10

#install.packages("tidyverse")
library(tidyr)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

method<-c(rep(1,6),rep(2,6),rep(3,6),rep(4,6))
method<-as.factor(method)
obs<-c(0.34,0.12,1.23,0.70,1.75,0.12,
  0.91,2.94,2.14,2.36,2.86,4.55,    
  6.31,8.37,9.75,6.09,9.82,7.24,
  17.15,11.82,10.97,17.20,14.35,16.82)
df<-data.frame(method,obs)

Model: Y_ij = mu + tau_i + e_ij

H0: tau_1 = tau_2 = tau_3 = tau_4 = 0 (all method means equal)

HA: at least one method mean is different

m0 <- aov(obs ~ method, data=df)
summary(m0)

##             Df Sum Sq Mean Sq F value Pr(>F)    
## method       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(m0)

The p-value is 4×10⁻¹¹ < 0.05, so we reject H₀.

#Conclusion: At least one method’s mean discharge is different.

boxplot(obs~method, data = df, main="boxplot of discharge by method", xlab = "Method", ylab = "Discharge")

Box-Cox

library("MASS")

## 
## Attaching package: 'MASS'

## The following object is masked from 'package:dplyr':
## 
##     select

bc<-boxcox(obs~method)

lambda=0.5
obs<-obs^(lambda)
boxcox(obs~method)

boxplot(obs~method, data = df, main="boxplot of discharge by method lambda 0.5", xlab = "Method", ylab = "Discharge")

# Kruskal–Wallis, α=0.05
kruskal.test(obs ~ method, data=df)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  obs by method
## Kruskal-Wallis chi-squared = 21.156, df = 3, p-value = 9.771e-05

Result: p < 0.05 → reject H0