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Incomplete Block Designs

Incomplete Block Designs. Randomized Block Design. We want to compare t treatments Group the N = bt experimental units into b homogeneous blocks of size t. In each block we randomly assign the t treatments to the t experimental units in each block.

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Incomplete Block Designs

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  1. Incomplete Block Designs

  2. Randomized Block Design • We want to compare t treatments • Group the N = bt experimentalunits into b homogeneous blocks of size t. • In each block we randomly assign the t treatments to the t experimental units in each block. • The ability to detect treatment to treatment differences is dependent on the within block variability.

  3. Comments • The within block variability generally increases with block size. • The larger the block size the larger the within block variability. • For a larger number of treatments, t, it may not be appropriate or feasible to require the block size, k, to be equal to the number of treatments. • If the block size, k, is less than the number of treatments (k < t)then all treatments can not appear in each block. The design is called an Incomplete Block Design.

  4. Commentsregarding Incomplete block designs • When two treatments appear together in the same block it is possible to estimate the difference in treatments effects. • The treatment difference is estimable. • If two treatments do not appear together in the same block it not be possible to estimate the difference in treatments effects. • The treatment difference may not be estimable.

  5. Example • Consider the block design with 6 treatments and 6 blocks of size two. 1 2 2 3 1 3 4 5 5 6 4 6 • The treatments differences (1 vs 2, 1 vs 3, 2 vs 3, 4 vs 5, 4 vs 6, 5 vs 6) are estimable. • If one of the treatments is in the group {1,2,3} and the other treatment is in the group {4,5,6}, the treatment difference is not estimable.

  6. Definitions • Two treatments i and i* are said to be connected if there is a sequence of treatments i0 = i, i1, i2, … iM = i* such that each successive pair of treatments (ij and ij+1) appear in the same block • In this case the treatment difference is estimable. • An incomplete design is said to be connected if all treatment pairs i and i* are connected. • In this case all treatment differences are estimable.

  7. Example • Consider the block design with 5 treatments and 5 blocks of size two. 1 2 2 3 1 3 4 5 1 4 • This incomplete block design is connected. • All treatment differences are estimable. • Some treatment differences are estimated with a higher precision than others.

  8. Definition An incomplete design is said to be a Balanced Incomplete Block Design. • if all treatments appear in exactly r blocks. • This ensures that each treatment is estimated with the same precision • The value of l is the same for each treatment pair. • if all treatment pairs i and i* appear together in exactly l blocks. • This ensures that each treatment difference is estimated with the same precision. • The value of l is the same for each treatment pair.

  9. Some Identities Let b = the number of blocks. t = the number of treatments k = the block size r = the number of times a treatment appears in the experiment. l = the number of times a pair of treatment appears together in the same block • bk = rt • Both sides of this equation are found by counting the total number of experimental units in the experiment. • r(k-1) = l (t – 1) • Both sides of this equation are found by counting the total number of experimental units that appear with a specific treatment in the experiment.

  10. BIB Design A Balanced Incomplete Block Design (b = 15, k = 4, t = 6, r = 10, l = 6)

  11. An Example • For this purpose: • subjects will be asked to taste and compare these cereals scoring them on a scale of 0 - 100. • For practical reasons it is decided that each subject should be asked to taste and compare at most four of the six cereals. • For this reason it is decided to use b = 15 subjects and a balanced incomplete block design to assess the differences in taste of the six brands of cereal. A food processing company is interested in comparing the taste of six new brands (A, B, C, D, E and F) of cereal.

  12. The design and the data is tabulated below:

  13. Analysis for the Incomplete Block Design Recall that the parameters of the design where b = 15, k = 4, t = 6, r = 10,l= 6 denotes summation over all blocks j containing treatment i.

  14. Anova Table for Incomplete Block Designs Sums of Squares SS yij2 = 234382 S Bj2/k = 213188 S Qi2 = 181388.88 Anova Sums of Squares SStotal =SS yij2 –G2/bk = 27640.6 SSBlocks =S Bj2/k – G2/bk = 6446.6 SSTr = (S Qi2 )/(r – 1) = 20154.319 SSError = SStotal - SSBlocks - SSTr = 1039.6806

  15. Anova Table for Incomplete Block Designs

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