library(DoE.base)
## Warning: package 'DoE.base' was built under R version 4.0.5
## Loading required package: grid
## Loading required package: conf.design
## Warning: package 'conf.design' was built under R version 4.0.3
## 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)
yield<-c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
dat<-data.frame(A,B,C,D,yield)
mod<-lm(yield~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 -8.543513e-17 
##           C:D         A:B:C         A:B:D         A:C:D         B:C:D 
##  1.110223e-16  5.000000e-01  3.750000e-01 -1.250000e-01 -3.750000e-01 
##       A:B:C:D 
##  5.000000e-01
halfnormal(mod)
## 
## Significant effects (alpha=0.05, Lenth method):
## [1] A   A:C A:D D

From the halfnormal plot , A, AD, AC, D are the significant one.

  1. testing for the significance of the other effects at the 0.05 level of significance
mod2<-aov(yield~C+A+D+A*D+A*C,data = dat)
summary(mod2)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## C            1  16.00   16.00   9.846 0.010549 *  
## A            1  81.00   81.00  49.846 3.46e-05 ***
## D            1  42.25   42.25  26.000 0.000465 ***
## A:D          1  64.00   64.00  39.385 9.19e-05 ***
## C:A          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

Conclusion

All the Factors : C,A,D,AD,and, AC have p-value less than alpha=0.05 , we fail to reject them. Our Final model will therefore consist of: C,A,D,AD,and, AC.