Untitled

The model equation is as follows

Yijk= \(\mu+\alpha_i+\beta_j(i)+\epsilon_ijk\)

where \(\alpha\) is process

and \(\beta\) is batch

Hypothesis to be tested

Null Hypothesis:- \(\alpha(i)\) = 0 for all i

alternate Hypothesis:- \(\alpha(i)\) Not equal to 0 for some i

Null Hypothesis:- \(\sigma^2\beta\) = 0

alternate Hypothesis:- \(\sigma^2\beta\) Not equal to 0

Process<-c(rep(1,12),rep(2,12),rep(3,12))
batch<-rep(c(rep(1,3),rep(2,3),rep(3,3),rep(4,3)),3)
Observation<-c(25,30,26,19,28,20,15,17,14,15,16,13,19,17,14,23,24,21,18,21,17,35,27,25,14,15,20,35,21,24,38,54,50,25,29,33)
dat<-data.frame(Process,batch,Observation)
library(GAD)

## Loading required package: matrixStats

## Loading required package: R.methodsS3

## R.methodsS3 v1.8.1 (2020-08-26 16:20:06 UTC) successfully loaded. See ?R.methodsS3 for help.

Process<-as.fixed(Process)
batch<-as.random(batch)
model<-lm(Observation~Process+batch%in%Process)
gad(model)

## Analysis of Variance Table
## 
## Response: Observation
##               Df  Sum Sq Mean Sq F value    Pr(>F)    
## Process        2  676.06  338.03  1.4643    0.2815    
## Process:batch  9 2077.58  230.84 12.2031 5.477e-07 ***
## Residual      24  454.00   18.92                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Conclusion:- As the value of Process:batch is 5.477e-07 which is lesser than \(\alpha\) = 0.05, hence we conclude that batch has significant effects on the process