AREA ADMINISTRATIVA FACULTAD DE CIENCIAS MÉDICAS

The possible design plans in th a t occasion are listed in th e right h alf of Table (4.1). The available degrees of freedom for the group by period in teractio n in the three period two sequence designs are (p — l) ( s — 1) = 2 allowing only the

estim ation of tre a tm e n t and first-order (or Fleiss) residual effects. In all these designs first-order carry-over is aliased w ith treatm en t by period in teractio n, while the assum ption of carry-over being dependent not only upon previous tre a tm e n t bu t also upon th e current one (treatm ent by first-order carry-over interaction) cannot be tested.

A way to overcome this problem is to allow for more sequences a n d /o r more periods in the design. Here the first possibility is only considered and th e two- sequence three-period designs presented in the previous section, are combined

in pairs giving four-sequence three-period designs. For exam ple, by com bining 3.2.i w ith 3.2.j th e three-period four-sequence design 3.4.ij is generated. T he three possibilities, labeled 3.4.12, 3.4.13 and 3.4.23, are given in Table (4.1). In this way we increase th e available degrees of freedom for th e group by period interaction from two to six. If we include treatm en t, first-order and second-order carry­ over term s th en two d.f rem ain for estim ating un in terestin g sequence by period interaction term s. Note th a t the 2 d .f of th e tre a tm e n t by period interaction are aliased w ith first and second order carry-over effects, as for th e 2x2 case. Under uniform or AR(1) w ithin-error structu re four m odels will be studied:

• M l : Inclusion of residual term s of any kind is n o t considered based on knowledge about the pharm acological effect of th e drug on hum ans. Only tre a tm e n t effect (r) is fitted.

• M 2 : F irst order carry-over effect added to m odel M l (sim ple carry-over

model).

• M 3 : A second order carry-over effect fu rth er added to th e sim ple carry­ over model (second order carry-over m odel).

• M 4 : A special type of treatm en t by first-order carry-over interaction is fitted in addition to the treatm en t term (Fleiss m odel).

It was concluded th a t under model M l and A R (1) error stru c tu re designs 3.4.12

and 3.4.23 are equally efficient for estim ating tre a tm e n t difference, while under the simple carry-over model the ideal choice is 3.4.13. In th e Fleiss m odel, the optim um decision depends upon the correlation coefficient. More specifically de­ sign 3.4.13 is preferred for small values of p, while for th e larger values 3.4.12 becomes the favored one. Finally under th e com pletely unrealistic m odel M3, design 3.4.23 is th e best choice. Sim ilar results for th e sim ple carry-over model are proven by M atthew s (see [61]).

If now uniform covariance structure is assum ed under m odel M l all designs are equally good for assessing treatm en t effect, whereas for th e Fleiss and second order carry-over model, optim um decision depends again upon p. For th e simple carry-over model, optim um design does not depend on th e assum ed covariance

Table 4.2: Optim um three period four sequence designs for r and A

U pper h a lf : A R (1) structure - B o tto m h a lf : U niform structure

O ptim um for r (3 .4 .index) O p tim u m for A (3 .4 .index)

M l M2 M3 M4 M2 M3 M4 .12 or .23 Vp .13 Vp .23 Vp .13 if p G (0,0.6] .12 if p G (0 .6 ,1 ) .13 Vp .13 Vp .12 Vp .12 or .13 or .23 Vp .13 Vp 23 if p G (0 ,0.8] 12 if p G (0 .8 ,1 ) .13 i f p G (0,0.5] .12 i f p G (0 .5 ,1 ) .13 i f p G (0,0.2] .12 if p G (0 .2 ,1 ) .13 Vp .12 Vp structure.

Turning now to the issue of estim ating efficiently the residual effect designs 3.4.12 and 3.4.13 are preferred in all the cases (AR(1) error stru ctu re), b u t in o ther oc­ casions the unknown value of p plays a key role in th e final choice (Uniform error structure). Results are shown in Table (4.2).

Obviously if the num ber of sequences used in th e trial is increased, then th e pre­ cision with which we estim ate tre a tm e n t or residual term s will be increased as well. This implies th a t four sequence designs should be preferred for running a cross-over study th a n two sequence ones. B ut a four sequence design is norm ally more expensive to conduct and requires the m anagem ent of four groups of p a­ tients. In conclusion, if th e experim enter decides to ru n a four sequence design then 3.4.13 is a good choice as it has good perform ance for estim ating tre a tm e n t difference when carry-over term s are included in th e model, irrespective of the covariance stru ctu re assumed.

4.3

U sing m ore periods

In th a t section designs m ade of four tre a tm e n t periods in either two, four or six treatm en t sequence groups are considered. Only designs m ade up of dual balanced treatm en t sequences are investigated. T he logistics of running such a study are far more com plicated from th e study-designs considered so far. If we

assume th a t th e experim enter keeps the com pletion tim e of th e tria l fixed, then sub-dividing th is tim e into four equal tim e-intervals (instead of three or two), may cause difficulties in collecting th e am ount of inform ation required for regu­ latory or other authorities. In addition th e cost for conducting such stu d y m ight not be negligible. From th e statistical point of view, by using m ore periods it is expected th a t all the effects of interest will be estim ated m ore precisely, b u t also non-estim able effects in two or three period plans become estim able in th e four period family. As in the previous section treatm en t, first-order and second-order residual effects will be included in th e model, b u t also th e best design plan when different carry-over types (e.g. Fleiss) assumed, will be presented.

The optim um design will be the one which estim ates tre a tm e n t (or carry-over) difference w ith m inim um variance. It is obvious th a t other functions could be considered to optim ize, b u t these choices depend upon th e interests of the exper­ im enter. For exam ple m inim izing th e variance of th e overall tre a tm e n t effect (i.e. treatm en t plus residual com ponent) or the to ta l stu d y cost are two such func­ tions. As before, b oth uniform and AR(1) w ithin error stru c tu re will be assumed throughout. Finally note th a t when Fleiss type of carry-over is incorporated into the analysis, second order carry-over of the same type is not included, because it is quite unlikely in practice to occur. The same argum ent can be p u t forward for the simple carry-over model, b u t th e reason for considering such a term here, is sim ply to stu d y the sensitivity of optim um plan when higher order carry-over term s are considered. T hird, fourth or higher order residual term s will no t bother us in w hat follows. There are seven different four period dual-sequence designs, listed in Table (4.3).

By allowing m ore periods, the set of estim able interactions increases, b u t some of them like th e tre a tm e n t by carry-over one (rA) are still no t estim able. U n­ der the A R (1) w ithin-error stru c tu re when tre a tm e n t and all carry-over term s

are included (M3), designs 4.2.6 and 4.2.7 are th e optim um ones for estim atin g treatm en t effect b u t the decision depends on intra-class correlation, while in the case of th e sim ple carry-over m odel (M2) design 4.2.3 is th e preferable one. If the Fleiss type of carry-over holds th en 4.2.1 and 4.2.6 are equally efficient for small values of p, b u t 4.2.3 is th e optim um for th e large ones. F inally if th e

tria list is confident enough th a t no residual term s should be present because an adequate w ash-out period has been allowed for, then th e advisable design is the 4.2.2. These results are also com hrm ed by M atthew s (see [61]). From the above

Table 4.3: Two-sequence, four-period designs

4.2.1 4.2.2 4.2.3 4.2.4 A A B B A B A B A B B A A B A A B B A A B A B A B A A B B A B B 4.2.5 4.2.6 4.2.7 A A B A A B B B A A A B B B A B B A A A B B B A

discussion it can be concluded th a t if a m odel w ith elaborated carry-over term s is used, th en optim um designs are m ade of sequences w ith non-equal replication of A ’s and B ’s (designs 4.2.6 or 4.2.7), while as residual term s are removed gradually from the m odel th en equal num ber of A ’s and B ’s app ear in each sequence for the optim um plan. T his is the price we have to pay for including carry-over term s. T he dangers from adm inistered the sam e drug in a num ber of adjacent periods is to bias th e clinician’s assessment of th e su b jec t’s response, as th e random ization code could be easily broken. More im p o rtan tly if one of the treatm en ts is placebo and design 4.2.6 (or 4.2.7) is used, th en a group of p atien ts will suffer discom fort for a long period and be willing to abandon the trial. All th e above shows th a t a lot of conflicting objectives have to be reconciled, one of which is the statistical efficiency, before a specific design is chosen. U nder uniform covariance stru ctu re sim ilar conclusions derived when com pared to th e A R (1) case.

T urning now to the optim um estim ation of carry-over effect design 4.2.6 seems to have good perform ance over the range of th e models studied and irrespective of the covariance stru c tu re assumed. G enerally speaking it is easier to find a robust plan for estim atin g carry-over ra th e r th a n tre a tm e n t effect.

By com bining two-sequence four-period generic designs in pairs we form 21 dis­ tin ct four-period four-sequence designs, each one referred to as 4.4.ab if designs 4.2.a and 4.2.b are joined together. C o n trast to the two sequence plans, stu d ­

U nder uniform covariance stru ctu re and when th e full set of carry-over term s is present (model M3), designs 4.4.12 and 4.4.14 estim ate r optim aly, b u t th e decision which one to use depends on p. For th e simple carry-over m odel design 4.4.13 is our b est choice, while 4.4.16 is th e favorite one for th e Fleiss type of carry-over, irrespective of the value of p. In th e absence of any residual term s any com bination of 4.2.1, 4.2.2 and 4.2.3 in pairs can be used to estim ate optim aly th e tre a tm e n t difference. From the above discussion the m ajor two-sequence design for constructing the optim um four-sequence plan is 4.2.1. If th e A R (1) stru c tu re

is assum ed and m odel M l (no-carryover) is used for analysis purposes th en 4.2.2 is th e m ajor building block for the optim um four-sequence design, while design 4.2.3 plays th a t role for th e simple and Fleiss type of carryover; for m ore details see Table (4.4). This family of plans has also been studied by M atthew s (see [61], [62]) and sim ilar conclusions were derived.

Table 4.4: O ptim um four-period designs for treatm en t effect

U pper:T w o sequences-M iddlerFour sequences-Lower : S ix sequences A R (1) w ith in -su b ject error structure assum ed

M l M2 M3 M4 .2 Vp .3 Vp .6 i f p G (0.0,0.4] .7 if p G (0 .4 ,1 .0 ) .1 or .6 i f p G (0 .0 ,0 .5] .3 i f p G (0 .5 ,1 .0 ) .23 or .24 or .25 Vp .13 i f p G (0.0,0.7] .35 if p G (0 .7 ,1 .0 ) .56 or .47 if p G (0.0,0.1] .47 i f p G (0.1,0.4] .12 i f p G (0 .4 ,1 .0 ) .16 i f p G (0 .0,0.5 ] .13 or .36 i f p G (0.5 ,0 .7 ] .34 i f p G (0 .7 ,1 .0 ) .235 or .245 .234 Vp .134 i f p G (0.0,0.2] .135 if p G (0 .2 ,1 .0 ) .126 i f p G (0.0,0.5] .127 if p G (0 .5 ,1 .0 ) .167 i f p G (0 .0 ,0 .2] .136 i f p G (0 .2 ,0 .8] .134 or .346 i f p G (0 .8 ,1 .0 )

Following the sam e principle, com bining any three two-sequence four-period de­ signs, a six-sequence four-period design is produced. There are 35 d istin ct designs in th a t family and the optim um com bination for estim ating r is presented again in Table (4.4). Note here th a t th e 35 distin ct designs produced in th a t way in two treatm en ts co n stitu te all the m em bers for th a t family. The strateg y of pro­ ducing more com plicated designs by combining generic ones can be extended to

th e situ atio n where more th a n two treatm en ts are com pared. W orth noting th a t the plans generated in th a t way constitute a new design family th e size of which grows too fast. Identification of subsets w ith high probability of containing plans w ith optim um properties is highly desirable. Form al proof th a t the optim um plan for th e subset is the optim um for the family as well, or a t least th a t the efficiency of the form er is quite high, could be difficult to derive. Efficient subset construction could considerably simplify the design search for th e original family. It should be noted th a t th e optim ality conclusions draw n so far do depend upon th e intra-class correlation coefficient p.

A related work by J.N.S M atthew s (see [61]) in which th e simple carry-over m odel w ith fixed subject effects and AR(1) w ithin-error stru ctu re is assum ed as the m odel generated the d a ta a t hand, manages to determ ine m athem atically the optim um design. This work restricts atten tio n on three and four-period de­ sign families. U nder these assum ptions, negative correlation between successive m easurem ents on a p atien t is possible. M atthew s concludes th a t the final de­ cision concerning the design to use is highly affected not only by the value of p, b u t also the proportion of patients allocated in each sequence group. This is an uninteresting result since p is unknown in practice while equal num ber of p atients are usually allocated to the sequence groups. Being aware of these facts, M atthew s goes even further and examines the robustness of various designs con­ sidered before. He deduces th a t over the full range of p and under the simple carry-over model a design w ith good perform ance for estim atin g tre a tm e n t and residual effect is 4.4.13 in our notation. U nfortunately our results do not suggest a specific design w ith good properties over the range of m odels studied. This is an indication th a t this line of research will be difficult (if a t all possible) to be taken any further.

4.4

M odel m is-specification

A wide range of different criteria have been proposed in the lite ra tu re for choosing the optim um design. In the cross-over set-up the assum ed carry-over effect is crucial in deciding the best design for the analysis. Types of carry-over, already

discussed, make this term depending upon current and previous treatm en t regime (Fleiss carryover). These ideas can be further extended in various directions, producing m ore elaborated carry-over schemes, although th e validity of those plans in real life problem s has been questioned a lot in the p ast (see M atthew s [62] and Fleiss [18]). One such direction allows th e current p a tie n t’s response to depend on th e whole treatm en t history of th a t p atien t, i.e th e residual effect at tim e Ms a function of all treatm en t effects up to and including tim e t. Although such a scenario assumes th a t carry-over from current tre a tm e n t is present in all (or some) subsequent treatm en t periods, it is extrem ely unlikely to be encountered in practical applications.

A more general scheme, can be described as follows: if carry-over from treatm ent A to tre a tm e n t B (or from B to A) is denoted by A, then carry-over from A to A (or from B to B) will be (f)X for some 0 < 0 < 1. This will be referred to as the