Two Way anova
fangya tan
July 21, 2017
Two way anova test for program instructor and supplement on sodium taking on students grade
Input = ("
Instructor Supplement Sodium
'Brendon Small' A 1200
'Brendon Small' A 1400
'Brendon Small' A 1350
'Brendon Small' A 950
'Brendon Small' A 1400
'Brendon Small' B 1150
'Brendon Small' B 1300
'Brendon Small' B 1325
'Brendon Small' B 1425
'Brendon Small' B 1500
'Brendon Small' C 1250
'Brendon Small' C 1150
'Brendon Small' C 950
'Brendon Small' C 1150
'Brendon Small' C 1600
'Brendon Small' D 1300
'Brendon Small' D 1050
'Brendon Small' D 1300
'Brendon Small' D 1700
'Brendon Small' D 1300
'Coach McGuirk' A 1100
'Coach McGuirk' A 1200
'Coach McGuirk' A 1250
'Coach McGuirk' A 1050
'Coach McGuirk' A 1200
'Coach McGuirk' B 1250
'Coach McGuirk' B 1350
'Coach McGuirk' B 1350
'Coach McGuirk' B 1325
'Coach McGuirk' B 1525
'Coach McGuirk' C 1225
'Coach McGuirk' C 1125
'Coach McGuirk' C 1000
'Coach McGuirk' C 1125
'Coach McGuirk' C 1400
'Coach McGuirk' D 1200
'Coach McGuirk' D 1150
'Coach McGuirk' D 1400
'Coach McGuirk' D 1500
'Coach McGuirk' D 1200
'Melissa Robins' A 900
'Melissa Robins' A 1100
'Melissa Robins' A 1150
'Melissa Robins' A 950
'Melissa Robins' A 1100
'Melissa Robins' B 1150
'Melissa Robins' B 1250
'Melissa Robins' B 1250
'Melissa Robins' B 1225
'Melissa Robins' B 1325
'Melissa Robins' C 1125
'Melissa Robins' C 1025
'Melissa Robins' C 950
'Melissa Robins' C 925
'Melissa Robins' C 1200
'Melissa Robins' D 1100
'Melissa Robins' D 950
'Melissa Robins' D 1300
'Melissa Robins' D 1400
'Melissa Robins' D 1100
")
data = read.table(textConnection(Input),header=TRUE)
data$Instructor = factor(data$Instructor,
levels=unique(data$Instructor))
data$Supplement = factor(data$Supplement,
levels=unique(data$Supplement))
summary(data)## Instructor Supplement Sodium
## Brendon Small :20 A:15 Min. : 900
## Coach McGuirk :20 B:15 1st Qu.:1100
## Melissa Robins:20 C:15 Median :1200
## D:15 Mean :1219
## 3rd Qu.:1325
## Max. :1700Check the cell sizes are balanced
library(FSA)## ## FSA v0.8.13. See citation('FSA') if used in publication.
## ## Run fishR() for related website and fishR('IFAR') for related book.Summarize(Sodium~ Supplement +Instructor, data=data, digits=3)## Supplement Instructor n mean sd min Q1 median Q3 max
## 1 A Brendon Small 5 1260 191.703 950 1200 1350 1400 1400
## 2 B Brendon Small 5 1340 132.994 1150 1300 1325 1425 1500
## 3 C Brendon Small 5 1220 238.747 950 1150 1150 1250 1600
## 4 D Brendon Small 5 1330 233.452 1050 1300 1300 1300 1700
## 5 A Coach McGuirk 5 1160 82.158 1050 1100 1200 1200 1250
## 6 B Coach McGuirk 5 1360 100.933 1250 1325 1350 1350 1525
## 7 C Coach McGuirk 5 1175 148.955 1000 1125 1125 1225 1400
## 8 D Coach McGuirk 5 1290 151.658 1150 1200 1200 1400 1500
## 9 A Melissa Robins 5 1040 108.397 900 950 1100 1100 1150
## 10 B Melissa Robins 5 1240 62.750 1150 1225 1250 1250 1325
## 11 C Melissa Robins 5 1045 116.458 925 950 1025 1125 1200
## 12 D Melissa Robins 5 1170 178.885 950 1100 1100 1300 1400Define the linear model
model <- lm(Sodium ~ Instructor+Supplement+ Instructor*Supplement, data=data)
summary(model)##
## Call:
## lm(formula = Sodium ~ Instructor + Supplement + Instructor *
## Supplement, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -310.00 -90.00 -17.50 86.25 380.00
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1260.00 69.44 18.145 <2e-16
## InstructorCoach McGuirk -100.00 98.20 -1.018 0.3136
## InstructorMelissa Robins -220.00 98.20 -2.240 0.0297
## SupplementB 80.00 98.20 0.815 0.4193
## SupplementC -40.00 98.20 -0.407 0.6856
## SupplementD 70.00 98.20 0.713 0.4794
## InstructorCoach McGuirk:SupplementB 120.00 138.88 0.864 0.3919
## InstructorMelissa Robins:SupplementB 120.00 138.88 0.864 0.3919
## InstructorCoach McGuirk:SupplementC 55.00 138.88 0.396 0.6938
## InstructorMelissa Robins:SupplementC 45.00 138.88 0.324 0.7473
## InstructorCoach McGuirk:SupplementD 60.00 138.88 0.432 0.6677
## InstructorMelissa Robins:SupplementD 60.00 138.88 0.432 0.6677
##
## (Intercept) ***
## InstructorCoach McGuirk
## InstructorMelissa Robins *
## SupplementB
## SupplementC
## SupplementD
## InstructorCoach McGuirk:SupplementB
## InstructorMelissa Robins:SupplementB
## InstructorCoach McGuirk:SupplementC
## InstructorMelissa Robins:SupplementC
## InstructorCoach McGuirk:SupplementD
## InstructorMelissa Robins:SupplementD
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 155.3 on 48 degrees of freedom
## Multiple R-squared: 0.3491, Adjusted R-squared: 0.2
## F-statistic: 2.341 on 11 and 48 DF, p-value: 0.02121Box plot for seperate factors
par(mfrow=c(1,2))
plot(Sodium ~Instructor +Supplement, data=data)
Judging by the boxplots, there appears to be differences in instructor as well as Supplement
Interaction plots
interaction.plot(data$Supplement,data$Instructor, data$Sodium)
interaction.plot(data$Instructor, data$Supplement,data$Sodium)
Although there are some interactions, but they are still roughly parallel, we will comfirm using the Anova model
Anova
anova(model)## Analysis of Variance Table
##
## Response: Sodium
## Df Sum Sq Mean Sq F value Pr(>F)
## Instructor 2 290146 145073 6.0173 0.004658 **
## Supplement 3 306125 102042 4.2324 0.009841 **
## Instructor:Supplement 6 24438 4073 0.1689 0.983880
## Residuals 48 1157250 24109
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Studying the output of ANOVA table, we see that there is no evidence of a significant interation since p-value=0.98. Furthermore, Supplement p-value=0.009841, it tells us there is not enough evidence to conclude there is significant supplement leve.
Check Anova model assupmtion, qq-plot and residual plots
qqnorm(model$residuals)
plot(model$fitted, model$residuals, xlab="fitted",ylab="residuals")
x<- residuals(model)
library(rcompanion)
plotNormalHistogram(x)
As a result, we can see it follows a normal distribution