Optimal%20Crossover%20Designs%20are%20they%20a%20good%20thing?

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Optimal Crossover Designsare they a good thing?

• John Matthews
• University of Newcastle upon Tyne

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Early developments

• As with much experimental design, early work
  motivated by attempts to achieve orthogonality
  Williams (1949), Quenouille (1953)
• Hedayat and Afsarinejad (1975, 1978) were the
  first to apply formal optimal design criteria
• Tool that is used is Universal Optimality, due to
  Kiefer (1975)

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• Usual model assumes continuous outcome subject
  i in period j yields yij and (for i1n j1p)
  yij ai ßj td(i,j) ?d(i,j-1)
  eijwhere t (?) is the direct (first order
  carryover) effect of treatment and a, ß are
  (fixed) subject and period effects. Also the
  error terms are independent with constant
  variance
• Information matrix of treatment effects from a
  design D is C(t,?D) and for t alone is C(tD)
  can be found from design matrices by standard
  methods

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• Suppose O(t,n,p) is the set of all crossover
  designs with given size
• A design D ?? ? O(t,n,p) can be established to be
  universally optimal for the estimation of t
  ifi) C(tD) is completely symmetric (c.s.
  (a-b)Ib11T) (in fact CT10, so a(t-1)b0)ii)
  tr(C(tD)) ? tr(C(tD)) for all D ??
• Early work in this approach was by Hedayat
  Afsarinejad (1975,1978), Cheng Wu (1980),
  Kunert (1983,1984)

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• Notions of uniformity and balance emerge as
  importantuniform on periods - equal
  replication of treatments within a perioduniform
  on subjects - equal replication within each
  subjectuniform if both of above - implies we
  need tn and tIp.Balanced each treatment
  precedes each other treatment equally
  oftenStrongly balanced each treatment
  preceded every treatment equally often
• One of the earliest results (Cheng and Wu, 1980)
  is that strongly balanced uniform designs are
  universally optimal (UO) over O(t,n,p).Not
  surprising everything orthogonal to everything
  else! Very restrictive and model dependent

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• Now C(tD) Ctt Ct?C-??C?t partition of
  C(t,?D)
• Strongly balanced designs gave Ct?0 so problems
  inverting C?? are avoided. Not the case with
  other types of design.
• Long series of papers looking at balanced uniform
  crossover designs (from Cheng and Wu 1980, Kunert
  , 1983,1984 to Hedayat and Yang 2004).
• Much attention focussed on tpEarly results
  (Kunert 1984) show thatBalanced Uniform Design
  (BUD) isUO over O(t,t,t) tgt2 and UO over
  O(t,2t,t) if tgt5Recent results (Hedayat and Yang
  2003,2004)UO over O(t,n,t) for tgt2 and
  n?½t(t-1)UO over O(t,n,t) for 4?t ?12 n ?
  ½t(t2)

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• However, if we make n large enough we can find
  designs in O(t,n,t) which are better than BUDs.
• Of course, while we know t and can prescribe pt
  the value of n is not determined by combinatorial
  niceties.
• Power considerations and availability of units
  have a central role.
• Convenient to put several BUDs from O(t,t,t)
  together this is a BUD but is a BUD good?
• Kunert answered this in 1984so BUDs have
  efficiency exceeding 96 (t3), 99 (t4) etc.

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• Other designs of note include those of Stufken
  (1991) which have recently been shown to be UO
  over O(t,n,p).
• Awkward to describe succinctly and are
  combinatorially demanding but do at least cover
  cases with p lt t.(Hedayat Yang 2004 extending
  Kushner, 1998, and Stufken 1991 see also Hedayat
  and Zhao, 1990, for p2)
• Technique has largely been to determine
  conditions on C(tD) such that a design yielding
  such a matrix will be UO. Then need to seek
  designs with this property.
• Works only for some combinations of (t,n,p)
• An alternative approach is more constructive

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• Illustrated for case t2 (Matthews,
  1990).Method considers dual-balanced designs,
  which allocate equally to sequences with A and B
  interchangedThere are 2p possible sequences of
  treatments of A and B but only N2p-1
  dual-balanced pairs of sequencesSuppose a
  proportion xi of patients are allocated to
  sequence pair i (1N)Variance of estimate of t
  can be writtenChoose x such that first term
  is maximal and q12Tx0
• Method gives a way of constructing optimal
  designs for given p, albeit with an approximate
  design formulation. Greatly extended by Kushner,
  1997, 1998, to tgt2.

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• BUT, designs still very model-dependent.
• Uniformity emerges from the row-column structure
  in the model
• Balance emerges from the nature of the carryover
  which is assumed in the model
• Several strands of work have emerged looking at
  optimal designs for alternative
  models.Modification of the unit effect -
  random effects
  (Carrière and Reinsel, p2,
  1993)Period effect
  - no real change here (see later)

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• Main changes to model - temporal aspects
• Dependence of error term
• Form of carryover term
• Former from repeated measures aspect Latter
  because of criticism of traditional form

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Dependent Errors

• Largely started with autocorrelated errors, as a
  tractable variation from independence
• Some progress on methodology for general
  dependence (Kushner, 1997)
• Designs do change with autocorrelation
• Optimal designs need dispersion parameters
  specified in advance some work on uncertainty
  in value (Donev, 1998)

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Carryover

• Traditional model seen as of methodological
  convenience rather than scientifically realistic
• Its use could mislead if it is thought to allow
  for carryover when it does not
• Reaction in design community is to consider a
  range of alternative modelsReaction in user
  community is to avoid crossovers
• Much attention paid to self-carryover, i.e. model
  which allowsA B B C to differ from A B C
  B ? ?

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• Interesting results (and different from results
  for traditional model)
• Recent contributions by Afsarinejad and Hedayat,
  2002 and Kunert and Stufken, 2002
• Also more general model considered by Bose
  Mukherjee, 2003
• BUT, are these modified models any more plausible
  scientifically than the traditional one?
• Given limited information on carryover terms, can
  we decide post-hoc which model to fit?
• If so, how do we decide which design to choose?
• Even if we can decide, there is likely to be a
  limited range of models under consideration, none
  of which may be suitable

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Some examples

• Choice of model and practical constraints on
  designs suggest that off the shelf designs may
  have limited application
• Design tools will be more useful than designs
• Complexity of area makes this quite challenging
  but computer algorithms may help (e.g. John et
  al. 2004)

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Example 1 bladder reconstruction

• Patients with rebuilt bladders have problems with
  mucous production in the new bladder
• Treatment to thin the mucous is required
• NDow et al. 2001 report a 4 period crossover
  comparing six treatments4 treatments are 2 2
  factorial oral treatmentsplus 2 instillations
• Each patient receives 4 treatments but unwise to
  include gt 1 instillation in the sequence
• Carryover eliminated by washout

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Example 1 bladder reconstruction

• Used 6 replications of a Latin square for 5
  treatments with last column omitted to give 4
  periods
• Oral treatments and one of the instillations used
  in three replicates and oral treatments plus the
  other instillation used in remaining replicates
• Adequate (?) and achieves practical objectives
  but is it as good as could have been done?
• Illustrates another aspect, namely design
  solutions are often needed quickly.
• Mature methodological development is often out of
  the question

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Example 2 paediatric dialysis

• Patients on haemo-dialysis have indwelling lines
  which are connected to a dialyser 2-3 times per
  week
• Must keep lumen of line clear of clot between
  dialysis sessions
• Trial compared two anti-clotting agents
• Few children have haemo-dialysis and protocols
  differ markedly between centres, so multi-centre
  study would be awkward
• Captive population so decided on a long crossover
  (30 periods) with few patients (9 in the end,
  planned for 6)

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Example 2 paediatric dialysis

• What is the true replication when repeatedly
  measuring the same patient?
• Anti-clotting agent instilled into lumen of line
  but volume titrated so that little will escape
  into system
• Dialysis will flush system so extensively that
  carryover is eliminated and (?) so is the
  autocorrelation.
• Assume inter-patient variation in propensity to
  clot is eliminated by patient effect in model
• Is a period effect realistic?
• Probably not but outcome is weight of aspirated
  clot and this may well depend on inter-dialytic
  interval

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Example 2 paediatric dialysis

• Suppose weight of clot for patient i in period j
  is yij
• Model isyij ai p(i,j) td(i,j) eij
• p(i,j) p1 if i?D3 and j is a Monday p2 if
  i?D3 and j is a Wednesday p3 if i?D3 and j is
  a Friday
• This is for thrice-weekly patients D3 extension
  needed to incorporate twice-weekly cases

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Example 2 paediatric dialysis

• Specifically derived optimal design for this
  model was used
• Design required equal replication of treatments
  i) within patientii) within dialysis day (Mon,
  Wed, Fri)
• Trial largely succeeded data looked rather
  different from model markedly heteroscedastic
• Model used allowed extra patients and regimen
  changes to be accommodated more readily than with
  standard model

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Example 3 Sub-clinical hypothyroidism

• Usual AB/BA design
• Diagnosis based on TSH and fT4 levels
• Treatment is with thyroxine, which has a short
  half-life
• Principal outcome is biochemical and it is easy
  to set an adequate washout
• BUT secondary (?) interest is in variables
  measuring quality of life and carryover cannot
  readily be eliminated here.
• Carryover is a genuine problem in many practical
  circumstances

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Other things

• Other kinds of models e.g. unequal error
  variances
• Missing data viewed from perspective of
  disconnection etc. e.g. Low et al. 1999, but what
  about interpretation and role of ITT?
• Computer search
• Row and column designs, perhaps with dependent
  errors

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Comments

• Much excellent theory is it widely used?
• Carryover is the methodological aspect which
  characterises the crossover
• Often not present in practical crossovers
• Although beware secondary variables!
• Models may need to be much more application
  specific
• Need to choose designs for such models