Factorial Designs - 1

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Factorial Designs - 1

• Intervention studies with 2 or more categorical
  explanatory variables leading to a numerical
  outcome variable are called Factorial Designs.
• A factor is simply a categorical variable with
  two or more values, referred to as levels.
• A study in which there are 3 factors with 2
  levels is called a 23 factorial Design.

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Factorial Designs - 2

• If BLOCKING has been used it is counted as one of
  the factors.
• Blocking helps to improve precision by raising
  homogeneity of response among the subjects
  comprising the block..

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Factorial Designs - 3

• Advantages of factorial Designs are
• A greater precision can be obtained in estimating
  the overall main factor effects.
• Interaction between different factors can be
  explored.
• Additional factors can help to extend validity of
  conclusions derived.

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Analysis of Factorial Designs - 1

• Procedure used is General Linear Modelling.
• To compare the effects of different types of
  protein on growth laboratory mice were fed diets
  based on either cereal protein, or beef or pork.
  There were 20 mice in each group, half on a low
  quantity of the particular protein and half on
  larger quantity.
• Thus we have a study with 3 factors and 2 levels
  a 23 Factorial Design.

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Analysis of Factorial Designs - 2

• Factor Type Levels Values
• Type PR fixed 3 1 2 3
• Lo/Hi fixed 2 1 2
• Analysis of Variance for Weight(g, using Adjusted
  SS for Tests
• Source DF Seq SS Adj SS Adj MS
  F P
• Type PR 2 266.5 266.5 133.3
  0.58 0.561
• Lo/Hi 1 3168.3 3168.3 3168.3
  13.90 0.000
• Error 56 12764.1 12764.1 227.9
• Total 59 16198.9

F statistic is not significant (P 0.561) for
Type of protein but significant (P 0.000) for
Amount of protein fed.
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Analysis of Factorial Designs - 2
The main effects plot shows difference in weight
gain with amount rather than type of protein
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Analysis of Factorial Designs - 3
The lines cross. It means that there is
interaction between the type of protein and the
quantity fed. When the amount of protein fed is
small, growth is better on cereal, but with
larger quantities animal protein does better.
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Multiple Regression Approach to Analysis of
factorial Designs

• The regression equation is
• Weight (g) 87.9 1.73 Cereal 1.23 Beef
  7.27 Pork 3.13 X1X3 3.13 X2X3
• The above equation is obtained with effect coding