1 4. Two-Sample T-Test (with VST) (By Ajala, Ponmile - Kininge, Rucha - Tejada, Omar )

[DESCRIPTION]

1.1 Assumptions

[ TBD ]

\[ x+\frac{1}{2}=\gamma_1 \]

2 Example

[This is where we state the problem, and repeat the steps we mentioned above]

We use ANOVA model to predict variance and normality using the Residuals vs Fitted plot

#Question b
library(tidyr)
library(dplyr)
data1<-c(0.34,0.12,1.23,0.70,1.75,0.12)
data2<-c(0.91,2.94,2.14,2.36,2.86,4.55)
data3<-c(6.31,8.37,9.75,6.09,9.82,7.24)
data4<-c(17.15,11.82,10.97,17.20,14.35,16.82)
data<-data.frame(data1,data2,data3,data4)
datapivot<-pivot_longer(data,c(data1,data2,data3,data4))
aov.model<-aov(value~name,data=datapivot)
summary(aov.model)
##             Df Sum Sq Mean Sq F value Pr(>F)    
## name         3  708.7   236.2   76.29  4e-11 ***
## Residuals   20   61.9     3.1                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
plot(aov.model)

Conclusion:

Looking at the Residuals vs Fitted plot, we cannot confirm a constant spread between the residuals at the different fitted values indicating unstable variance.

For validation, we may also use the boxplot to visually check for variance as shown below. The width of the boxplot varies too much with respect to each different estimation method indicating unstable variance.

#visual way
dataplot<-c(data1,data2,data3,data4)
x<-c(rep(1,6),rep(2,6),rep(3,6),rep(4,6))
meanx<-c(rep(mean(data1),6),rep(mean(data2),6),rep(mean(data3,6),6),rep(mean(data4,6),6))
boxplot(dataplot~x,xlab="Estimation Method",ylab="observation",main="Boxplot of Observations")

Performimg Kruskal-Wallace test

#Question c
kruskal.test(value~name,data=datapivot)
## 
##  Kruskal-Wallis rank sum test
## 
## data:  value by name
## Kruskal-Wallis chi-squared = 21.156, df = 3, p-value = 9.771e-05

With a p-value of 9.771e-05 & 0.05 level of significance, we will reject the null hypothesis.

#Question d
library(MASS)

Based on the above plot, we can say the value of lambda is approximately 0.5

boxcox(dataplot~x)

lambda=.5  # only if 1 is not in CI on lambda
dataplot<-dataplot^(lambda) # if lambda is not zero
#pop<-log(pop) # if lambda is equal to zero 
boxcox(dataplot~x)

The above plot now shows a value of lambda close to 1, indicating the transformation was effective.

boxplot(dataplot~x,xlab="Method Type",ylab="Flow Frequency",main="Boxplot of Observations")