Getting in data

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(12,18,13,20,17,25,15,25,10,24,13,24,19,21,17,23)
dat<-data.frame(A,B,C,D,yield)

Question a

library(DoE.base)
model<-lm(yield~A*B*C*D,data=dat)
coef(model)
## (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(model)

From the half normal plot we can see that

factor A, Factor C and interaction factor A:C:D appears to be significant

Question b

Question 2

Stating hypothesis

WE would test the highest order term first

alpha(i) beta(j) gamma(k) = 0

alpha(i) beta(j) gamma(k) is not equals to zero

model2<-lm(yield~A+C+D+A+C+A+D+C+D+A*C*D,data=dat)
summary(model2)
## 
## Call:
## lm.default(formula = yield ~ A + C + D + A + C + A + D + C + 
##     D + A * C * D, data = dat)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
##   -1.5   -1.0    0.0    1.0    1.5 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  18.5000     0.3187  58.050 8.61e-12 ***
## A             4.0000     0.3187  12.551 1.52e-06 ***
## C             1.7500     0.3187   5.491  0.00058 ***
## D             0.3750     0.3187   1.177  0.27314    
## A:C          -0.7500     0.3187  -2.353  0.04643 *  
## A:D           0.1250     0.3187   0.392  0.70513    
## C:D          -0.6250     0.3187  -1.961  0.08550 .  
## A:C:D        -1.3750     0.3187  -4.315  0.00256 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.275 on 8 degrees of freedom
## Multiple R-squared:  0.9645, Adjusted R-squared:  0.9334 
## F-statistic: 31.03 on 7 and 8 DF,  p-value: 3.475e-05

Since the higher order interaction factors is significant A:C:D(0.00256) at alpha is 0.05

We could stop our hypothesis here