Problem 11

Entering and sorting data:

library(DoE.base)
## Warning: package 'DoE.base' was built under R version 4.1.3
## 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
A <- rep(c(-1,1),8)
B <- rep(c(-1,-1,1,1),4)
C <- rep(c(rep(-1,4), rep(1,4)),2)
D <- c(rep(-1,8), rep(1,8))
obs11 <- c(12,
           18,
           13,
           20,
           17,
           25,
           15,
           25,
           10,
           24,
           13,
           24,
           19,
           21,
           17,
           23
)

Problem 11(a)

The halfnormal plot is shown as follows:

model11 <- lm(obs11~A*B*C*D)
halfnormal(model11)
## 
## Significant effects (alpha=0.05, Lenth method):
## [1] A     C     A:C:D

From the half-normal plot, we observe that the main factor A and C are significant. Also, the 3-way interaction between the factors A,C, and D is also important.

Problem 11(b)

So, based on our previous calculation, we create the following model:

yijkl = mu + ai +bj + ck +abcijk + eijkl

Where, mu = grand mean

ai = main effect of factor A

bj = main effect of factor C

ck = main effect of factor D

abcijk = 3-way interaction between factor A,C,D

eijkl = random error

Null hypothesis: Ho: ai = 0,

bj = 0,

ck = 0,

abcijk = 0

Alternative hypothesis: Ha: ai ≠ 0,

bj ≠ 0,,

ck ≠ 0,,

abcijk ≠ 0

From ANOVA analysis we have:

model11mod <- aov(obs11~A+C+D+A*C*D)
summary(model11mod)
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## A            1 256.00  256.00 157.538 1.52e-06 ***
## C            1  49.00   49.00  30.154  0.00058 ***
## D            1   2.25    2.25   1.385  0.27314    
## A:C          1   9.00    9.00   5.538  0.04643 *  
## A:D          1   0.25    0.25   0.154  0.70513    
## C:D          1   6.25    6.25   3.846  0.08550 .  
## A:C:D        1  30.25   30.25  18.615  0.00256 ** 
## Residuals    8  13.00    1.63                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

From ANOVA, we have:

main effect p-values for A and C less than the level of signifcance of alpha = 0.05. Hence we reject the null hypothesis and conclude that A and C have significant effects.

Also, we conclude that Two-way interaction A and C and 3-way interaction between A,C,D are significant as their p-value is less than the level of significance of alpha = 0.05.

All other main factors and interactions are insignificant.

Complete code chunk:

library(DoE.base)
A <- rep(c(-1,1),8)
B <- rep(c(-1,-1,1,1),4)
C <- rep(c(rep(-1,4), rep(1,4)),2)
D <- c(rep(-1,8), rep(1,8))
obs11 <- c(12,
           18,
           13,
           20,
           17,
           25,
           15,
           25,
           10,
           24,
           13,
           24,
           19,
           21,
           17,
           23
)

model11 <- lm(obs11~A*B*C*D)
halfnormal(model11)

model11mod <- aov(obs11~A+C+D+A*C*D)
summary(model11mod)