library(car)
## Loading required package: carData
setwd("C:/Users/lhomm/OneDrive/Documents/R")
Dat <- read.table("https://users.stat.ufl.edu/~rrandles/sta4210/Rclassnotes/data/textdatasets/KutnerData/Chapter%2022%20Data%20Sets/CH22PR11.txt")
colnames(Dat) <- c("Days", "Fitness", "Count", "Age")
attach(Dat)
Fitness <- as.factor(Dat$Fitness)
levels(Fitness) <- c("Below Average", "About Average", "Above Average")
Res.Age <- Dat$Age - mean(Dat$Age)
Lm1 <- lm(Days ~ Res.Age + Fitness)
Lm1$residuals
## 1 2 3 4 5
## 2.069714e-01 -4.502794e-01 -3.647770e-01 -2.324352e-01 8.999065e-01
## 6 7 8 9 10
## 1.177507e-01 -5.439582e-01 3.668211e-01 1.361341e-01 7.397358e-05
## 11 12 13 14 15
## -6.690716e-01 -9.300368e-01 3.190336e-01 -3.813382e-01 4.201491e-01
## 16 17 18 19 20
## 5.837172e-01 -1.634941e-01 6.848327e-01 7.938041e-01 -2.099143e-01
## 21 22 23 24
## 2.294924e-01 2.800501e-01 -1.038910e+00 -5.452268e-02
Lm1Res1 <-Lm1$residuals[1:8]
Lm1Res2 <-Lm1$residuals[9:18]
Lm1Res3 <-Lm1$residuals[19:24]
Lm1Fit1 <- Lm1$fitted.values[1:8]
Lm1Fit2 <- Lm1$fitted.values[9:18]
Lm1Fit3 <- Lm1$fitted.values[19:24]
PlotRF1 <- plot(Lm1Res1, Lm1Fit1)
PlotRF2 <- plot(Lm1Res2, Lm1Fit2)
PlotRF3 <- plot(Lm1Res3, Lm1Fit3)
StandardLM1 = rstandard(Lm1)
qqnorm(StandardLM1, datax = TRUE)
qqline(StandardLM1, datax = TRUE)
shapiro.test(StandardLM1)
##
## Shapiro-Wilk normality test
##
## data: StandardLM1
## W = 0.98277, p-value = 0.9409
### The coefficient of correlation is .98277 we can conclude that there is a high level of correlation and that the data is normally distributed. ###
### Y_i_j = Mu + Tau_1I_i_j_1 + Tau_2I_i_j_2 + Beta_1X^*_i_j + Beta_2I_i_j_1X^*_i_j + Beta_3I_i_j_2X^*_i_j + e_i_j ###
### Ho: Beta_2 = Beta_3 = 0 H1: ∃i ∈ 2, 3 : βi ̸= 0 ###
Lm2 <- lm(Days ~ Res.Age*Fitness)
Lm2
##
## Call:
## lm(formula = Days ~ Res.Age * Fitness)
##
## Coefficients:
## (Intercept) Res.Age
## 34.87779 1.19510
## FitnessAbout Average FitnessAbove Average
## -1.79582 -8.70822
## Res.Age:FitnessAbout Average Res.Age:FitnessAbove Average
## -0.05017 -0.05821
Anova(Lm1)
## Anova Table (Type II tests)
##
## Response: Days
## Sum Sq Df F value Pr(>F)
## Res.Age 409.83 1 1329.39 < 2.2e-16 ***
## Fitness 246.08 2 399.11 < 2.2e-16 ***
## Residuals 6.17 20
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(Lm2)
## Anova Table (Type II tests)
##
## Response: Days
## Sum Sq Df F value Pr(>F)
## Res.Age 409.83 1 1241.1044 < 2.2e-16 ***
## Fitness 246.08 2 372.6086 2.257e-15 ***
## Res.Age:Fitness 0.22 2 0.3359 0.7191
## Residuals 5.94 18
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Decsion_Rule <- qf(0.05, 2, 18, lower.tail=FALSE) ### Decision Rule ###
Decsion_Rule
## [1] 3.554557
### With a small F-Value and a large P-Value we fail to reject Ho. ###
### No the general linera model uses a linear model and does not work for non-linear models. ###
summary(Lm2)
##
## Call:
## lm(formula = Days ~ Res.Age * Fitness)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.0379 -0.3681 0.0652 0.3090 0.8252
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.87779 0.23816 146.445 < 2e-16 ***
## Res.Age 1.19510 0.04757 25.123 1.82e-15 ***
## FitnessAbout Average -1.79582 0.30563 -5.876 1.45e-05 ***
## FitnessAbove Average -8.70822 0.35766 -24.348 3.15e-15 ***
## Res.Age:FitnessAbout Average -0.05017 0.07980 -0.629 0.537
## Res.Age:FitnessAbove Average -0.05821 0.08186 -0.711 0.486
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5746 on 18 degrees of freedom
## Multiple R-squared: 0.9945, Adjusted R-squared: 0.993
## F-statistic: 655.4 on 5 and 18 DF, p-value: < 2.2e-16
### The interaction terms are not signifigant so we will refit. But it seems that all but the interaction terms are signifigant. ###
summary(Lm1)
##
## Call:
## lm(formula = Days ~ Res.Age + Fitness)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.03891 -0.36892 0.05891 0.33098 0.89991
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 34.95046 0.21338 163.794 < 2e-16 ***
## Res.Age 1.16729 0.03201 36.461 < 2e-16 ***
## FitnessAbout Average -1.84738 0.28694 -6.438 2.8e-06 ***
## FitnessAbove Average -8.72289 0.33296 -26.198 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5552 on 20 degrees of freedom
## Multiple R-squared: 0.9943, Adjusted R-squared: 0.9935
## F-statistic: 1170 on 3 and 20 DF, p-value: < 2.2e-16
### All of the terms are signigigant. ###