Nested Design

Model equation

\[ y_{ijk} = \mu +\tau_i + \beta_{j(i)} + \gamma_{k(ij)} + \epsilon_{(ijk)l}\\ \tau_i = Effect \; of \;machine\\ \beta_{j(i)} = Effect \; of \; spindle\\ \gamma_{k(ij)} = Effect \; of \; k^{th} \; spindle \; with \; the \; i^{th} \; machine\\ \epsilon_{(ijk)l} = Error \; term \; NID(0,\sigma^2) \]

Hypothesis to be tested

\[ H_0: \mu_{\gamma_{k(ij)}}=0 \] \[ H_0: \mu_{\gamma_{k(ij)}} \not= 0 \]

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.
values<- c(12,8,14,12,14,16,9,9,15,10,10,15,11,10,13,11,12,15,12,8,14,13,11,14)
spindle <- as.random(rep(seq(1,2),12))
obervations<- as.fixed(c(rep(1,8),rep(2,8),rep(3,8)))
machine<-as.fixed(rep(c(rep(1,2),rep(2,2),rep(3,2)),4))
dat<-data.frame(machine,spindle,obervations,values)
model<- lm(values~machine+spindle%in%machine)
gad(model)
## Analysis of Variance Table
## 
## Response: values
##                 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
plot(model)

With an assumed \(\alpha=0.05\), there is a significant effects on dimensional variability due to the machine and spindle factors.

The plots the residuals shows a constant variance and fairly follows a normal distribution.