Data

library(GAD)
## Loading required package: matrixStats
## Loading required package: R.methodsS3
## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.
process_temp <- c(rep(1,4),rep(2,4),rep(3,4))

process <- rep(process_temp,3)

batch <- rep(seq(1,4),9)

d <- c(25,19,15,15,29,23,28,35,24,35,38,25,
       30,28,17,16,27,24,21,27,25,21,34,29,
       26,20,14,13,24,21,27,25,20,24,30,33)

data2 <- data.frame(process,batch,d)

data2$batch <- as.random(data2$batch)
data2$process <- as.fixed(data2$process)

Model Equation

\[ y_{ijk}=\mu+\alpha_j+\beta_{j(i)}+\epsilon_{ijk} \]

Hypothesis

Here, we want to test if there is a difference between the Process or not. Therefore,

\[ Ho: \alpha_i=0 \\Ha:\alpha_i\neq0 \]

Test

model3 <- lm(d~process+batch%in%process, data=data2)
gad(model3)
## Analysis of Variance Table
## 
## Response: d
##               Df Sum Sq Mean Sq F value   Pr(>F)   
## process        2 446.06 223.028  3.5365 0.073563 . 
## process:batch  9 567.58  63.065  4.1965 0.002349 **
## Residual      24 360.67  15.028                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

From the results above, it is possible to assume that, since the value of the process p-value is higher than \(\alpha=0.05\), we can not reject Ho. Therefore, there is no significant difference between the process.