ANOVA Two Factor Models

1. ANOVA Two Factor Models

2. 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

3. Partitioning of SS and DF Factor A SSA DFA = a -1 Treatment SSTr DFTr = ab - 1 Factor B SSB DFB = b -1 Interaction (I) SSI = SSTr – (SSA+SSB) DFI = (ab-1)-((a-1)+(b-1)) =(a-1)(b-1) Error SSE DFE = (n-1)-(ab-1) =ab(r-1) TOTAL SST DFT = n-1 = rab - 1

4. ANOVA TABLE rab-1 ab(r-1) SST-SSA-SSB-SSI • 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 ErrorSSE(n-1) - Sabove SSE/DFE

5. 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

6. 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.

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

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

9. 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 replicationsfrom each of the 12 diet-exercise treatments, thus n = (4)(3)(4) = 48 observations The response variable is weight loss over 3 months

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

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

12. 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