Question

Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows: Treatment Replicate ABCCombination I II III (1) 22 31 25 a 32 43 29 b 35 34 50 ab 55 47 46 c 44 45 38 ac 40 37 36 bc 60 50 54 abc 39 41 47.

Several times we have used the hierarchy principle in selecting a model; that is, we have included nonsignificant lower order terms in a model because they were factors involved in significant higher order terms. Hierarchy is certainly not an absolute principle that must be followed in all cases. To illustrate, consider the model resulting from Problem 6.1, which required that a nonsignificant main effect be included to achieve hierarchy. Using the data from Problem 6.1.

(a) Fit both the hierarchical and the nonhierarchical models.

(b) Calculate the PRESS statistic, the adjusted R2, and the mean square error for both models.

(c) Find a 95 percent confidence interval on the estimate of the mean response at a cube corner (x1 = x2 = x3 = +- 1)

(d) Based on the analyses you have conducted, which model do you prefer?

Answer 1

We use the R software to obtain the factor with largest effect.

> tt <- read.csv("clipboard",sep="\t",stringsAsFactors=TRUE)
> head(tt)
Cutting Geometry Angle Life
1      No       No    No   22
2      No       No    No   31
3      No       No    No   25
4     Yes       No    No   32
5     Yes       No    No   43
6     Yes       No    No   29
> sapply(tt,class)
Cutting Geometry     Angle      Life
"factor" "factor" "factor" "integer"
> summary(aov(Life~Cutting*Geometry*Angle,tt))
                       Df Sum Sq Mean Sq F value   Pr(>F)   
Cutting                 1    0.7     0.7   0.022 0.883680   
Geometry                1 770.7   770.7 25.547 0.000117 ***
Angle                   1 280.2   280.2   9.287 0.007679 **
Cutting:Geometry        1   16.7    16.7   0.552 0.468078   
Cutting:Angle           1 468.2   468.2 15.519 0.001172 **
Geometry:Angle          1   48.2    48.2   1.597 0.224475   
Cutting:Geometry:Angle 1   28.2    28.2   0.934 0.348282   
Residuals              16 482.7    30.2                    
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

We can thus wee that the Geometry is having the highest influence/effect.