Randomized Block Design In chapter on paired ttests

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 H 0 because it removes sources of variance that inflate 2. 1

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 2

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? 3

4 Where does SSB come from? SST SSB SSTotal SSE Residual SSE

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

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

Computational Formulas CM = (ΣX)2/n SSTotal = ΣX 2 – CM SST = ΣTi 2/b – CM SSB = ΣBi 2/p – CM SSE = SSTotal – SST – SSB p = # of samples b = # of blocks Ti = Total for ith treatment Bi = Total for ith block 7

Randomized Block Design – Example 1 H 0: 1 = 2 = 3 HA: At least two differ significantly Statistical test: F = MST/MSE Rej. region: Fobt > F(2, 8, . 05) = 4. 46 8

Randomized Block Design – Example 1 CM = 104834. 4 SSTotal = ΣX 2 – CM = 782 + 812 + … + 942 – 104834. 4 = 105198 – 104834. 4 = 363. 6 9

Randomized Block Design – Example 1 SST = Σ(Ti 2)/b – CM = 4012/5 + 4212/5 + 4322/5 – 104834. 4 = 104933. 2 – 104834. 4 = 98. 8 SSB = 2442/3 + … + 2712/3 – 104834. 4 = 105075. 33 – 104834. 4 = 240. 93 10

Randomized Block Design – Example 1 SSE = SSTotal – SST – SSB = 363. 6 – 98. 8 – 240. 93 = 23. 87 11

Randomized Block Design – Example 1 Source df SS MS F Treat Blocks Error Total 2 4 8 14 98. 8 240. 93 23. 87 363. 6 49. 4 60. 23 2. 98 16. 55 20. 18 12

Randomized Block Design – Example 1 H 0: SN = GO HA: SN ≠ GO (Note: this is a post-hoc test. We’ll do N-K. ) Statistical test: Q = X i – Xj √MSE/n 13

Randomized Block Design – Example 1 Rank order sample means: Sleepy 90. 3 Sneezy 86 Grumpy 81. 3 Qcrit = Q(4, 8, . 05) = 4. 53 Dopey 80. 6 Goofy 79. 67 14

Randomized Block Design – Example 1 Qobt: 86 – 79. 67 √(2. 984)/3 = 6. 33 0. 997 Reject H 0. Goofy & Sneezy differ significantly. = 6. 35 15