ANOVA

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• ANOVA
• Two Factor Models

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2 Factor Experiments

• Two factors can either independently or together
  interact to affect the average response levels.
• Factor A -- a levels
• Factor B -- b levels
• Thus total treatments (combinations) ab
• replications for each A/B treatment -- r
• Thus total number of observations, n rab
• Assumptions
• Each treatment has a normal distribution
• Standard deviations equal
• Sampling random and independent

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Partitioning of SS and DF

TOTAL SST DFT n-1 rab - 1
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ANOVA TABLE

• Now, SST SSTr SSE
• But SSTr broken down into SSA, SSB, SSI
• SS DF MS
• Factor A SSA a-1 SSA/DFA
• Factor B SSB b-1 SSB/DFB
• Interaction SSI (a-1)(b-1)
  SSI/DFI
• Total SST n-1

Error SSE (n-1) - Sabove SSE/DFE
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Approach

• FIRST
• Can we conclude Interaction affects mean values?
• F Test -- Compare F MSI/MSE to F.05,DFI,DFE
• IF YES -- STOP
• IF NO, DO BELOW
• Can we conclude Factor A alone affects mean
  values?
• F Test -- Compare F MSA/MSE to F.05,DFA,DFE
• Can we conclude Factor B alone affects mean
  values?
• F Test -- Compare F MSB/MSE to F.05,DFB,DFE

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Example 1

• Can we conclude that diet and exercise affect
  weight loss in men?
• The factorial experiment used has
• 2 factors diet and exercise programs
• a 4 levels for diets
• none, low cal, low carb, modified liquid
• b 3 levels for exercise programs
• none, 3 times/wk, daily
• r 4 replications from each of the 12
  diet-exercise treatments, thus n (4)(3)(4)
  48 observations
• The response variable is weight loss over 3
  months.

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Excel Approach -- Men

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Excel Output -- Men

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Example 2

• Can we conclude that diet and exercise affect
  weight loss in women?
• Again, the factorial experiment used has
• 2 factors diet and exercise programs
• a 4 levels for diets
• none, low cal, low carb, modified liquid
• b 3 levels for exercise programs
• none, 3 times/wk, daily
• r 4 replications from each of the 12
  diet-exercise treatments, thus n (4)(3)(4)
  48 observations
• The response variable is weight loss over 3 months

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Excel Approach -- Women

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Excel Output -- Women

STOP!
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Review

• Two Factor Designs
• 2 Factors (A and B) and Interaction
• Assumptions
• Degrees of Freedom
• Sum of Squares
• Mean Squares
• Approach
• F-test for interaction first if detect
  interaction, STOP
• Else F-tests for individual factors
• Excel Two Factor With Replication