1 Question 1:

a)Display the halfnormal plot for this data and determine which factors appear to be significant. 

b)Pull terms that do not appear to be significant into error and test for the significance of the other effects at the 0.05 level of significance.   

1.1 Solution:

PART A:

Reading the Data:

Obs <- c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
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  16
## 5  -1 -1  1 -1  17
## 6   1 -1  1 -1  15
## 7  -1  1  1 -1  20
## 8   1  1  1 -1  15
## 9  -1 -1 -1  1  10
## 10  1 -1 -1  1  25
## 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 
##  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

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  81.00   81.00  49.846 3.46e-05 ***
## C            1  16.00   16.00   9.846 0.010549 *  
## D            1  42.25   42.25  26.000 0.000465 ***
## A:C          1  72.25   72.25  44.462 5.58e-05 ***
## A:D          1  64.00   64.00  39.385 9.19e-05 ***
## Residuals   10  16.25    1.62                     
## ---
## 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,col = c("red","blue"))

interaction.plot(A,C,Obs,col = c("red","blue"))

2 Source Code:

getwd()

#Part 1A:
install.packages("DoE.base")
library(DoE.base)

Obs <- c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
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

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

halfnormal(Model)

Model1 <- aov(Obs~A+C+D+A*C+A*D,data = Data)
summary(Model1)
library(ggfortify)
library(ggplot2)
autoplot(Model1)


#Interaction Plots:
interaction.plot(A,D,Obs,col = c("red","blue"))
interaction.plot(A,C,Obs,col = c("red","blue"))