Anova

#Anova # Null Hypothesis: Population means are equal # Alternative Hypothesis: At least two population means are not equal # Alpha : 0.05

dataoneway <- read.table("https://rstatisticsandresearch.weebly.com/uploads/1/0/2/6/1026585/onewayanova.txt",h=T)
head(dataoneway)

##   Group Length
## 1     1   19.0
## 2     1   18.6
## 3     1   18.3
## 4     1   18.0
## 5     1   18.2
## 6     1   18.6

Assumption 1: All samples are independent, and collected in >2 independent categorical groups Label groups and set as categorical factors

#convert to factor variable
dataoneway$Group <- as.factor(dataoneway$Group)

table(dataoneway$Group)

## 
##  1  2  3 
## 35 35 35

#Assumption 2: Dependent variable is continuous

str(dataoneway)

## 'data.frame':    105 obs. of  2 variables:
##  $ Group : Factor w/ 3 levels "1","2","3": 1 1 1 1 1 1 1 1 1 1 ...
##  $ Length: num  19 18.6 18.3 18 18.2 18.6 18.5 18.2 18.4 18.9 ...

#Assumption 3: Normal distributions of each group, no major outliers

dataoneway$Group = factor(dataoneway$Group,labels = c("Wall lizard", "Viviparous lizard", "Snake-eyed lizard"))

Group1 <- subset(dataoneway, Group == "Wall lizard")
Group2 <- subset(dataoneway, Group == "Viviparous lizard")
Group3 <- subset(dataoneway, Group == "Snake-eyed lizard")

qqnorm(Group1$Length)
qqline(Group1$Length)

qqnorm(Group2$Length)
qqline(Group2$Length)

qqnorm(Group2$Length)
qqline(Group2$Length)

#shapiro wilk test #Ho : Data is normal #Ha : Data is not normal #if the p-value is greater than 0.05 we accept the null hypothesis, means data is statistically normal

shapiro.test(Group1$Length)

## 
##  Shapiro-Wilk normality test
## 
## data:  Group1$Length
## W = 0.98269, p-value = 0.8425

shapiro.test(Group2$Length)

## 
##  Shapiro-Wilk normality test
## 
## data:  Group2$Length
## W = 0.97817, p-value = 0.6986

shapiro.test(Group3$Length)

## 
##  Shapiro-Wilk normality test
## 
## data:  Group3$Length
## W = 0.97219, p-value = 0.5063

#Assumption 4: Homogeneneity of variances #equal variance among the three groups -

Null Hypothesis : Equal variance among the groups

Alternative Hypothesis: Unequal variance

bartlett.test(Length ~ Group, data = dataoneway)

## 
##  Bartlett test of homogeneity of variances
## 
## data:  Length by Group
## Bartlett's K-squared = 0.43292, df = 2, p-value = 0.8054

#Since the p-value is greater tha 0.05, therefore accept the null hypothesis hence statistically equal variance among the group

#One Way ANOVA - Test if the means of the k populations are equal

model = lm(Length ~ Group, data = dataoneway)
anova(model)

## Analysis of Variance Table
## 
## Response: Length
##            Df Sum Sq Mean Sq F value Pr(>F)   
## Group       2 10.615  5.3074  7.0982 0.0013 **
## Residuals 102 76.267  0.7477                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

#command 2
model_1 = aov(Length ~ Group, data=dataoneway)
summary(model_1)

##              Df Sum Sq Mean Sq F value Pr(>F)   
## Group         2  10.61   5.307   7.098 0.0013 **
## Residuals   102  76.27   0.748                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Since the p-value of 0.0013 which is less than 0.05 therefore we reject the Ho

#Post-hoc test TukeyHSD - Test which of the groups have different means
#tukey's honest significant difference
TukeyHSD(aov(model))

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = model)
## 
## $Group
##                                           diff        lwr        upr     p adj
## Viviparous lizard-Wall lizard       -0.7200000 -1.2116284 -0.2283716 0.0020955
## Snake-eyed lizard-Wall lizard       -0.1028571 -0.5944855  0.3887713 0.8726158
## Snake-eyed lizard-Viviparous lizard  0.6171429  0.1255145  1.1087713 0.0098353

#command 2#
TukeyHSD(model_1)

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = Length ~ Group, data = dataoneway)
## 
## $Group
##                                           diff        lwr        upr     p adj
## Viviparous lizard-Wall lizard       -0.7200000 -1.2116284 -0.2283716 0.0020955
## Snake-eyed lizard-Wall lizard       -0.1028571 -0.5944855  0.3887713 0.8726158
## Snake-eyed lizard-Viviparous lizard  0.6171429  0.1255145  1.1087713 0.0098353

plot(TukeyHSD(model_1))

library("ggplot2")

## Warning: package 'ggplot2' was built under R version 4.0.5

ggplot(dataoneway, aes(x = Group, y = Length)) +
  geom_boxplot(fill = "grey80", colour = "black") +
  scale_x_discrete() + xlab("Treatment Group") +
  ylab("Length (cm)")