Re-Estimation at Multiple Interim Looks

Why Multiple Looks Need a Carry-Forward Rule

A sample-size re-estimation rule may be applied more than once before the final analysis. The first update changes the enrollment plan, but it does not erase the participants already enrolled or reset the evidence accumulated so far. At a later look, the design must use the information then available and the target that is currently in force. Without an explicit carry-forward convention, it is easy to count participants twice, use an obsolete target, or inadvertently reduce enrollment after an earlier increase.

This tutorial treats each target as a total sample size: the cumulative number of participants intended by the end of the trial, not the number to recruit after the current look. The example rule below allows the target to increase but not decrease. That is one clear operational choice, not a universal requirement; a design that permits decreases must specify its own lower limits and stopping rules in advance.

Definition: Let $N_0$ be the initial total sample-size target. At interim look $k$, let $n_k$ be the cumulative number of participants with usable outcomes and let $N_{k-1}$ be the total target in force just before the look. A re-estimation rule produces a recommended total target $M_k$. Under an increase-only rule with maximum $U$, the updated total target is $$ N_k=\min\{U,\ \max(N_{k-1},M_k)\}. $$ The number still to enroll after look $k$ is $N_k-n_k$, not $N_k-N_{k-1}$.

This rule assumes that the cap is at least as large as the target already in force and that the look occurs before the current target is reached. The design must also ensure that a proposed target is not smaller than the number already enrolled. An increase-only convention means that a lower recommendation at a later look does not undo an earlier increase.

Specify the Looks Before the Trial

A multiple-look design needs a schedule as well as a re-estimation mapping. For example, a protocol might set looks at fixed cumulative enrollment counts based on the initial target. If the initial target is $N_0=100$, it could schedule assessments after 40 and 70 participants have usable outcomes. If the first update increases the target, the second look still occurs at 70 under this schedule; its information fraction is then calculated relative to the target currently in force, not automatically treated as 70% of that larger target.

Another design can schedule a later look as a prespecified fraction of the revised target. That changes when the later assessment occurs, so the schedule must say how the fraction is rounded and what happens if a recalculated look would occur too soon after the previous one. Either approach can be operationally coherent. The important point is to state the rule in advance and apply the same convention throughout.

At each look, the analysis should use the cumulative information through that look. Under a model with independent participant-level outcomes, the observations since the previous look are new information; earlier observations remain part of the cumulative estimate, but must not be added again as if they were new participants. A conditional-power calculation is also tied to a specified candidate final target and planning assumption. After a target changes, later calculations must use the updated target and the evidence at the new look.

Conditional Power at a Later Look

The following normal-endpoint calculation illustrates how the evidence and candidate total target enter a conditional-power assessment. Suppose independent outcomes have known standard deviation $\sigma$, and the treatment effect is measured relative to a null value of zero. Write $\delta^*=\Delta^*/\sigma$ for the standardized effect used for planning. For a one-sided final test with boundary $c$, define the cumulative interim statistic at sample size $n$ as $Z_n=\sqrt n\,\bar X_n/\sigma$.

For a candidate final total $N>n$, the remaining $N-n$ outcomes are independent of the observed interim data. Conditional power is the probability, under the planning effect for the unobserved outcomes, that the final statistic exceeds $c$, conditional on the observed interim statistic $z_n$. This calculation assumes the stated normal model, known variance, independent increments, a fixed planning effect, and a specified final boundary.

Z_N=\sqrt{\frac{n}{N}}\,z_n+ \sqrt{\frac{N-n}{N}}\,Z_{\mathrm{future}}, \qquad Z_{\mathrm{future}}\sim N\!\left(\delta^*\sqrt{N-n},1\right).

The first term is the contribution of the observed data to the final statistic; the second is the contribution of future data. Consequently, the conditional distribution of the final statistic has variance $(N-n)/N$, and its conditional mean includes the observed interim contribution. If $\Phi$ denotes the standard normal cumulative distribution function, conditional power is

\operatorname{CP}(N\mid n,z_n;\delta^*)= \Phi\!\left( \frac{\sqrt{n/N}\,z_n+\delta^*(N-n)/\sqrt N-c} {\sqrt{(N-n)/N}} \right).

The formula is a model-based calculation, not a guarantee of the eventual result. In a multiple-look design, the next look uses the current cumulative statistic and the target then in force. It does not simply carry forward the earlier conditional-power value.

Worked Example: The First Update Sets a Larger Target

Consider a hypothetical one-sided normal-endpoint trial with known $\sigma$, planning effect $\delta^*=0.30$, final boundary $c=1.96$, initial target $N_0=100$, and first look at $n_1=40$. Suppose the observed interim statistic is $z_{40}=0.20$. For illustration, the protocol has a predeclared mapping from conditional power to a recommended total target: below 0.30, recommend 160; from 0.30 to below 0.50, recommend 140; from 0.50 to below 0.70, recommend 120; and at least 0.70, recommend 100. Assume the maximum is $U=160$.

First calculate conditional power for the current total target $N=100$. The conditional mean of the final statistic is $$ \sqrt{\frac{40}{100}}(0.20)+\frac{0.30(100-40)}{\sqrt{100}} =0.1265+1.800=1.9265. $$ The conditional standard deviation is $\sqrt{60/100}=0.7746$. Therefore, $$ \operatorname{CP}=\Phi\!\left(\frac{1.9265-1.96}{0.7746}\right) =\Phi(-0.0433)\approx 0.4827. $$

The illustrative mapping recommends $M_1=140$. Applying the increase-only update gives $N_1=\min(160,\max(100,140))=140$. After the first look, the total target is 140, so 100 participants remain to be enrolled if no later update changes that target. The recommendation is not 140 additional participants.

Reassess Using the Target Currently in Force

At a later look, the design first follows its prespecified schedule and determines the cumulative sample size and interim statistic. It then calculates conditional power using that current evidence and the total target currently in force. A deterministic mapping produces a new recommendation, which is combined with the existing target under the prespecified carry-forward rule.

Reach the scheduled look.
Record the cumulative number $n_k$ with usable outcomes and calculate the interim statistic from all data available through that look.

Recalculate the interim assessment.
Use the current evidence, the planning assumptions, and the target $N_{k-1}$ in force immediately before the look. Do not substitute the original target if it has already been revised.

Apply the predeclared mapping.
Translate the assessment into a recommended total target $M_k$, using the specified rounding, lower-bound, and cap conventions.

Carry the target forward.
For an increase-only rule, set $N_k=\min\{U,\max(N_{k-1},M_k)\}$. Enroll $N_k-n_k$ more participants, unless a separate prespecified stopping rule applies.

Worked Example: A Second Look Raises the Target Again

Continue the hypothetical design above. The second look is fixed at $n_2=70$, and the target in force before it is $N_1=140$. Suppose the cumulative interim statistic is now $z_{70}=-0.50$. Use the same planning effect, boundary, illustrative mapping, and cap.

For $N=140$, the conditional mean is $$ \sqrt{\frac{70}{140}}(-0.50)+\frac{0.30(140-70)}{\sqrt{140}} =-0.3536+1.7748=1.4212. $$ The conditional standard deviation is $\sqrt{70/140}=0.7071$, giving $$ \operatorname{CP}=\Phi\!\left(\frac{1.4212-1.96}{0.7071}\right) =\Phi(-0.762)\approx 0.223. $$

The mapping recommends $M_2=160$, so the updated total target is $N_2=\min(160,\max(140,160))=160$. At the second look, 70 participants have already contributed outcomes; 90 more are required to reach 160. The first increase is retained, and the second recommendation adds 20 to the total target, not 20 to the 70 already accrued.

Worked Example: A Lower Later Recommendation Does Not Reverse an Increase

Suppose instead that at the second look $z_{70}=1.00$, while $N_1=140$ remains the current target. The conditional mean is $$ \sqrt{\frac{70}{140}}(1.00)+\frac{0.30(70)}{\sqrt{140}} =0.7071+1.7748=2.4819. $$ With conditional standard deviation $0.7071$, the conditional power is $$ \Phi\!\left(\frac{2.4819-1.96}{0.7071}\right) =\Phi(0.738)\approx 0.770. $$

The mapping recommends $M_2=100$, but the increase-only carry-forward rule gives $N_2=\min(160,\max(140,100))=140$. The design therefore retains a total target of 140 and requires 70 more participants after this look. A later, more favorable assessment does not shrink the target under this rule.

Why the Repeated Rule Matters

A second assessment can respond to information that was not available at the first one. It can therefore lead to a further increase, no change, or—if the protocol permits—another action. But repeated assessment also creates more opportunities for noisy interim estimates to affect enrollment. The conditional-power calculation describes a conditional probability under its assumptions; it does not by itself ensure the desired unconditional power or preserve the type I error rate of the final analysis.

In particular, an unblinded sample-size decision based on observed treatment differences can affect the distribution of a conventional final test. The test and any group-sequential monitoring boundaries must be justified for the full prespecified adaptive design. Do not infer that a fixed-sample cutoff remains valid merely because every re-estimation step was written down in advance.

A fixed-sample design is simpler to operate and does not use interim results to revise enrollment, but it cannot respond to those results. A single-look re-estimation rule has fewer decision points and a simpler schedule. Multiple looks can make decisions more responsive, but require more coordination, more opportunities for operational error, and careful calibration of the complete design. These approaches are not interchangeable without considering their inferential consequences.

Useful diagnostics include checking that each look happened at the prespecified cumulative count, that the current target used in the calculation matches the target in the trial records, and that the recommended target was rounded and capped as specified. Also verify that $n_k<N_{k-1}$ when computing future conditional power, that the sample-size mapping is applied to the current look rather than copied from an earlier one, and that enrollment remaining is calculated as $N_k-n_k$. Large or erratic changes can signal unstable interim estimates, an inconsistent implementation, or a mismatch between the planning assumptions and the data. They are reasons to investigate, not automatic evidence that the rule should be changed mid-trial.

Common Mistakes and Limits

Adding the revised target to accrued enrollment. If the new target is 140 total and 40 participants have already contributed outcomes, the remaining enrollment is 100, not 140.
Using the original target at every look. After an update, later conditional power must use the target currently in force, unless the protocol explicitly defines a different calculation.
Reusing the first look’s conditional power. A later look has new cumulative evidence. Recalculate from that evidence; do not treat the earlier conditional-power value as new data.
Silently lowering an earlier increase. Whether later recommendations can reduce the target is a design choice. The increase-only rule here retains the largest target reached so far, subject to the cap.
Assuming repeated conditional-power checks protect error rates. Conditional power is a planning quantity. Valid final inference requires a procedure appropriate to the complete adaptive and monitoring design.

The normal calculation also depends on its assumptions: known variance, independent increments, a correctly specified planning effect, and a suitable normal model. If those assumptions do not fit the endpoint or analysis, the conditional-power calculation and the sample-size mapping must be adapted and justified rather than used mechanically. The bookkeeping rule for carrying total targets forward remains useful, but it cannot correct a misspecified statistical model.

Check Your Understanding

Answer the questions using the increase-only convention unless a question states otherwise. Brief solutions follow.

A trial has accrued 55 participants and its revised total target is 130. How many more participants are needed to reach the target?
The current target is 120, the next recommendation is 150, and the cap is 160. What is the updated target under the rule?
The current target is 150 and a later recommendation is 125. What target applies under an increase-only rule, and why?
Why must conditional power at a second look use the evidence at that look rather than simply reuse the first look’s conditional power?
Name two assumptions used in the normal conditional-power formula above.

Solutions. (1) $130-55=75$ more participants. (2) $\min(160,\max(120,150))=150$, so the updated total target is 150. (3) The target remains 150 because $\max(150,125)=150$; this rule does not undo an earlier increase. (4) The second look contains additional observed information, so its conditional distribution of the final statistic differs; conditional power must be recalculated using the current cumulative statistic and current target. (5) Examples include independent participant outcomes, known standard deviation, a normal endpoint model, and a specified planning effect for future outcomes.