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MANE 6313

Week 6, 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 factorial designs.

Introduction

  • We want to extend the analysis of variance to consider two or more treatment factors

  • In general, the most efficient type of experiment is a factorial design.

  • Suppose factor \(A\) has \(a\) levels and factor \(B\) has \(b\) levels. Each replicate of the factorial design contains all \(ab\) treatment combinations.

  • This arrangement of treatments is said to be a crossed design

  • Examine two-factor example in figure 5-1 & 5-2 on page 180

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Notation and other details

  • If each factor has two levels, we can label the observations at a high level as (+) and observations at a low level as (-)

  • Thus \(A^+\), represent those experiments in which factor \(A\) is at its high level.

  • The effect of switching from the high level to level of a factor can be found by \(A^+-A^-\) where \(A^+\) is the average of all observations at the high level of \(A\) and \(A^-\) is the average of all observations at the low level of \(A\)

  • When referring to one of the primary factors of an experiment, such as \(A\), the difference due to changing levels is often called a main effect

Interactions

  • We may observe differences in response between the levels of one factor are not the same at all levels of the other factor.
  • When these differences occur, an interaction is said to have occurred
  • Study Figures 5-3 and 5-4 on page 181 to understand interactions
  • We will be investigating this phenomenon with new models and analysis

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Advantages of Factorial Designs

  • More efficient (less experiments with same precision) than one-at-a-time experiments

  • Inherent replication

  • When interactions are present, one-at-a-time experiments may produce wrong results

  • We are making estimates at several levels of each of the factors yielding results that are valid over a range of experimental conditions.

One-Factor At a Time Experiments

  • Read Supplemental article "One-Factor-at-a-Time versus Designed Experiments"

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Source: Montgomery, Introduction to Statistical Quality Control.