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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
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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
Crossover Designs COD • Useful for comparing a limited number of treatments (from 2 to 6) • Usually not used for factorial designs (other than simple 22 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
· 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
A is preceded by B in second group, but never by A B is preceded by A in first group, but never by B A is preceded by B in second group second period and by A in third period B is preceded by A in first group second period and by B in third period
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 tt Latin square · If carryover effects are assumed Williams showed a balanced design for direct and carryover treatment effects can be created using one particular tt Latin square if t is an even number, and two particular Latin squares if t is an odd number
1 2 3 2 3 1 3 1 2 1 3 2 1 1 3 2 2 1 3 2 3 1 3 2 2 3 1 2 1 3 3 1 2 3 2 1 1 2 3
1 2 3 4 2 3 4 1 3 4 1 2 4 1 2 3 1 4 3 2 1 1 4 3 2 2 1 4 3 3 2 1 4 2 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 3
Jonathan Chipman 2006 Blocks: Subjects Factor: Running Surface - BYU rubberized track - grass - asphalt Response: Time to sprint 40 yards Latin square column factor: Trial (to account for exhaustion effect) Willams’ design used to account for carryover effects
procglm; class subject period treat carry; model y=subject period treat carry; means treat; lsmeans treat; run;
Nonorthogonality of direct and treatment effects The GLM Procedure Least Squares Means treat y LSMEAN asphalt Non-est grass Non-est track Non-est
procglm; class subject period treat carry; model y=subject period carry treat ; run; Source DF Type I SS Mean Square F Value Pr > F subject 1124.208388892.2007626385.85 <.0001 period 23.206505561.6032527862.54 <.0001 carry 20.021681110.010840560.420.6615 treat 20.639163330.3195816712.470.0004 Source DF Type III SS Mean Square F Value Pr > F subject 1124.254710002.2049736486.01 <.0001 period 10.001837500.001837500.070.7920 carry 20.233205560.116602784.550.0252 treat 20.639163330.3195816712.470.0004
Solution Lucas - Add an extra period • Group • 1 1 3 2 • 2 1 3 • 3 2 1 • 4 2 3 1 • 5 3 1 2 • 6 1 2 3 • Group • 1 1 3 2 2 • 2 1 3 3 • 3 2 1 1 • 4 2 3 1 1 • 5 3 1 2 2 • 6 1 2 3 3
The objective is to compare the trend over time in the response between treatment groups.
Diet fixed effect week fixed effect Cow random effect
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
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 σ2(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
Subject time 1 time 2 time 3 time 4 time 5 summary 1 y11y12y13y14y15 f(y11, …,y15) 2 y21y22y23y24y25 f(y21, …,y25) · · · · · · · · · · · · · · · · · · · · · nyn1yn2yn3yn4yn5 f(yn1, …,yn5) Summarizing with function over time removes correlation “Growth Curve Approach”