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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

Describe general factorial designs and additional topics

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

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

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).

Judging Models

Fitting the correct regression model can be as much art as it is a science.

Source: https://blog.minitab.com/en/adventures-in-statistics-2/when-should-you-fit-a-non-hierarchical-regression-model#:~:text=Topics%3A%20Design%20of%20Experiments%20-%20DOE%2C%20Regression%20Analysis,terms%3A%20A%2C%20B%2C%20C%2C%20A%2AB%2C%20A%2AC%2C%20and%20B%2AC.

  • Parsimonious model
  • Hierarchical Model
  • Example Problem

Parsimonious Model

A parsimonious model is a model that achieves a desired level of goodness of fit using as few explanatory variables as possible

Source: https://www.statology.org/parsimonious-model/

  • Occam's Razor states that the simplest explanation is most likely the right one
  • Statistical Reasons:
    • Parsimonious models are easier to interpret and understand
    • Parsimonious models tend to have more predictive ability
    • Parsimonious models are less likely to be impacted by multicollinearity

Hierarchical Model

In the world of linear models, a hierarchical model contains all lower-order terms that comprise the higher-order terms that also appear in the model. For example, a model that includes the interaction term ABC is hierarchical if includes these terms: A, B, C, AB, AC, and B*C

Source: https://blog.minitab.com/en/adventures-in-statistics-2/when-should-you-fit-a-non-hierarchical-regression-model#:~:text=Topics%3A%20Design%20of%20Experiments%20-%20DOE%2C%20Regression%20Analysis,terms%3A%20A%2C%20B%2C%20C%2C%20A%2AB%2C%20A%2AC%2C%20and%20B%2AC.

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.

  • Generally, blocks are considered to be a random effect

Problem 5.28 (9th Edition)

image

Create Data Frame

Data Frame for Problem 5.28

Analyze Data

Full Model

Second Model

Second Model