Question 1:

Reading the Data:

y <- c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
a <- c(-1,1)
b <- c(-1,-1,1,1)
c<- c(-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)
A<- c(rep(a,8))
B<-c(rep(b,4))
C<-c(rep(c,2))
D<-c(rep(d,1))
Dat <-data.frame(A,B,C,D,y)
Dat
##     A  B  C  D  y
## 1  -1 -1 -1 -1 12
## 2   1 -1 -1 -1 18
## 3  -1  1 -1 -1 13
## 4   1  1 -1 -1 16
## 5  -1 -1  1 -1 17
## 6   1 -1  1 -1 15
## 7  -1  1  1 -1 20
## 8   1  1  1 -1 15
## 9  -1 -1 -1  1 10
## 10  1 -1 -1  1 25
## 11 -1  1 -1  1 13
## 12  1  1 -1  1 24
## 13 -1 -1  1  1 19
## 14  1 -1  1  1 21
## 15 -1  1  1  1 17
## 16  1  1  1  1 23

a) Displaying the half normal plot:

Model <- lm(y~A*B*C*D, data=Dat)
coef(Model)
##   (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
library(DoE.base)
## Warning: package 'DoE.base' was built under R version 4.5.2
## Loading required package: grid
## Loading required package: conf.design
## Warning: package 'conf.design' was built under R version 4.5.2
## 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
halfnormal(Model)

## 
## Significant effects (alpha=0.05, Lenth method):
## [1] A   A:C A:D D

Comment : Effect A,D,AD and AC are significant here as they don’t lie close to the normal line.

b) Non significant factors:

Writing the hypothesis :

Null: \(H_{\circ}\)

\[(\alpha \gamma)_{i,k} =0 \] \[(\alpha \lambda)_{i,l} = 0\] \[(\alpha)_{i}=0 \] \[ (\gamma)_{k}=0\] \[ (\lambda)_{l}=0\] Alternate : \(H_{a}\) \[(\alpha \gamma)_{i,k} \neq 0 \] \[(\alpha \lambda)_{i,l} \neq 0\] \[(\alpha)_{i} \neq 0 \] \[ (\gamma)_{k} \neq 0\] \[ (\lambda)_{l} \neq 0\] As AD and AC are also significant following factors will have to be considered A,C,D,AC,AD

Model1<- aov(y~A+B+C+D+A*C+A*D+A*B+B*C+B*D+C*D,data=Dat)
summary(Model1)
##             Df Sum Sq Mean Sq F value  Pr(>F)   
## A            1  81.00   81.00  31.765 0.00244 **
## B            1   1.00    1.00   0.392 0.55864   
## C            1  16.00   16.00   6.275 0.05416 . 
## D            1  42.25   42.25  16.569 0.00963 **
## A:C          1  72.25   72.25  28.333 0.00313 **
## A:D          1  64.00   64.00  25.098 0.00407 **
## A:B          1   2.25    2.25   0.882 0.39068   
## B:C          1   0.25    0.25   0.098 0.76684   
## B:D          1   0.00    0.00   0.000 1.00000   
## C:D          1   0.00    0.00   0.000 1.00000   
## Residuals    5  12.75    2.55                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Conlusion: Based on the above values tested main effects A,D,AC,AD appear to be significant at alpha =0.05 as p values are less than 0.05

Interaction Plots for Analysis:

interaction.plot(A,D,y,col = c("red","blue"))

interaction.plot(A,C,y,col = c("red","blue"))