Probelm 14.3
library(GAD)
## Loading required package: matrixStats
## Loading required package: R.methodsS3
## R.methodsS3 v1.8.1 (2020-08-26 16:20:06 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)
obs <- 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)
Dat <- data.frame(Machine,Spindle,obs)
Dat
## Machine Spindle obs
## 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
Machine <- as.fixed(Machine)
Spindle <- as.random(Spindle)
Model <- lm(obs~Machine+Spindle%in%Machine)
gad(Model)
## Analysis of Variance Table
##
## Response: obs
## 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
Yi,j,k=μ+αi+βj(i)+ϵi,j,k
From the given problem we can see that this is a Nested design in which we can see that machine is the principal factor and spindle is the nested factor within the machine. So we are considering machine as a fixed factor and spindle as a random factor.
From the analysis we can see that the p-value for spindle is 0.0004428 which is less than 0.05 so, we reject the null hypothesis for the nested factor and we can conclude that spindle has a significant effect on dimensional variability of the component. On the other hand as visible from the p-value of the machine is 0.2915630 which is more than 0.05 so we can say that machine does not have a significant effect.