#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(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)

#17a

#factorial model
model <- aov(Yield ~ factor(A) * factor(B) * factor(C) * factor(D))
summary(model)
##                                         Df Sum Sq Mean Sq
## factor(A)                                1  42.25   42.25
## factor(B)                                1  16.00   16.00
## factor(C)                                1   1.00    1.00
## factor(D)                                1  81.00   81.00
## factor(A):factor(B)                      1   0.00    0.00
## factor(A):factor(C)                      1   0.00    0.00
## factor(B):factor(C)                      1   0.25    0.25
## factor(A):factor(D)                      1  64.00   64.00
## factor(B):factor(D)                      1  72.25   72.25
## factor(C):factor(D)                      1   2.25    2.25
## factor(A):factor(B):factor(C)            1   2.25    2.25
## factor(A):factor(B):factor(D)            1   0.25    0.25
## factor(A):factor(C):factor(D)            1   2.25    2.25
## factor(B):factor(C):factor(D)            1   4.00    4.00
## factor(A):factor(B):factor(C):factor(D)  1   4.00    4.00

Answer:

From the full factorial at α = 0.05, three effects are statistically significant. Factor A, factor D, and the interaction of factors a and D have a significant effect. All other main effects (B and C) and all other interactions have p-values greater than 0.05 and are not significant. Therefore, the significant factors are A, D, and the A×D interaction, which should be retained in the model while all non-significant terms can be pooled into error

#17b

model_reduced <- aov(Yield ~ factor(A) + factor(D) + factor(A):factor(D))
summary(model_reduced)
##                     Df Sum Sq Mean Sq F value Pr(>F)  
## factor(A)            1  42.25   42.25   4.852 0.0479 *
## factor(D)            1  81.00   81.00   9.301 0.0101 *
## factor(A):factor(D)  1  64.00   64.00   7.349 0.0189 *
## Residuals           12 104.50    8.71                 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Answer: After pooling non-significant terms into the error term, the reduced model retains only the significant effects: A, D, and A×D. The reduced model confirms that all three effects remain significant at α = 0.05. This indicates that both time and temperature independently affect yield, and importantly, the effect of time depends on the temperature level, as evidenced by the significant interaction. Factors B and C can be eliminated from consideration as they do not significantly affect yield in this process

{r eval=FALSE}