Model Equation

\[Y_{ijk}=\mu+\tau_i+\beta_{j(i)}+\epsilon_{ijk}\]

Where \(\mu\) is grand mean; \(\tau_i\) is effect of process; \(\beta_{j(i)}\) is effect of batch in process; \(\epsilon_{ijk}\) is effect of error

Process Effect:

\[H_0:\tau_1=\tau_2=\tau_3=0 \]\[ H_a:\text{at least one }\tau_i\neq0\]

Batch Effect: \[H_0:\sigma_B^2=0 \]\[ H_a:\sigma_B^2>0\]

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)
Obs <- 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, Obs)
library(GAD)
Process <- as.fixed(Process)
Batch <- as.random(Batch)
model <- lm(Obs ~ Process + Batch %in% Process)
gad(model)
## $anova
## Analysis of Variance Table
## 
## Response: Obs
##               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 ***
## Residuals     24  454.00   18.92                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

At \(\alpha=0.05\), there is no significant difference in mean burning rate among the three production processes \((p=0.2815)\). However, the batch-within-process effect is significant, indicating substantial variability among batches within the same production process.