Homework 17
Derrell Dunn
2025-11-22
1 Problem
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<-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_lbs<-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_lbs)
mod<-lm(yield_lbs~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-011.1 part a
Display the halfnormal plot for this data and determine which factors appear to be significant.
halfnormal(mod)
##
## Significant effects (alpha=0.05, Lenth method):## [1] A A:C A:D D1.2 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.
test_not_significant<-lm(yield_lbs~(A+D+(A*C)+(A*D)),data = dat)
halfnormal(test_not_significant)## Warning in halfnormal.lm(test_not_significant): halfnormal not recommended for
## models with more residual df than model df
##
## Significant effects (alpha=0.05, Lenth method):## [1] A A:C A:D Danova(test_not_significant)## Analysis of Variance Table
##
## Response: yield_lbs
## Df Sum Sq Mean Sq F value Pr(>F)
## A 1 81.00 81.000 49.8462 3.456e-05 ***
## D 1 42.25 42.250 26.0000 0.0004647 ***
## C 1 16.00 16.000 9.8462 0.0105485 *
## A:C 1 72.25 72.250 44.4615 5.583e-05 ***
## A:D 1 64.00 64.000 39.3846 9.193e-05 ***
## Residuals 10 16.25 1.625
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 11.3 Source code
library(DoE.base)
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_lbs<-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_lbs)
mod<-lm(yield_lbs~A*B*C*D, data=dat)
coef(mod)
halfnormal(mod)
test_not_significant<-lm(yield_lbs~(A+D+(A*C)+(A*D)),data = dat)
halfnormal(test_not_significant)
anova(test_not_significant)