Factorial Designs - 1Factorial Designs - 1

Intervention studies with 2 or more Intervention studies with 2 or more categorical explanatory variables leading categorical explanatory variables leading to a numerical outcome variable are to a numerical outcome variable are called Factorial Designs.called Factorial Designs.

A A factorfactor is simply a categorical variable is simply a categorical variable with two or more values, referred to as with two or more values, referred to as levelslevels..

A study in which there are 3 factors with A study in which there are 3 factors with 2 levels is called a 22 levels is called a 233 factorial Design. factorial Design.

Factorial Designs - 2Factorial Designs - 2

If If BLOCKINGBLOCKING has been used it is has been used it is counted as one of the factors.counted as one of the factors.

Blocking helps to improve precision Blocking helps to improve precision by raising homogeneity of response by raising homogeneity of response among the subjects comprising the among the subjects comprising the block..block..

Factorial Designs - 3Factorial Designs - 3

Advantages of factorial Designs are:Advantages of factorial Designs are: A greater precision can be obtained A greater precision can be obtained

in estimating the overall main factor in estimating the overall main factor effects.effects.

Interaction between different factors Interaction between different factors can be explored.can be explored.

Additional factors can help to extend Additional factors can help to extend validity of conclusions derived.validity of conclusions derived.

Analysis of Factorial Analysis of Factorial Designs - 1Designs - 1

Procedure used is General Linear Procedure used is General Linear Modelling.Modelling.

To compare the effects of different types of To compare the effects of different types of protein on growth laboratory mice were protein on growth laboratory mice were fed diets based on either fed diets based on either cerealcereal protein, protein, or or beefbeef or or porkpork. There were 20 mice in . There were 20 mice in each group, half on a each group, half on a lowlow quantity of the quantity of the particular protein and half on particular protein and half on largerlarger quantity.quantity.

Thus we have a study with 3 factors and 2 Thus we have a study with 3 factors and 2 levels – a levels – a 2233 Factorial Design Factorial Design..

Analysis of Factorial Analysis of Factorial Designs - 2Designs - 2

Factor Type Levels Values Factor Type Levels Values Type PR fixed 3 1 2 3Type PR fixed 3 1 2 3 Lo/Hi fixed 2 1 2Lo/Hi fixed 2 1 2

Analysis of Variance for Weight(g, using Adjusted SS for TestsAnalysis of Variance for Weight(g, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F PSource DF Seq SS Adj SS Adj MS F P Type PR 2 266.5 266.5 133.3 0.58 0.561Type PR 2 266.5 266.5 133.3 0.58 0.561 Lo/Hi 1 3168.3 3168.3 3168.3 13.90 0.000Lo/Hi 1 3168.3 3168.3 3168.3 13.90 0.000 Error 56 12764.1 12764.1 227.9Error 56 12764.1 12764.1 227.9 Total 59 16198.9Total 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.

Analysis of Factorial Analysis of Factorial Designs - 2Designs - 2

Lo/HiType PR

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ght(

g)Main Effects Plot - LS Means for Weight(g)

The main effects plot shows difference in weight gain with amount rather than type of protein

Analysis of Factorial Analysis of Factorial Designs - 3Designs - 3

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EastWestNorth

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Interaction Plot - Data Means for Weight(g)

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.

Multiple Regression Multiple Regression Approach to Analysis of Approach to Analysis of

factorial Designsfactorial DesignsThe regression equation isThe regression equation is

Weight (g) = 87.9 + 1.73 Cereal + Weight (g) = 87.9 + 1.73 Cereal + 1.23 Beef + 7.27 Pork + 3.13 1.23 Beef + 7.27 Pork + 3.13 X1*X3+ 3.13 X2*X3X1*X3+ 3.13 X2*X3

The above equation is obtained with The above equation is obtained with effect codingeffect coding