Flipped Assignment 16 - Unreplicated 2^k Factorial Designs
Jeffrey Goens, Ben Herndon, Gabriel Garza, Scott Michaelis
11/14/2021
Problem
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 | B | C | D | yield |
|---|---|---|---|---|
| - | - | - | - | 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)
yield <- c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
dat <- data.frame(A,B,C,D,yield)
mod1 <- lm(yield~A*B*C*D,data=dat)Part A
Display the half-normal plot for this data and determine which factors appear to be significant.
halfnormal(mod1)##
## Significant effects (alpha=0.05, Lenth method):## [1] A A:C A:D D
Based on the half normal plot, factors D,AxD,AxC, and A are significant since higher order effect terms include C, will be pulled into model as well, but not going to explicitly state it.
Part 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.
#make them factors so we can do anova
A <- as.factor(c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1))
B <- as.factor(c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1))
C <- as.factor(c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1))
D <- as.factor(c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1))
dat <- data.frame(A,B,C,D,yield)
aov1 <- aov(yield~A+D+A*C+A*D,data = dat)
summary(aov1)## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 81.00 81.00 49.846 3.46e-05 ***
## D 1 42.25 42.25 26.000 0.000465 ***
## C 1 16.00 16.00 9.846 0.010549 *
## 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#just to prove that these are the same
aov2 <- aov(yield~A+C+D+A*C+A*D,data = dat)
summary(aov2)## 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 1.63
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1All of the remaining effects are significant at the alpha = .05 level
All Code Used
Below you will find a block containing all of the code used to generate this document
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)
yield <- c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
dat <- data.frame(A,B,C,D,yield)
mod1 <- lm(yield~A*B*C*D,data=dat)
#Part A
halfnormal(mod1)
#Part B
A <- as.factor(c(-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1,-1,1))
B <- as.factor(c(-1,-1,1,1,-1,-1,1,1,-1,-1,1,1,-1,-1,1,1))
C <- as.factor(c(-1,-1,-1,-1,1,1,1,1,-1,-1,-1,-1,1,1,1,1))
D <- as.factor(c(-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,1,1,1,1,1))
dat <- data.frame(A,B,C,D,yield)
aov1 <- aov(yield~A+D+A*C+A*D,data = dat)
summary(aov1)
aov2 <- aov(yield~A+C+D+A*C+A*D,data = dat)
summary(aov2)