getwd()## [1] "C:/Users/n0910803/Downloads"# 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)
#}# read the CSV with headers
regression1<-read.csv("incidents.csv", header=T,sep =",")
#we tell that the first row is a header with the weader=t
#as it is related to the previous file, in this case incidents = infections.#View(regression1)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#how many data points do I have in this dataset= 16
#data type of area+ Character
#population = #character (this is no good)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 ...regression1$population <- as.numeric(gsub(",","",regression1$population))
regression1$population## [1] 107353 326534 444752 750000 64403 2744878 1600000 2333000 1572816
## [10] 712091 6900000 2700000 4900000 4200000 5200000 7100000#replaces the limitor to convert the population in a data type = numericstr(regression1$population)## num [1:16] 107353 326534 444752 750000 64403 ...#to know more about the structure.
#to know more about the distribution I use Histogram.
#to know more about the values I use summary.regression2<-regression1[,-1]#new data frame with the deletion of column 1
#we are taking out "area".head(regression2)reg.fit1<-lm(regression1$incidents ~ regression1$population)summary(reg.fit1)##
## Call:
## lm(formula = regression1$incidents ~ regression1$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 *
## regression1$population 8.462e-05 5.804e-05 1.458 0.1669
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 534.9 on 14 degrees of freedom
## Multiple R-squared: 0.1318, Adjusted R-squared: 0.0698
## F-statistic: 2.126 on 1 and 14 DF, p-value: 0.1669Based on the output obtained above please answer the following question: Is Population significant at a 5% significance level? What is the adjusted-R squared of the model?
#H0: Population does not affect incidents.
#We fail to reject the Null Hypothesis.
#pop is not significant because p-value is 0.1669 (greater than 0.05)
#Pop explains only 6,98% of the incidents. reg.fit2<-lm(incidents ~ zone+population, data = regression1)summary(reg.fit2)##
## Call:
## lm(formula = incidents ~ zone + population, data = regression1)
##
## Residuals:
## Min 1Q Median 3Q Max
## -537.21 -273.14 -57.89 188.17 766.03
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.612e+02 1.675e+02 0.962 0.35363
## zonewest 7.266e+02 1.938e+02 3.749 0.00243 **
## population 6.557e-05 4.206e-05 1.559 0.14300
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 384.8 on 13 degrees of freedom
## Multiple R-squared: 0.5828, Adjusted R-squared: 0.5186
## F-statistic: 9.081 on 2 and 13 DF, p-value: 0.003404Based on the output obtained above please answer the following question: Are Population and/or Zone significant at a 5% significance level? What is the adjusted-R squared of the model?
#Population is not significant.
#However if the zone is West, we are likely to have more incidents.
#Zone is critical at a 5% critical level.
#The model is significant: p-value is 0.003404
#This model explains 51.86% of the incidents. It improved from the previous model.#How about about the interaction between zone and pop?regression1$zone <- ifelse(regression1$zone == "west", 1, 0)#Please explain the syntax and the output.
#here we are converting this variable in numerical.#View(regression1)str(regression1)## 'data.frame': 16 obs. of 4 variables:
## $ area : chr "Boulder" "California-lexington" "Huntsville" "Seattle" ...
## $ zone : num 1 0 0 1 1 0 1 1 0 0 ...
## $ population: num 107353 326534 444752 750000 64403 ...
## $ incidents : int 605 103 161 1703 1003 527 721 704 105 403 ...#regression1$zone<-as.integer((regression1$zone),replace=TRUE) was not necessaryinteraction<-regression1$zone*regression1$population#Please explain the syntax.reg.fit3<-lm(regression1$incidents~interaction+regression1$population+regression1$zone)summary(reg.fit3)##
## Call:
## lm(formula = regression1$incidents ~ interaction + regression1$population +
## regression1$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
## regression1$population 6.352e-05 7.868e-05 0.807 0.4352
## regression1$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? Is Zone significant at a 5% significance level? Is the interaction term significant at a 5% significance level?
Let us now run a model where the only feature is the interaction term. Is the interaction term significant at a 5% significance level? What is the adjusted-R squared of the model?
#The interaction is not good, it is not significant
#Population is not significant.
#However if the zone is significant. Zone is critical at a 5% critical level.
#The model is critical: 0.01237 p-value.
#This model explains 47.86% of the incidents.
#is not as good as the previous model.reg.fit4<-lm(regression1$incidents~interaction)summary(reg.fit4)##
## Call:
## lm(formula = regression1$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: 0.01093Which of the modes run above would you choose to make predictions? Why?
#it is significant in this case. p-value is 0.01093
# The adjusted R-Squared is lower, but the interaction term is significant.
# Here you have the model with only one variable, and it is significant.