In a process development study on yield, four factors were studied, each at two levels: time (A), concentration (B), pressure (C), and temperature (D).

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 <- rep(c(rep(-1,8),rep(1,8)))
results <- c(12,    18, 13, 16, 17, 15, 20, 15, 10, 25, 13, 24, 19, 21, 17, 23)

a)

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

library(DoE.base)
halfnormal(aov(results~A*B*C*D))

According to this, our significant factors should be A, AC, AD, and D.

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.

Null hypothesis: \(\alpha=0\) vs \(\alpha\neq0\)

\(\beta=0\) vs \(\beta\neq0\)

\(\gamma=0\) vs \(\gamma\neq0\)

\(\alpha\beta=0\) vs \(\alpha\beta\neq0\)

\(\alpha\gamma=0\) vs \(\alpha\gamma\neq0\)

\(\beta\gamma=0\) vs \(\beta\gamma\neq0\)

\(\alpha\beta\gamma=0\) vs \(\alpha\beta\gamma\neq0\)

Where each term represents the amout of error given by the factor.

summary(aov(results~A+C+D+A*C+A*D))
##             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

We can see here that A,C,D,AC, and AD are significant at an alpha value of .05.

All Code Used:

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 <- rep(c(rep(-1,8),rep(1,8)))
results <- c(12,    18, 13, 16, 17, 15, 20, 15, 10, 25, 13, 24, 19, 21, 17, 23)

library(DoE.base)
halfnormal(aov(results~A*B*C*D))

summary(aov(results~A*C*D))