Problem 14.3

library(GAD)
## Loading required package: matrixStats
## Loading required package: R.methodsS3
## R.methodsS3 v1.8.2 (2022-06-13 22:00:14 UTC) successfully loaded. See ?R.methodsS3 for help.
machine <- c(rep(1,8),rep(2,8),rep(3,8))
spindle <- rep(c(rep(1,4),rep(2,4)),3)
observation <- c(12,9,11,12,8,9,10,8,14,15,13,14,12,10,11,13,14,10,12,11,16,15,15,14)
machine <- as.fixed(machine)
spindle <- as.random(spindle)
dat <- data.frame(machine,spindle,observation)
dat
##    machine spindle observation
## 1        1       1          12
## 2        1       1           9
## 3        1       1          11
## 4        1       1          12
## 5        1       2           8
## 6        1       2           9
## 7        1       2          10
## 8        1       2           8
## 9        2       1          14
## 10       2       1          15
## 11       2       1          13
## 12       2       1          14
## 13       2       2          12
## 14       2       2          10
## 15       2       2          11
## 16       2       2          13
## 17       3       1          14
## 18       3       1          10
## 19       3       1          12
## 20       3       1          11
## 21       3       2          16
## 22       3       2          15
## 23       3       2          15
## 24       3       2          14
model <- lm(observation~machine+spindle%in%machine)
gad(model)
## Analysis of Variance Table
## 
## Response: observation
##                 Df Sum Sq Mean Sq F value    Pr(>F)    
## machine          2  55.75 27.8750  1.9114 0.2915630    
## machine:spindle  3  43.75 14.5833  9.9057 0.0004428 ***
## Residual        18  26.50  1.4722                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Model Equation: Y-ijk = mu + alpha_i + beta_j(i) + e_ijk, where spindle is nested within machine. Hypothesis Test:

Null H_0: alpha_i = 0 for all i Alternative H_a: alpha_i != 0 for some i

Null H_0: beta_j(i) = 0 for all i,j Alternative H_a: beta_j(i) != 0 for some i,j

From the summary we see that, the factor machine is not significant since P value (0.2915630) is higher than \(\alpha = 0.05\) level of significance thus we fail to reject \(H_0\). On the other hand, the factor spindle is nested within the factor machine is significant since P value (0.0004428) is lower than \(\alpha = 0.05\) level of significance thus we reject \(H_0\).

Source Code:

library(GAD)
machine <- c(rep(1,8),rep(2,8),rep(3,8))
spindle <- rep(c(rep(1,4),rep(2,4)),3)
observation <- c(12,9,11,12,8,9,10,8,14,15,13,14,12,10,11,13,14,10,12,11,16,15,15,14)
machine <- as.fixed(machine)
spindle <- as.random(spindle)
dat <- data.frame(machine,spindle,observation)
dat
model <- lm(observation~machine+spindle%in%machine)
gad(model)