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

2. BIBD Example ExampleBlock 1234 AAAB BBCC CDDD Verify that  Verify that 

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

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

5. Least Squares estimator of  The solution for  i : The solution for  i :

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

7. Properties of LS estimators

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

10. Example Day 1234567 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

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

12. Interblock Estimators

13. Some properties: Some properties:

14. Interblock Estimators For the random block model, For the random block model, SourcedfSSEMS Trta-1SS Trt Block adj b-1SS Block(adj) ErrorN-a-b+1 TotalN-1

15. Interblock Estimators

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

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