FLIPPED ASSIGNMENT 17

PART A:

obs <-c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
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)
dat <- data.frame(A,B,C,obs)
dat
##     A  B  C obs
## 1  -1 -1 -1  12
## 2   1 -1 -1  18
## 3  -1  1 -1  13
## 4   1  1 -1  16
## 5  -1 -1  1  17
## 6   1 -1  1  15
## 7  -1  1  1  20
## 8   1  1  1  15
## 9  -1 -1 -1  10
## 10  1 -1 -1  25
## 11 -1  1 -1  13
## 12  1  1 -1  24
## 13 -1 -1  1  19
## 14  1 -1  1  21
## 15 -1  1  1  17
## 16  1  1  1  23
Mod <- lm(obs~A*B*C*D,data = dat)
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 -6.595890e-16 
##           C:D         A:B:C         A:B:D         A:C:D         B:C:D 
## -2.984437e-16  5.000000e-01  3.750000e-01 -1.250000e-01 -3.750000e-01 
##       A:B:C:D 
##  5.000000e-01
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
halfnormal(Mod)

## 
## Significant effects (alpha=0.05, Lenth method):
## [1] A   A:C A:D D

COMMENT: From the plot, we know that the significant factors are A, D, AD and AC since they are far away from the normal line.

PART B:

Null Hypothesis:

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

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

Alternate Hypothesis:

\[ (\alpha \gamma )_{i,k}\neq 0 \]

\[ (\alpha \lambda ) _{i,j}\neq 0 \]

Since AD and AC are significant, A, C, D, AC, AD have to be considered as well.

Model2<- aov(obs~A+C+D+A*C+A*D,data=dat)
summary(Model2)
##             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

Comment - Based on above values, the main effects A, D, AC, AD appear to be significant at alpha =0.05 since the p values are less than 0.05.

Interaction Plots:

interaction.plot(A,D,obs,col = c("red","blue"))

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

SOURCE CODE:

obs <-c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
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)
dat <- data.frame(A,B,C,obs)
dat
Mod <- lm(obs~A*B*C*D,data = dat)
coef(Mod)
library(DoE.base)
halfnormal(Mod)
Model2<- aov(obs~A+C+D+A*C+A*D,data=dat)
summary(Model2)
interaction.plot(A,D,obs,col = c("red","blue"))
interaction.plot(A,C,obs,col = c("red","blue"))