Considerations for Selecting a Crossover Design

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Considerations for Selecting a Crossover Design

• John Stufken
• National Science Foundation
• Justus Seely Memorial Conference

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Considerations for Selecting a Crossover Design

• John Stufken
• University of Georgia
• Justus Seely Memorial Conference

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How to Survive at the NSF ?

• Kunert and Stufken, 2002, JASA
• Hedayat and Stufken, 2003, JBS
• Bose and Stufken, 2003, in preparation
• Presentation will draw from all three of these

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What is a Crossover Design Notation

• t treatments n subjects p periods design is a
  p by n array
• A crossover design for t4, p3,
  n12 A C B D A B A D C D B C B A C B D A C A
  D C D B C B A A B D D C A B C D
• Yij random variable for the observation in
  period i for subject j assumed to be continuous.
• d(i,j) the treatment assigned to subject j for
  period i by design d (after randomization)

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What is a Crossover Design Notation

• t treatments n subjects p periods design is a
  p by n array
• A crossover design for t4, p3,
  n12 A C B D A B A D C D B C B A C B D A C A
  D C D B C B A A B D D C A B C D
• Yij random variable for the observation in
  period i for subject j assumed to be continuous.
• d(i,j) the treatment assigned to subject j for
  period i by design d (after randomization)d(2,3)
  C

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Why Use a Crossover Design?

• Practical considerations a crossover design
  requires fewer subjects for the same number of
  observations than a parallel trial
• Increased precision of treatment comparisons due
  to the availability of within-subject information
• But ...

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• Possible Concerns When Using a Crossover Design

• The duration of the experiment tends to be longer
  than with a parallel trial typically only a
  small number of periods is used
• Possibility of dropouts missing data
• Carry-over effects (wash-out periods modeling of
  carry-over effects) treatment-period interaction
• Within-subject covariance structure

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Simple Models When Using a Crossover Design

• Assumes a continuous dependent variable fixed
  effects
• Yij µ pi ßj td(i,j) eij no carry-over
  effects model
• Yij µ pi ßj td(i,j) ?d(i-1,j) eij
  traditional model
• Yij µ pi ßj td(i,j) (1-dij) ?d(i-1,j)
  dij fd(i-1,j) eij self and mixed carry-over
  effects model
• Yij µ pi ßj td(i,j) ?td(i-1,j) eij
  proportional carry-over effects model

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• Optimal Designs

• Just one of the considerations that should play a
  role when selecting a crossover design
• Will focus on the models just shown, with all
  effects fixed, and error terms iid normal
• Results shown will focus on two treatments (t2),
  except for model 4
• Universal optimality criterion design is a
  discrete probability measure over the possible
  sequences
• When t2, a sequence and its dual are used
  equally often (e.g., the dual of ABB is BAA)

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Optimal designs under model 4

• For a fixed ?, a design d is universally optimal
  for treatment effects if
• d is uniform over the periods
• Cd11, Cd12 Cd21 and Cd22 are completely
  symmetric matrices and
• d uses only sequences that are admissible for
  this ?
• Only iii. depends on ? and needs additional work.
  It is completely solved for p2,3,4 in Bose and
  Stufken, 2003

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• Example 1 p 2, t 2
• For any ? ? 2, only AB is admissible
• t2 AB, BA t3 AB, BC, CA t4 AB, AC, AD,
  BA, BC, BD, CA, CB, CD, DA, DB, DC
• Example 2 p 3, t 2
• If ? 0, only ABB is admissible ABB, BAA is
  optimal
• If ? optimal
• Example 3 p 3, t 3
• ABC ABA ABC
  ABB ABC
• -4.73 -1.27
  0.52 11.48
• ?

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• In Summary
• There are many considerations for deciding
  whether to use a crossover design and, if so,
  which one to use
• Optimal design considerations are but one of
  these
• Different models can lead to vastly different
  optimal designs
• It may be prudent to select a design that is
  efficient under a variety of possible models
• Mathematical tools are now available to compute
  such efficiencies under many different models