DF, p-value: 0.01237Based on the output obtained above, please answer the following question:
Is Population significant at a 5% significance level? Ans: It not Not significant cause (0.4352 > 0.05)
Is Zone significant at a 5% significance level? Ans: Zone is a statistically significant predictor of incidents
Is the interaction term significant at a 5% significance level? What is the adjusted-R squared of the model? Ans: the interaction term are not statistically significant cause (0.9755<0.05). The adjusted R-squared of the model is 0.4786, indicating that approximately 48% of the variation in incidents is explained by the model.
reg.fit4<-lm(regression2$incidents~interaction)
summary(reg.fit4)
Call:
lm(formula = regression2$incidents ~ interaction)
Residuals:
Min 1Q Median 3Q Max
-650.28 -301.09 -83.71 123.23 1103.76
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.951e+02 1.320e+02 3.751 0.00215 **
interaction 1.389e-04 4.737e-05 2.932 0.01093 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 451.9 on 14 degrees of freedom
Multiple R-squared: 0.3804, Adjusted R-squared: 0.3361
F-statistic: 8.595 on 1 and 14 DF, p-value: 0.01093Is the interaction term significant at a 5% significance level? What is the adjusted-R squared of the model? Ans: Yes, it is significant at the 5% level (p = 0.01093). The adjusted R-squared of the model is 0.3361, indicating that approximately 34% of the variation in incidents is explained by the model.
Which of the models run above would you choose to make predictions? Why?? Ans: The model reg.fit2 is preferred for prediction because it has the highest adjusted R-squared (0.5186), indicating the strongest explanatory power. It provides the best trade-off between explanatory power and simplicity, maximizing adjusted R-squared while avoiding unnecessary interaction terms.