Question 14.3
A manufacturing engineer is studying the dimensional variability of a particular component that is produced on three machines. Each machine has two spindles, and four components are randomly selected from each spindle. The results follow. Analyze the data, assuming that machines and spindles are fixed factors.
This is a Nested design, in which machine is the principal factor and spindle is the nested factor within the machine.
We’re considering machine as the fixed factor over here while the nested spindle as the random factor as advised by Dr. Matis to be able to run gad model on it. In this case this is a mixed effects model.
Model Equation:
\(Y_{i,j,k}\)=\(\mu+\alpha_i+\beta_{j(i)}+\epsilon_{i,j,k}\)
Loading Data
Machine <- c(rep(1,8),rep(2,8),rep(3,8))
Spindle <- rep(c(rep(1,4),rep(2,4)),3)
Response <- 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)
library(GAD)
Data <- data.frame(Machine,Spindle,Response)
Data
## Machine Spindle Response
## 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
Analysis
Machine <- as.fixed(Machine)
Spindle <- as.random(Spindle)
Model <- lm(Response~Machine+Spindle%in%Machine)
gad(Model)
## Analysis of Variance Table
##
## Response: Response
## 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
Since the p-value for spindle is 0.0004428, thus at any reasonable level of significance we reject the null hypothesis for the nested factor and thus 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, machine does not have a significant effect.