1. ANOVA Randomized Block Models Randomized Block Models 2-Factor Without Replication

2. Reducing Variability Remember -- we are more sure of our results if we can reduce variability. Also we may feel that more than one factor affects the outcome of the experiment. Both of these concerns are tackled by looking at multifactor designs. In this module we will only look at “block” designs. –Excel: 2-Factor Without Replication

3. “Pairing Things Up” Recall that for the difference in population means, one design was a matched pairs design which could be used in an attempt to reduce variability in certain situations. Consider the situation where we are trying to decide if the learning mode affects the score on the final exam. –We could have been unlucky and by chance selected people that were “good in math” in one group and not so good in math in another group, clouding the issue of whether it was the learning mode that affected the final exam grades.

4. Randomized Blocks EXCEL: 2 Factor Without Replication Like a matched pairs experiment for more than two populations Assumptions b k 1.Experiment consists of a series of b blocks for each of k levels of the treatments. 2.One observation is selected randomly and independently from each block for each treatment level each of which is assumed to have normal distribution. 3.The standard deviations for each treatment- block combination are equal.

5. Example A modeler wishes to create an experimental design so that each teaching format has one person who scored on the Math portion of the SAT: ≥700 600-700 500-600 400-500 < 400 Begin by calculating: Treatment Means Block Means Grand Mean

6. LectureText Videotape Internet SAT ≥ 70094 808890 600-70085 758076 500-60090 726680 400-50064 607577 <40067 585172 DATA: Exam Grade Blocked By SAT Score x Tr 80 69 72 79  x Tr 80 69 72 79  x Bl 8879776962 75 Grand mean Grand mean  x Treatments Blocks

7. Sums Of Squares/DF Degrees of Freedom (DF) --Degrees of Freedom (DF) -- –Total = n-1 –Treatments = #Levels - 1 = k-1 –Blocks = #Blocks -1 = b-1 –Error = DFT-(DFTr + DFBl) = n-k-b+1 (SS)Sum of Squares (SS) is divided into: –Total -- SST =  (x ij -  x) 2 –Treatment -- SSTr =  b ∙ (  x Tj -  x) 2 –Block -- SSBl =  k ∙ (  x Bi -  x) 2 –Error -- SSE = SST – (SSTr + SSBl)

8. Partitioning the Sums of Squares and the Degrees of Freedom Number DF SS Levels of the factor: k k-1 SSTr Different blocks: b b-1 SSBl Thus Total observations n=kb n-1 SST Subtracting Error SSE = SST – (SSTr+ SSBl) (n-1)- ((k-1)+(b-1))

9. MS and F tests MSTr = SSTr/DFTr MSBl = SSBl/DFBl MSE = SSE/DFE Can we conclude treatment means differ? –F test with F = MSTr/MSE compared to F.05,DFTr,DFE Can we conclude block means differ? –F test with F = MSBl/MSE compared to F.05,DFBl,DFE

10. Hand Calculations Sum of Squares:Sum of Squares: SST = (94-75) 2 +  ) 2 +…+(72-75) 2 = 2534 SSTr =  ((80-75) 2 +(69-75) 2 +(72-75) 2 +(79-75) 2 )= 430 SSBl =  ) 2 +  ) 2 +  ) 2 +  ) 2 +  ) 2 )= 1576 SSE = SST - SSTr - SSBl = 2534 – (430 + 1576) = 528 DF --DF -- Total = DFT= n-1 = 19 Treatments = DFTr = #Levels - 1 = k-1 = 3 Blocks = DFBl = #Blocks -1 = b-1 = 4 Error = DFT- (DFTr + DFBl) = 19-3-4 = 12

11. Can We Conclude the Treatment Means Differ? H 0 : µ T1 = µ T2 = µ T3 = µ T4 H A : At least on of these µ Tj ’s differs from the others Select α =.05 Reject H 0 (Accept H A ) if: F = MSTr/MSE > F.05, 3,12 = 3.49 Calculations: MSTr = 430/3 =143.33 MSE = 528/12 = 44 F = 143.33/44 =3.26 3.26 < 3.49 Cannot conclude a difference in the treatment means

12. Can We Conclude the Block Means Differ? H 0 : µ B1 = µ B2 = µ B3 = µ B4 = µ B5 H A : At least on of these µ Bi ’s differs from the others Select α =.05 Reject H 0 (Accept H A ) if: F = MSBl/MSE > F.05, 4,12 = 3.26 Calculations: MSBl = 1576/4 = 394 MSE = 528/12 = 44 F = 394/44 = 8.95 8.95 > 3.26 Can conclude a difference in the block means

13. EXCEL for Randomized Blocks

14. Randomized Block Output BlocksTreatmentsp-value for Blocks p-value for Treatments Low -- Can conclude SAT affects grade High -- Cannot conclude teaching format affects grade

15. Review Blocks tend to reduce variability Two Factor Without Replication (Randomized Block) –Assumptions –Degrees of Freedom –Sum of Squares –Mean Squares F tests for –Treatments –Blocks Excel