---

Problem

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<-rep(c(-1,1),8)
A

##  [1] -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1

B<-rep(c(-1,-1,1,1),4)
B

##  [1] -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1

C<-rep(c(-1,-1,-1,-1,1,1,1,1),2)
C

##  [1] -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1

D<-c(rep(-1,8),rep(1,8))
D

##  [1] -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)

d11<-data.frame(A,B,C,D,Yield)

mod<-lm(Yield~ A * B * C * D , data=d11)

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)

## 
## Significant effects (alpha=0.05, Lenth method):

## [1] A   A:C A:D D

Based on the half normal plot, A, A:D, A:C and D appear to be significant because they are plotted away from the other effects.

mod1<-lm(Yield~ A + C + D +  A : C + A : D  , data=d11)

anova(mod1)

## Analysis of Variance Table
## 
## Response: Yield
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## A          1  81.00  81.000 49.8462 3.456e-05 ***
## C          1  16.00  16.000  9.8462 0.0105485 *  
## D          1  42.25  42.250 26.0000 0.0004647 ***
## A:C        1  72.25  72.250 44.4615 5.583e-05 ***
## A:D        1  64.00  64.000 39.3846 9.193e-05 ***
## Residuals 10  16.25   1.625                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The effects A, C, D, A:C and A:D are significant because their p-values are less than 0.05 and C was retained because A:C was included.