2. Crossover Trials Useful when runs are blocked by human subjects or large animals To increase precision of treatment comparisons all treatments are administered to each subject or animal in a sequence Primary purpose is to compare effect of treatments Secondary purpose is to protect against bias from carryover effects and to estimate carryover effects Used extensively in pharmaceutical research, sensory evaluation of food products, animal feeding trials and psychological research

3. Crossover Designs COD Useful for comparing a limited number of treatments (from 2 to 6) Usually not used for factorial designs (other than simple 2 2 factorial) Since treatments are applied sequentially in time, COD are only useful for comparing temporary treatments of chronic conditions Special designs and models for testing and estimating carryover effects are required

5. Compartment Model

12. Treatment effect is confounded with the carryover effect

23. · Design choice dependent on assumptions · Assumption of first-order carryover effects · Variance balance as a design criteria - variance (or standard error of difference in direct treatment means or carryover means) is the same regardless of the pair of treatments - this balance is achieved if every treatment is preceded by every treatment

24. A is preceded by B in second group, but never by A B is preceded by A in first group, but never by B B is preceded by A in first group second period and by B in third period A is preceded by B in second group second period and by A in third period

25. Designs for t treatments and p periods where p = t · If no carryover effects are assumed a balanced design for direct treatment effects can be created using any t  t Latin square · If carryover effects are assumed Williams showed a balanced design for direct and carryover treatment effects can be created using one particular t  t Latin square if t is an even number, and two particular Latin squares if t is an odd number

27. 1 2 3 2 3 1 3 1 2 3 2 1 1 3 2 2 1 3 1 3 2 2 3 1 2 1 3 3 1 2 3 2 1 1 2 3 1 2 3

28. 1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 4 3 2 1 1 4 3 2 2 1 4 3 3 2 1 4 1 4 2 3 2 1 3 4 3 2 4 1 4 3 1 2 3 2 4 1 4 3 1 2 1 4 2 3 2 1 3 4 1 2 3

33. Jonathan Chipman 2006 Factor: Running Surface - BYU rubberized track - grass - asphalt Blocks: Subjects Response: Time to sprint 40 yards Latin square column factor: Trial (to account for exhaustion effect) Willams’ design used to account for carryover effects

38. proc glm; class subject period treat carry; model y=subject period treat carry; means treat; lsmeans treat; run;

39. Nonorthogonality of direct and treatment effects The GLM Procedure Least Squares Means treat y LSMEAN asphalt Non-est grass Non-est track Non-est

40. proc glm; class subject period treat carry; model y=subject period carry treat ; run; Source DF Type I SS Mean Square F Value Pr > F subject 11 24.20838889 2.20076263 85.85 <.0001 period 2 3.20650556 1.60325278 62.54 <.0001 carry 2 0.02168111 0.01084056 0.42 0.6615 treat 2 0.63916333 0.31958167 12.47 0.0004 Source DF Type III SS Mean Square F Value Pr > F subject 11 24.25471000 2.20497364 86.01 <.0001 period 1 0.00183750 0.00183750 0.07 0.7920 carry 2 0.23320556 0.11660278 4.55 0.0252 treat 2 0.63916333 0.31958167 12.47 0.0004

41. Group 1 1 3 2 2 2 1 3 3 3 2 1 4 2 3 1 5 3 1 2 6 1 2 3 Group 1 1 3 2 2 2 2 1 3 3 3 3 2 1 1 4 2 3 1 1 5 3 1 2 2 6 1 2 3 3 Solution Lucas - Add an extra period

42. The objective is to compare the trend over time in the response between treatment groups.

44. Diet fixed effect Cow random effect week fixed effect

45. Usual Assumptions of Univariate Model Experimental Error is independent, has equal variance across treatment×time combinations, and is normally distributed with mean zero Similar assumptions for random effects Independence assumption is justified by randomization In Split-Plot Type Experiments, sub-plots are not independent because they are measured within the same whole-plot. Randomization of subplot treatments to subplots equalizes the correlation between all possible pairs of subplots – this creates a condition called compound symmetry, which justifies the normal univariate analysis

46. Usual Assumptions of Univariate Model In repeated measures designs, you can’t randomize levels of time within a subject or cow! Huyuh and Feldt (1970) showed that if σ 2 (yi-yj) = 2λ for i ≠ j (Huyuh-Feldt condition) then univariate analysis is justified. The Mauchly (1940) sphericity test can be used to determine if the Huyuh- Feldt condition holds. This can be performed by proc glm

50. Subject time 1 time 2 time 3 time 4 time 5 summary 1 y 11 y 12 y 13 y 14 y 15 f(y 11, …,y 15 ) 2 y 21 y 22 y 23 y 24 y 25 f(y 21, …,y 25 ) · · · · · · · n y n1 y n2 y n3 y n4 y n5 f(y n1, …,y n5 ) Summarizing with function over time removes correlation “Growth Curve Approach”

53. · ·

54. sums x i = day, n =5 y i = log(concentration) f = k slope

57. Linear orthogonal polynomial over time If you don’t have a summary function, proc glm can summarize with orthogonal polynomials over time.

58. Quadratic orthogonal polynomial over time