Balanced Incomplete Block Design BIBD n Notation anumber

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

BIBD n Example 1 A B C n Verify that l=2 Block 2 3 A A B C D D 4 B C D

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

Least Squares critetion for a n These are the adjusted means that one obtains from LSMEANS in SAS

Least Squares estimator of a n The solution for ai:

Intrablock estimator for a n These estimators are called intrablock estimators because they can be expressed as contrasts of within block observations.

Properties of LS estimators

Variance of LS estimators n 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).

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

Data Table 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 A=Ascorbic Acid B=Citric Acid C=Control D=Hesperidin E=Lemon Juice F=Black Pepper G=Piperine

Interblock Model n When Block is a random effect, alternative estimators of treatment effects can be found

Interblock Estimators

Interblock Estimator Properties n Some properties:

Interblock EMS n For the random block model, Source Trt Blockadj Error Total df SS EMS a-1 SSTrt b-1 SSBlock(adj) N-a-b+1 N-1

Variance Compoents

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

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