MANE 6313
Week 7, Module A
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
Describe two-level factorial designs.
Introduction
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Let us consider an experiment with \(k\) factors. If only 2 levels (hi,low) are observed the experiment has \(2\times 2\times\cdots\times 2\) or \(2^k\) treatment combinations. Such an experiment is called a \(2^k\) factorial design.
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Assumptions used throughout chapter 6: the factors are fixed, designs are completely randomized and the normality assumption is satisfied
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\(2^k\) design are very useful in the early stages of experimentation. This design and (fractional factorial designs) are referred to as screening experiment.
\(2^2\) Design
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Examine Figure 6--1 on page 195.
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New notation: a, b, ab, (1).
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\(a\) represents the total of all \(n\) replicates when Factor A is at its hi level and Factor B is at its low level.
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\(ab\) represents the total of all \(n\) replicates when both Factors A and B are at their hi levels
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(1) represents the total of all \(n\) replicates when Factors A and B are both at their low levels.
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Standard order for a \(2^2\) experiment is (1), a, b, ab
Estimating Effects
- Main effect \(A\)
- AB interaction
| Treatment | I | A | B | AB |
|---|---|---|---|---|
| (1) | + | - | - | + |
| a | + | + | - | - |
| b | + | - | + | - |
| ab | + | + | + | + |
Design Matrix in standard order for a \(2^2\) Design
Contrasts and Sum of Squares
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The main effect equations are very similar to a contrast
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The contrast for \(A\) is \(ab + a - b - (1)\)
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To convert the information to \(SS_A\),