1 Problem:

In a process development study on yield, four factors were studied, each at two levels: time (A), concentration (B), pressure (C), and temperature (D).

  1. Display the halfnormal plot for this data and determine which factors appear to be significant.
  2. Pull terms that do not appear to be significant into error and test for the significance of the other effects at the 0.05 level of significance.

1.1 Solution a:

Loading the data into R

library(DoE.base)
A<-c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
B<-c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
C<-c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
D<-c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
obs<-c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
dat<-data.frame(A,B,C,D,obs)

Using DoE package and finding the factors that appear to be significant

mod<-lm(obs~A*B*C*D,data=dat)
coef(mod)
##   (Intercept)             A             B             C             D 
##  1.737500e+01  2.250000e+00  2.500000e-01  1.000000e+00  1.625000e+00 
##           A:B           A:C           B:C           A:D           B:D 
## -3.750000e-01 -2.125000e+00  1.250000e-01  2.000000e+00 -1.027824e-16 
##           C:D         A:B:C         A:B:D         A:C:D         B:C:D 
## -1.595946e-16  5.000000e-01  3.750000e-01 -1.250000e-01 -3.750000e-01 
##       A:B:C:D 
##  5.000000e-01
halfnormal(mod)

Based on the above plot, we can observe that the significant factors are A, D, A:D & A:C (On this last term, We have to keep the “C” term because its embedded.)

1.2 Solution b:

The significant factors are A, D, A:D & A:C (On this last term, We have to keep the “C” term because its embedded.) to analyze the interaction. Therefore the interaction should be between A, C, D, A:D & A:C.

pullmod<- aov(obs~A+C+D+A*D+A*C,data = dat)
summary(pullmod)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## A            1  81.00   81.00  49.846 3.46e-05 ***
## C            1  16.00   16.00   9.846 0.010549 *  
## D            1  42.25   42.25  26.000 0.000465 ***
## A:D          1  64.00   64.00  39.385 9.19e-05 ***
## A:C          1  72.25   72.25  44.462 5.58e-05 ***
## Residuals   10  16.25    1.62                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

ANOVA analysis tell us that factors A,C,D,AC & AD are all significant at a 95% confidence interval

1.3 Complete R Code

#A
library(DoE.base)
A<-c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1)
B<-c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1)
C<-c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1)
D<-c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1)
obs<-c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
dat<-data.frame(A,B,C,D,obs)
mod<-lm(obs~A*B*C*D,data=dat)
coef(mod)
halfnormal(mod)

#B
pullmod<- aov(obs~A+C+D+A*D+A*C,data = dat)
summary(pullmod)