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Optimal Crossover Designs are 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 y ij and (for i=1…n; j=1…p) y ij = α i + β j + τ d(i,j) + ρ d(i,j-1) + ε ij 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

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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+b11 T ) (in fact C T 1=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)

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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

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Now C(τ| D ) = C ττ – C τρ C - ρρ C ρτ {partition of C(τ,ρ| D )} Strongly balanced designs gave C τρ =0 so problems inverting C ρρ 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)

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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: so BUDs have efficiency exceeding 96% (t=3), 99% (t=4) etc.

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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

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Illustrated for case t=2 (Matthews, 1990). Method considers dual-balanced designs, which allocate equally to sequences with A and B interchanged There are 2 p possible sequences of treatments of A and B but only N=2 p-1 dual- balanced pairs of sequences Suppose a proportion x i of patients are allocated to sequence pair i (=1…N) Variance of estimate of τ can be written Choose x such that first term is maximal and q 12 T x=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.

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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, p=2, 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 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

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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 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

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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 y ij Model is : y ij = α i + π (i,j) + τd(i,j) + ε ij π (i,j)= π 1 if i D 3 and j is a Monday = π 2 if i D 3 and j is a Wednesday = π 3 if i D 3 and j is a Friday This is for thrice-weekly patients D 3 – 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 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

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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

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