Sample size requirements to estimate key design parameters from external pilot randomised controlled trials: a simulation study
 M Dawn Teare^{1}Email author,
 Munyaradzi Dimairo^{1},
 Neil Shephard^{1},
 Alex Hayman^{1},
 Amy Whitehead^{1} and
 Stephen J Walters^{1}
DOI: 10.1186/1745621515264
© Teare et al.; licensee BioMed Central Ltd. 2014
Received: 13 December 2013
Accepted: 20 June 2014
Published: 3 July 2014
Abstract
Background
External pilot or feasibility studies can be used to estimate key unknown parameters to inform the design of the definitive randomised controlled trial (RCT). However, there is little consensus on how large pilot studies need to be, and some suggest inflating estimates to adjust for the lack of precision when planning the definitive RCT.
Methods
We use a simulation approach to illustrate the sampling distribution of the standard deviation for continuous outcomes and the event rate for binary outcomes. We present the impact of increasing the pilot sample size on the precision and bias of these estimates, and predicted power under three realistic scenarios. We also illustrate the consequences of using a confidence interval argument to inflate estimates so the required power is achieved with a prespecified level of confidence. We limit our attention to external pilot and feasibility studies prior to a twoparallelbalancedgroup superiority RCT.
Results
For normally distributed outcomes, the relative gain in precision of the pooled standard deviation (SD_{ p }) is less than 10% (for each five subjects added per group) once the total sample size is 70. For true proportions between 0.1 and 0.5, we find the gain in precision for each five subjects added to the pilot sample is less than 5% once the sample size is 60. Adjusting the required sample sizes for the imprecision in the pilot study estimates can result in excessively large definitive RCTs and also requires a pilot sample size of 60 to 90 for the true effect sizes considered here.
Conclusions
We recommend that an external pilot study has at least 70 measured subjects (35 per group) when estimating the SD_{ p } for a continuous outcome. If the event rate in an intervention group needs to be estimated by the pilot then a total of 60 to 100 subjects is required. Hence if the primary outcome is binary a total of at least 120 subjects (60 in each group) may be required in the pilot trial. It is very much more efficient to use a larger pilot study, than to guard against the lack of precision by using inflated estimates.
Keywords
sample size feasibility studies pilot studies binary outcomes continuous outcomes, RCTsBackground
In 2012/13, the National Institute for Health Research (NIHR) funded £208.9 million of research grants across a broad range of programmes and initiatives to ensure that patients and the public benefit from the most costeffective uptodate health interventions and treatments as quickly as possible [1]. A substantial proportion of these research grants were randomised controlled trials (RCTs) to assess the clinical effectiveness and costeffectiveness of new health technologies. Welldesigned RCTs are widely regarded as the least biased research design for evaluating new health technologies and decisionmakers, such as the National Institute for Health and Care Excellence (NICE), are increasingly looking to the results of RCTs to guide practice and policy.
RCTs aim to provide precise estimates of treatment effects and therefore need to be well designed to have good power to answer specific clinically important questions. Both overpowered and underpowered trials are undesirable and each poses different ethical, statistical and practical problems. Good trial design requires the magnitude of the clinically important effect size to be stated in advance. However, some knowledge of the population variation of the outcome or the event rate in the control group is necessary before a robust sample size calculation can be done. If the outcome is well established, these key population or control parameters can be estimated from previous studies (RCTs or cohort studies) or through metaanalyses. However, in some cases finding robust estimates can pose quite a challenge if reliable data, for the proposed trial population under investigation, do not already exist.
A systematic review of published RCTs with continuous outcomes found evidence that the population variation was underestimated (in 80% of reported endpoints) in the sample size calculations compared to the variation observed when the trial was completed [2]. This study also found that 25% of studies were vastly underpowered and would have needed five times the sample size if the variation observed in the trial had been used in the sample size calculation. A more recent review of trials with both binary and continuous outcomes [3] found that there was a 50% chance of underestimating key parameters. However, they too found large differences between the estimates used in the sample size calculation compared to the estimates derived from the definitive trial. This suggests that many RCTs are indeed substantially underpowered or overpowered. A systematic review of RCT proposals reaching research ethics committees [4] found more than half of the studies included did not report the basis for the assumed values of the population parameters. So the values assumed for the key population parameters may be the weakest part of the RCT design.
A frequently reported problem with publicly funded RCTs is that the recruitment of participants is often slower or more difficult than expected, with many trials failing to reach their planned sample size within the originally envisaged trial timescale and trialfunding envelope. A review of a cohort of 122 trials funded by the United Kingdom (UK) Medical Research Council and the NIHR Health Technology Assessment programme found that less than a third (31%) of the trials achieved their original patient recruitment target, 55/122 (45.1%) achieved less than 80% of their original target and half (53%) were awarded an extension [5]. Similar findings were reported in a recently updated review [6]. Thus, many trials appear to have unrealistic recruitment rates. Trials that do not recruit to the target sample size within the time frame allowed will have reduced power to detect the prespecified target effect size.
Thus the success of definitive RCTs is mainly dependent on the availability of robust information to inform the design. A welldesigned, conducted and analysed pilot or feasibility trial can help inform the design of the definitive trial and increase the likelihood of the definitive trial achieving its aims and objectives. There is some confusion about terminology and what is a feasibility study and what is a pilot study. UK public funding bodies within the NIHR portfolio have agreed definitions for pilot and feasibility studies [7]. Other authors have argued against the use of the term ‘feasibility’ and distinguish three types of preclinical trial work [8].
Distinguishing features of pilot and feasibility studies
NIHR guidance states:
 i)
standard deviation of the outcome measure, which is needed in some cases to estimate sample size;
 ii)
willingness of participants to be randomised;
 iii)
willingness of clinicians to recruit participants;
 iv)
number of eligible patients over a specific time frame;
 v)
characteristics of the proposed outcome measure and in some cases feasibility studies might involve designing a suitable outcome measure;
 vi)
followup rates, response rates to questionnaires, adherence/compliance rates, intracluster correlation coefficients in cluster trials, etc.
Feasibility studies for randomised controlled trials may themselves not be randomised. Crucially, feasibility studies do not evaluate the outcome of interest; that is left to the main study.
If a feasibility study is a small RCT, it need not have a primary outcome and the usual sort of power calculation is not normally undertaken. Instead the sample size should be adequate to estimate the critical parameters (e.g. recruitment rate) to the necessary degree of precision.
Pilot trials are a version of the main study that is run in miniature to test whether the components of the main study can all work together[9]. It will therefore resemble the main study in many respects, including an assessment of the primary outcome. In some cases this will be the first phase of the substantive study and data from the pilot phase may contribute to the final analysis; referred to as an internal pilot. Or at the end of the pilot study the data may be analysed and set aside, a socalled external pilot[10].
For the purposes of this paper we will use the term pilot study to refer to the pilot work conducted to estimate key parameters for the design of the definitive trial. There is extensive but separate literature on twostage RCT designs using an internal pilot study [11–14].
There is disagreement over what sample size should be used for pilot trials to inform the design of definitive RCTs [15–18]. Some recommendations have been developed although there is no consensus on the matter. Furthermore, the majority of the recommendations focus on estimating the variability of a continuous outcome and relatively little attention is paid to binary outcomes. The disagreement stems from two competing pressures. Small studies can be imprecise and biased (as defined here by comparing the median of the sampling distribution to the true population value), so larger sample sizes are required to reduce both the magnitude of the bias and the imprecision. However, in general participants measured in an external pilot or feasibility trial do not contribute to the estimation of the treatment effect in the final trial, so our aim should be to maintain adequate power while keeping the total number of subjects studied to a minimum. Recently some authors have promoted the practice of taking account of the imprecision in the estimate of the variance for a continuous outcome. Several suggest the use of a onesided confidence interval approach to guarantee that power is at least what is required more than 50% of the time [15, 18, 19].
This paper aims to provide recommendations and guidelines with respect to two considerations. Firstly, what is the number of subjects required in an external pilot RCT to estimate the uncertain critical parameters (SD for continuous outcomes; and consent rates, event rates and attrition rates for binary outcomes) needed to inform the design of the definitive RCT with a reasonable degree of precision? Secondly, how should these estimates from the pilot study be used to inform the sample size (and design) for the definitive RCT? We shall assume that the pilot study (and the definitive RCT) is a twoparallelbalancedgroup superiority trial of a new treatment versus control.
For the purposes of this work we assume that the sample size of the definitive RCT is calculated using a level of significance and power argument. This is the approach that is currently commonly employed in RCTs; however, alternative methods to calculate sample size have been proposed, such as using the width of confidence intervals [20] and Bayesian approaches to allow for uncertainty [21–23].
Methods
Our aim is to demonstrate the variation in estimates of population parameters taken from small studies. Though the sampling distributions of these parameters are well understood from statistical theory, we have chosen to present the behaviours of the distributions through simulation rather than through the theoretical arguments as the visual representation of the resulting distributions makes the results accessible to a wider audience.
Randomisation is not a necessary condition for estimating all parameters of interest. However, it should be noted that some parameters of interest during the feasibility phase are related to the randomisation procedure itself, such as the rate of willingness to be randomised, and the rate of retention or dropout in each randomised arm. In addition, randomisation ensures the equal distribution of known and unknown covariates on average across the randomised groups. This ensures that we can estimate parameters within arms without the need to worry about confounding factors. In this work we therefore decided to allow for the randomisation of participants to mimic the general setting for estimating all parameters, although it is acknowledged that some parameters are independent of randomisation.
We first consider a normally distributed outcome measured in two groups of equal size. We considered study groups of from 10 to 80 subjects using increments of five per group. For each pilot study size, 10,000 simulations were performed. Without loss of generality, we assumed the true population mean of the outcome is 0 and the true population variance is 1 (and that these are the same in the intervention and control groups). We then use the estimate of the SD, along with other information, such as the minimum clinically important difference in outcomes between groups, and Type I and Type II errors levels, to calculate the required sample size (using the significance thresholds approach) for the definitive RCT.
The target difference or effect size that is regarded as the minimum clinically important difference is usually the difference in the means when comparing continuous outcomes for the intervention with those of the control group. This difference is then converted to a standardised effect size by dividing by the population SD. More details of the statistical hypothesis testing framework in RCTs can be found in the literature [24, 25].
For a twogroup pilot RCT we can use the SD estimate from the new treatment group or the control/usual care group or combine the two SD estimates from the two groups and use a pooled standard deviation (SD_{ p }) estimated from the twogroup specific sample SDs. For sample size calculations, we generally assume the variability of the outcome is the same or equal in both groups, although this assumption can be relaxed and methods are available for calculating sample sizes assuming unequal SDs in each group [26, 27]. This is analogous to using the standard ttest with two independent samples (or multiple linear regression), which assumes equal variances, to analyse the outcome data compared with using versions of the ttest that do not assume equal variances (e.g. Satterthwaite’s or Welch’s correction).
We assume binary outcomes are binomially distributed and consider a number of different true population proportions as the variation of proportion estimator is a function of the true proportion. When estimating an event rate, it may not always be appropriate to pool the two arms of the study so we study the impact of estimating a proportion from a single arm where the study size increases in steps of five subjects. We considered true proportions in the range 0.1 to 0.5 in increments of 0.05. For each scenario and sample size, we simulated the feasibility study at least 10,000 times depending on the assumed true proportion. For the binary outcomes, the number of simulations was determined by requiring the proportion to be estimated within a standard error of 0.001. Hence, the largest number of simulations required was 250,000 when the true proportion was equal to 0.5. Simulations were performed in Stata version 12.1 [28] and R version 13.2 [29].
Normally distributed outcomes
To quantify the relative change in precision, we compared the average width of the 95% confidence intervals (WCI_{2n }) for the SD_{ p } for study sizes of 2n with the average width when the study size was increased to 2(n + 5). We use the width of the confidence interval as this provides a measure of the precision of the estimate.
Bias is assessed by subtracting the true value from each estimate and taking the mean of these differences.
To consider the impact on power and planned sample size, we need to state reasonable specific alternative hypotheses. In trials, it is uncommon to see large differences between treatments so we considered small to medium standardised effect sizes (differences between the group means) of 0.2, 0.35 and 0.5 [30]. For each true effect size of 0.2, 0.35 or 0.5, we divide by the SD_{ p } estimate for each replicate, and use this value to calculate the required sample size. For each simulated pilot study, we calculate the planned sample size for the RCT assuming either the unadjusted or adjusted SD_{ p } estimated from the pilot. Using this planned sample size (where the SD_{ p } has been estimated) we then calculate the true power of the planned study assuming that we know that the true population SD_{ p } is in fact 1.
Binary outcomes
As for the continuous outcomes, bias is assessed by subtracting the true population value from each estimate and taking the signed mean of these. We also report the 95% coverage probability [32].
Results and discussion
Normally distributed outcomes
Binary outcomes
Conclusions
Our simulated data visually demonstrate the large sampling variation that is the main weakness when estimating key parameters from small sample sizes. Small samples sizes do lead to biased estimates, but the bias is negligible compared to the sampling variation. When we examine the relative percentage gain in precision by adding more subjects to the sample, our data suggest that a total of at least 70 may be necessary for estimating the standard deviation of a normally distributed variable with good precision, and 60 to 100 subjects in a single group for estimating an event rate seems reasonable. Treatmentindependent parameters may be estimated by pooling the two groups, so in many cases our recommended sample size will be the total sample size. On average when the definitive RCT is planned using an estimate from a pilot study there will be a tendency for the planned study to be underpowered. However, if the definitive RCT is planned for a continuous outcome requiring a power of 90% then the true power will be 80% with at least 76% assurance provided the estimates come from a pilot with at least 20 subjects. We considered three realistic effect sizes of 0.2, 0.35 and 0.5 of a standard deviation to evaluate the impact of adjusting for the anticipated uncertainty in the estimate from the pilot when calculating the sample size for the planned RCT as was recently suggested [18]. For all of the effect sizes considered, it is not efficient to use small pilots and apply the inflation adjustment, as this will result in larger sample sizes (pilot plus main study) in total. Further, we only considered sample sizes planned when requiring 90% power, and examine the conditional power assuming we know the true alternative. On average using imprecise estimates but requiring high power will result in acceptable power with much less ‘cost’ as measured by total sample size. Hence, it is actually more efficient to use a large external pilot study to reduce the variation around the target power for the definitive RCT.
The implication of using estimates of key parameters from small pilot studies is the risk of both over and underpowered studies. While overpowered studies may not seem such an acute problem, they are potentially a costly mistake and may result in a study being judged as prohibitively large. This would seem to be an argument in favour of utilising internal pilot studies, but an internal pilot requires the key design features of the trial to be fixed, so any change in measurement of the treatment effect following an internal pilot will lead to analysis difficulties.
A major and welldocumented problem with published trials is under recruitment, where there is a tendency to recruit fewer subjects than targeted. One reason for under recruitment may well be that event rates such as recruitment and willingness to be randomised cannot be accurately estimated from small pilots, and in fact increasing the pilot size to between 60 and 100 per group may give much more reliable data on the critical recruitment parameters.
In reality, when designing external pilot trials, there is a need to balance two competing issues: maximising the precision (of the critical parameters you wish to estimate) and minimising the size of the external pilot trial, which impacts on resources, time and costs. Thus there is a tradeoff between the precision (of the estimates of the critical parameters) and size (number of subjects) of the pilot study. When designing external pilot trials, researchers need to understand that they are trading off the precision of the estimates against the total sample size of the definitive study when they decide to have an external pilot study with a small sample size.
Abbreviations
 NICE:

National Institute for Health and Care Excellence
 NIHR:

National Institute for Health Research
 RCT:

randomised control trial
 SD:

standard deviation
 UK:

United Kingdom.
Declarations
Acknowledgements
MDT, SJW, AW and NS are funded by the University of Sheffield. MD is fully funded by NIHR as part of a doctoral research fellowship (DRF201205182). AH was funded by NIHRResearch Design Service and the University of Sheffield. The views expressed are those of the authors and not necessarily those of the National Health Service, the NIHR, the Department of Health or organisations affiliated to or funding them.
The authors thank the three reviewers for their detailed critical comments, which substantially improved the manuscript. We also thank members of the Medical Statistics Group at the School of Health and Related Research, University of Sheffield, for constructive discussions and input to the project. We acknowledge the University of Sheffield for supporting this research.
Authors’ Affiliations
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