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#install.packages("GAD")
library(GAD)

RCBD (Randomized Complete Block Design)

# Chemical in 4 groups of 4
chemical <- c(rep(1,4), rep(2,4), rep(3,4), rep(4,4))

# Bolt runs 1..4 inside each chemical group 
bolt <- c(seq(1,4), seq(1,4), seq(1,4), seq(1,4))

# Observations 
obs <- c(73,68,74,71,
         73,67,75,72,
         75,68,78,73,
         73,71,75,75)

# Declare fixed/random for GAD
chemical <- as.fixed(chemical)  # treatments fixed
bolt <- as.random(bolt)     # blocks random

model<- lm(obs ~ chemical+ bolt)
gad(model)

## $anova
## Analysis of Variance Table
## 
## Response: obs
##           Df  Sum Sq Mean Sq F value    Pr(>F)    
## chemical   3  14.187   4.729  2.7349 0.1057217    
## bolt       3 104.188  34.729 20.0843 0.0002515 ***
## Residuals  9  15.563   1.729                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Linear effects model (additive RCBD):
Yij = μ + αi + βj + εij
αi = effect of chemical i (fixed)
βj = effect of bolt j (random)
εij = random error

Hypotheses for chemicals:
H0: α1 = α2 = α3 = α4 = 0
HA: At least one αi ≠ 0

Decision (α = 0.15):

p < 0.15 → Reject H0.

The analysis shows that both the chemical and bolt effects are significant. This means different chemicals affect cloth strength differently, and bolts of cloth vary in baseline strength. Blocking by bolt reduced error variance and improved precision

CRD (no blocks: one-way ANOVA)

# Completely Randomized Design (CRD)

chemical <- c(rep(1,4), rep(2,4), rep(3,4), rep(4,4))
obs <- c(
  73, 73, 75, 73,
  68, 67, 68, 71,
  74, 75, 78, 75,
  71, 72, 73, 75
)

# Declare factor as fixed BEFORE fitting the model
chemical <- as.fixed(chemical)

# Fit the one-way ANOVA model
model_crd <- lm(obs ~ chemical)

# Now run GAD safely
gad(model_crd)

## $anova
## Analysis of Variance Table
## 
## Response: obs
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## chemical   3 104.19  34.729  14.008 0.0003168 ***
## Residuals 12  29.75   2.479                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Model:
Yij = μ + αi + εij
αi = effect of chemical i (fixed)
εij = random error

Hypotheses for chemicals:
H0: α1 = α2 = α3 = α4 = 0
HA: At least one αi ≠ 0

Decision (α = 0.15):

• Chemical: p = 0.000316 < 0.15 → Reject H0. Chemicals differ significantly.

Comments:
The CRD shows that chemical type has a significant effect on cloth strength. However, without blocking, the residual mean square (error) is larger (about 2.48), meaning there is more unexplained variation compared to the RCBD.