The model equation is:
Yijk =μ + αi + βj(i) + εijk
μ = grand mean αi = Higher level Factor (Process)
βj(i) = Lower level Factor (Batch)
εijk = error
State the Hypothesis:
Null Hypothesis, Ho: αi = 0
βj(i) = 0
Alternative hypothesis, Ha: αi ≠ 0
βj(i) ≠ 0
library(GAD)## Warning: package 'GAD' was built under R version 4.2.2## 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.P<-c(rep(1,12), rep(2,12), rep(3,12))
B<-rep(c(rep(1,3), rep(2,3), rep(3,3), rep(4,3)),3)
Obs<-c(25,30,26,19,28,20,15,17,14,15,16,13,29,27,24,23,24,21,28,21,27,35,27,25,24,25,20,35,21,24,38,34,30,25,29,33)
Data<-data.frame(P,B,Obs)
Data$P<-as.fixed (Data$P)
Data$B<-as.random(Data$B)
model<-lm(Obs~P+B%in%P, data = Data)
GAD::gad(model)## Analysis of Variance Table
##
## Response: Obs
## Df Sum Sq Mean Sq F value Pr(>F)
## P 2 446.06 223.028 3.5365 0.073563 .
## P:B 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 ' ' 1Comment: P value for Batch nested in Process is lower than Alpha value (0.05) so batch has significant effect.
So we reject the null hypothesis for the nested lower level factor. Thus the lower level factor (batch nested in process is significant).