swiss.dt <- swiss
plot(Agriculture ~ Education, data = swiss.dt)
#Linear model
swiss.lm <- lm(Agriculture ~ Education, data = swiss.dt)
coef(swiss.lm)
## (Intercept) Education
## 67.243236 -1.510527
plot(Agriculture ~ Education, data = swiss.dt)
abline(coef(swiss.lm))
#Access fitted values:
fitted(swiss.lm)[1:5]
## Courtelary Delemont Franches-Mnt Moutier Neuveville
## 49.11691 53.64849 59.69060 56.66954 44.58533
#Residuals
residuals(swiss.lm)[1:5]
## Courtelary Delemont Franches-Mnt Moutier Neuveville
## -32.116908 -8.548490 -19.990599 -20.169545 -1.085326
summary(swiss.lm)
##
## Call:
## lm(formula = Agriculture ~ Education, data = swiss.dt)
##
## Residuals:
## Min 1Q Median 3Q Max
## -42.927 -9.633 2.188 12.762 26.720
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 67.2432 3.9321 17.10 < 2e-16 ***
## Education -1.5105 0.2707 -5.58 1.3e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 17.65 on 45 degrees of freedom
## Multiple R-squared: 0.409, Adjusted R-squared: 0.3959
## F-statistic: 31.14 on 1 and 45 DF, p-value: 1.305e-06
confint(swiss.lm)
## 2.5 % 97.5 %
## (Intercept) 59.323529 75.1629429
## Education -2.055715 -0.9653397
#Graphing the confience and Prediction bands
library(HH)
## Loading required package: lattice
## Loading required package: grid
## Loading required package: latticeExtra
## Loading required package: RColorBrewer
## Loading required package: multcomp
## Loading required package: mvtnorm
## Loading required package: survival
## Loading required package: TH.data
## Loading required package: MASS
##
## Attaching package: 'TH.data'
## The following object is masked from 'package:MASS':
##
## geyser
## Loading required package: gridExtra
ci.plot(swiss.lm)
#Hypothesis testing
summary(swiss.lm)
##
## Call:
## lm(formula = Agriculture ~ Education, data = swiss.dt)
##
## Residuals:
## Min 1Q Median 3Q Max
## -42.927 -9.633 2.188 12.762 26.720
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 67.2432 3.9321 17.10 < 2e-16 ***
## Education -1.5105 0.2707 -5.58 1.3e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 17.65 on 45 degrees of freedom
## Multiple R-squared: 0.409, Adjusted R-squared: 0.3959
## F-statistic: 31.14 on 1 and 45 DF, p-value: 1.305e-06
#Pr is sig less than .05
#Simple coefficient of determination
anova(swiss.lm)
## Analysis of Variance Table
##
## Response: Agriculture
## Df Sum Sq Mean Sq F value Pr(>F)
## Education 1 9704 9704.0 31.141 1.305e-06 ***
## Residuals 45 14023 311.6
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
swisSum<- summary(swiss.lm)
swisSum$r.squared
## [1] 0.4089891
#Residual Analysis
#Normality assumption
hist(residuals(swiss.lm))
library(RcmdrMisc)
## Loading required package: car
## Loading required package: carData
##
## Attaching package: 'car'
## The following objects are masked from 'package:HH':
##
## logit, vif
## Loading required package: sandwich
qqPlot(swiss.lm)
## La Chauxdfnd ValdeTravers
## 40 44
#Shapiro test: if p-value is less than .05 then normality assumption is upheld
shapiro.test(residuals(swiss.lm))
##
## Shapiro-Wilk normality test
##
## data: residuals(swiss.lm)
## W = 0.95732, p-value = 0.08418
#Constant Variance Assumption
plot(swiss.lm, which = 3)
#Data appears to somewhate of a slight fanning out of the residuals
library(lmtest)
## Loading required package: zoo
##
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric
bptest(swiss.lm)
##
## studentized Breusch-Pagan test
##
## data: swiss.lm
## BP = 0.22039, df = 1, p-value = 0.6387
#Indepedence Assumption
#There is no obvious cyclcial wave pattern or structure, looking for random and no structure
plot(swiss.lm, which = 1)
#Durbin-watson test
dwtest(swiss.lm, alternative = "two.sided")
##
## Durbin-Watson test
##
## data: swiss.lm
## DW = 1.0969, p-value = 0.0007551
## alternative hypothesis: true autocorrelation is not 0