Question 1

A randomized complete block design (RCBD) is used, where Chemical is the treatment factor and Bolt is the blocking factor. Chemical is considered a fixed effect, while Bolt is considered a random effect.

Linear Effects Model

The linear effects model is

\[Y_{ij}=\mu+\alpha_i+\beta_j+\epsilon_{ij}\]

where:

  • \(Y_{ij}\) = tensile strength for chemical \(i\) on bolt \(j\)
  • \(\mu\) = overall mean tensile strength
  • \(\alpha_i\) = effect of chemical \(i\)
  • \(\beta_j\) = effect of bolt \(j\)
  • \(\epsilon_{ij}\) = random error

Here, \(\alpha_i\) is a fixed effect and \(\beta_j\) is a random effect.

The hypotheses for the chemical effect are

\[H_0:\alpha_1=\alpha_2=\alpha_3=\alpha_4=0\]

or equivalently,

\[H_0:\mu_1=\mu_2=\mu_3=\mu_4\]

versus

\[H_a:\text{At least one chemical mean is different}\]

The significance level is

\[\alpha=0.15\]

chemical <- factor(rep(1:4, each = 5))
bolt <- factor(rep(1:5, times = 4))
strength <- c(73,68,74,71,67,73,67,75,72,70,75,68,78,73,68,73,71,75,75,69)
data1 <- data.frame(chemical, bolt, strength)
data1
##    chemical bolt strength
## 1         1    1       73
## 2         1    2       68
## 3         1    3       74
## 4         1    4       71
## 5         1    5       67
## 6         2    1       73
## 7         2    2       67
## 8         2    3       75
## 9         2    4       72
## 10        2    5       70
## 11        3    1       75
## 12        3    2       68
## 13        3    3       78
## 14        3    4       73
## 15        3    5       68
## 16        4    1       73
## 17        4    2       71
## 18        4    3       75
## 19        4    4       75
## 20        4    5       69

RCBD Analysis Using GAD

library(GAD)

chemical <- as.fixed(data1$chemical)
bolt <- as.random(data1$bolt)

model1 <- lm(strength ~ chemical + bolt)
gad(model1)
## $anova
## Analysis of Variance Table
## 
## Response: strength
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## chemical   3  12.95   4.317  2.3761    0.1211    
## bolt       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

For the chemical effect, the ANOVA gives approximately

\[F=2.376\]

with a p-value of approximately

\[p=0.121\]

Since

\[0.121 < 0.15,\]

we reject \(H_0\). Therefore, at the 0.15 significance level, there is sufficient evidence that the mean tensile strength differs among the four chemical agents.

The Bolt effect is also highly significant, indicating that there is substantial variability among the bolts.

Question 2

In this case, the experiment is treated as a completely randomized design (CRD). Bolt is no longer used as a blocking factor, so Chemical is the only factor included in the model.

Linear Effects Model

The linear effects model is

\[Y_{ij}=\mu+\alpha_i+\epsilon_{ij}\]

where:

  • \(Y_{ij}\) = observed tensile strength
  • \(\mu\) = overall mean tensile strength
  • \(\alpha_i\) = effect of chemical \(i\)
  • \(\epsilon_{ij}\) = random error

The hypotheses are

\[H_0:\alpha_1=\alpha_2=\alpha_3=\alpha_4=0\]

or equivalently,

\[H_0:\mu_1=\mu_2=\mu_3=\mu_4\]

versus

\[H_a:\text{At least one chemical mean is different}\]

with

\[\alpha=0.15.\]

CRD Analysis Using GAD

chemical <- as.fixed(data1$chemical)
model2 <- lm(strength ~ chemical)
gad(model2)
## $anova
## Analysis of Variance Table
## 
## Response: strength
##           Df Sum Sq Mean Sq F value Pr(>F)
## chemical   3  12.95  4.3167  0.3863 0.7644
## Residuals 16 178.80 11.1750

For the chemical effect, the ANOVA gives approximately

\[F=0.386\]

with a p-value of approximately

\[p=0.764\]

Since

\[0.764 > 0.15,\]

We fail to reject \(H_0\). Therefore, when the Bolt blocking factor is ignored, there is insufficient evidence to conclude that the four chemical agents have different effects on mean tensile strength.

Question 3

The conclusions from Questions 1 and 2 are different. In the RCBD analysis, where Bolt is included as a blocking factor, the chemical effect is significant at \(\alpha=0.15\) (\(p\approx0.121\)). However, when Bolt is ignored in the completely randomized design, the chemical effect is not significant (\(p\approx0.764\)).

The Bolt effect in the RCBD is highly significant, indicating that Bolt represents an important source of nuisance variability. Blocking on Bolt removes much of this variability from the experimental error, making it easier to detect differences among the chemical treatments. Therefore, blocking by Bolt is beneficial for this experiment.

Complete R Code

library(GAD)
chemical <- factor(rep(1:4, each = 5))
bolt <- factor(rep(1:5, times = 4))
strength <- c(73, 68, 74, 71, 67,73, 67, 75, 72, 70,75, 68, 78, 73, 68,73, 71, 75, 75, 69)
data1 <- data.frame(chemical, bolt, strength)
data1


# Question 1: Randomized Complete Block Design

chemical <- as.fixed(data1$chemical)
bolt <- as.random(data1$bolt)
model1 <- lm(strength ~ chemical + bolt)
gad(model1)

# Question 2: Completely Randomized Design
chemical <- as.fixed(data1$chemical)
model2 <- lm(strength ~ chemical)
gad(model2)