A manufacturing engineer is studying the dimension- al 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.
Ho: \(\alpha_{i} = 0\) - Null Hypothesis
Ha: \(\alpha_{i} \ne 0\) - Alternative Hypothesis
Ho: \(\beta_{j} = 0\) - Null Hypothesis
Ha: \(\beta_{j} \ne 0\) - Alternative Hypothesis
Model Equation \(y_{ijk} = \mu + \alpha_i + \beta_{j(i)}+\epsilon_{ijk}\)
## 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 | spindle | obs |
|---|---|---|
| 1 | 1 | 12 |
| 1 | 2 | 8 |
| 1 | 1 | 9 |
| 1 | 2 | 9 |
| 1 | 1 | 11 |
| 1 | 2 | 10 |
| 1 | 1 | 12 |
| 1 | 2 | 8 |
| 2 | 1 | 14 |
| 2 | 2 | 12 |
| 2 | 1 | 15 |
| 2 | 2 | 10 |
| 2 | 1 | 13 |
| 2 | 2 | 11 |
| 2 | 1 | 14 |
| 2 | 2 | 13 |
| 3 | 1 | 14 |
| 3 | 2 | 16 |
| 3 | 1 | 10 |
| 3 | 2 | 15 |
| 3 | 1 | 12 |
| 3 | 2 | 15 |
| 3 | 1 | 11 |
| 3 | 2 | 14 |
Analysis of Variance
## 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 ' ' 1From the result values of “Prob > F” less than 0.0500 indicate model terms are significant. In this case the nested spindle in machine is the significant model term Hence, there is a significant effect on dimensional variability due to the nested spindle in machine factor.
machine<-rep(c(rep(seq(1:3),each=8)),1)
spindle<-c(rep(seq(1:2),12))
obs<-c(12,8,9,9,11,10,12,8,14,12,15,10,13,11,14,13,14,16,10,15,12,15,11,14)
machine<-as.fixed(machine)
spindle<-as.random(spindle)
length(spindle)
library(GAD)
model<-lm(obs~machine+spindle%in%machine)
gad(model)
data.frame(machine,spindle,obs)