Background

Properly conducted, the RCT is generally considered to be the gold standard for assessing the comparative clinical efficacy and effectiveness of healthcare interventions, as well as providing a key source of data for estimating cost-effectiveness [1]. These trials are routinely used to evaluate a wide range of treatments and have been successfully used in a variety of health and social care settings. Central to the design of a RCT is an a-priori sample size calculation, which ensures the study has a high probability of achieving its pre-specified objectives.

The difference between groups used to calculate a sample size for the trial, the “target difference”, is the magnitude of difference in the outcome of interest that the RCT is designed to reliably detect. Reassurance in this regard is typically confirmed by having a sample size which has a sufficiently high level of statistical power (typically 80 or 90%) for detecting a difference as big as the target difference, while setting the statistical significance at the level planned for the statistical analysis (usually this is the 2-sided 5% level). A comprehensive methodological review conducted by the original DELTA (Difference ELicitation in TriAls) group [2, 3] highlighted the available methods and limitations in current practice. It showed that despite there being many different approaches available, some are used only rarely in practice [4]. The initial DELTA guidance does not fully meet the needs of funders and researchers. The overall aim of the DELTA2 project, commissioned by the UK Medical Research Council (MRC)/National Institute for Health Research (NIHR) Methodology Research Programme (MRP), and described here, was to produce updated guidance for researchers and funders on specifying and reporting the target difference (“effect size”) in the sample size calculation of a RCT. In this article, we summarise the process of developing the new guidance, as well as the relevant considerations, key messages and recommendations for determining and reporting a RCT’s sample size calculation (Tables 1 and 2). This article on choosing the target difference for a randomised controlled trial (RCT) and undertaking and reporting the sample size calculatio has been dak published in tge BMJ and BMC Trials journals.

Table 1 DELTA2 recommendations undertaking a sample size calculation and choosing the target difference for a RCT
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Table 2 DELTA2 recommended reporting items for the sample size calculation of a RCT with a superiority question
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Development of the DELTA2 guidance

The DELTA2 guidance is the culmination of a five stage process to meet the stated project objectives (see Fig. 1) which included two literature reviews of existing funder guidance and recent methodological literature, a Delphi process to engage with a wider group of stakeholders, a 2 day workshop and finalising the core guidance.

Fig. 1
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DELTA2 project components of work

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The literature review was conducted between April and December 2016 (searching up to April 2016). The Delphi study had two rounds: one held in 2016 before a two-day workshop in Oxford (September 2016) and another between August and November 2017. The general structure of the guidance was devised at the workshop. It was substantially revised based upon feedback from stakeholders received through the Delphi study. In addition, stakeholder engagement events were held at various meetings throughout the development of the guidance: the Society for Clinical Trials (SCT) meeting, and Statisticians in the Pharmaceutical Industry (PSI) conferences both in May 2017, Joint Statistical Meeting (JSM) in August 2017 and a Royal Statistical Society (RSS) Reading local group meeting in September 2017. These interactive sessions provided feedback on the scope (in 2016) and then draft guidance (in 2017). The core guidance was provisionally finalised in October 2017 and reviewed by the funders’ representatives for comment (MRP advisory group). The guidance was further revised and finalised in February 2018. The full guidance document incorporating case studies and relevant appendices is available here [5]. Further details on the findings of the Delphi study and the wider engagement with stakeholders are reported elsewhere [6]. The guidance and key messages are summarised in the remainder of the paper.

The target difference and sample size calculations in RCTs

The role of the sample size calculation is to determine how many patients are required for the planned analysis of the primary outcome to be informative. It is typically achieved by specifying a target difference for the key (primary) outcome which can be reliably detected and the required sample size calculated. In this summary paper we restrict considerations to the most common trial design looking at a superiority question (one which assumes no difference and looks for a difference), although the full guidance considers equivalence and non-inferiority designs which invert the hypothesis and how the use of the target difference differs for such designs [5].

The precise research question that the trial is primarily set up to answer will determine what needs to be estimated in the planned primary analysis, this is known formally as the ‘estimand’. A key part of deciding this is choosing the primary outcome, which requires careful consideration. The target difference should be a difference that is appropriate for that estimand [7,8,9,10]. Typically (for superiority trials), an “intention to treat” or treatment policy estimand - that is, according to the randomised groups irrespective of subsequent compliance with the treatment allocation - is used. Other analyses that address different estimands [8, 9, 11] of interest (e.g. those based on the effect upon receipt of treatment and the absence of non-compliance) could also inform the choice of sample size. Different stakeholders can have somewhat differing perspectives on the appropriate target difference [12]. However, a key principle is that the target difference should be one that would be viewed as important by at least one (and preferably more) key stakeholder groups that is, patients, health professionals, regulatory agencies, and healthcare funders. In practice, the target difference is not always formally considered and in many cases appears, at least from trial reports, to be determined upon convenience, the research budget, or some other informal basis [13]. The target difference can be expressed as an absolute difference (e.g., mean difference or difference in proportions) or a relative difference (e.g., hazard or risk ratio), and it is also often referred to, rather imprecisely, as the trial “effect size”.

Statistical calculation of the sample size is far from an exact science [14]. Firstly, investigators typically make assumptions that are a simplification of the anticipated analysis. For example, the impact of adjusting for baseline factors is very difficult to quantify upfront, and even though the analysis is intended to be an adjusted one (such as when randomisation has been stratified or minimised), [15] the sample size calculation is often conducted based on an unadjusted analysis. Secondly, the calculated sample size can be sensitive to the assumptions made in the calculations such that a small change in one of the assumptions can lead to substantial change in the calculated sample size. Often a simple formula can be used to calculate the required sample size. The formula varies according to the type of outcome, how the target difference is expressed (e.g. a risk ratio versus a difference in proportions), and somewhat implicitly the design of the trial and the planned analysis. Typically, a sample size formula can be used to calculate the required number of observations in the analysis set, which varies depending on the outcome and the intended analysis. In some situations, ensuring the sample size is sufficient for more than one planned analysis may be appropriate.

When deciding upon the sample size for a RCT, it is necessary to balance the risk of incorrectly concluding there is a difference when no actual difference between the treatments exists, with the risk of failing to identify a meaningful treatment difference when the treatments do differ. Under the conventional approach, referred to as the statistical hypothesis testing framework [16], the probabilities of these two errors are controlled by setting the significance level (Type I error) and statistical power (1 minus Type II error) at appropriate levels (typical values are 2 sided 5% significance and 80% or 90% power respectively). Once these two inputs have been set, the sample size can be determined given the magnitude of the between group difference in the outcome it is desired to detect (the target difference). The calculation (reflecting the intended analysis) is conventionally done on the basis of testing for a difference of any magnitude. As a consequence, it is essential when interpreting the analysis of a trial to consider the uncertainty in the estimate, which is reflected in the confidence interval. A key question of interest is what magnitude of difference can be ruled out. The expected (predicted) width of the confidence interval can be determined for a given target difference and sample size calculation which is a helpful further aid in making an informed choice about this part of a trial’s design [17]. Other statistical and economic approaches to calculating the sample size have been proposed such as precision and Bayesian based approaches, [16, 18,19,20,20] and value of information analysis, [21] though they are not at present commonly applied [22].

The required sample size is very sensitive to the target difference. Under the conventional approach, halving the target difference quadruples the sample size for a two arm 1:1 parallel group superiority trial with a continuous outcome [23]. Appropriate sample size formulae vary depending upon the proposed trial design and statistical analysis, although the overall approach is consistent. In more complex scenarios, simulations may be used but the same general principles hold. It is prudent to undertake sensitivity calculations to assess the potential effect of misspecification of key assumptions (such as the control response rate for a binary outcome or the anticipated variance of a continuous outcome).

The sample size calculation and the target difference, if well specified, help provide reassurance that the trial is likely to detect a difference at least as large as the target difference in terms of comparing the primary outcome between treatments. Failure to clarify sufficiently what is important and realistic at the design stage can lead to subsequent sample size revisions, an unnecessarily inconclusive trial due to lack of statistical precision, or to ambiguous interpretation of the findings [24, 25]. When specifying the target difference with a definitive trial in mind, the following guidance should be considered.

Specifying the target difference for a randomised controlled trial

Different statistical approaches can be taken to specify the target difference and calculate the sample size but the general principles are the same. To aid those new to the topic and to encourage better practice and reporting regarding the specification of the target difference for a RCT, a series of recommendations is provided in Tables 1 and 2. Seven broad types of methods can be used to justify the choice of a particular value as the target difference: these are summarised in Table 3.

Table 3 Methods that can be used to inform the choice of the target difference
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Broadly speaking, two different approaches can be taken to specify the target difference for a RCT. A difference that is considered to be:

  • important to one or more stakeholder groups

  • realistic (plausible), based on either existing evidence, or expert opinion.

A very large literature exists on defining and justifying a (clinically) important difference, particularly for quality of life outcomes [26,27,28]. In a similar manner, discussions of the relevance of estimates from existing studies are also common; there are a number of potential pitfalls to their use, which requires careful consideration of how they should inform the choice of the target difference [2]. It has been argued that a target difference should always be both important and realistic [29], which would seem particularly apt when designing a definitive (Phase III) superiority RCT. In a sample size calculation for a RCT, the target difference between the treatment groups, strictly relates to a group level difference for the anticipated study population. However, the difference in an outcome that is important to an individual might differ from the corresponding value at the population level. More extensive consideration of the variations in approach is provided elsewhere [3, 30].

Reporting the sample size calculation

The approach taken when determining the sample size and the assumptions made should be clearly specified. This information should include all the inputs and formula or simulation results, so that it is clear what the sample size was based upon. This information is critical for reporting transparency, allows the sample size calculation to be replicated, and clarifies the primary (statistical) aim of the study. Under the conventional approach with a standard (1:1 allocation two arm parallel group superiority) trial design and unadjusted statistical analysis, the core items needed to be stated are the primary outcome, the target difference appropriately specified according to the outcome type, the associated “nuisance” parameter (that is, a parameter that, together with the target difference, uniquely specifies the difference on the original outcome scale—eg, the event rate in the control group for a binary primary outcome), and the statistical significance and power. More complicated designs can have additional inputs that also need considered, like the intra-cluster correlation for a cluster randomised design.

A set of core items should be reported in all key trial documents (grant applications, protocols and main results papers) to ensure reproducibility and plausibility of the sample size calculation. The full list of recommended core items are given in Table 2 which is an update of the previously-proposed list [31]. When the sample size calculation deviates from the conventional approach, whether by research question or statistical framework, the core reporting set may be modified to provide sufficient detail to ensure the sample size calculation is reproducible and the rationale for choosing the target difference is transparent. However, the key principles remain the same. If the sample size is determined based upon a series of simulations, this would need to be described in sufficient detail to enable equivalent level of transparency and assessment. Additional items to give more explanation of the rationale should be provided where space allows (e.g. grant applications and trial protocols). Trial result publications can then reference these documents if sufficient space is not available to provide a full description.

Discussion

Researchers are faced with a number of difficult decisions when designing a RCT, the most important of which are the choice of trial design, primary outcome and sample size. The latter is largely driven by the choice of the target difference, although other aspects of sample size determination also contribute.

The DELTA2 guidance provides help on specifying a target difference and undertaking and reporting the sample size calculation for a RCT. The guidance was developed in response to a growing recognition from funders, researchers, as well as other key stakeholders (such as patients and the respective clinical communities) of a real need for practical and accessible advice to inform a difficult decision. The new guidance document therefore aims to bridge the gap between the existing (limited) guidance and this growing need.

The key message for researchers is the need to be more explicit about the rationale and justification of the target difference when undertaking and reporting a sample size calculation. Increasing focus is being placed upon the target difference in the clinical interpretation of the trial result, whether statistically significant or not. Therefore the specification and reporting of the target difference, and other aspects of the sample size calculatio, needs to be improved.