Protocol amendments are often necessary in clinical trials. Sometimes a change in the inclusion and/or exclusion criteria is required. There are various reasons for a change of the inclusion and exclusion criteria, some of them are mentioned in the ICH E9 guideline [1] and by Cleophas et al. [2]. On the one hand, newly emerging medical knowledge can be one reason especially for long-term trials, on the other hand, regular violations of entry criteria and too low recruitment rates could also make changes necessary. In any case, changes of the inclusion/exclusion criteria have to be described in a protocol amendment. Moreover, according to Cleophas et al. [2], the "amendment should also cover any statistical consequences ... and alterations to the planned statistical analysis".

When entry criteria are changed during the trial, the populations before and after the amendment may differ. Chow and Shao [3] presented an example of a placebo-controlled clinical trial in patients with asthma. Because patient enrolment was slow the inclusion criteria were relaxed. To be precise, according to the original protocol, patients with a baseline FEV_{1} (forced expiratory volume in liter per second) between 1.5 and 2.0 could be included. The first amendment extended this range up to an upper bound of 2.5, and the second amendment to an upper bound of 3.0. Please note that the change in FEV_{1}, i.e. the difference between the value after treatment and that at baseline, was the primary endpoint in this study.

Other examples of amendments that change entry criteria were presented [4, 5]. Svolba and Bauer's [4] example is a two-armed, long-term trial that investigated the time until relapse of cutaneous melanoma. In this study an amendment to the protocol increased the inclusion limit for cholesterol. Dubertret et al. [5] report the results of a phase III placebo-controlled trial in patients with moderate-to-severe plaque psoriasis. In this trial efalizumab, a humanized monoclonal antibody, was compared with placebo. During the study the protocol was amended to modify the inclusion criteria.

These examples show that amendments that relax or modify the entry criteria occur in different indications. Moreover, in the examples the entry criteria were changed once or at least twice. In practice, the inclusion and exclusion criteria are rarely changed very often.

As mentioned above, the actual patient population after an amendment may deviate from the originally targeted population [3]. For example, a modification of the acceptable range of baseline values can change the variance of the difference value after treatment minus baseline [6]. Usually, the difference in the populations before and after the amendment is ignored in the statistical analysis. As a consequence, the data are pooled; a procedure that can introduce a bias and can decrease the power of the study, maybe beneath the necessary power which was fixed when planning the study.

Chow and Shao [3] proposed a method that takes the potential differences in the populations before and after an amendment into account. The main idea is to divide the trial data according to the different treatment groups and phases. A new phase is started after each amendment. Thus, when there are *K* amendments *K* + 1 phases result (*K* ≥ 1). For each combination of group (T for treatment and C for control group) and phase (for phases 0, 1,..., *K*), a single value for the endpoint *y* and value(s) the predictor *x* (which may be of dimension greater than 1) are determined by e.g. computing the sample means. In the case of *K* amendments one gets the points
for the control group and
for the treatment group. Then, weighted linear regression analysis should be performed on the points for the control and treatment group, respectively.

As mentioned above, changing inclusion and exclusion criteria can change the target population. A change in the target population, however, can cause a change of the efficacy parameter. Therefore, we have *K* + 1 possibly different null hypotheses, one for each phase. Let
be the population mean of a normally distributed endpoint in phase *i* in the treatment (control) group. Then, the *i*-th null hypothesis is
, *i* = 0, 1,..., *K*. The global null hypothesis is the intersection of the different null hypotheses:
. Note that even if efficacy is the same in all phases under the intersection null hypothesis,
for *i* ≠ *j* is possible as long as
holds for every *i*, i.e. when the mean difference is constant over the phases, the population means may differ between the phases under H_{0}. The variances are assumed to be equal for the two groups within each phase. However, there may be differences in variability between the different phases.

In case of one single amendment the procedure proposed by Chow and Shao [3] reduces to a weighted regression for two data points for the study and control group respectively. If one then decides to model the effect for the original target population (or the amended one) alone the procedure yields the maximum likelihood estimate for the mean for the first (second) population. One would probably not use this estimate to test the intersection hypothesis. For testing the intersection null, one would have to use some other contrast as briefly mentioned by Chow and Shao [3] on page 661.

As an alternative we propose a test procedure based on a combination test. We suggest analysing the subpopulations before and after the change, or, in general, the *K* + 1 subpopulations, separately and then combining the *p*-values of the test statistics. We use Fisher's combination test, one of the methods that can be recommended according to an extensive simulation study [7]. This test uses that, under the null hypothesis, -2log(*p*
_{1} ⋯ *p*
_{
k
}) has a
– distribution if the *p*-values *p*
_{
i
}are independent [8]. When the intersection hypothesis *H*
_{0} is true and the whole study is divided into two or more phases, then each
should be true and the *p*-values *p*
_{
i
}are independent because each patient contributes to one *p*-value only. Furthermore, it is assumed here that the protocol amendment is independent from the *p*-values and hence cannot be based on any type of unblinded data. This limitation is further discussed below.

In practice, it may or may not be appropriate to test only the intersection null hypothesis, depending on the goal of the trial. Rejecting the intersection null hypothesis does not automatically permit to identify the population(s) where the treatment is efficient. In order to justify to which population (= phase) a proof of efficacy refers a multiple testing procedure can be used [9–11]. A multiple testing procedure that controls the multiple level *α* can easily implemented when Fisher's combination test is used as proposed above. In case of one single amendment, when the combination test rejects the intersection hypothesis, those individual null hypotheses
(*i* = 0, 1) can be rejected for which the corresponding test gives a *p*-value not exceeding *α* [12] (page 1034).