Input Data

library(DoE.base)
## Warning: package 'DoE.base' was built under R version 4.2.2
## 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 <- 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)
obs <- c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)

data <- data.frame(A,B,C,D,obs)

mod <- lm(obs~A*B*C*D,data=data)

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

Item a

From the plots, the factors A, C, D, A:C, and A:D are significant since they do not follow a normal line. In other words, they do not behave like a random error.

Item b

Hypothesis

\[ H_o: \alpha\delta_{im}=0 \\ H_a: \alpha\delta_{im}\neq0 \]

\[ H_o: \alpha\gamma_{il}=0 \\ H_a: \alpha\gamma_{il}\neq0 \]

Where, \(\alpha\) is the main effect of A, \(\gamma\) is the main effect of C, \(\delta\) is the main effect of D.

Hypothesis Testing

mod2 <- lm(obs~A+C+D+A*C+A*D,data=data)

summary(mod2)
## 
## Call:
## lm.default(formula = obs ~ A + C + D + A * C + A * D, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.6250 -0.9375  0.1250  0.8750  1.3750 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  17.3750     0.3187  54.520 1.04e-13 ***
## A             2.2500     0.3187   7.060 3.46e-05 ***
## C             1.0000     0.3187   3.138 0.010549 *  
## D             1.6250     0.3187   5.099 0.000465 ***
## A:C          -2.1250     0.3187  -6.668 5.58e-05 ***
## A:D           2.0000     0.3187   6.276 9.19e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.275 on 10 degrees of freedom
## Multiple R-squared:  0.9443, Adjusted R-squared:  0.9165 
## F-statistic: 33.91 on 5 and 10 DF,  p-value: 5.856e-06

From the test, we can assure now that the terms that remain the linear model equation, whiche are only five out of sixteen (A,C,D,A:C,A:D), are significant. All the p-values are lesser than alpha (\(\alpha=0.05\)).