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MANE 6313

Week 6, Module F

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

General Factorial Design

Resources for the Week 6, Module F micro-lecture are:

General Factorial Design

  • We will assume \(n\geq 2\) so we can include all two-factor interactions and estimate SS-error

  • For a fixed model with 3 factors we use the following model

\[ \begin{aligned} y_{ijkl}&=&\mu+\tau_i+\beta_j+\gamma_k+(\tau\beta)_{ij}+(\tau\gamma)_{ik}+(\beta\gamma)_{jk}\\ &+&(\tau\beta\gamma)_{ijk}+\varepsilon_{ijkl} \left\{\begin{array}{l}i=1,2,\ldots,a\\ j=1,2,\ldots,b\\ k=1,2,\ldots,c\\ l=1,2,\ldots,n\\ \end{array}\right.\end{aligned} \]
  • Sum of squares equations given on pages 201-202 (no surprises)

Important Point

  • The ANOVA and analysis is always the same for experiments with fixed factors

  • The presence of random factors complicates the design

  • The expected mean squares must be calculated and the divisor will not always be MS(error)!

  • Discussed in chapter 12 (not covered in class).

Blocking in a Factorial Design

  • Consider the two-factor factorial design conducted as a randomized block design

  • The statistical model is

\[ \begin{aligned} y_{ijk}&=&\mu+\tau_i+\beta_j+(\tau\beta)_{ij}+\delta_k+\varepsilon_{ijk} \left\{\begin{array}{l}i=1,2,\ldots,a\\ j=1,2,\ldots,b\\ k=1,2,\ldots,n\\ \end{array}\right.\end{aligned} \]

where \(\delta_k\) is the block effect.

  • The model assumes that interactions between blocks and treatments is negligible.

  • If these interactions exist, they can not be separated from the error component.

  • Sum of squares formulas and an example ANOVA are given in table 5-20 on page 215.

Problem 5.28

problem 5.28