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ANOVA Two Factor Models 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 Error SSE DFE = (n-1)-(ab-1) =ab(r-1) Factor A SSA DFA = a -1 Factor B SSB DFB = b -1 Interaction (I) SSI = SSTr – (SSA+SSB) DFI = (ab-1)-((a-1)+(b-1)) =(a-1)(b-1) Treatment SSTr DFTr = ab - 1 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 rab-1 Error SSE (n-1) - above SSE/DFE SST-SSA-SSB-SSI ab(r-1)

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ApproachFIRST 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 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 2 factors – diet and exercise programs a = 4 levels a = 4 levels for diets – none, low cal, low carb, modified liquid b = 3 levels b = 3 levels for exercise programs – none, 3 times/wk, daily r = 4 replications r = 4 replications from each of the 12 diet-exercise treatments, thus n = (4)(3)(4) = 48 observations response variable The response variable is weight loss over 3 months.

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Excel Approach -- Men MUST have 1 row and 1 column of labels! Number of replications in each diet-exercise treatment

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Excel Output -- Men 1. High p-value for interaction Cannot conclude interaction DietExercise 3. Low p-value for exercise Can conclude exercise alone affects weight loss 2. High p-value for diet Cannot conclude diet alone affects weight loss Error

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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 2 factors – diet and exercise programs a = 4 levels a = 4 levels for diets – none, low cal, low carb, modified liquid b = 3 levels b = 3 levels for exercise programs – none, 3 times/wk, daily r = 4 replications r = 4 replications from each of the 12 diet-exercise treatments, thus n = (4)(3)(4) = 48 observations response variable The response variable is weight loss over 3 months

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Excel Approach -- Women MUST have 1 row and 1 column of labels! Number of replications in each diet-exercise combination

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Excel Output -- Women Low p-value for interaction Can conclude diet and exercise interact to affect weight loss DietExercise Error 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

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ANalysis Of VAriance can be used to test for the equality of three or more population means. H 0 : 1 = 2 = 3 = ... = k H a : Not all population.

ANalysis Of VAriance can be used to test for the equality of three or more population means. H 0 : 1 = 2 = 3 = ... = k H a : Not all population.

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