Designs for Estimating

1. Designs for Estimating Carry-over (or Residual) Effects of Treatments Ref: “Design and Analysis of Experiments” Roger G. Petersen, Dekker 1985

2. The Cross-over or Simple Reversal Design An Example • A clinical psychologist wanted to test two drugs, A and B, which are intended to increase reaction time to a certain stimulus. • He has decided to use n = 8 subjects selected at random and randomly divided into two groups of four. • The first group will receive drug A first then B, while • the second group will receive drug B first then A.

3. To conduct the trial he administered a drug to the individual, waited 15 minutes for absorption, applied the stimulus and then measured reaction time. The data and the design is tabulated below:

4. The Switch-back or Double Reversal Design An Example • A following study was interested in the effect of concentrate type on the daily production of fat-corrected milk (FCM) . • Two concentrates were used: • A - high fat; and • B - low fat. • Five test animals were then selected for each of the two sequence groups • ( A-B-A and B-A-B) in a switch-back design.

5. The data and the design is tabulated below: One animal in the first group developed mastitis and was removed from the study.

6. The Incomplete Block Switch-back Design An Example • An insurance company was interested in buying a quantity of word processing machines for use by secretaries in the stenographic pool. • The selection was narrowed down to three models (A, B, and C). • A study was to be carried out , where the time to process a test document would be determined for a group of secretaries on each of the word processing models. • For various reasons the company decided to use an incomplete block switch back design using n = 6 secretaries from the secretarial pool.

7. The data and the design is tabulated below: BIB incomplete block design with t = 3 treatments – A, B and block size k = 2. A B A C B C

8. Designs for Estimating Carry-over (or Residual) Effects of Treatments Ref: “Design and Analysis of Experiments” Roger G. Petersen, Dekker 1985

9. The Latin Square Change-Over (or Round Robin) Design Selected Latin Squares Change-Over Designs (Balanced for Residual Effects) Period = Rows Columns = Subjects

10. The Latin Square Change-Over (or Round Robin) Design Selected Latin Squares Change-Over Designs (Balanced for Residual Effects) Period = Rows Columns = Subjects

11. Four Treatments

12. An Example • An experimental psychologist wanted to determine the effect of three new drugs (A, B and C) on the time for laboratory rats to work their way through a maze. • A sample of n= 12 test animals were used in the experiment. • It was decided to use a Latin square Change-Over experimental design.

13. The data and the design is tabulated below:

14. Analysis : The Latin Square Change-Over (or Round Robin) Design Assume that we have q p × p Latin Squares (Balanced for Residual Effects) Period = Rows Columns = Subjects

15. Notation

16. Square 1

17. Square 2

18. Square q

19. Some other sums

20. Adjusted Effects

21. ANOVA table entries

23. Some additional S.S.

24. The ANOVA Table

25. Means and their sample variances after adjustment 1. Adjusted treatment mean 2. Adjusted residual effect 3. Permanent treatment mean

26. An Example • An experimental psychologist wanted to determine the effect of three new drugs (A, B and C) on the time for laboratory rats to work their way through a maze. • A sample of n= 12 test animals were used in the experiment. • It was decided to use a Latin square Change-Over experimental design.

27. The data and the design is tabulated below:

28. The ANOVA Table

29. Summary Statistics

30. Table: Adjusted mean run times and their standard errors

31. Next Topic: Orthogonal Linear Contrasts