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Balanced Incomplete Block Design (BIBD)

Balanced Incomplete Block Design (BIBD). Notation a-number of treatments m-number of treatments/block r-number of treatment reps b-number of blocks N=ar=bm r(m-1)= l (a-1). BIBD. Example Block 1 2 3 4 A A A B B B C C C D D D Verify that l=2. BIBD.

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Balanced Incomplete Block Design (BIBD)

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  1. Balanced Incomplete Block Design (BIBD) • Notation • a-number of treatments • m-number of treatments/block • r-number of treatment reps • b-number of blocks • N=ar=bm • r(m-1)=l(a-1)

  2. BIBD • Example Block 1234 A A A B B B C C C D D D • Verify that l=2

  3. BIBD • One way to generate BIBD’s (assuming a and m have been selected) is to choose • This worked well in the previous example • It can be inefficient for large a or m≈a/2

  4. Least Squares estimator of a • These are the adjusted means that one obtains from LSMEANS in SAS

  5. Least Squares estimator of a • The solution for ai:

  6. Least Squares estimator of a • These estimators are called intrablock estimators because they can be expressed as contrasts of within block observations.

  7. Properties of LS estimators

  8. Properties of LS estimators • The variance estimate is not the square of the standard error for the LSMEANS that one obtains in SAS (it’s not corrected for the overall mean).

  9. Example • Region II Science Fair Winner • Treatment is organic chemical repellent • Block is Day • Response is average number of flies on underside of lid (smaller is better) • Originally presented as a Latin Square (additional block was Fly Group)

  10. Day 1 2 3 4 5 6 7 C=19.8 E=7.8 G=13.0 D=16.0 F=11.0 G=5.3 A=11.7 E=5.3 F=12.3 B=11.2 D=10.0 E=6.0 A=13.2 C=17.3 D=16.2 A=16.0 B=17.2 G=10.8 B=15.7 C=18.0 F=12.7 Example A=Ascorbic Acid B=Citric Acid C=Control D=Hesperidin E=Lemon Juice F=Black Pepper G=Piperine

  11. Interblock Estimators • When Block is a random effect, alternative estimators of treatment effects can be found

  12. Interblock Estimators

  13. Interblock Estimators • Some properties:

  14. Interblock Estimators • For the random block model, SourcedfSSEMS Trt a-1 SSTrt Blockadj b-1 SSBlock(adj) Error N-a-b+1 Total N-1

  15. Interblock Estimators

  16. BIBD Estimators • Suppose we want to minimize the variance of estimators of the form: • The best weighted estimator is:

  17. Power Analysis • The noncentrality parameters look similar to all other one factor designs we have so far seen

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