library(GAD)## Loading required package: matrixStats## Warning: package 'matrixStats' was built under R version 4.2.2## Loading required package: R.methodsS3## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.proc <- c(rep(1,4),rep(2,4),rep(3,4),rep(1,4),rep(2,4),rep(3,4),rep(1,4),rep(2,4),rep(3,4))
batch <- c(rep(seq(1,4),9))
obs <- 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)
proc <- as.fixed(proc)
batch <- as.random(batch)
dat1 <- data.frame(proc,batch,obs)
dat1## proc batch obs
## 1 1 1 25
## 2 1 2 19
## 3 1 3 15
## 4 1 4 15
## 5 2 1 29
## 6 2 2 23
## 7 2 3 28
## 8 2 4 35
## 9 3 1 24
## 10 3 2 35
## 11 3 3 38
## 12 3 4 25
## 13 1 1 30
## 14 1 2 28
## 15 1 3 17
## 16 1 4 16
## 17 2 1 27
## 18 2 2 24
## 19 2 3 21
## 20 2 4 27
## 21 3 1 25
## 22 3 2 21
## 23 3 3 34
## 24 3 4 29
## 25 1 1 26
## 26 1 2 20
## 27 1 3 14
## 28 1 4 13
## 29 2 1 24
## 30 2 2 21
## 31 2 3 27
## 32 2 4 25
## 33 3 1 20
## 34 3 2 24
## 35 3 3 30
## 36 3 4 33\(y_{ijk}=\alpha_i+\beta_{j(i)}+\epsilon_{ijk}\)
\(H_{0A}: \alpha_i=0,H_{1A}: \alpha_i\ne0\)
\(H_{0B}:\sigma^2_{j(i)}=0,H_{1B}:\sigma^2_{j(i)}\ne0\)
model <- lm(obs~proc+batch%in%proc)
gad(model)## Analysis of Variance Table
##
## Response: obs
## Df Sum Sq Mean Sq F value Pr(>F)
## proc 2 446.06 223.028 3.5365 0.073563 .
## proc: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 ' ' 1From this analysis we can see that only the nested factor (batch) is significant (\(\alpha=0.05\)) and the process (factor A) does not significantly affect the burning rate of propellant (observeation).