QUESTION 11

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


First we start with the data input from the table given to us in the question:

library(DoE.base)
## Loading required package: grid
## Loading required package: conf.design
## Registered S3 method overwritten by 'DoE.base':
##   method           from       
##   factorize.factor conf.design
## 
## Attaching package: 'DoE.base'
## The following objects are masked from 'package:stats':
## 
##     aov, lm
## The following object is masked from 'package:graphics':
## 
##     plot.design
## The following object is masked from 'package:base':
## 
##     lengths
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(12,18,13,20,17,25,15,25,10,24,13,24,19,21,17,23)
dat<-data.frame(A,B,C,D,obs)

Part A:

For calculating the halfnormal plotting :

mod<-lm(obs~A*B*C*D,data = dat)
coef(mod)
## (Intercept)           A           B           C           D         A:B 
##      18.500       4.000       0.250       1.750       0.375       0.250 
##         A:C         B:C         A:D         B:D         C:D       A:B:C 
##      -0.750      -0.500       0.125       0.125      -0.625       0.500 
##       A:B:D       A:C:D       B:C:D     A:B:C:D 
##      -0.125      -1.375       0.125       0.375
halfnormal(mod)
## 
## Significant effects (alpha=0.05, Lenth method):
## [1] A     C     A:C:D

We observe that the significant factors are A, C and A:C:D, therefore the interaction should be between A, C, D, AC, AD.

Part B:

Hypothesis :

Null : \(\alpha_i=0\)

Alternative: \(\alpha_i\not=0\)

Null: \(\beta_j=0\)

Alternative: \(\beta_j\not=0\)

Null: \(\gamma_k=0\)

Alternative: \(\gamma_k\not=0\)

Null: \(\lambda_l=0\)

Alternative: \(\lambda_l\not=0\)

model<-aov(obs~A+C+D+A*C+A*D)
model
## Call:
##    aov.default(formula = obs ~ A + C + D + A * C + A * D)
## 
## Terms:
##                      A      C      D    A:C    A:D Residuals
## Sum of Squares  256.00  49.00   2.25   9.00   0.25     49.50
## Deg. of Freedom      1      1      1      1      1        10
## 
## Residual standard error: 2.22486
## Estimated effects are balanced

We can observe the degrees of freedom above.

interaction.plot(A, D, obs)

interaction.plot(A, C, obs)

Conclusion:

We can conclude from this that the interaction do not take place but are really close and therefore the significant point i.e. A, C, D show what effects are there with respect to the observations.