ANOVAANOVA

Two Factor ModelsTwo Factor Models

2 Factor Experiments2 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 DFPartitioning of SS and DF

ErrorSSE

DFE = (n-1)-(ab-1)=ab(r-1)

Factor ASSA

DFA = a -1

Factor BSSB

DFB = b -1

Interaction (I)SSI = SSTr – (SSA+SSB)DFI = (ab-1)-((a-1)+(b-1))

=(a-1)(b-1)

TreatmentSSTr

DFTr = ab - 1TOTALSST

DFT = n-1 = rab - 1

ANOVA TABLEANOVA TABLE• Now, SST = SSTr + SSE–But SSTr broken down into SSA, SSB, SSI

SS DF MSFactor A SSA a-1 SSA/DFAFactor B SSB b-1 SSB/DFBInteraction SSI (a-1)(b-1) SSI/DFI

Total SST n-1rab-1

ErrorError SSESSE (n-1) - (n-1) - aboveabove SSE/DFESSE/DFE

SST-SSA-SSB-SSI

ab(r-1)

ApproachApproachFIRSTFIRST• Can we conclude Interaction affects mean values?– F Test -- Compare F = MSI/MSE to F.05,DFI,DFE

IF YES -- STOP IF YES -- STOP IF NO, DO BELOWIF 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 1Example 1• Can we conclude that diet and exercise affect weight loss in

men?• The factorial experiment used has:

2 factors2 factors – diet and exercise programsa = 4 levelsa = 4 levels for diets – • none, low cal, low carb, modified liquid

b = 3 levelsb = 3 levels for exercise programs – • none, 3 times/wk, daily

r = 4 replicationsr = 4 replications from each of the 12 diet-exercise treatments, thus n = (4)(3)(4) = 48 observations

The response variableresponse variable is weight loss over 3 months.

Excel Approach -- Excel Approach -- Men MUST have 1 row and 1 column of labels!

Number of replications ineach diet-exercise treatment

Excel Output -- MenExcel Output -- Men

1. High p-value for interactionCannot conclude interaction

Diet

Exercise3. Low p-value for exerciseCan conclude exercisealone affects weight loss

2. High p-value for dietCannot conclude diet aloneaffects weight loss

Error

Example 2Example 2• Can we conclude that diet and exercise affect weight loss

in women?• Again, the factorial experiment used has:

2 factors2 factors – diet and exercise programsa = 4 levelsa = 4 levels for diets – • none, low cal, low carb, modified liquid

b = 3 levelsb = 3 levels for exercise programs – • none, 3 times/wk, daily

r = 4 replicationsr = 4 replications from each of the 12 diet-exercise treatments, thus n = (4)(3)(4) = 48 observations

The response variableresponse variable is weight loss over 3 months

Excel Approach -- Excel Approach -- Women MUST have 1 row and 1 column of labels!

Number of replications ineach diet-exercise combination

Excel Output -- WomenExcel Output -- Women

Low p-value for interactionCan conclude diet and exerciseinteract to affect weight loss

Diet

Exercise

Error

STOP!STOP!

ReviewReview• 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