Problem 1

Load Data

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
A<-rep(c(-1,1),8)
B<-rep(c(-1,1),each=2,times=4)
C<-rep(c(-1,1),each=4,times=2)
D<-rep(c(-1,1),each=8)

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

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

a.

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 -1.027824e-16 
##           C:D         A:B:C         A:B:D         A:C:D         B:C:D 
## -1.595946e-16  5.000000e-01  3.750000e-01 -1.250000e-01 -3.750000e-01 
##       A:B:C:D 
##  5.000000e-01
halfnormal(mod)

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

The effects A, AC, AD, and D appear to be significant because they are separated from the other effects on the half-normal plot. C is also relatively large and may be significant.

b.

The effects that appear insignificant are pulled into error. The reduced model includes A, C, D, AC, and AD.

mod2<-aov(obs~A+C+D+A:C+A:D,data=dat)
summary(mod2)
##             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

Hypotheses

For each effect:

\[ H_0:\text{Effect}=0 \]

\[ H_a:\text{Effect}\neq0 \]

The significance level is \(\alpha=0.05\).

Conclusion

The p-values for A, C, D, AC, and AD are all less than 0.05, thus, we reject the null hypothesis for these effects.

Time (A), pressure (C), temperature (D), and the interactions AC and AD have statistically significant effects on yield. Concentration (B) and the remaining interactions do not appear to be significant.

R Script

library(DoE.base)

A<-rep(c(-1,1),8)
B<-rep(c(-1,1),each=2,times=4)
C<-rep(c(-1,1),each=4,times=2)
D<-rep(c(-1,1),each=8)

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

dat<-data.frame(A,B,C,D,obs)
dat
mod<-lm(obs~A*B*C*D,data=dat)
coef(mod)
halfnormal(mod)
mod2<-aov(obs~A+C+D+A:C+A:D,data=dat)
summary(mod2)