IE-5342 Assignment 17
Bradley Solomon
Last compiled on November 01, 2026 at 8:31 PM - CST
1 Introduction
In a process development study on yield, four factors were studied, each at two levels: time (A), concentration (B), pressure (C), and temperature (D).
runnum<-c(1:16)
runact<-c(5,9,8,13,3,7,14,1,6,11,2,15,4,16,10,12)
design<- expand.grid(A = c(-1, 1),
B = c(-1, 1),
C = c(-1, 1),
D = c(-1, 1))
yield<-c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
df<-data.frame(runnum,runact,design,yield)
df## runnum runact A B C D yield
## 1 1 5 -1 -1 -1 -1 12
## 2 2 9 1 -1 -1 -1 18
## 3 3 8 -1 1 -1 -1 13
## 4 4 13 1 1 -1 -1 16
## 5 5 3 -1 -1 1 -1 17
## 6 6 7 1 -1 1 -1 15
## 7 7 14 -1 1 1 -1 20
## 8 8 1 1 1 1 -1 15
## 9 9 6 -1 -1 -1 1 10
## 10 10 11 1 -1 -1 1 25
## 11 11 2 -1 1 -1 1 13
## 12 12 15 1 1 -1 1 24
## 13 13 4 -1 -1 1 1 19
## 14 14 16 1 -1 1 1 21
## 15 15 10 -1 1 1 1 17
## 16 16 12 1 1 1 1 232 Part a
Display the half normal plot for this data and determine which factors appear to be significant.
library(DoE.base)
model1<-lm(yield~A*B*C*D, data = df)
halfnormal(model1)
The results of the half normal plot show that the two factor interactions between A:D and A:C appear to be significant. The main effects A and D are also significant but these main effects are dependent of each other due to the presence of the significant two factor interactions.
3 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.
The half normal plot results in Part a show that there are no significant four factor or 3 factor interactions; therefore, these will be pulled into the error of our model.
model2<-lm(yield~A*B+A*C+A*D+B*C+B*D+C*D,data = df)
summary(model2)##
## Call:
## lm.default(formula = yield ~ A * B + A * C + A * D + B * C +
## B * D + C * D, data = df)
##
## Residuals:
## 1 2 3 4 5 6 7 8 9 10 11
## 0.125 0.625 0.125 -0.875 -0.875 0.125 0.625 0.125 -1.125 0.375 0.875
## 12 13 14 15 16
## -0.125 1.875 -1.125 -1.625 0.875
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.738e+01 3.992e-01 43.523 1.21e-07 ***
## A 2.250e+00 3.992e-01 5.636 0.00244 **
## B 2.500e-01 3.992e-01 0.626 0.55864
## C 1.000e+00 3.992e-01 2.505 0.05416 .
## D 1.625e+00 3.992e-01 4.070 0.00963 **
## A:B -3.750e-01 3.992e-01 -0.939 0.39068
## A:C -2.125e+00 3.992e-01 -5.323 0.00313 **
## A:D 2.000e+00 3.992e-01 5.010 0.00407 **
## B:C 1.250e-01 3.992e-01 0.313 0.76684
## B:D 1.874e-16 3.992e-01 0.000 1.00000
## C:D -2.776e-16 3.992e-01 0.000 1.00000
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.597 on 5 degrees of freedom
## Multiple R-squared: 0.9563, Adjusted R-squared: 0.8689
## F-statistic: 10.94 on 10 and 5 DF, p-value: 0.008256The interaction effects B:D and C:D will both be pulled into the model error since they are the most insignificant two factor interactions with p-values equal to 1.
model3<-lm(yield~A*B+A*C+A*D+B*C,data = df)
summary(model3)##
## Call:
## lm.default(formula = yield ~ A * B + A * C + A * D + B * C, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.625 -0.875 0.125 0.625 1.875
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.3750 0.3374 51.497 2.73e-10 ***
## A 2.2500 0.3374 6.669 0.000286 ***
## B 0.2500 0.3374 0.741 0.482829
## C 1.0000 0.3374 2.964 0.020991 *
## D 1.6250 0.3374 4.816 0.001930 **
## A:B -0.3750 0.3374 -1.111 0.303091
## A:C -2.1250 0.3374 -6.298 0.000405 ***
## A:D 2.0000 0.3374 5.928 0.000583 ***
## B:C 0.1250 0.3374 0.370 0.721982
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.35 on 7 degrees of freedom
## Multiple R-squared: 0.9563, Adjusted R-squared: 0.9064
## F-statistic: 19.15 on 8 and 7 DF, p-value: 0.0004218The interaction effect B:C is the next insignificant two factor interaction that will be pulled into the model error since it has the largest p-value = 0.721982.
model4<-lm(yield~A+B+C+D+A*B+A*C+A*D,data = df)
summary(model4)##
## Call:
## lm.default(formula = yield ~ A + B + C + D + A * B + A * C +
## A * D, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.500 -1.000 0.125 0.750 1.750
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.3750 0.3187 54.520 1.42e-11 ***
## A 2.2500 0.3187 7.060 0.000106 ***
## B 0.2500 0.3187 0.784 0.455366
## C 1.0000 0.3187 3.138 0.013850 *
## D 1.6250 0.3187 5.099 0.000931 ***
## A:B -0.3750 0.3187 -1.177 0.273139
## A:C -2.1250 0.3187 -6.668 0.000158 ***
## A:D 2.0000 0.3187 6.276 0.000239 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.275 on 8 degrees of freedom
## Multiple R-squared: 0.9554, Adjusted R-squared: 0.9165
## F-statistic: 24.51 on 7 and 8 DF, p-value: 8.447e-05The interaction effect A:B is the next insignificant two factor interaction that will be pulled into the model error since it has the largest p-value = 0.273139.
model5<-lm(yield~A+B+C+D+A*C+A*D,data = df)
summary(model5)##
## Call:
## lm.default(formula = yield ~ A + B + C + D + A * C + A * D, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.3750 -0.9375 0.1250 0.9375 1.3750
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 17.3750 0.3254 53.391 1.43e-12 ***
## A 2.2500 0.3254 6.914 6.96e-05 ***
## B 0.2500 0.3254 0.768 0.462036
## C 1.0000 0.3254 3.073 0.013292 *
## D 1.6250 0.3254 4.993 0.000746 ***
## A:C -2.1250 0.3254 -6.530 0.000108 ***
## A:D 2.0000 0.3254 6.146 0.000170 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.302 on 9 degrees of freedom
## Multiple R-squared: 0.9477, Adjusted R-squared: 0.9129
## F-statistic: 27.2 on 6 and 9 DF, p-value: 2.796e-05Using an alpha = 0.05 we have two significant two factor interactions, A:C and A:D. The p-values for these interactions is the following:
A:C p-value = 0.000108
A:D p-value = 0.000170
4 Complete R Code
# Analysis Data
runnum<-c(1:16)
runact<-c(5,9,8,13,3,7,14,1,6,11,2,15,4,16,10,12)
design<- expand.grid(A = c(-1, 1),
B = c(-1, 1),
C = c(-1, 1),
D = c(-1, 1))
yield<-c(12,18,13,16,17,15,20,15,10,25,13,24,19,21,17,23)
df<-data.frame(runnum,runact,design,yield)
df
# Part a
library(DoE.base)
model1<-lm(yield~A*B*C*D, data = df)
halfnormal(model1)
# Part b
model2<-lm(yield~A*B+A*C+A*D+B*C+B*D+C*D,data = df)
summary(model2)
model3<-lm(yield~A*B+A*C+A*D+B*C,data = df)
summary(model3)
model4<-lm(yield~A+B+C+D+A*B+A*C+A*D,data = df)
summary(model4)
model5<-lm(yield~A+B+C+D+A*C+A*D,data = df)
summary(model5)