1. The linear effects model is y(ij) = μ + α(i) + β(j) + ε(ij), where μ is the overall mean, α(i) is the fixed chemical effect, β(j) is the random bolt effect, and ε(ij) is the random error.

The null hypothesis is that all chemical means are equal, while the alternative is that at least one differs. For bolt, the null hypothesis is that its variance component is zero, while the alternative is that it is greater than zero.

library(GAD)

chemical <- as.fixed(factor(rep(1:4, each = 5)))
bolt <- as.random(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)

dat <- data.frame(chemical, bolt, strength)

model <- aov(strength ~ chemical + bolt, data = dat)

gad(model)
## $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

At α = 0.15, the chemical effect is significant (p = 0.1211). Therefore, we reject H0 and conclude that at least one chemical has a different mean tensile strength. The bolt effect is also significant (p < 0.001), indicating substantial variability among bolts.

  1. The linear effects model is y(ij) = μ + α(i) + ε(ij), where μ is the overall mean, α(i) is the chemical effect, and ε(ij) is the random error.

The null hypothesis is that all chemical means are equal, while the alternative is that at least one differs.

model2 <- aov(strength ~ chemical, data = dat)
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

At α = 0.15, the chemical effect is not significant (p = 0.7644). Therefore, we fail to reject H0. There is insufficient evidence that the mean tensile strengths differ among the four chemicals.

  1. The RCBD showed a significant chemical effect (p = 0.1211), while the CRD did not (p = 0.7644). The MSE decreased from 11.175 in the CRD to 1.817 in the RCBD. Also, the bolt effect was significant (p < 0.001). Therefore, Bolt represents a significant amount of nuisance variability, and blocking improved the ability to detect differences among chemicals.