1 Randomized Complete Block Design

Linear effect model :

\[ y_{ij} = \mu + \tau_i + \beta_j + e_{ij} \]

Hypothesis:

\[ H_0 : \tau_i = 0 , \forall i \\ H_1 : \tau_i \ne 0, \text{some } i \]

with chemical as a fixed effect, and bolts as random effect.

GAD:

## $anova
## Analysis of Variance Table
## 
## Response: obs_rcbd
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## chem_rcbd  3  12.95   4.317  2.3761    0.1211    
## bolt_rcbd  4 157.00  39.250 21.6055 2.059e-05 ***
## Residuals 12  21.80   1.817                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Since p-value is less than \(\alpha =\) 0.15, there is enough evidence to reject \(H_0\). Therefore, there is a significant difference in mean tensile strength among the four chemicals.

2 Completely Randomized Desing

Linear effect model :

$$ y_{ij} = \mu + \tau_i + e_{ij} $$

Hypothesis: $$ H_0 : \tau_i = 0 , \forall i \\ H_1 : \tau_i \ne 0, \text{some } i $$

ANOVA:

##             Df Sum Sq Mean Sq F value Pr(>F)
## chem_aov     3  12.95   4.317   0.386  0.764
## Residuals   16 178.80  11.175

Since p-value is greater than $\alpha = $ 0.15, there is not enough evidence to reject \(H_0\). Therefore, there is no significant difference in mean tensile strength among the four chemicals.

3 Disscussion

There is a difference in the findings between the RCBD and CRD. In the RCBD, the chemical effect is significant at \(\alpha\)=0.15, while in the CRD, the chemical effect is not significant. The MSE decreases from 11.175 in the CRD to 1.817 in the RCBD. Therefore, Bolt represents a substantial amount of nuisance variability, and blocking on Bolt improves the ability to detect differences among the chemicals.

4 Executed Code

library(GAD)

#rcbd
library(GAD)
chem_rcbd <- c(rep(1,5), rep(2,5), rep(3,5), rep(4,5))
bolt_rcbd <- c(seq(1,5), seq(1,5), seq(1,5), seq(1,5))
obs_rcbd <- c(73, 68, 74, 71, 67,
              73, 67, 75, 72, 70, 
              75, 68, 78, 73, 68, 
              73, 71, 75, 75, 69 )

chem_rcbd<- as.fixed(chem_rcbd)
bolt_rcbd <- as.random(bolt_rcbd)
rcbd_model <- lm(obs_rcbd~chem_rcbd + bolt_rcbd)
gad(rcbd_model)

#crd
chem_aov <- c(rep(1,5), rep(2,5), rep(3,5), rep(4,5))
obs_aov <- c(73, 68, 74, 71, 67,
             73, 67, 75, 72, 70, 
             75, 68, 78, 73, 68, 
             73, 71, 75, 75, 69 )

chem_aov <- as.factor(chem_aov)
aov_model <- aov(obs_aov~chem_aov)
summary(aov_model)