1. Randomized Block Design
In chapter on paired t-tests, we learned to “match” subjects on variables that:
influence performance
but are not of interest.
Matching gives a more sensitive test of H0 because it removes sources of variance that inflate 2.
Lecture 17

2. Randomized Block Design
In the analysis of variance, the matched subjects design is called the Randomized Block Design.
subjects are first put into blocks
a block is a group matched on some variable
subjects in a block are then randomly assigned to treatments
for p treatments, you need p subjects per block
Lecture 17

3. Question: does SSB come from SSE or from SST?
Sums of squares
In the RBD, we compute SST as before. Compute SSB (SS for Blocks) analogously:
Compute deviations of block means from grand mean.
Square deviations, then add them up.
Question: does SSB come from SSE or from SST?
Lecture 17

4. Where does SSB come from?
SST
SSB
SSTotal
SSE
Residual
SSE
Lecture 17

5. Conceptual Formulas SST = Σb(XTi – XG)2 p-1 SSB = Σp(XBi – XG)2 b-1
SSTotal = Σ(Xi – XG)2 n-1
SSE = SSTotal – SST – SSB (b-1)(p-1)
= n-b-p+1
MST = SST/(p-1)
MSB = SSB/(b-1)
MSE = SSE/(b-1)(p-1) = SSE/(n-b-p+1)
Lecture 17

6. Summary table Source df SS MS F Treat p-1 SST SST/(p-1) MST/MSE
Blocks b-1 SSB SSB/(b-1) MSB/MSE
Error n-p-b+1 SSE SSE/(n-b-p+1)
Total n-1 SSTotal
Lecture 17

7. Computational Formulas
CM = (ΣX)2/n
SSTotal = ΣX2 – CM
SST = ΣTi2/b – CM
SSB = ΣBi2/p – CM
SSE = SSTotal – SST – SSB
p = # of samples b = # of blocks
Ti = Total for ith treatment
Bi = Total for ith block
Lecture 17

8. Randomized Block Design – Example 1
H0: 1 = 2 = 3
HA: At least two differ significantly
Statistical test: F = MST/MSE
Rej. region: Fobt > F(2, 8, .05) = 4.46
Lecture 17

9. Randomized Block Design – Example 1
CM =
SSTotal = ΣX2 – CM
= … –
= –
= 363.6
Lecture 17

10. Randomized Block Design – Example 1
SST = Σ(Ti2)/b – CM
= 4012/ / /5 –
= –
= 98.8
SSB = 2442/3 + … /3 –
= – =
Note how much SSB is removing from SSE here.
Lecture 17

11. Randomized Block Design – Example 1
SSE = SSTotal – SST – SSB
= – 98.8 –
= 23.87
Lecture 17

12. Randomized Block Design – Example 1
Source df SS MS F
Treat
Blocks
Error
Total
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13. Randomized Block Design – Example 1
H0: SN = GO
HA: SN ≠ GO
(Note: this is a post-hoc test. We’ll do N-K.)
Statistical test: Q = Xi – Xj
√MSE/n
Lecture 17

14. Randomized Block Design – Example 1
Rank order sample means:
Sleepy Sneezy Grumpy Dopey Goofy
Qcrit = Q(4, 8, .05) = 4.53
Lecture 17

15. Randomized Block Design – Example 1
Qobt:
86 – = = 6.35
√(2.984)/
Reject H0. Goofy & Sneezy differ significantly.
Lecture 17