ANOVA Two Factor Models Two Factor Models

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

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

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)

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

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.

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

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

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

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

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

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