Two-trials rule

The two trial rule, for example in a placebo controlled setting, effectively translates into a very stringent requirement for statistical significance. In a single trial the conventional two-sided type I error rate is 0.05. It follows that in order... 

Both rules are illustrated in Figure 12.1, which shows the portion of the space for the z-statistics from the two trials which leads to overall significance. For the two-trials rule, it is the area of the graph to the right of the vertical dashed line and also above the horizontal dashed line. For the pooled-trials rule it is the area to north-east of the solid diagonal line. It can be seen that for this rule, large values of the z-statistic for one trial can compensate for values in the other. 

The question then arises. Why should we prefer the two-trial rule to the pooled-trial rule, especially since, as can be demonstrated, the latter is more powerful or, for a given... 

Figure 12.1 Illustration of the two-trials rule and the pooled-trials rule.

As explained in Chapter 12, however, the two-trials rule is not particularly logical. The alternative pooled-trials rule would actually require only 4/5 of the number of patients of the two-trials rule for an overall power of 80% and an overall size of 1/1600, which is what, effectively, the two-trials rule implies. 

An efficient meta-analysis is supposed to summarize and use all the information from a series of trials. If this claim is true, then once the meta-analysis is available the results from the individual trials are no longer relevant. Suppose now that we have two meta-analyses each based on two trials and that the P-values in each case are highly significant and that the point estimates and confidence intervals are identical. Does it make any difference to know that in the one case (case A) both contributing trials were just significant and that in the second case (case B) one trial was more highly significant but the other not quite. If not, then the two-trials rule is superfluous. 

It would thus seem that the sort of serious and life-threatening diseases where sequential trials are run ought to be an exception to the two-trials rule. However, this raises a problem with the standard of evidence. As we saw in Chapter 12, one interpretation of the two-trials rule is that it reflects the fact that higher standards of evidence are required in practice than suggested by the conventional 5% significance level. If we do not have a two-trials rule, then it simply means that our standards of evidence are lower for the sorts of indications in which sequential trials are run than for other areas of drug development. This may seem to be appropriate for serious diseases, where more is at stake and we intuitively feel that it is unreasonable to carry on in the search for proof of efficacy where belief is already strong. 

A discussion of the two pivotal trial rule and under what conditions sponsors may be allowed to deviate from that requirement is included in the CPMP (2001) Points to Consider on Application with 1. Meta-Analysis 2. One Pivotal Study paper that covers meta-analysis, as there are some common issues. 

Now, if the reason for this rule is simply that we want a type I error rate of 1/1600, we can, in fact, replace it by a more efficient one. This is (assuming the trials are of equal precision) to require that the mean of the two z-statistics is greater than 2.28 (or, equivalently to require that their sum is greater than 4.56). This mean will have a variance of 1/2 and hence a standard error of l/V, and hence the standardized value of the Normal distribution corresponding to the critical value of 2.28 for the mean is 2.28/(l/.y/2) = 2.28 x a 2 = 3.23 which corresponds to a tail area of the Normal distribution of 1/1600. Thus, this test also has the required type I error rate. Let us call a requirement that this test be significant the pooled- trials rule. 

Protocol rules, also known as business rules, are the rules that reflect the trial design. For examples, a protocol rule states that up to 99 interim visits can be scheduled between two protocol visits, visits may occur before or after their expected dates given a predefined leeway, and if the trial has multiple... 

From the Descartes rule of signs, since there is one change in the sign of the coefficients in (C), there is only one positive real root. (The same rule applied to - fA in (C) indicates that there may be two negative real roots for fA, but these are not allowable values.) Solution of (C) by trial or by means of the E-Z Solve software (file exl4-6.msp) gives... 

The proposed sample size was 600 patients and two interims were planned after 200 and 400 patients (completing 3 months follow-up) using the O Brien and Fleming scheme with adjusted two-sided significance levels of 0.00052, 0.014 and 0.045. A futility rule was also introduced, based on conditional power (under the current trend) being below 30 per cent for the trial to be stopped. 

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