Question 1:  

PART A:

Reading the Data:

Obs <- c(12,18,13,20,17,25,15,25,10,24,13,24,19,21,17,23)
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))

Data <- data.frame(A,B,C,D,Obs)
Data
##     A  B  C  D Obs
## 1  -1 -1 -1 -1  12
## 2   1 -1 -1 -1  18
## 3  -1  1 -1 -1  13
## 4   1  1 -1 -1  20
## 5  -1 -1  1 -1  17
## 6   1 -1  1 -1  25
## 7  -1  1  1 -1  15
## 8   1  1  1 -1  25
## 9  -1 -1 -1  1  10
## 10  1 -1 -1  1  24
## 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

Running the Model:

Model <- lm(Obs~A*B*C*D,data = Data)
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

Plotting Half Normal Plot:

library(DoE.base)
halfnormal(Model)

Comment:

--> We can see from above that significant factors in model are A,D,AD and AC .As these are fall far away from the normal line.

PART B:

Writing the Hypothesis for two interaction and 3 main effects:

Null: \(H_o\)\[ (\alpha\gamma)_{i,k}=0 \]

\[ (\alpha\lambda)_{i,l}=0 \]

\[ \alpha_{i}=0 \]

\[ \gamma_{k}=0 \]

\[ \lambda_{l}=0 \]

Alternate: \(H_a\)

\[ (\alpha\gamma)_{i,k}\neq0 \]

\[ (\alpha\lambda)_{i,l}\neq0\]

\[ \alpha_{i}\neq0 \]

\[ \gamma_{k}\neq0 \]

\[ \lambda_{l}\neq0 \]

As AD and AC are also significant hence following are factors that we will have to consider A, C , D, AC , AD

Model1 <- aov(Obs~A+C+D+A*C+A*D,data = Data)
summary(Model1)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## A            1 256.00  256.00  51.717 2.96e-05 ***
## C            1  49.00   49.00   9.899   0.0104 *  
## D            1   2.25    2.25   0.455   0.5155    
## A:C          1   9.00    9.00   1.818   0.2073    
## A:D          1   0.25    0.25   0.051   0.8267    
## Residuals   10  49.50    4.95                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
library(ggfortify)
library(ggplot2)
autoplot(Model1)

Analysis of the residuals indicates that the model is viable (constant variance, normality).

Conclusion:

--> All the tested main effects (A,C,D) appears to be significant at alpha =0.05. Also the interaction term (AC & AD) appears to be significant at alpha = 0.05

P-Value for A = 3.45 e^-5

P-Value for C = 0.01054

P-Value for D = 0.0004647

P-Value for AC = 5.83 e^-5

P-Value for AD = 9.19 e^-5

Plotting Interaction Graphically for Analysis:

Interaction Plots:

interaction.plot(A,D,Obs)

interaction.plot(A,C,Obs)