STATISTICS IN MEDICINE, VOL. 9, 1121-1129 (1990)
FACTORIAL DESIGNS FOR CROSSOVER CLINICAL TRIALS D. J. FLETCHER Department of Biometry, School of Crop Sciences, University of Sydney, NSW 2006. Australia
S . M. LEWIS Department of Mathematics, The University. Southampton SO9 5NH, U.K.
AND J. N. S . MATTHEWS Division of Medical Statistics, The Medical School, Framlington Place, Newcastle upon Tyne NE2 4HH, U.K.
SUMMARY When measuring the joint effect of two factors it is advantageous to use a factorial design. If the application is suitable, efficiency may be further improved by using a crossover design. This paper presents a flexible method for amalgamating these two devices. Designs are constructed from smaller designs, known as bricks, generated cyclically from tabulated initial sequences. The bricks have known efficiencies for estimation of direct treatment main effects and interactions; the efficiencies can be simply combined to approximate the efficiencies of the whole design. This allows the user to build a design that is tailored to the particular objectives of the experiment. Three and four periods, and two factors with up to four levels are considered.
1. INTRODUCTION In medical research it is often required to measure the effect on some response of two or more factors. It has long been known' that statistically it is more efficient and informative in this situation to use a factorial design, than to run separate experiments for each factor. If used appropriately, this type of design not only needs fewer patients or experimental subjects to answer questions about each factor, but can address the issue of interactions between the factors. Another method that allows more efficient use of existing experimental resources is to measure each subject under different treatments in successive treatment periods, that is to use a crossover design (see Jones and Kenward'). Obviously this is only possible if the nature of the experiment lends itseif to such a design; for example, crossover designs are not appropriate if the treatments are intended to cure the condition being investigated. However, if it is possible to use a crossover design, then the potential advantages of comparing treatment effects with respect to withinsubject variation can be considerable.
* Please address correspondence to the third author. 0277-67 12/90/101121-O9$05.OO 0 1990 by John Wiley & Sons, Ltd.
Received October 1989 Revised February I990
1122
D. J. FLETCHER., S. M. LEWIS A N D J. N. S. MATTHEWS
An application that is typical of many found in the pharmaceutical industry is the testing of a combination of drugs. Here two or more compounds that, when given singly, have an effect on some condition, are given together. If the separate effects of the compounds are already known, much of the interest in the trial will often centre on their interaction. Examples include diuretics and /3-blockers for the control of moderate hypertension, antacids and anti-spasmodics in the treatment of gastrointestinal disorders and analgesics based on opioids and NSAID. A slight variant is when one of the compounds is intended to alleviate side-effects of the other compound, as with aspirin and cimetidine for long term analgesia. A different type of example is an investigation of the management of fluid and electrolyte balance in neonates. Over the second to fifth days of life, these patients, being well apart from their prematurity, are sufficiently stable for a crossover trial to be both feasible and beneficial. This paper is concerned with amalgamating crossover and factorial designs. Some earlier work3*' has concentrated on producing designs that allow efficient estimation of main effects. In some cases these designs estimate interactions inefficiently, and as these are often of importance in medical work, a new selection of designs was con~tructed.~ The purpose of the present paper is both to increase the range of available designs and to give a more accessible account of them for those statisticians unfamiliar with design terminology. In Section 2 some relevant theory is briefly outlined and the notation used in the remainder of the paper is explained. Section 3 presents the designs and explains how their tabulation should be used. A numerical example is given in Section4; advantages ,and limitations are discussed in Section 5.
2. GENERALIZED CYCLIC CROSSOVE,R DESIGNS
When treatments have a factorial nature, the total number of treatment combinations, u, is likely to be larger than the number of periods, p, that are at the experimenter's disposal. Consequently we only consider designs where p < u. Also, as it is the most common case in practice, only designs for two factors at two, three or four levels are given. The designs used in this paper are generalized cyclic crossovers introduced by Fletcher4 and based on the generalized cyclic designs of John.6 These designs are produced by cyclic generation from one or more suitably chosen initial sequences. The initial sequences, and hence all the sequences, have length p ; these form the treatment sequences for the crossover design, with the order of application corresponding to the order in the sequence. A treatment combination from an rn x n experiment (that is two factors with rn and n levels, respectively) is denoted by x y where x and y are integers, 0 < x GIrn - 1 and 0 < y < n - 1. For a four-period, 2 x 3 design, a possible initial sequence is (00 81 12 01): further sequences are generated by repeatedly adding 1, separately for the two factors, and reducing the results modulo 2 in the first place and modulo 3 in the second, until six distinct sequences are obtained, namely:
(00 11 12 01)
(01 12 10 02)
(02 10 11 00)
(10 01 02 11)
(11 02 00 12)
(12 cm 01 10).
Thus, for example (01 12 10 02) is obtained from (00 11 12 01) by adding 1 (mod 3) to the second digit of each treatment label; similarly (10 01 02 11) is found by adding 1 (mod 2) to the first digit of each label in the initial sequence. For an rn x n design, arithmetic for the first factor is modulo rn and for the second modulo n, producing rnn distinct sequences. Clearly the same set of rnn sequences would be obtained if any of
FACTORIAL DESIGNS FOR CROSSOVER CLINICAL TRIALS
1123
the sequences were chosen as the initial sequence. For definiteness all our initial sequences start with the combination labelled 00.Apart from this there are no restrictions on what constitutes a possible initial sequence; it is one of the main objects of this paper to determine which sequences are good in some sense. The designs are chosen so that they have desirable properties for the estimation of direct treatment effects in the following model, where yij is the response on subject j in period i:
+ P i + S j + Tt(i, j ) + Ytci-1.j) + E i j .
(1) Here pi is the effect of the ith period, s j is the effect of the jth subject, t(i,j ) is the treatment given in period i to subject j, zk is the direct effect of treatment k and ya is its first-order carryover effect; yt(o, J1 = 0 for all j . The error terms are independent with constant variance. Thus our designs are intended for the estimation of direct treatment effects, adjusted for firstorder carryover effects of treatment. When the persistence of a treatment effect beyond the period of application cannot be excluded a priori then a carryover term should be included in the model, regardless of its statistical ~ignificance.~ Only first-order effects of carryover are considered; if a more persistent carryover is anticipated then it is probably unwise to use a crossover design. Designs appropriate for the experimental purpose are constructed using the bricks in Table I. Bricks, a term first used by Pearce* in the context of block designs, are simply small designs, here generated by one or two initial sequences. Table I1 describes the efficiencies of each brick for each main effect and interaction; for each treatment structure and number of periods, Table I gives various alternative bricks. Designs are built by putting together bricks so as to give a design with desirable properties. The rules for approximating the efficiencies of the combined design from those of the constituent bricks, are given in Section 3. The bricks in Table I1 were chosen by computer search, using criteria given in Section 3. Essentially these criteria ensured that the chosen bricks had high canonical efficiency factors for some of the direct factorial effects. Canonical efficiency factors (c.e.f.s) for the direct effects are the non-zero eigenvalues of the information matrix (divided by the treatment replication) for these parameters. A full description of c.e.f.s can be found in Johng (p. 25) but, briefly, they give the efficiency that the design achieves for certain contrasts, with respect to a fully orthogonal design. In general a more concrete interpretation is hampered because these contrasts are defined by the corresponding eigenvectors, and these do not necessarily represent meaningful contrasts in terms of the experiment (some may even be complex). However, for cyclic designs the eigenvectors, and hence the eigenvalues, can be identified with specific main effects and interactions and it is this property that makes Table I1 possible. Consequently, designs with higher c.e.f.s are more desirable, as this means the corresponding effect is estimated more efficiently. However, increasing the c.e.f. for one effect may reduce it for another, so in practice it will often be necessary to reach some compromise between the relative efficiencies achieved for the different effects. The number of eigenvalues that are identified with a particular factorial effect is equal to the degrees of freedom of that effect. For most of the designs in this paper, all the eigenvalues corresponding to an effect have a common value and that is the c.e.f.; this property is known as factorial balance. For the other designs the harmonic mean of the associated eigenvalues is used as the measure of efficiency. A final property of generalized cyclic crossover designs isfactorial structure. This means that a particular direct factorial effect is orthogonal to all the other factorial effects, direct or carryover, except the corresponding carryover effect. This simplifies the analysis, because it is necessary to adjust a direct main effect (for example), only for the corresponding carryover effect, and not for all the other effects.This feature may also have some advantages when explaining the analysis to non-statistical colleagues. Yij
=P
Table I. Initial sequences for bricks for rn x n designs with p periods Brick 2~2;p=3 1 2
Initial sequences (00 10 01) (00 10 11) (00 11 01) (00 11 10)
2 x 2;p=4 1 2 3 4
2 x 3;p=3 1 2 3 4
5
Brick 3 x 3;p==3 1 2 3 5
(00 10 01) (00 20 21) (00 10 12) (00 20 02) (00 10 02) (00 20 22) (00 10 11) (00 20 01) (00 10 21) (0020 11)
6 7 8 9 10
(00 10 22) (00 20 12) (00 11 01) (00 22 20) (00 11 10) (00 22 02) (00 12 02) (00 21 20) (00 12 10) (00 21 01)
11 12 13 14
(00 11 20) (00 22 (00 11 21) (00 22 (00 12 20) (00 21 (00 12 22) (00 21
4
(00 01 (00 01 (00 01 (00 01
10 11) (00 18 01 11) 10 11) (00 10 11 01) 10 11) (00 11 01 10) 11 10) (00 11 10 01)
(0001 10) (00 02 12) (00 01 11) (00 02 10) (00 01 11) (00 11 10) (00 02 12) (00 12 10) (00 10 01) (00 10 11)
Initial sequences
3 x 3;p==4
(00 10 11) (00 10 12) (00 10 11) (00 11 01) (00 10 12) (00 12 02) (00 11 01) (00 12 10)
5
(00 11 20 12) (00 12 20 11) (00 21 10 22) (00 22 10 21) (00 10 21 11)
6 7 8 9 10
(00 11 21 10) (00 20 12 22) (00 22 12 20) (00 10 22 12) (00 12 22 10)
11 12 13 14 15
(00 20 11 21) (00 21 11 20) (00 01 11 10) (00 02 22 20) (00 10 11 01)
16 17 18 19 20
(00 20 22 02) (00 01 21 20) (00 02 12 10) (00 10 12 02) (00 20 21 01)
21 22 23 24 25
(00 01 10 12) (00 02 20 21) (00 12 10 01) (00 21 20 02) (00 01 20 22)
26 21 28
(00 02 10 11) (00 11 10 02) (00 22 20 01)
1 2 3 4
2~3;p=4 1 2 3 4
(0001 11 02) (00 02 12 01) (00 11 12 01) (00 12 11 02)
2x4;p=3
(00 10 01) (00 12 11) (00 10 03) (00 12 13) (00 10 11) (00 12 01) (00 10 13) (00 12 03) (00 10 01) (00 12 13) 6 7 8
(00 10 03) (00 12 11) (00 10 11) (00 12 03) (00 10 13) (00 12 01)
2x4;p=4 1 2 3 5
(00 02 01 11) (00 02 03 13) (000201 12) (00 02 03 12) (00 02 11 01)
6 7 8 9 10
(00 02 13 03) (00 02 11 12) (0002 13 12) (00 02 01 13) (00 02 03 11)
11 12
(00 02 11 03) (00 02 13 01)
4
12) 10) 11) 10)
FACTORIAL DESIGNS FOR CROSSOVER CLINICAL TRIALS
Table 11. Efficiencies for the bricks given in Table I. For designs without factorial balance the minimum and maximum efficiencies are given Direct efficiencies ( x 100)
A
B
A3
2x2;p=3 1 2
72 72
67 72
72 67
2x2;p=4 1 2 3 4
91 100 91 100
91 91 64
91
2 x 3;p=3 1, 2 3, 4 5 6 7, 8 9
67 86 72 89 72 72
55 52 50 54 60 55
72 59 72
55 91
67 69
82 78
5-8
72 72
57 (53, 67) 50 (44,67)
51 (44,72) 58 (53, 72)
2x4;p=4 1, 2 3, 4 5, 6 7, 8 9, 10 11,12
67 67 55 91 67 55
71 (64,91) 66 (55, 73) 71 (64,91) 66 (55, 73) 63 (55, 91) 63 (55,91)
55 54 (45,91) 58 (55, 67) 51 (45, 67) 60 (5564) 65 (64,67)
3 x 3;p=3 1 4 5, 6 7-10 11-14
55 80 55 80
50 50 55 55
65 (55, 80) 55 62 (50, 80) 52 (50, 55)
3 x 3;p=4 1-4 5-12 13-20 21-28
69 69 67 67
50 67 67
67 57 (50, 67) 58 (50,69) 68 (67, 69)
Bricks ~
2 x 3;p=4 1, 2 3, 4 2x4;p=3 1 4
64
50
64 100 91
51
54 69
1125
1126
D. J. FLETCHER, S. M. LEWIS AND J. N. S. MATTHEWS
Those designs which have factorial balance have the useful property that, in the analysis, the sums of squares for each factorial effect can be split up orthogonally into single degree of freedom components corresponding to any set of orthogonal contrasts (for example, orthogonal polynomials) within the factorial effect. 3. TABULATION O F BRICKS
For a given number of periods (three or four) and treatment structure, all possible initial sequences were generated by a computer. The c.e.f.s of the direct treatment effects in the corresponding generalized cyclic crossover brick were computed. Bricks with any c.e.f. less than 0.5 were discarded, as were bricks that were less efficient than another brick on all effects. For some period and treatment combinations, no single sequence gave a brick with all c.e.f.s exceeding 0.5. In this case, all pairs of initial sequences were searched, giving rise to bricks comprising two initial sequences. The results of this search are given in Table I, with the corresponding efficiencies for the main effects A and B, and their interaction AB, in Table 11. The experimenter can combine bricks freely, producing a design that is suitable for a specific application. For a 2 x 3 design in three periods, brick 1 has c.e.f.s 067,055and 0.72 for A, B and AB, respectively. If the user wishes to bolster the estimation of B, then brick 1 may be combined with brick 8, which has c.e.f.s 0.72, 0.60 and 054. The resulting design has c.e.f.s that are approximately the means of the c.e.f.s of the constituent bricks, namely (0.70, 0.58; 0.63). If a design is composed of b , copies of one brick and b, copies of a second brick, then the resulting design has c.e.f.s that are approximately the weighted means of the constituent c.e.f.s, with weights proportional to b , and b,. These approximations are actually lower boundsY5although the discrepancy is usually small. This lower bound is achieved when a brick is combined with itself, so it can make sense to combine two equally efficient, but distinct, bricks. By combining bricks, using these results to give some guidance, designs with the necessary number of sequences and appropriate efficiencies can be obtained. When the two factors have the same number of levels, the bricks are always given with A having an efficiency at least that of B. In this case, reversing the order of the digits in each label will produce a design with the c.e.f.s for A and B interchanged. Most of the 3 x 3 and 2 x 4 designs in Table I do not possess factorial balance. For these bricks the efficienciesin Table I1 are the harmonic means of two or three eigenvalues; we suggest that designs are constructed using this average figure. However, some contrasts within these effects will be estimated with higher efficiency and some with lower efficiency. The lowest and highest efficiencies that apply to a contrast within a particular effect are given by the minimum and maximum eigenvalues, and these are given in Table 11. When constructing a design including one of these bricks, it would be prudent to run a ‘dummy’ analysis to ensure that no practically important contrast is poorly estimated.
4. AN EXAMPLE Some features of these designs can be illustrated with the followingsmall example. The data have been extracted from a larger randomized study of the effects of two anti-hypertensive agents, A and B, that are thought to act by different mechanisms. The full study did not use one of the designs in this paper, as they had not been developed when this trial was carried out. The two drugs were given at two levels, the lower being a placebo in each case; the treatment periods lasted three weeks, with no intervening washout periods. There was a run-in period of three weeks on
1I27
FACTORIAL DESIGNS FOR CROSSOVER CLINICAL TRIALS
Table 111. Data from the anti-hypertension study, with treatment means. Adjusted means are adjusted for subject, period and the corresponding carryover treatment effect: standard errors for adjusted means are 3.0 (drug A), 3.1 (drug B) Diastolic blood pressure (mmHg) Patient
Treatment sequence (drug A first)
visit 1
visit 2
visit 3
mean
1 2 3 4
00 01 10 11
10 11 00 01
11 10 01 00
101 100 84 74
76 80 100 92
66 97 81 108
81.0 923 88-3 91.3
5 6 7 8
00 01 10 11
10 11 00 01
01 00 11 10
105 97 86 87
85 87 94 91
104 87 87 92
98-0 903 890 90.0
Mean
91.8
88.1
Drug B Absent Present Drug A
Absent Present
99.2 86.7
94.2 80.2
Marginal mean
Raw Adjusted
92-9 93.2
87.2 869
90-3
Marginal mean Raw Adjusted 96.7 83.4
97.3 82.8
placebo, and only patients with a diastolic blood pressure exceeding 95 mmHg were entered into the trial. The patients visited the clinic at the end of each of three treatment periods, when the sitting diastolic blood pressure was recorded. The data given in Table I11 are for eight patients treated using the sequences from brick 1 for three-period, 2 x 2 designs; the label for drug A is given first. The analysis of variance is given in Table IV. As pointed out in Section 2, the factorial structure of the design means that each direct treatment sum of squares only need to be adjusted for the corresponding carryover sum of squares. Of course, if some data are missing, then this structure will be lost, but provided there are not too many missing values, the resulting non-orthogonality should not be great. From Table IV there is little evidence that there are any carryover effects but, as Abeyasekera and Curnow’ point out, simply omitting such terms because they are not significant can lead to bias in the direct estimates. As there is no evidence of an interaction between A and B, the treatment effects are summarized in Table 111, where adjusted and raw means are given for the direct effects of A and B. It can be seen from Table IV that the mean square for patients is actually less than the residual. This is very unusual in this sort of trial; it may be an artefact of both the small size of this dataset and the rather artificial way the data were selected out of a larger study or possibly a consequence of the restriction of the diastolic blood pressure at entry.
1128
D. J. FLETCHER, S. M. LEWIS AND J. N. S. MATTHEWS
Table IV. Analysis of variance of data from anti-hypertension study. The sums of squares for patients and periods are not adjusted for treatments. Each treatment effect is adjusted for all other effects in the model; since the design has factorial structure (see text) this amounts to adjusting for patients, periods and the corresponding carryover or direct effect
ss
MS
468.3 53.1
66.9 266
9 13.5 27.0
Direct B (adj carryover B) Carryover B (adj direct B) Direct AB (adj carryover AB) Carryover AB (adj direct AB) Residual
Source of variation
D.F.
Patients Periods Direct A (adj carryover A) Carryover A (adj direct A)
Total
23
F
P
913.5 27.0
11.6 0.3
0.0 1 058
162.6 10.6
162.6 10.6
2-1 0.1
0.19 0.73
5.2 78.7
5.2 78.7
0.1 1.o
0.80 0.35
631.7
79.0
2511.0
Suppose that the following parameterization is adopted for the direct treatment effects: Drug B Present
Absent
Absent Present
‘A
+
TAB
- TA
- ‘B - TAB
?A
- 7A - 7 B
Drug A
Using a similar scheme for the carryover parameters, yA, yB, (and standard errors): 7A
- 7.26 (2.13)
YA
7B
-
3.19 (2.22) - 0.55 (2.13)
YB
TAB
YAB
YAB,
+
TB
- TAB
+ ZB + TAB
gives the following estimates
1.44 (2.46) 1.62 (4.44) - 2.46 (2.46). -
The relationship between the c.e.f.s and the standard errors of the estimates can be seen here. The variances of 7 A and rABare equal, as are the corresponding c.e.f.s for brick 1. The ratio between the variances of zA and zB is the same as the ratio between the corresponding c.e.f.s, namely 0.67/0.72 (to within rounding error). 5. DISCUSSION
While crossover designs have been used with factorial treatments for some time, there has been little guidance from the design literature until r e ~ e n t l y . ~This - ~ paper has presented a flexible approach, whereby the experimenter can build suitable designs from the given bricks. Thus it is possible to adjust the efficiency on the main effects and interactions, so that effects that are important in a particular application are not estimated inefficiently. The designs have been derived by adapting known types of factorial designs to the crossover setting. This approach ensures that our designs have desirable properties, such as factorial
FACTORIAL DESIGNS FOR CROSSOVER CLINICAL TRIALS
1129
structure. Another possible approach, which we have not considered, would have been to try and impose a factorial treatment structure on a known class of crossover designs, such as the cyclic designs of Davis and Hall.” However, there seems no reason to suppose that this approach would lead to more efficient designs. The efficiencies in Table I1 are calculated assuming that a carryover term is included in the model. It has been suggested that a crossover design should estimate the direct treatment effect efficiently, whether or not a carryover term is included in the model. Although our designs are highly efficient when the carryover effect is omitted, it is in fact unnecessary to insist that designs are efficient in both cases. If it is thought that there may be a carryover effect, then it should be included at the design stage. When it comes to the analysis, this term should be retained because, as Abeyasekera and Curnow7 have shown, it can introduce bias to omit a carryover term simply because it fails to reach statistical significance. If carryover effects can be excluded a priori, then a crossover design is merely an example of a row and column design, and these are available for factorial treatments.’ Our designs are based on the model (1) which is widely used for the analysis of data from crossover trials. In particular this means that only interactions between two direct or two carryover effects are considered. While this model is plausible for many applications, it has been criticizedI2 particularly when the design allows a treatment to follow itself, for example, as with some two-treatment three-period designs. Although no treatment combination follows itself in any of the present designs, a single factor is often kept at the same level over adjacent periods. It would be interesting to investigate different models for carryover, perhaps by including interactions between direct and carryover effects, but this is beyond the scope of the present paper. A common situation to which our designs do not apply occurs when the lowest levels of each factor correspond to no treatment, or possibly a placebo. It may then be unethical to give the 00 treatment; work is in progress to develop a class of designs that do not use this combination.
’
ACKNOWLEDGEMENTS
We are very grateful to Dr. John Lewis and ICI for permission to use the data in the example, to Professor John Thompson for his help and to the referees for their comments.
REFERENCES 1. Fisher, R. A. The Design of Experiments, 1st edn, Oliver and Boyd, Edinburgh, 1935. 2. Jones, B. and Kenward, M. G . Design and Analysis of Cross-Over Trials, Chapman and Hall, London, 1989. 3. Fletcher, D. J. and John, J. A. ‘Changeover designs and factorial structure’, Journal of the Royal Statistical Society, Series B, 47, 117-124 (1985). 4. Fletcher, D. J. ‘A new class of change-over designs for factorial experiments’, Biometrika, 74, 649654 (1987). 5. Lewis, S. M., Fletcher, D. J. and Matthews, J. N . S. ‘Factorial cross-over designs in clinical trials’ in Dodge, Y., Fedorov, V. V. and Wynn, H. P. (eds.), Optimal Design and Analysis of Experiments, North Holland, Amsterdam, 1988. 6. John, J. A. ‘Generalised cyclic designs in factorial experiments’, Biometrika, 60, 55-63 (1973). 7. Abeyasekera, S. and Curnow, R. N. ‘The desirability of adjusting for residual effects in a crossover design’, Biometrics, 40, 1071-1078 (1984). 8. Pearce, S. C. The Agricultural Field Experiment, Wiley, Chichester, 1983. 9. John, J. A. Cyclic Designs, Chapman and Hall, London, 1987. 10. Davis, A. W. and Hall, W. B. ‘Cyclic change-over designs’, Biometrika, 56, 283-293 (1969). 11. John, J. A. and Lewis, S. M. ‘Factorial experiments in generalized cyclic row-column designs’, Journal of the Royal Statistical Society, Series B, 45, 245-251 (1983). 12. Fleiss, J. L. ‘Letter to the Editor’, Biometrics, 42, 449-450 (1986).
FACTORIAL DESIGNS FOR CROSSOVER CLINICAL TRIALS D. J. FLETCHER Department of Biometry, School of Crop Sciences, University of Sydney, NSW 2006. Australia
S . M. LEWIS Department of Mathematics, The University. Southampton SO9 5NH, U.K.
AND J. N. S . MATTHEWS Division of Medical Statistics, The Medical School, Framlington Place, Newcastle upon Tyne NE2 4HH, U.K.
SUMMARY When measuring the joint effect of two factors it is advantageous to use a factorial design. If the application is suitable, efficiency may be further improved by using a crossover design. This paper presents a flexible method for amalgamating these two devices. Designs are constructed from smaller designs, known as bricks, generated cyclically from tabulated initial sequences. The bricks have known efficiencies for estimation of direct treatment main effects and interactions; the efficiencies can be simply combined to approximate the efficiencies of the whole design. This allows the user to build a design that is tailored to the particular objectives of the experiment. Three and four periods, and two factors with up to four levels are considered.
1. INTRODUCTION In medical research it is often required to measure the effect on some response of two or more factors. It has long been known' that statistically it is more efficient and informative in this situation to use a factorial design, than to run separate experiments for each factor. If used appropriately, this type of design not only needs fewer patients or experimental subjects to answer questions about each factor, but can address the issue of interactions between the factors. Another method that allows more efficient use of existing experimental resources is to measure each subject under different treatments in successive treatment periods, that is to use a crossover design (see Jones and Kenward'). Obviously this is only possible if the nature of the experiment lends itseif to such a design; for example, crossover designs are not appropriate if the treatments are intended to cure the condition being investigated. However, if it is possible to use a crossover design, then the potential advantages of comparing treatment effects with respect to withinsubject variation can be considerable.
* Please address correspondence to the third author. 0277-67 12/90/101121-O9$05.OO 0 1990 by John Wiley & Sons, Ltd.
Received October 1989 Revised February I990
1122
D. J. FLETCHER., S. M. LEWIS A N D J. N. S. MATTHEWS
An application that is typical of many found in the pharmaceutical industry is the testing of a combination of drugs. Here two or more compounds that, when given singly, have an effect on some condition, are given together. If the separate effects of the compounds are already known, much of the interest in the trial will often centre on their interaction. Examples include diuretics and /3-blockers for the control of moderate hypertension, antacids and anti-spasmodics in the treatment of gastrointestinal disorders and analgesics based on opioids and NSAID. A slight variant is when one of the compounds is intended to alleviate side-effects of the other compound, as with aspirin and cimetidine for long term analgesia. A different type of example is an investigation of the management of fluid and electrolyte balance in neonates. Over the second to fifth days of life, these patients, being well apart from their prematurity, are sufficiently stable for a crossover trial to be both feasible and beneficial. This paper is concerned with amalgamating crossover and factorial designs. Some earlier work3*' has concentrated on producing designs that allow efficient estimation of main effects. In some cases these designs estimate interactions inefficiently, and as these are often of importance in medical work, a new selection of designs was con~tructed.~ The purpose of the present paper is both to increase the range of available designs and to give a more accessible account of them for those statisticians unfamiliar with design terminology. In Section 2 some relevant theory is briefly outlined and the notation used in the remainder of the paper is explained. Section 3 presents the designs and explains how their tabulation should be used. A numerical example is given in Section4; advantages ,and limitations are discussed in Section 5.
2. GENERALIZED CYCLIC CROSSOVE,R DESIGNS
When treatments have a factorial nature, the total number of treatment combinations, u, is likely to be larger than the number of periods, p, that are at the experimenter's disposal. Consequently we only consider designs where p < u. Also, as it is the most common case in practice, only designs for two factors at two, three or four levels are given. The designs used in this paper are generalized cyclic crossovers introduced by Fletcher4 and based on the generalized cyclic designs of John.6 These designs are produced by cyclic generation from one or more suitably chosen initial sequences. The initial sequences, and hence all the sequences, have length p ; these form the treatment sequences for the crossover design, with the order of application corresponding to the order in the sequence. A treatment combination from an rn x n experiment (that is two factors with rn and n levels, respectively) is denoted by x y where x and y are integers, 0 < x GIrn - 1 and 0 < y < n - 1. For a four-period, 2 x 3 design, a possible initial sequence is (00 81 12 01): further sequences are generated by repeatedly adding 1, separately for the two factors, and reducing the results modulo 2 in the first place and modulo 3 in the second, until six distinct sequences are obtained, namely:
(00 11 12 01)
(01 12 10 02)
(02 10 11 00)
(10 01 02 11)
(11 02 00 12)
(12 cm 01 10).
Thus, for example (01 12 10 02) is obtained from (00 11 12 01) by adding 1 (mod 3) to the second digit of each treatment label; similarly (10 01 02 11) is found by adding 1 (mod 2) to the first digit of each label in the initial sequence. For an rn x n design, arithmetic for the first factor is modulo rn and for the second modulo n, producing rnn distinct sequences. Clearly the same set of rnn sequences would be obtained if any of
FACTORIAL DESIGNS FOR CROSSOVER CLINICAL TRIALS
1123
the sequences were chosen as the initial sequence. For definiteness all our initial sequences start with the combination labelled 00.Apart from this there are no restrictions on what constitutes a possible initial sequence; it is one of the main objects of this paper to determine which sequences are good in some sense. The designs are chosen so that they have desirable properties for the estimation of direct treatment effects in the following model, where yij is the response on subject j in period i:
+ P i + S j + Tt(i, j ) + Ytci-1.j) + E i j .
(1) Here pi is the effect of the ith period, s j is the effect of the jth subject, t(i,j ) is the treatment given in period i to subject j, zk is the direct effect of treatment k and ya is its first-order carryover effect; yt(o, J1 = 0 for all j . The error terms are independent with constant variance. Thus our designs are intended for the estimation of direct treatment effects, adjusted for firstorder carryover effects of treatment. When the persistence of a treatment effect beyond the period of application cannot be excluded a priori then a carryover term should be included in the model, regardless of its statistical ~ignificance.~ Only first-order effects of carryover are considered; if a more persistent carryover is anticipated then it is probably unwise to use a crossover design. Designs appropriate for the experimental purpose are constructed using the bricks in Table I. Bricks, a term first used by Pearce* in the context of block designs, are simply small designs, here generated by one or two initial sequences. Table I1 describes the efficiencies of each brick for each main effect and interaction; for each treatment structure and number of periods, Table I gives various alternative bricks. Designs are built by putting together bricks so as to give a design with desirable properties. The rules for approximating the efficiencies of the combined design from those of the constituent bricks, are given in Section 3. The bricks in Table I1 were chosen by computer search, using criteria given in Section 3. Essentially these criteria ensured that the chosen bricks had high canonical efficiency factors for some of the direct factorial effects. Canonical efficiency factors (c.e.f.s) for the direct effects are the non-zero eigenvalues of the information matrix (divided by the treatment replication) for these parameters. A full description of c.e.f.s can be found in Johng (p. 25) but, briefly, they give the efficiency that the design achieves for certain contrasts, with respect to a fully orthogonal design. In general a more concrete interpretation is hampered because these contrasts are defined by the corresponding eigenvectors, and these do not necessarily represent meaningful contrasts in terms of the experiment (some may even be complex). However, for cyclic designs the eigenvectors, and hence the eigenvalues, can be identified with specific main effects and interactions and it is this property that makes Table I1 possible. Consequently, designs with higher c.e.f.s are more desirable, as this means the corresponding effect is estimated more efficiently. However, increasing the c.e.f. for one effect may reduce it for another, so in practice it will often be necessary to reach some compromise between the relative efficiencies achieved for the different effects. The number of eigenvalues that are identified with a particular factorial effect is equal to the degrees of freedom of that effect. For most of the designs in this paper, all the eigenvalues corresponding to an effect have a common value and that is the c.e.f.; this property is known as factorial balance. For the other designs the harmonic mean of the associated eigenvalues is used as the measure of efficiency. A final property of generalized cyclic crossover designs isfactorial structure. This means that a particular direct factorial effect is orthogonal to all the other factorial effects, direct or carryover, except the corresponding carryover effect. This simplifies the analysis, because it is necessary to adjust a direct main effect (for example), only for the corresponding carryover effect, and not for all the other effects.This feature may also have some advantages when explaining the analysis to non-statistical colleagues. Yij
=P
Table I. Initial sequences for bricks for rn x n designs with p periods Brick 2~2;p=3 1 2
Initial sequences (00 10 01) (00 10 11) (00 11 01) (00 11 10)
2 x 2;p=4 1 2 3 4
2 x 3;p=3 1 2 3 4
5
Brick 3 x 3;p==3 1 2 3 5
(00 10 01) (00 20 21) (00 10 12) (00 20 02) (00 10 02) (00 20 22) (00 10 11) (00 20 01) (00 10 21) (0020 11)
6 7 8 9 10
(00 10 22) (00 20 12) (00 11 01) (00 22 20) (00 11 10) (00 22 02) (00 12 02) (00 21 20) (00 12 10) (00 21 01)
11 12 13 14
(00 11 20) (00 22 (00 11 21) (00 22 (00 12 20) (00 21 (00 12 22) (00 21
4
(00 01 (00 01 (00 01 (00 01
10 11) (00 18 01 11) 10 11) (00 10 11 01) 10 11) (00 11 01 10) 11 10) (00 11 10 01)
(0001 10) (00 02 12) (00 01 11) (00 02 10) (00 01 11) (00 11 10) (00 02 12) (00 12 10) (00 10 01) (00 10 11)
Initial sequences
3 x 3;p==4
(00 10 11) (00 10 12) (00 10 11) (00 11 01) (00 10 12) (00 12 02) (00 11 01) (00 12 10)
5
(00 11 20 12) (00 12 20 11) (00 21 10 22) (00 22 10 21) (00 10 21 11)
6 7 8 9 10
(00 11 21 10) (00 20 12 22) (00 22 12 20) (00 10 22 12) (00 12 22 10)
11 12 13 14 15
(00 20 11 21) (00 21 11 20) (00 01 11 10) (00 02 22 20) (00 10 11 01)
16 17 18 19 20
(00 20 22 02) (00 01 21 20) (00 02 12 10) (00 10 12 02) (00 20 21 01)
21 22 23 24 25
(00 01 10 12) (00 02 20 21) (00 12 10 01) (00 21 20 02) (00 01 20 22)
26 21 28
(00 02 10 11) (00 11 10 02) (00 22 20 01)
1 2 3 4
2~3;p=4 1 2 3 4
(0001 11 02) (00 02 12 01) (00 11 12 01) (00 12 11 02)
2x4;p=3
(00 10 01) (00 12 11) (00 10 03) (00 12 13) (00 10 11) (00 12 01) (00 10 13) (00 12 03) (00 10 01) (00 12 13) 6 7 8
(00 10 03) (00 12 11) (00 10 11) (00 12 03) (00 10 13) (00 12 01)
2x4;p=4 1 2 3 5
(00 02 01 11) (00 02 03 13) (000201 12) (00 02 03 12) (00 02 11 01)
6 7 8 9 10
(00 02 13 03) (00 02 11 12) (0002 13 12) (00 02 01 13) (00 02 03 11)
11 12
(00 02 11 03) (00 02 13 01)
4
12) 10) 11) 10)
FACTORIAL DESIGNS FOR CROSSOVER CLINICAL TRIALS
Table 11. Efficiencies for the bricks given in Table I. For designs without factorial balance the minimum and maximum efficiencies are given Direct efficiencies ( x 100)
A
B
A3
2x2;p=3 1 2
72 72
67 72
72 67
2x2;p=4 1 2 3 4
91 100 91 100
91 91 64
91
2 x 3;p=3 1, 2 3, 4 5 6 7, 8 9
67 86 72 89 72 72
55 52 50 54 60 55
72 59 72
55 91
67 69
82 78
5-8
72 72
57 (53, 67) 50 (44,67)
51 (44,72) 58 (53, 72)
2x4;p=4 1, 2 3, 4 5, 6 7, 8 9, 10 11,12
67 67 55 91 67 55
71 (64,91) 66 (55, 73) 71 (64,91) 66 (55, 73) 63 (55, 91) 63 (55,91)
55 54 (45,91) 58 (55, 67) 51 (45, 67) 60 (5564) 65 (64,67)
3 x 3;p=3 1 4 5, 6 7-10 11-14
55 80 55 80
50 50 55 55
65 (55, 80) 55 62 (50, 80) 52 (50, 55)
3 x 3;p=4 1-4 5-12 13-20 21-28
69 69 67 67
50 67 67
67 57 (50, 67) 58 (50,69) 68 (67, 69)
Bricks ~
2 x 3;p=4 1, 2 3, 4 2x4;p=3 1 4
64
50
64 100 91
51
54 69
1125
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D. J. FLETCHER, S. M. LEWIS AND J. N. S. MATTHEWS
Those designs which have factorial balance have the useful property that, in the analysis, the sums of squares for each factorial effect can be split up orthogonally into single degree of freedom components corresponding to any set of orthogonal contrasts (for example, orthogonal polynomials) within the factorial effect. 3. TABULATION O F BRICKS
For a given number of periods (three or four) and treatment structure, all possible initial sequences were generated by a computer. The c.e.f.s of the direct treatment effects in the corresponding generalized cyclic crossover brick were computed. Bricks with any c.e.f. less than 0.5 were discarded, as were bricks that were less efficient than another brick on all effects. For some period and treatment combinations, no single sequence gave a brick with all c.e.f.s exceeding 0.5. In this case, all pairs of initial sequences were searched, giving rise to bricks comprising two initial sequences. The results of this search are given in Table I, with the corresponding efficiencies for the main effects A and B, and their interaction AB, in Table 11. The experimenter can combine bricks freely, producing a design that is suitable for a specific application. For a 2 x 3 design in three periods, brick 1 has c.e.f.s 067,055and 0.72 for A, B and AB, respectively. If the user wishes to bolster the estimation of B, then brick 1 may be combined with brick 8, which has c.e.f.s 0.72, 0.60 and 054. The resulting design has c.e.f.s that are approximately the means of the c.e.f.s of the constituent bricks, namely (0.70, 0.58; 0.63). If a design is composed of b , copies of one brick and b, copies of a second brick, then the resulting design has c.e.f.s that are approximately the weighted means of the constituent c.e.f.s, with weights proportional to b , and b,. These approximations are actually lower boundsY5although the discrepancy is usually small. This lower bound is achieved when a brick is combined with itself, so it can make sense to combine two equally efficient, but distinct, bricks. By combining bricks, using these results to give some guidance, designs with the necessary number of sequences and appropriate efficiencies can be obtained. When the two factors have the same number of levels, the bricks are always given with A having an efficiency at least that of B. In this case, reversing the order of the digits in each label will produce a design with the c.e.f.s for A and B interchanged. Most of the 3 x 3 and 2 x 4 designs in Table I do not possess factorial balance. For these bricks the efficienciesin Table I1 are the harmonic means of two or three eigenvalues; we suggest that designs are constructed using this average figure. However, some contrasts within these effects will be estimated with higher efficiency and some with lower efficiency. The lowest and highest efficiencies that apply to a contrast within a particular effect are given by the minimum and maximum eigenvalues, and these are given in Table 11. When constructing a design including one of these bricks, it would be prudent to run a ‘dummy’ analysis to ensure that no practically important contrast is poorly estimated.
4. AN EXAMPLE Some features of these designs can be illustrated with the followingsmall example. The data have been extracted from a larger randomized study of the effects of two anti-hypertensive agents, A and B, that are thought to act by different mechanisms. The full study did not use one of the designs in this paper, as they had not been developed when this trial was carried out. The two drugs were given at two levels, the lower being a placebo in each case; the treatment periods lasted three weeks, with no intervening washout periods. There was a run-in period of three weeks on
1I27
FACTORIAL DESIGNS FOR CROSSOVER CLINICAL TRIALS
Table 111. Data from the anti-hypertension study, with treatment means. Adjusted means are adjusted for subject, period and the corresponding carryover treatment effect: standard errors for adjusted means are 3.0 (drug A), 3.1 (drug B) Diastolic blood pressure (mmHg) Patient
Treatment sequence (drug A first)
visit 1
visit 2
visit 3
mean
1 2 3 4
00 01 10 11
10 11 00 01
11 10 01 00
101 100 84 74
76 80 100 92
66 97 81 108
81.0 923 88-3 91.3
5 6 7 8
00 01 10 11
10 11 00 01
01 00 11 10
105 97 86 87
85 87 94 91
104 87 87 92
98-0 903 890 90.0
Mean
91.8
88.1
Drug B Absent Present Drug A
Absent Present
99.2 86.7
94.2 80.2
Marginal mean
Raw Adjusted
92-9 93.2
87.2 869
90-3
Marginal mean Raw Adjusted 96.7 83.4
97.3 82.8
placebo, and only patients with a diastolic blood pressure exceeding 95 mmHg were entered into the trial. The patients visited the clinic at the end of each of three treatment periods, when the sitting diastolic blood pressure was recorded. The data given in Table I11 are for eight patients treated using the sequences from brick 1 for three-period, 2 x 2 designs; the label for drug A is given first. The analysis of variance is given in Table IV. As pointed out in Section 2, the factorial structure of the design means that each direct treatment sum of squares only need to be adjusted for the corresponding carryover sum of squares. Of course, if some data are missing, then this structure will be lost, but provided there are not too many missing values, the resulting non-orthogonality should not be great. From Table IV there is little evidence that there are any carryover effects but, as Abeyasekera and Curnow’ point out, simply omitting such terms because they are not significant can lead to bias in the direct estimates. As there is no evidence of an interaction between A and B, the treatment effects are summarized in Table 111, where adjusted and raw means are given for the direct effects of A and B. It can be seen from Table IV that the mean square for patients is actually less than the residual. This is very unusual in this sort of trial; it may be an artefact of both the small size of this dataset and the rather artificial way the data were selected out of a larger study or possibly a consequence of the restriction of the diastolic blood pressure at entry.
1128
D. J. FLETCHER, S. M. LEWIS AND J. N. S. MATTHEWS
Table IV. Analysis of variance of data from anti-hypertension study. The sums of squares for patients and periods are not adjusted for treatments. Each treatment effect is adjusted for all other effects in the model; since the design has factorial structure (see text) this amounts to adjusting for patients, periods and the corresponding carryover or direct effect
ss
MS
468.3 53.1
66.9 266
9 13.5 27.0
Direct B (adj carryover B) Carryover B (adj direct B) Direct AB (adj carryover AB) Carryover AB (adj direct AB) Residual
Source of variation
D.F.
Patients Periods Direct A (adj carryover A) Carryover A (adj direct A)
Total
23
F
P
913.5 27.0
11.6 0.3
0.0 1 058
162.6 10.6
162.6 10.6
2-1 0.1
0.19 0.73
5.2 78.7
5.2 78.7
0.1 1.o
0.80 0.35
631.7
79.0
2511.0
Suppose that the following parameterization is adopted for the direct treatment effects: Drug B Present
Absent
Absent Present
‘A
+
TAB
- TA
- ‘B - TAB
?A
- 7A - 7 B
Drug A
Using a similar scheme for the carryover parameters, yA, yB, (and standard errors): 7A
- 7.26 (2.13)
YA
7B
-
3.19 (2.22) - 0.55 (2.13)
YB
TAB
YAB
YAB,
+
TB
- TAB
+ ZB + TAB
gives the following estimates
1.44 (2.46) 1.62 (4.44) - 2.46 (2.46). -
The relationship between the c.e.f.s and the standard errors of the estimates can be seen here. The variances of 7 A and rABare equal, as are the corresponding c.e.f.s for brick 1. The ratio between the variances of zA and zB is the same as the ratio between the corresponding c.e.f.s, namely 0.67/0.72 (to within rounding error). 5. DISCUSSION
While crossover designs have been used with factorial treatments for some time, there has been little guidance from the design literature until r e ~ e n t l y . ~This - ~ paper has presented a flexible approach, whereby the experimenter can build suitable designs from the given bricks. Thus it is possible to adjust the efficiency on the main effects and interactions, so that effects that are important in a particular application are not estimated inefficiently. The designs have been derived by adapting known types of factorial designs to the crossover setting. This approach ensures that our designs have desirable properties, such as factorial
FACTORIAL DESIGNS FOR CROSSOVER CLINICAL TRIALS
1129
structure. Another possible approach, which we have not considered, would have been to try and impose a factorial treatment structure on a known class of crossover designs, such as the cyclic designs of Davis and Hall.” However, there seems no reason to suppose that this approach would lead to more efficient designs. The efficiencies in Table I1 are calculated assuming that a carryover term is included in the model. It has been suggested that a crossover design should estimate the direct treatment effect efficiently, whether or not a carryover term is included in the model. Although our designs are highly efficient when the carryover effect is omitted, it is in fact unnecessary to insist that designs are efficient in both cases. If it is thought that there may be a carryover effect, then it should be included at the design stage. When it comes to the analysis, this term should be retained because, as Abeyasekera and Curnow7 have shown, it can introduce bias to omit a carryover term simply because it fails to reach statistical significance. If carryover effects can be excluded a priori, then a crossover design is merely an example of a row and column design, and these are available for factorial treatments.’ Our designs are based on the model (1) which is widely used for the analysis of data from crossover trials. In particular this means that only interactions between two direct or two carryover effects are considered. While this model is plausible for many applications, it has been criticizedI2 particularly when the design allows a treatment to follow itself, for example, as with some two-treatment three-period designs. Although no treatment combination follows itself in any of the present designs, a single factor is often kept at the same level over adjacent periods. It would be interesting to investigate different models for carryover, perhaps by including interactions between direct and carryover effects, but this is beyond the scope of the present paper. A common situation to which our designs do not apply occurs when the lowest levels of each factor correspond to no treatment, or possibly a placebo. It may then be unethical to give the 00 treatment; work is in progress to develop a class of designs that do not use this combination.
’
ACKNOWLEDGEMENTS
We are very grateful to Dr. John Lewis and ICI for permission to use the data in the example, to Professor John Thompson for his help and to the referees for their comments.
REFERENCES 1. Fisher, R. A. The Design of Experiments, 1st edn, Oliver and Boyd, Edinburgh, 1935. 2. Jones, B. and Kenward, M. G . Design and Analysis of Cross-Over Trials, Chapman and Hall, London, 1989. 3. Fletcher, D. J. and John, J. A. ‘Changeover designs and factorial structure’, Journal of the Royal Statistical Society, Series B, 47, 117-124 (1985). 4. Fletcher, D. J. ‘A new class of change-over designs for factorial experiments’, Biometrika, 74, 649654 (1987). 5. Lewis, S. M., Fletcher, D. J. and Matthews, J. N . S. ‘Factorial cross-over designs in clinical trials’ in Dodge, Y., Fedorov, V. V. and Wynn, H. P. (eds.), Optimal Design and Analysis of Experiments, North Holland, Amsterdam, 1988. 6. John, J. A. ‘Generalised cyclic designs in factorial experiments’, Biometrika, 60, 55-63 (1973). 7. Abeyasekera, S. and Curnow, R. N. ‘The desirability of adjusting for residual effects in a crossover design’, Biometrics, 40, 1071-1078 (1984). 8. Pearce, S. C. The Agricultural Field Experiment, Wiley, Chichester, 1983. 9. John, J. A. Cyclic Designs, Chapman and Hall, London, 1987. 10. Davis, A. W. and Hall, W. B. ‘Cyclic change-over designs’, Biometrika, 56, 283-293 (1969). 11. John, J. A. and Lewis, S. M. ‘Factorial experiments in generalized cyclic row-column designs’, Journal of the Royal Statistical Society, Series B, 45, 245-251 (1983). 12. Fleiss, J. L. ‘Letter to the Editor’, Biometrics, 42, 449-450 (1986).
