Motivating example

In the late 1980s a major new trial in colorectal cancer was being planned under the auspices of the UK Coordinating Committee on Cancer Research [23]. For patients with rectal cancer, the standard treatment at the time was surgery alone. Two treatments held promise for improvements in survival - peri-operative radiotherapy (XRT), and loco-regional chemotherapy namely portal vein infusion of fluorouracil (FU). There were a number of possible trial designs. Comparing surgery + XRT and surgery + FU directly in a two-arm trial is a simple approach, but one which fails to provide any comparison with standard treatment. Conducting two separate trials, one comparing surgery alone to surgery plus FU, the second comparing surgery alone to surgery + XRT was an alternative. Although simple in approach, having trials running simultaneously which are competing for patients is not a good idea, and may lead to neither achieving sufficient patients to give useful results. Conducting the same two trials sequentially leads to what is likely to be a long delaybefore answers to both questions are obtained, given the need for adequate follow-up. A more attractive option was a three-arm trial, surgery alone versus surgery+XRT versus surgery + FU. This is more efficient, in that the control patients are effectively used twice, once in the comparison with XRT and once in the comparison with FU. It enables both questions to be addressed simultaneously and also allows a direct comparison of surgery + XRT and surgery + FU (if adequately powered). However, perhaps surprisingly, the most efficient design of all is a four-arm trial, including the three arms above plus the fourth in which patients receive both XRT and FU in addition to surgery. The four arms can be thought of as arising by subjecting a single patient to two, simultaneous, randomizations (see Fig. 4.4), hence its other name the 2 x 2 design: in the first they are randomized to radiotherapy or no radiotherapy, in the second they are randomized between chemotherapy and no chemotherapy. Thinking of the design this way shows where the increase in efficiency comes from - now all patients, not just those in the control arm, are used twice; they are randomized twice and so contribute to two questions.

We estimate the benefit to FU by comparing all the patients allocated no FU with all the patients allocated FU. Importantly though, the analysis is stratified by the radiotherapy allocation such that we only directly compare arms that differ only by the addition of FU. Thus arm A would be compared with arm C, and arm B would be compared with arm D. This gives two estimates of the treatment effect which we then combine to give

Fig. 4.4 Factorial randomization in the AXIS trial.

an overall estimate. Similarly, comparing arm A with arm B, and comparing arm C with arm D would assess the impact of radiotherapy. The assumption here is that the effect of adding FU will be approximately the same when it is added to surgery alone, and when it is added to radiotherapy. If this is not the case, then combining the two estimates to give one overall estimate is inappropriate. This is known as an additivity assumption, and is one of the key requirements for a factorial trial discussed below.

General requirements Although these designs are appealing, and should perhaps be considered more, there are three practical considerations which must be met:

♦ it must be practically possible to combine the treatments,

♦ the toxicity of combined treatment must be acceptable,

♦ the anticipated treatment effects must be approximately additive.

With respect to the last point, the analysis of a factorial trial in the manner described above is only appropriate when the individual treatment effects are approximately additive. In the AXIS example, we would say the treatment effects were approximately additive if the estimate of the effect of FU on survival was approximately the same amongst those patients who do and do not receive radiotherapy. Where this is not the case, there is said to be an interaction; the effect of the combined treatments compared with the control group will be very different from the effect that would be expected, based on the comparison of the individual treatment effects. It could be much less than the individual effects, suggesting antagonism between the treatments, or much greater, suggesting synergy. Designing a trial as a factorial design provides the opportunity to investigate evidence for interaction. This will often be of interest, but unfortunately where it exists, the trial must be analysed in a way which loses the efficiency for which the design was chosen. Taking the AXIS example, if we found synergy between radiotherapy and FU, such that the effect of FU in patients receiving radiotherapy in addition to surgery was very much greater than the effect in patients receiving surgery alone, it would be best to present the treatment effect in these subgroups separately. However, the subgroups, having half the number of patients anticipated for the main effect, will have much lower power to detect true treatment differences.

Figure 4.5 illustrates possible sets of survival curves from a factorial trial in which patients are randomized between radiotherapy or no radiotherapy and between chemotherapy or no chemotherapy. In Fig. 4.5(a), two possible scenarios are illustrated in which there is no evidence of interaction. On the left-hand side of the figure, both treatments are effective on their own, and the effect when the two are combined is as expected under the additivity assumption. On the right-hand side of the figure, chemotherapy is effective, but radiotherapy is not; therefore the combined effect of radiotherapy plus chemotherapy compared with control is similar to that of chemotherapy alone versus control.

In Fig. 4.5(b) are examples suggesting interaction. On the left-hand side panel, neither treatment is effective on its own, but rather than having no effect as would be expected under additivity, the combination in fact shows substantially improved survival. On the right-hand side panel the opposite applies. Each treatment is, on its own, effective, but the combination does no better than the individual treatments. This might be seen for example if two different treatments are effectively attacking the same target.

Fig. 4.5 (a) Hypothetical factorial trials with no interaction. (b) Hypothetical factorial trials with interaction.

------No CT, No RT ---No CT, RT - CT, No RT - CT + RT

Fig. 4.5 (a) Hypothetical factorial trials with no interaction. (b) Hypothetical factorial trials with interaction.

AXIS was considered an appropriate setting for a factorial trial. FU was given directly into the hepatic vein, and was primarily aimed at preventing liver metastases. Radiotherapy was given either pre-operatively, before any chemotherapy, or 4-6 weeks post-operatively after completion of radiotherapy with the primary aim of preventing local recurrence. Therefore it was practical to combine treatments, there was no expectation of enhanced toxicity, and it was considered clinically unlikely that the treatments would interact.

In general, where there is a reasonable expectation that two treatments may interact, a factorial design may still be appropriate, but the sample size should be calculated to anticipate the possibility, and to provide sufficient power for treatment comparisons in subgroups if necessary.

Because each patient is 'used' twice, the sample size required for a factorial design with two treatments is very similar, or even identical, to that for a simple 2-arm trial. This economy can be extended further when there are three or more (say, n) treatments of interest; again each patient is randomized to receive or not receive each treatment and consequently can be allocated any one of 2n treatment combinations.

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