MANE 6313

Week 13, Module E

Student Learning Outcome

• Select an appropriate experimental design with one or more factors,
• Select an appropriate model with one or more factors,
• Evaluate statistical analyses of experimental designs,
• Assess the model adequacy of any experimental design, and
• Interpret model results.

Module Learning Outcome

Generate Box-Behnken Design in R

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bbd() Function

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Example Problem

• Taken from Dean, Voss, and Drajuljic (2016)¹

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Example Problem, Data Table

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Coding Formula

\[ \begin{aligned} x_1&\sim (temp-185)/35\\ x_2&\sim (humidity-50)/50\\ x_3&\sim (pressure-5)/4 \end{aligned} \]

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R Chunk for Box-Behnken Design

library(rsm)
bbd11.df <- bbd(3,n0=5,coding=list(x1~(temp-185)/35,x2~(humidity-50)/50,x3~(pressure-5)/4),randomize = FALSE)
y <- c(83,36,98,87,103,153,94,107,51,106,48,108,80,81,77,80,82)
bbd11.df$y <- y
print(bbd11.df)

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R Chunk for Box-Behnken Design Output

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R Chunk for Fitting Model

bbd11.model1 <- rsm(y~SO(x1,x2,x3),data=bbd11.df)
summary(bbd11.model1)

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Model Output, part 1

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Model Output, part 2

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Model Output, part 3

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Model Refinement

• For this example, we will use \(\alpha=0.1.\)
• Start with pure quadratic and work back to first order
• For pure quadratic, x2 and x3 are statistically significant using \(\alpha=0.1\)
• For two-factor interactions, no factors are statistically significant
• For first order, x2 is statistically significant
• Using hierarchy, the model will be FO(x2,x3)+PQ(x2,x3)

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Second Model, R Chunk

bbd11.model2 <- rsm(y~FO(x2,x3)+PQ(x2,x3),data=bbd11.df)
summary(bbd11.model2)

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Second Model, Output

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Partial F- Test

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1. Dean, Voss, and Drajuljic (2017). Design and Analysis of Experiments, 2nd edition. Springer Texts in Statistics. ↩