Randomized Block Design 1 2 3 4 5
Randomized Block Design 1. 2. 3. 4. 5. 6. 7. Randomized block design Sums of squares in the RBD Where does SSB come from? Conceptual formulas Summary table Computational formulas Examples Lecture 17 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 H 0 because it removes sources of variance that inflate 2. Lecture 17 2
Randomized Block Design (RBD) • In the analysis of variance, the matched subjects/within 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
Sums of squares in the RBD • We compute SSTreat as before. Compute SSB (SS for Blocks) analogously: • Compute deviations of block means from grand mean. • Square deviations, then add them up. Lecture 17 4
Sums of squares in the RBD • SSTotal is composed of SSTreat + SSE • SSB must come out of SSTotal • Does SSB come from SSTreat or from SSE? Lecture 17 5
Where does SSB come from? SST SSB SSTotal SSE Residual SSE Lecture 17 6
Conceptual Formulas SST = Σb(XTi – XG)2 SSB = Σp(XBi – XG)2 SSTotal = Σ(Xi – XG)2 SSE = SSTotal – SST – SSB p-1 b-1 n-1 (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 7
8 Summary table Source df Treat Blocks Error Total p-1 b-1 n-p-b+1 n-1 SS MS F SSTreat SSB SSE SSTotal SST/(p-1) SSB/(b-1) SSE/(n-b-p+1) MST/MSE MSB/MSE Lecture 17
Computational Formulas CM = (ΣX)2 n SSE = SSTotal – SST – SSB SSTotal = ΣX 2 – CM Lecture 17 9
Computational Formulas SSTreat = ΣT 2 i – CM b SSB = ΣB 2 i – CM p Ti=Total for ith treatment b=# of blocks Bi=Total for ith block p=# of samples Lecture 17 10
11 Randomized Block Design – Example 1 a 1. Five Grade 10 high school students in an advanced math program are tested at the beginning of the term. Later in the term, they write a midterm and then a final exam. Each test/exam contains similar mathematics problems, and a comparison will be done to see whether significant differences exist between mean scores on the 3 exams. The students’ scores on the exams are: Student First exam Midterm Final Grumpy 78 84 82 Sneezy 81 86 91 Dopey 79 80 83 Goofy 77 80 82 Sleepy 86 91 94 Lecture 17
12 Randomized Block Design – Example 1 a a. Is there an overall significant difference between mean scores on the 3 exams ( =. 05). b. Although no specific predictions were made beforehand, after inspecting the data it could be seen that Sneezy consistently obtained higher exam scores than Goofy. Regardless of the results of your analysis in part (a), perform a post-hoc test to determine whether Sneezy and Goofy differ significantly on their overall average on the 3 exams ( =. 05). Lecture 17
13 Randomized Block Design – Example 1 a 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 Lecture 17
14 Randomized Block Design – Example 1 a CM = 104834. 4 SSTotal = ΣX 2 – CM = 782 + 812 + … + 942 – 104834. 4 = 105198 – 104834. 4 = 363. 6 Lecture 17
15 Randomized Block Design – Example 1 a SSTreat = Σ(Ti 2) – CM b = 4012 + 4212 + 4322 – 104834. 4 5 5 5 = 104933. 2 – 104834. 4 = 98. 8 Lecture 17
16 Randomized Block Design – Example 1 a SSB = ΣB 2 i – CM p SSB = 2442 + … + 2712 – 104834. 4 3 3 = 105075. 33 – 104834. 4 = 240. 93 Lecture 17
17 Randomized Block Design – Example 1 a SSE = SSTotal – SSTreat – SSB = 363. 6 – 98. 8 – 240. 93 = 23. 87 Lecture 17
18 Randomized Block Design – Example 1 a Source df SS MS F Treat Blocks 4 Error Total 2 98. 8 49. 4 16. 55 240. 93 60. 23 20. 18 8 23. 87 2. 98 14 363. 6 Decision: Reject HO – average scores do differ across exams. Lecture 17
19 Randomized Block Design – Example 1 b H 0 : w = A HA : W ≠ A (Note: this is a post-hoc test. We’ll do N-K. ) Statistical test: Q= X i – Xj √MSE/n Lecture 17
20 Randomized Block Design – Example 1 b Rank order sample means: Sleepy 90. 3 Sneezy 86 Grumpy 81. 3 r = 4 Qcrit = Q(4, 8, . 05) = 4. 53 Lecture 17 Dopey 80. 6 Goofy 79. 67
21 Randomized Block Design – Example 1 b Qobt: 86 – 79. 67 √(2. 984)/3 = 6. 33 0. 997 = 6. 35 Reject HO. Sneezy & Goofy differ significantly in their overall average on the 3 exams. Lecture 17
22 Randomized Bloc Design – Example 2 2. People in a weight loss program are weighed at the beginning of the program, 3 weeks after starting, and 3 months after starting. The following are the weights (in pounds) of a random sample of 5 participants at each of these time periods. Person Start 3 Wks 3 Months Mickey 210 201 193 Minnie 245 240 242 Hewey 236 228 200 Dewey 197 190 167 Louie 340 328 290 Lecture 17
23 Randomized Bloc Design – Example 2 a. Are there significant differences between weights across the 3 time periods? ( = . 05) b. Regardless of your answer in part a. , perform the appropriate tests to determine at which time periods the participants’ mean weights differ significantly. Lecture 17
24 Randomized Block Design – Example 2 a H 0 : 1 = 2 = 3 HA: At least two differ significantly Statistical test: F= Rejection region: Fobt > F(2, 8, . 05) = 4. 46 Lecture 17 MST MSE
25 Randomized Block Design – Example 2 a CM = 35072 15 = 819936. 6 SSTotal = ΣX 2 – CM = 2102 + 2452 + … + 2902 – 819936. 6 = 855701 – 819936. 6 = 35764. 4 Lecture 17
26 Randomized Block Design – Example 2 a SSTreat = Σ(Ti 2) – CM b = 12282 + 11872 + 10922 – 819936. 6 5 5 5 = 821883. 4 – 819936. 6 = 1946. 8 Lecture 17
27 Randomized Block Design – Example 2 a SSB = ΣB 2 i – CM p SSB = 6042 + 7272 … + 9582 – 819936. 6 3 3 3 = 852973. 67 – 819936. 6 = 33037. 07 Lecture 17
28 Randomized Block Design – Example 2 a SSE = SSTotal – SSTreat – SSB = 35764. 4 – 1946. 8 – 33037. 07 = 780. 5 Lecture 17
29 Randomized Block Design – Example 2 a Source df SS MS F Treat Blocks Error Total 2 4 8 14 1946. 8 33037. 07 780. 5 35764. 4 973. 4 8259. 27 97. 563 9. 977 84. 656 Decision: Reject HO – weights do differ across the 3 time periods. Lecture 17
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