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We next discuss the concepts that give rise to early stopping and interim design adaptations. As we explain next, the latter motivate the development of sequential and adaptive designs.

5.2.1 Early stopping

For studies, such as those performed in agriculture, where typically all the results are made simultaneously available after a certain period of time, it is reasonable to analyse data and derive conclusions only after the full dataset is collected. However, in biomedical studies, as well as in industrial applications, where data is accumulated gradually over a period of time, it is more natural to monitor results as observed [Jennison and Turnbull, 2000; Whitehead, 1997].

Especially for clinical trials, not only it is natural, but several administrative, economic and ethical reasons necessitate data monitoring. In this setting, it is im- portant to check that the study is being conducted as planned (for example eligibility criteria are satisfied), while it is also crucial to ensure the safety of the administrated

treatments. Furthermore, for ethical and economic reasons, it is important to take full advantage of the available human and monetary resources. Therefore, it is often advantageous to terminate the study earlier than planned, if this is suggested by interim results. Such early stopping allows the remaining resources to be allocated to another study. Furthermore, if the new treatment is proved unsafe or ineffective, early termination ensures that the remaining subjects can switch to a better treat- ment, while if the new treatment is proved safe and more effective, the time for the treatment to become publicly available is shortened [Jennison and Turnbull, 2000]. The latter ethical and economical issues were the main motivations for ex- tending sequential design methodology initially developed for industrial applications [Wald, 1945] to the medical field [Armitage, 1954, 1958]. However, the use of early sequential designs in medical studies was initially restrained by their demand to analyze data and decide for early stopping continuously, that is, after every ob- servation. Group-sequential design (GSD) methodology developed later by Pocock [1977] and O’Brien and Fleming [1979] enhanced application of sequential designs, particularly in clinical trials. In group-sequential designs, interim analyses are per- formed only after every time a number of observations (group) is collected. This makes them practically less demanding, while most of the benefits of continuous sequential designs are retained [Jennison and Turnbull, 2000].

Interim analysis and early stopping are also permitted under adaptive design methodology. However, as we explain next, the main motivation for the adaptive designs, developed after the work of Bauer and K¨ohne [1994] and Proschan and Hunsberger [1995], was to permit for interim design modifications.

5.2.2 Design modifications

As we briefly discussed above, in the standard fixed single-stage studies, the investi- gators rely entirely on the information available at the planning stage to select the experimental design. The latter defines how the study is to be conducted as well

as how to analyze the results and derive statistical inference. Regarding the study conduct, amongst other things, the investigators define the target population, the number of participating subjects (sample size), the rule for allocating subjects to different treatment groups and the measurements to be taken. Considering statis- tical inference, the primary and secondary hypotheses to be studied are defined as well as the corresponding measures (often called endpoints) and statistical method- ology to be used for their evaluation. For the conclusions of the study to remain valid under the traditional SSD, the study and the subsequent data analysis needs to be conducted following in every step the initially specified design [Armitage et al., 2002; Friedman et al., 2010].

However, in some cases, the information available at the planning stage is not sufficiently precise and reliable to define all aspects of the design. This is a problem arising even when primary design parameters, such as the sample size of the study, are to be determined. For example, to compute the sample size of a clinical trial with normally distributed responses, investigators are typically required to provide an estimate of the response variance as well as the value of the treatment effect described by regulatory authorities as the “minimal effect which has clinical relevance” [ICH, 1998]. However, in some cases, these values cannot be precisely defined, even after the imperative careful planning. A similar problem is also confronted in linear combination tests, in cases where the effect direction is not precisely known at the planning stage. In such circumstances, investigators may be reluctant to design the whole study based on imprecise estimates, as this might be considered as unethical and can proved to be inefficient [Bauer and K¨ohne, 1994; Kirby and Chuang-Stein, 2010].

One approach to this problem is to first perform a small external pilot study. This pilot study can be used to obtain information for various aspects of the design of the main study. In the tests developed in chapter 4, we follow this approach for performing linear combination tests. However, as discussed in section 4.6, this

approach has obvious limitations, especially in cases where only a restricted sample size is available.

A more satisfying solution is provided by adaptive designs as well as various developments of the classical group-sequential designs [Denne and Jennison, 2000; Jennison and Turnbull, 2003; Wittes and Brittain, 1990]. This can potentially miti- gate the ethical and economic issues arising with external pilot studies and provide type I error control under certain design modifications. To deal with the issue of insufficient information at the planning stage, these designs suggest performing an internal rather than an external pilot study. That is, to consider the pilot study as the first stage of a two-stage or, generally, a multi-stage study. The pilot or first- stage data are then used for deriving interim decisions (for example early stopping with rejection/acceptance of the null hypothesis), but they can also be used for re-assessing and possibly modifying various aspects of the design. In a multi-stage design, interim analysis and design modifications can be performed sequentially each time a number of observations are collected.

Such designs can be seen as a method of allowing for modification, at interim analyses, of the initially planned design. Various authors suggest interim design modifications, under the above framework, to deal with new or unexpected results becoming available during the study conduction [Chi et al., 1999; Proschan and Hunsberger, 1995]. Such results may originate from other studies, but it can also arise from the collected data. For example, Chi et al. [1999] describe a single-stage study where at an interim analysis the observed treatment effects were substantially lower than expected but still clinically significant. This suggested an increase in the initially planned sample size as the latter was most likely not sufficient to derive a statistically significant outcome. However, if such a sample size re-calculation was not initially planned, it can be controversial and as various authors have shown (for example Chi et al. [1999]; Proschan and Hunsberger [1995]), if ignored in the final analysis, may substantially inflate the type I error. On the other hand, if the

observed results are ignored and the study continues as planned, the resources of the study might be wasted without reaching a convincing outcome. In the example provided by Chi et al. [1999], the study progressed as originally planned and the final analysis led to a statistically insignificant outcome. The above design framework which permits for interim design modifications provides a solution to these issues.

Note that this design framework conceptually consists of two sequential parts [Brannath et al., 2007]. The design of the first part is fixed, while the design of the second part can be changed based on the information that becomes available at the interim analysis, that is, interim data and possibly external information. Each part may consist of a single-stage study, as in a single group of observations, but, it is not unusual to conduct more than a single stage in the second part.

As we mentioned earlier, several authors proposed methodologies for per- forming interim design modifications within the group-sequential design framework. These approaches are typically characterized by a pre-planned adaptivity, that is, adaptation rules are completely pre-specified at the planning stage. These pre- planned adaptation rules are restricted to ensure that the form of the test statistics remains as in SSD. For example, modifications are often required to be indepen- dent of effect estimates and based solely on nuisance parameter estimates. If design modifications are more flexible, often using the observed effects estimates, adaptive design methodology is required to ensure type I error control.

In the next section, we describe the methodology for performing testing pro- cedures within group-sequential and adaptive designs. We mainly focus on the issues related to the methodology developed in later chapters. Special attention is given to type I error control while power analysis of group sequential and adaptive tests is conducted in chapter 8.