Transcript Optimal Crossover Designs are they a good thing?

Optimal Crossover Designs
are they a good thing?
John Matthews
University of Newcastle upon Tyne
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)
• Usual model assumes continuous outcome – subject i in
period j yields yij and (for i=1…n; j=1…p)
yij = αi + βj + τd(i,j) + ρd(i,j-1) + εij
where τ (ρ) is the direct (first order carryover) effect of
treatment and α, β 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(τ,ρ|D) and for τ alone is C(τ|D): can be found from
design matrices by standard methods
• Suppose Ω(t,n,p) is the set of all crossover designs with
given size
• A design D   Ω(t,n,p) can be established to be
universally optimal for the estimation of τ if
i) C(τ|D) is completely symmetric (c.s. = (a-b)I+b11T)
(in fact CT1=0, so a+(t-1)b=0)
ii) tr(C(τ|D))  tr(C(τ|D*)) for all D* 
• Early work in this approach was by Hedayat &
Afsarinejad (1975,1978), Cheng & Wu (1980), Kunert
(1983,1984)
• Notions of uniformity and balance emerge as important:
uniform on periods - equal replication of treatments within a period
uniform on subjects - equal replication within each subject
uniform if both of above - implies we need t|n and tIp.
Balanced – each treatment precedes each other treatment equally often
Strongly 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 Ω(t,n,p).
Not surprising – everything orthogonal to everything else! Very
restrictive and model dependent
• Now C(τ|D) = Cττ – CτρC-ρρCρτ {partition of C(τ,ρ|D)}
• Strongly balanced designs gave Cτρ=0 so problems
inverting Cρρ 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 t=p
Early results (Kunert 1984) show that
Balanced Uniform Design (BUD) is
UO over Ω(t,t,t) t>2 and UO over Ω(t,2t,t) if t>5
Recent results (Hedayat and Yang 2003,2004)
UO over Ω(t,n,t) for t>2 and n½t(t-1)
UO over Ω(t,n,t) for 4t 12 & n  ½t(t+2)
• However, if we make n large enough we can find designs in Ω(t,n,t)
which are better than BUDs.
• Of course, while we know t and can prescribe p=t 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 Ω(t,t,t) together – this is a
BUD but is a BUD good?
• Kunert answered this in 1984:
tr {C(τ | BUD )}
1
 1 2
sup tr {C(τ | D  Ω(t , n, t ))}
(t  t  1)2
so BUDs have efficiency exceeding 96% (t=3), 99% (t=4) etc.
• Other designs of note include those of Stufken (1991)
which have recently been shown to be UO over Ω(t,n,p).
• Awkward to describe succinctly and are combinatorially
demanding but do at least cover cases with p < t.
(Hedayat & Yang 2004 extending Kushner, 1998, and Stufken
1991: see also Hedayat and Zhao, 1990, for p=2)
• Technique has largely been to determine conditions on
C(τ|D) 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
•
Illustrated for case t=2 (Matthews, 1990).
Method considers dual-balanced designs, which allocate equally to sequences with
A and B interchanged
There are 2p possible sequences of treatments of A and B but only N=2p-1 dualbalanced pairs of sequences
Suppose a proportion xi of patients are allocated to sequence pair i (=1…N)
Variance of estimate of τ can be written
T
q11
x
T
(q12
x )2
qT22 x
Choose x such that first term is maximal and q12Tx=0
•
Method gives a way of constructing optimal designs for given p, albeit
with an approximate design formulation. Greatly extended by Kushner,
1997, 1998, to t>2.
• 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, p=2, 1993)
Period effect
-
no real change here (see later)
• 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
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)
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 models
Reaction in user community is to avoid crossovers
• Much attention paid to self-carryover, i.e. model which
allows
A B B C to differ from A B C B


• 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
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)
Example 1: bladder reconstruction
• Patients with rebuilt bladders have problems with
mucous production in the new bladder
• Treatment to thin the mucous is required
• N’Dow et al. 2001 report a 4 period crossover comparing
six treatments
4 treatments are 2 × 2 factorial oral treatments
plus 2 instillations
• Each patient receives 4 treatments but unwise to include
> 1 instillation in the sequence
• Carryover eliminated by washout
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
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)
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
Example 2: paediatric dialysis
• Suppose weight of clot for patient i in
period j is yij
• Model is:
yij = αi + π(i,j) + τd(i,j) + εij
• π(i,j)
= π1 if iD3 and j is a Monday
= π2 if iD3 and j is a Wednesday
= π3 if iD3 and j is a Friday
• This is for thrice-weekly patients D3 – extension
needed to incorporate twice-weekly cases
Example 2: paediatric dialysis
• Specifically derived optimal design for this model was used
• Design required equal replication of treatments
i) within patient
ii) 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
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
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
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