Download - ANOVA Two Factor Models
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 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-1rab-1
ErrorError SSESSE (n-1) - (n-1) - aboveabove SSE/DFESSE/DFE
SST-SSA-SSB-SSI
ab(r-1)
ApproachApproach
FIRSTFIRST• 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 programs
a = 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 -- MenMen
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 programs
a = 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 -- WomenWomen
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