A hypothesis test of feasibility for external pilot trials assessing recruitment, follow-up and adherence rates

Amanda J. Farrin; Duncan T. Wilson; Julia Brown; Rebecca E. A. Walwyn

REVIEW 2 major objections 4 minor 28 references

A hypothesis test of feasibility for external pilot trials assessing recruitment, follow-up and adherence rates

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proposes that an external pilot trial's go/no-go decision be a formal hypothesis test on recruitment, follow-up and adherence, with type I and II error rates controlled by pilot sample size and critical value.

desk verdict A genuinely useful new framework for pilot trial progression decisions, but the advertised error rates rest on an unproven boundary assumption and a heuristic optimizer, so treat the numbers with caution. read the letter →

arxiv 1908.05562 v1 pith:Q3AILATL submitted 2019-08-15 stat.ME

Duncan T. Wilson , Rebecca E. A. Walwyn , Julia Brown , Amanda J. Farrin This is my paper · ORCID

classification stat.ME MSC 62F0362P10

keywords externalpilottrialprogressioncriteriahypothesistestdefinitivepowerrecruitmentratefollow-upadherencesamplesize

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

External pilot trials estimate recruitment, follow-up and adherence rates to decide whether a planned definitive trial is feasible. This paper argues that the go/no-go decision should be a formal hypothesis test rather than a checklist of independent progression criteria. Feasibility is defined by the statistical power the definitive trial would have under the true rates, and the pilot's estimates are condensed into a single test statistic. By choosing the pilot sample size and critical value, the researcher controls explicit type I and type II error rates, something the conventional checklist does not do. In the paper's analyses, conventional progression criteria perform no better than a coin toss, while the formal test gives reasonable error rates at around 50 participants per arm.

What carries the argument

The central object is the function $x(\varphi)=\varphi_a\mu\sqrt{\varphi_f E[N\mid\varphi_r]}/\sqrt{4\sigma^2+2\mu^2\varphi_a(1-\varphi_a)}$, the standardized signal that determines definitive-trial power, where $E[N\mid\varphi_r]$ is the expected number of participants recruited under the definitive design. The method uses $x(\hat\varphi)$ as the pilot test statistic, so hypotheses are level sets of $x$ and the decision is 'go' when $x(\hat\varphi)>c$. The sampling machinery is the pilot power function $h(n_p,c,\varphi)=\Pr[x(\hat\varphi)>c\mid n_p,\varphi]$, built from a negative-binomial distribution for recruitment refusals and binomial or multinomial distributions for follow-up and adherence; error rates are the maxima of $h$ and $1-h$ over null and alternative regions. That reduction turns the choice of pilot sample size and critical value into a multi-objective optimization problem rather than a rule of thumb.

What would settle it

For the TIGA-CUB setting with $n_p=50$ and $c=2.6422$, evaluate $h(n_p,c,\varphi)$ at parameter triples strictly inside the null region $\Phi_0$ rather than only on the boundary $x(\varphi)=x_0$; if any interior triple gives a go probability larger than the reported type I error, the claimed error control is not guaranteed.

Watch

Extended reading notes

Core claim

The paper claims that an external pilot trial's progression decision can be redesigned as a test of feasibility, with hypotheses defined by the power of the planned definitive trial. The definitive trial's power is $g(\varphi)=\Phi(x(\varphi)-z_{1-\alpha})$, where $\varphi=(\varphi_r,\varphi_f,\varphi_a)$ are the recruitment, follow-up and adherence rates. Choosing power thresholds $p_0$ and $p_1$ partitions the parameter space into a null region $\Phi_0=\{x(\varphi)\le x_0\}$ of infeasible trials and an alternative region $\Phi_1=\{x(\varphi)\ge x_1\}$ of feasible trials. The pilot proceeds if and only if $x(\hat\varphi)>c$, where $\hat\varphi$ is the pilot estimate and $c$ a critical value. The paper then computes the type I and II error rates $\alpha(n_p,c)=\max_{\varphi\in\Phi_0}\Pr[x(\hat\varphi)>c\mid\varphi,n_p]$ and $\beta(n_p,c)=\max_{\varphi\in\Phi_1}\Pr[x(\hat\varphi)\le c\mid\varphi,n_p]$, so $n_p$ and $c$ can be chosen prospectively to balance sampling cost against both errors. Re-designing TIGA-CUB shows the original 30-per-arm pilot has poor operating characteristics, roughly 50 per arm gives type I around 0.09 with type II around 0.23, and independent progression criteria, even with unlimited pilot size, have error rates no better than a coin toss. Extending the test to estimate an unknown outcome standard deviation raises both error rates, so a larger pilot is needed to maintain the same guarantees.

Load-bearing premise

The entire design rests on the assumption that the worst combinations of recruitment, follow-up and adherence rates sit exactly on the boundary where the planned trial's power equals the chosen threshold; if they sit inside the region instead, the reported error rates are too low.

Editorial extensions

If this is right

A pilot team can pre-specify the decision rule as 'go if $x(\hat\varphi)>c$' and choose $n_p$ and $c$ from the calculated type I and type II error curves, making sample-size justification part of the same calculation.
In the settings modelled, increasing the pilot from 30 to 50 participants per arm materially improves error rates, and values around 50 per arm keep type II error close to 0.2 while holding type I error near 0.1 when $p_1=0.8$ and $p_0\le0.65$.
Conventional independent progression criteria, which require all three estimated rates to pass their own thresholds, had error rates no better than a coin toss in the authors' scenarios, and larger pilot samples did not fix this.
If the pilot must also estimate the outcome standard deviation, both error rates rise; maintaining the same error control requires increasing sample size, for example from 50 to 70 per arm in the illustrative setting.
The formulation extends to designs whose power can be written as a function of the pilot-estimated parameters, including binary outcomes by normal approximation and cluster-randomised trials with known variance components.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The authors' critique of independent progression criteria is a caution about conjunctive decision rules generally: requiring every estimated rate to clear a threshold creates a reverse-multiplicity effect, so a natural design heuristic is to define the decision on the downstream quantity of interest, power, rather than on each process estimate separately.
Because the error rates are explicit, funders and trial oversight committees could specify an acceptable probability of investing in an underpowered trial, and the pilot size would follow from that tolerance; this gives a concrete way to translate risk appetite into a sample size.
Extending the binary stop/go rule to a stop/modify/go decision could be done with two critical values, and the same error-rate calculations would show whether the intermediate decision actually improves long-run decisions or just adds a third action with unmeasured consequences.

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Signed reviews

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this paper has a good idea—formalize external pilot progression as a hypothesis test on the planned definitive trial's power—and it works through the consequences in a careful, honest way. But the central numerical recommendations depend on error-rate calculations that are not fully established: the maxima over composite hypotheses are found with a heuristic, and the text has an internal inconsistency about the negative binomial recruitment model.

What's new: previous warnings about hypothesis tests in pilots were about tests of effectiveness; this paper defines hypotheses on the power scale, which is a real shift. The power derivation for the definitive trial (truncated binomial recruitment, follow-up/adherence with correlation) is careful, and the pilot statistic's sampling distribution is computed by enumeration. The comparison with independent progression criteria is useful—showing that conjunctive PCs can have error rates near coin-toss level is an important practical message.

The bigger worry is the unstated boundary assumption in Section 3.4. The type I and II errors are defined as suprema over Φ0 and Φ1, but the paper only says (in Section 3.5, for the conventional PC method) that the solutions are assumed to lie on the boundaries. For the proposed test, no monotonicity of h is proved, and the optimization is handed to NSGA-II, which has no global optimality guarantee. So the curves in Figures 2–4 are plausibly lower bounds rather than true error rates. The stress-test note is right on both counts. That said, the issue is fixable: a proof of monotonicity or a certified global optimization would do; a grid search over the boundary (as they already do for the PC method) would at least be transparent.

Second soft spot: Section 2 defines S~NB(np, φr), but Section 3.3 uses φ̂r = 2np/(2np+S), which only makes sense if the total pilot target is 2np and S~NB(2np, φr). The code may resolve this, but the code statement is itself incomplete ('freely available at' with no URL). This needs clarification before the numbers can be reproduced.

Overall: worth a serious referee. The conceptual contribution is solid, the worked example is instructive, and the critique of independent PCs is valuable. The soft spots are real but addressable, and they don't undercut the main idea. I'd send it to review, but I'd want the authors to fix the optimization justification and the NB inconsistency before publication.

Referee Report

2 major / 4 minor

Summary. The paper proposes a formal hypothesis-testing framework for progression decisions in external pilot trials. Feasibility of the planned definitive trial is quantified by its power, expressed as a function of recruitment, follow-up, and adherence rates. The null and alternative hypotheses correspond to the definitive trial power being below p0 or above p1. The test statistic is the plug-in estimate of the feasibility measure, x(φ̂), obtained from the pilot data. The authors show how the type I and II error rates of this test can be computed by enumerating the pilot sampling distribution and solving a bi-objective optimization for pilot sample size and critical value. The method is illustrated by re-designing TIGA-CUB and compared with conventional independent progression criteria, which are shown to have error rates no better than a coin toss in the considered scenarios. An extension incorporates an unknown outcome standard deviation. The paper includes a derivation of the definitive trial power and a reproducible implementation in R.

Significance. If the error-rate calculations were certified, this would be a valuable contribution to the design of external pilot trials, which currently rely on ad hoc progression criteria with little formal justification. Strengths of the paper include a careful modelling of the pilot data distribution, an explicit link between feasibility and definitive trial power, a complete implementation provided in the supplementary materials, and a concrete demonstration that conventional independent PCs can behave poorly. The main limitation is that the reported worst-case error rates rely on a heuristic optimizer without a proof that the optima are attained, which leaves the central design recommendation conditional.

major comments (2)

[Section 3.4, Eq. (2)] The advertised type I and II error rates are defined as suprema of h(np,c,φ) over the composite null and alternative hypotheses, but the numerical solution uses NSGA-II, a stochastic metaheuristic with no global optimality guarantee. The paper does not prove that h(np,c,φ) is monotone in φ, nor that the suprema are attained on the boundary surfaces x(φ)=x0 and x(φ)=x1; the boundary-attainment assumption is stated only for the conventional PC method in Section 3.5, not for the proposed test. If the true maxima lie in the interior of Φ0 or Φ1, the curves in Figures 2–4 are lower bounds rather than worst-case error rates, in which case the recommendation that np≈50 per arm is sufficient is not supported. Please either prove monotonicity of h in each component of φ (which would justify restricting the search to the boundary surfaces and allow a grid-based certified search as in Section 3.5) or use a certified global optimization method and report the resulting maxima.
[Section 2 and Section 3.3] The sampling model for the pilot is internally inconsistent. Section 2 defines S ∼ NB(np, φr) for the number of eligible patients who decline while recruiting to a target pilot sample size of np, whereas Section 3.3 defines the estimated recruitment rate as φ̂r = 2np/(2np + S) and states that np is the sample size per arm. These two statements are compatible only if S ∼ NB(2np, φr), i.e., if the pilot recruits until 2np consenting participants are obtained. Since the sampling distribution of S enters every probability calculation, the ambiguity must be resolved: please define clearly whether np is per arm or total and align the negative binomial specification in Section 2 with the estimator used in Section 3.3 and in the code.

minor comments (4)

[Section 3.1] The term 'one-sided type I error rate' is confusing because the definitive trial is described as a two-arm z-test and Section 4 uses a two-sided test at the 0.05 level; since the power formula uses z_{1-α}, the precise meaning of α (one-sided or per-sided) should be stated.
[Section 3.3] The densities pf(.) and pa(.) in the simplified formula for h are not defined explicitly; please state that they are binomial densities for the total number followed up across both arms and the number of adherers, respectively.
[Data availability statement] The data availability statement says the code is 'freely available at' but no URL is printed in the manuscript; please include the repository link.
[Section 6] In the formula for h(np,c,φ,σ), the conditioning set of the sample variance density should be defined; as written, p̂σ²(σ̂² | s, a, f, φ) appears to depend on s and a, whereas the preceding text indicates that σ̂²|f follows a scaled chi-square distribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the test statistic is the plug-in estimator of the power function used to define the hypotheses, but the error rates are computed from the model's sampling distribution rather than fitted, and the sole self-citation is a non-load-bearing pointer to future work.

full rationale

The paper's derivation chain is self-contained. Feasibility is defined through the definitive-trial power g(φ)=Φ(x(φ)−z_{1−α}) with x(φ) derived from the trial model in the appendix; the hypotheses Φ0={x(φ)≤x0} and Φ1={x(φ)≥x1} are fixed by user-specified thresholds p0 and p1; the test statistic is the plug-in x(φ̂), and the pilot operating characteristic h(np,c,φ) is obtained by enumerating the sampling distribution of the pilot estimators (Eq. 1) rather than by fitting any parameter to pilot data. The reported type I and II error rates therefore follow from the stated model, not from the inputs by definition. The only self-citation, reference [26], appears in the Discussion as a mention that treatment effect could be a future extension and is not used to justify any central claim. The heuristic NSGA-II maximization and the inconsistency between S∼NB(np,φ_r) and φ̂_r=2np/(2np+s) are correctness or robustness concerns about whether the advertised error rates are exact, but they are not circularity: the paper does not rename a fitted quantity as a prediction or import a load-bearing conclusion from its own prior work.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The method itself introduces no fitted parameters or invented entities. It does rely on a set of user-specified design inputs (p0, p1, mu, sigma^2, ne, nt) for the illustrative scenarios, and on several domain assumptions about the trial model, plus two ad hoc assumptions used to make the optimization and the unknown-variance extension tractable.

free parameters (6)

p0 = 0.65 (illustrative)
Power threshold dividing infeasible from indeterminate trials; user-specified. The paper's example sets p0=0.65; error rates depend on this choice.
p1 = 0.8 (illustrative)
Power threshold dividing feasible from indeterminate trials; user-specified. The paper sets p1=0.8.
mu (treatment effect) = 0.3 (illustrative)
Assumed minimal clinically important effect size in the TIGA-CUB re-design; user-specified.
sigma^2 (outcome variance) = 1 (illustrative)
Assumed outcome variance; user-specified.
ne (eligible patients) = 1000 (illustrative)
Assumed size of the eligible patient pool; user-specified.
nt (target sample size) = 514 (illustrative)
Assumed target sample size of the definitive trial; user-specified.

assumptions (6)

domain assumption Per-arm sample size of the definitive trial exceeds 30, so the sampling distributions of group means are normal.
Invoked in Section 3.1 to justify the power formula g(phi)=Phi(x(phi)-z_{1-alpha}).
domain assumption Follow-up rate is constant across intervention and control arms.
Assumed in Section 2; the paper notes it can be relaxed at the cost of an extra dimension.
domain assumption Non-adherence is absolute, so non-adherers receive no treatment effect.
Stated in Section 2; the paper notes violation under-estimates definitive trial power.
domain assumption The definitive trial primary analysis is a complete-case intention-to-treat z-test.
Assumed in Section 2. The label 'complete-case ITT' is internally inconsistent, since a true ITT analysis includes all randomized participants.
ad hoc to paper The maximum error rates over the composite hypotheses occur on the boundary surfaces x(phi)=x0 and x(phi)=x1.
Needed for the NSGA-II search in Section 3.4 to find the true suprema; stated explicitly only for the conventional approach in Section 3.5.
ad hoc to paper In the unknown-variance extension, sigma is restricted to sigma>sigma_star to avoid degenerate error rates.
Section 6 imposes the lower limit; without it, error rates tend to 1 as sigma tends to zero.

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Cite this review

Pith. "Pith review of A hypothesis test of feasibility for external pilot trials assessing recruitment, follow-up and adherence rates." pith.science (2026). https://pith.science/paper/Q3AILATL

@misc{pith2026190805562,
  author       = {Pith},
  title        = {Pith review of: A hypothesis test of feasibility for external pilot trials assessing recruitment, follow-up and adherence rates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q3AILATL}},
  note         = {Machine review of arXiv:1908.05562}
}

read the original abstract

The power of a large clinical trial can be adversely affected by low recruitment, follow-up and adherence rates. External pilot trials estimate these rates and use them, via pre-specified decision rules, to determine if the definitive trial is feasible and should go ahead. There is little methodological research underpinning how these decision rules, or the sample size of the pilot, should be chosen. In this paper we propose a hypothesis test of the feasibility of a definitive trial, to be applied to the external pilot data and used to make progression decisions. We quantify feasibility by the power of the planned trial, as a function of recruitment, follow-up and adherence rates. We use this measure to define hypotheses to test in the pilot, propose a test statistic, and show how the error rates of this test can be calculated for the common scenario of a two-arm parallel group definitive trial with a single normally distributed primary endpoint. We use our method to re-design TIGA-CUB, an external pilot trial comparing a psychotherapy with treatment as usual for children with conduct disorders. We then extend our formulation to include using the pilot data to estimate the standard deviation of the primary endpoint. and incorporate this into the progression decision.

Figures

Figures reproduced from arXiv: 1908.05562 by the authors.

**Figure 2.** Type I (α) and type II (β) error rates obtained for a range of critical values c and pilot sample sizes np when using the proposed method (solid lines). Error rates available when using conventional progression criteria are shown for comparison (dashed lines). (possibly biased) coin. Increasing the pilot sample size does not improve error rates, but actually makes them worse. These counter-intuitive results can be e… view at source ↗

**Figure 3.** Type I error rates for pilot trials of different sample sizes [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Type I (α) and type II (β) error rates obtained for a range of critical values c and pilot sample sizes np when using the proposed method (solid lines) and the conventional approach (dashed line). The power used to define the null hypothesis increases from left to righ…

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Reference graph

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Ben G. O. Sully, Steven A. Julious, and Jon Nicholl. A reinvestigation of recruitment to randomised, controlled, multicenter trials: a review of trials funded by two uk funding agencies. Trials, 14(1):166, Jun 2013

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[2] [2]

Michael P. Fay, M. Elizabeth Halloran, and Dean A. Follmann. Accou nting for variability in sample size estimation with applications to nonadheren ce and estimation of variance and eﬀect size. Biometrics, 63(2):465–474, dec 2006

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[3] [3]

Sample size requirements to estimate k ey de- sign parameters from external pilot randomised controlled trials: a simula- tion study

M Teare, Munyaradzi Dimairo, Neil Shephard, Alex Hayman, Amy W hite- head, and Stephen Walters. Sample size requirements to estimate k ey de- sign parameters from external pilot randomised controlled trials: a simula- tion study. Trials, 15(1):264, 2014

work page 2014

[4] [4]

Developing and evaluating complex intervention s: the 20 new medical research council guidance

Peter Craig, Paul Dieppe, Sally Macintyre, Susan Michie, Irwin Naz areth, and Mark Petticrew. Developing and evaluating complex intervention s: the 20 new medical research council guidance. BMJ: British Medical Journal , 337, 9 2008

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Eldridge, Gillian A

Sandra M. Eldridge, Gillian A. Lancaster, Michael J. Campbell, Leha na Thabane, Sally Hopewell, Claire L. Coleman, and Christine M. Bond. Deﬁning feasibility and pilot studies in preparation for randomised con - trolled trials: Development of a conceptual framework. PLOS ONE , 11(3):e0150205, mar 2016

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Informing eﬃcient randomised controlled trials: exp lo- ration of challenges in developing progression criteria for internal p ilot stud- ies

Kerry N L Avery, Paula R Williamson, Carrol Gamble, Elaine O’Connell Francischetto, Chris Metcalfe, Peter Davidson, Hywel Williams, and Jane M Blazeby. Informing eﬃcient randomised controlled trials: exp lo- ration of challenges in developing progression criteria for internal p ilot stud- ies. BMJ Open , 7(2):e013537, feb 2017

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