When the Maximum Sample Size Binds

What It Means for the Maximum to Bind

A conditional-power rule may call for a final sample size larger than the design permits. In that case, the maximum is not just a planning detail: it limits the conditional power the design can attain for the observed interim data under the chosen assumptions. The useful report is therefore not only “the target sample size exceeds the cap,” but also the conditional power at the cap and its distance from the target.

The calculation below uses a simple normal endpoint so that the effect of the cap is explicit. Suppose \(n_1\) independent observations have been collected at an interim analysis, and the observed interim mean is \(\bar{x}_1\). The outcome has known standard deviation \(\sigma\). A total of \(N\) observations will be available at the final analysis, with \(N-n_1\) new observations after the interim. Let \(\delta\) denote the effect assumed for planning the future observations. The final test rejects for a sufficiently large value of the standardized statistic, using a prespecified upper-tail boundary \(c\). In an actual design, \(c\) must be the boundary specified by that design; it should not be selected afresh just because a sample-size re-estimation is being considered.

Definition: For a proposed total sample size \(N>n_1\), the conditional power is the probability, conditional on the interim data observed so far, that the final test will cross its efficacy boundary, under a specified model for the future observations. In this tutorial, that model assumes independent normal observations with known standard deviation \(\sigma\) and mean \(\delta\) for observations not yet collected.

A maximum sample size \(N_{\max}\) is binding for a conditional-power target \(\gamma\) when the rule’s target-driven sample size exceeds the permitted maximum. Operationally, compare the target with the conditional power attainable at \(N_{\max}\). If that power is below \(\gamma\), the design cannot meet the target at the cap under the stated interim data and planning assumptions. The cap limits the total sample size: the largest number of additional observations is \(N_{\max}-n_1\).

Conditional Power at a Proposed Total Sample Size

Let \(S_1=n_1\bar{x}_1\) be the observed interim sum. Under the stated model, the sum \(S_2\) of the \(N-n_1\) future observations is conditionally distributed as \(N((N-n_1)\delta,(N-n_1)\sigma^2)\). The final standardized statistic is \(Z_N=(S_1+S_2)/(\sigma\sqrt{N})\). Thus, conditional on the interim data, it is normal with mean \(m_N\) and standard deviation \(s_N\), where

m_N=\frac{n_1\bar{x}_1+(N-n_1)\delta}{\sigma\sqrt{N}}, \qquad s_N=\sqrt{\frac{N-n_1}{N}}.

For an upper-tail efficacy event \(Z_N\ge c\), the conditional power is the upper tail of this conditional normal distribution. Equivalently, standardizing the future sum gives the calculation in the second line:

\begin{aligned} \operatorname{CP}(N\mid \bar{x}_1,\delta) &=P(Z_N\ge c\mid \bar{x}_1)\\ &=\Phi\!\left( \frac{n_1\bar{x}_1+(N-n_1)\delta-c\sigma\sqrt{N}} {\sigma\sqrt{N-n_1}} \right). \end{aligned}

Here \(\Phi\) is the standard normal cumulative distribution function. This formula depends on the observed interim mean, the assumed future effect, the known variability, and the final boundary. It is a probability conditional on the interim data and the assumed model—not a probability that the treatment effect is true, nor the unconditional power of the whole design.

Key result: If the conditional-power target is \(\gamma\), calculate \(\operatorname{CP}(N_{\max}\mid\bar{x}_1,\delta)\). If it is below \(\gamma\), the target is unattainable at the cap for these inputs. Report both values and the shortfall \(\gamma-\operatorname{CP}(N_{\max}\mid\bar{x}_1,\delta)\). The target sample size is a total; the remaining enrollment at the cap is \(N_{\max}-n_1\).

A sample-size search should respect the design’s permitted integer totals. If the conditional power is not monotone over the candidate range, do not infer the smallest eligible sample size from a single crossing or from a continuous approximation: evaluate the permitted candidates according to the design rule. In the examples below, the stated integer searches identify the first total meeting the target.

Worked Example: The Cap Leaves the Target Unmet

Hypothetical data. At an interim analysis, \(n_1=20\) observations have mean \(\bar{x}_1=0.20\) outcome units. Assume \(\sigma=1\) outcome unit, plan future observations under \(\delta=0.30\) outcome units, use the upper-tail boundary \(c=1.96\), and set the target to \(\gamma=0.80\). The maximum total sample size is \(N_{\max}=60\).

At the cap, 40 additional observations can be collected. Substituting \(N=60\) in the conditional-power formula gives

\begin{aligned} \operatorname{CP}(60\mid 0.20,0.30) &=\Phi\!\left( \frac{20(0.20)+40(0.30)-1.96(1)\sqrt{60}} {(1)\sqrt{40}} \right)\\ &=\Phi\!\left(\frac{16-15.1821}{6.3246}\right) =\Phi(0.1293) \approx 0.5515. \end{aligned}

The attainable conditional power at the maximum is about 55.2%, below the 80.0% target by about 24.9 percentage points. The cap therefore binds for this target under these assumptions. This does not mean that the study has a 55.2% unconditional probability of success; it is the conditional probability of crossing the final boundary, given this interim mean and the assumed future effect.

For comparison, the target-driven integer search without the cap first meets 80% at \(N=95\). At \(N=94\), the argument of \(\Phi\) is \((26.2-1.96\sqrt{94})/\sqrt{74}\approx0.8367\), giving conditional power about 0.7986. At \(N=95\), it is \((26.5-1.96\sqrt{95})/\sqrt{75}\approx0.8541\), giving about 0.8034. Thus the target-driven total is 95, but the permitted total is 60. The distinction matters: the cap allows 40 more observations, not 60 more.

A Cap Can Bind in One Interim Scenario but Not Another

The maximum is a fixed design constraint, but whether it binds depends on the interim data and the assumed future effect. A more favorable interim mean can make the target attainable without changing the cap. Conversely, a weak interim result can leave conditional power well below target even after using every permitted observation.

Worked Example: The Target Is Attainable Below the Cap

Hypothetical data. Keep \(n_1=20\), \(\sigma=1\) outcome unit, \(\delta=0.30\) outcome units, \(c=1.96\), \(\gamma=0.80\), and \(N_{\max}=60\), but suppose the interim mean is \(\bar{x}_1=0.50\) outcome units.

At \(N=42\), the conditional-power argument is \((20(0.50)+22(0.30)-1.96\sqrt{42})/\sqrt{22} = (16.6-12.7023)/4.6904\approx0.8310\). Therefore the conditional power is approximately \(\Phi(0.8310)=0.7971\), just below the target. At \(N=43\), the argument is \((10+23(0.30)-1.96\sqrt{43})/\sqrt{23} =(16.9-12.8526)/4.7958\approx0.8439\), so the conditional power is approximately 0.8006. The integer search first meets the target at a total of 43.

At the cap of 60, the conditional-power argument is \((10+40(0.30)-1.96\sqrt{60})/\sqrt{40} =(22-15.1821)/6.3246\approx1.0780\), giving conditional power about 0.8594. The cap does not bind: the target is met at 43, leaving 17 observations of headroom below the maximum. The calculation uses the same assumed future effect and boundary as in the previous example; the change in the interim mean changes the result.

Finding the Interim Mean Needed at the Cap

The cap calculation can also be rearranged to show how favorable the interim mean must be for a given target to remain attainable. Write \(z_\gamma=\Phi^{-1}(\gamma)\), the standard-normal quantile for the target. Since \(\Phi\) is increasing, \(\operatorname{CP}(N_{\max}\mid\bar{x}_1,\delta)\ge\gamma\) exactly when the argument of \(\Phi\) is at least \(z_\gamma\). Solving this inequality for \(\bar{x}_1\) gives

\bar{x}_1\ge \frac{c\sigma\sqrt{N_{\max}}+ z_\gamma\sigma\sqrt{N_{\max}-n_1} -(N_{\max}-n_1)\delta}{n_1}.

This threshold is specific to the assumed effect, boundary, variability, interim sample size, cap, and target. It is not a universal cutoff for the observed effect. It is a concise way to check whether the cap could meet the target for the particular design inputs.

Worked Example: Checking a Cap Threshold

Hypothetical data. Use \(n_1=20\), \(\sigma=1\) outcome unit, \(\delta=0.30\) outcome units, \(c=1.96\), \(N_{\max}=60\), and \(\gamma=0.80\). Since \(z_{0.80}\approx0.8416\), the required interim mean at the cap is

\begin{aligned} \bar{x}_1 &\ge \frac{1.96\sqrt{60}+0.8416\sqrt{40}-40(0.30)}{20}\\ &=\frac{15.1821+5.323\ldots-12}{20} \approx0.4252\text{ outcome units}. \end{aligned}

If the interim mean is 0.40 outcome units, it falls below this threshold. Direct calculation confirms the implication: \(\operatorname{CP}(60\mid0.40,0.30) =\Phi((20(0.40)+40(0.30)-1.96\sqrt{60})/\sqrt{40}) =\Phi(0.7618)\approx0.7769\). The cap misses the 0.80 target by about 0.0231, or 2.31 percentage points. The threshold calculation is a quick diagnostic; the direct conditional-power calculation makes the attainable probability explicit.

Assumptions, Diagnostics, and Alternatives

The calculation is only as appropriate as its model and inputs. It assumes independent future observations from the specified normal model, a known \(\sigma\), the stated planning effect \(\delta\), and the specified final efficacy boundary. If the variance is estimated, the endpoint is not adequately modeled by this normal setup, or the final boundary differs from \(c\), use the design’s appropriate conditional-power calculation instead. For a two-sided test, the efficacy event includes both tails; its conditional probability must include the lower-tail probability as well as the upper-tail probability rather than treating an upper-tail calculation as the exact two-sided answer.

Several checks help catch errors before making a decision:

Check the sample-size scale. Confirm that \(N_{\max}\) is a total and subtract \(n_1\) to obtain the maximum additional enrollment.
Check the conditional distribution. The uncertainty in the future sum is based on \(N-n_1\), so the denominator in the formula is \(\sigma\sqrt{N-n_1}\), not \(\sigma\sqrt{N}\).
Check the boundary and direction. The formula shown is for crossing an upper boundary. Use the design’s specified boundary and rejection event.
Check the assumed effect. Conditional power can change substantially when \(\delta\) changes. State the planning value explicitly; do not present the result as if it were determined by the interim data alone.
Check the cap directly. Report \(\operatorname{CP}(N_{\max}\mid\bar{x}_1,\delta)\), the target, and their difference. Do not report only an uncapped sample-size estimate.
Check rounding and admissible totals. A continuous solution or rounded sample size may cross the target differently from the permitted integer search. Evaluate the actual candidate totals and the cap.

The main alternative is to retain a fixed sample size, which avoids interim sample-size changes but does not tailor the planned total to the interim result. A capped re-estimation rule allows an increase when indicated, but a binding cap explicitly accepts that the chosen conditional-power target may remain unmet in some interim scenarios. A design team may revisit the cap or the target during planning, but lowering the target after seeing interim data is not the same as meeting the original target. Decisions about the cap involve consequences beyond this probability calculation; the conditional-power calculation itself does not resolve them.

Finally, conditional power is a planning quantity, not a guarantee of success. It is conditional on the interim data and depends on assumptions about future outcomes. Re-estimation based on interim data does not, by itself, establish that the final inferential procedure controls its error rate. The final boundary and analysis must remain those specified by a valid design. In particular, a conditional power below target is not evidence that the null hypothesis is true, and it does not by itself determine whether the study should stop or continue.

Common pitfall: “The uncapped calculation requires 95 participants, so the study will have 80% power at its maximum of 60.” This is incorrect: the cap prevents reaching 95, and the conditional power at 60 must be calculated separately. In the first worked example it is about 55.2%, not 80%. Report the target-driven total, the permitted total, and the attainable conditional power as distinct quantities.

Check Your Understanding

Answer each question using the stated normal model and upper-tail boundary.

Why does the conditional-power formula use \(\sqrt{N-n_1}\) in its denominator rather than \(\sqrt{N}\)?
Suppose \(n_1=10\), \(\bar{x}_1=0.30\) outcome units, \(\sigma=1\) outcome unit, \(\delta=0.20\) outcome units, \(N_{\max}=25\), and \(c=1.96\). Calculate conditional power at the cap and the shortfall from a target of 0.80.
For a fixed cap and target, what does the interim-mean threshold tell you, and what does it not tell you?
If conditional power at the cap is 0.55, does that mean the unconditional probability of trial success is 0.55 or that \(H_0\) has probability 0.45 of being true? Explain.

Solutions.

1. Conditional on the interim data, only the \(N-n_1\) future observations are random in this calculation. Their sum has variance \((N-n_1)\sigma^2\), so its standard deviation is \(\sigma\sqrt{N-n_1}\). The final statistic is scaled by \(\sqrt{N}\), but after standardizing the conditional future sum to calculate the probability, the denominator in the \(\Phi\) argument is \(\sigma\sqrt{N-n_1}\).

2. There are 15 additional observations at the cap. Substitution gives \(\operatorname{CP}(25)=\Phi((10(0.30)+15(0.20)-1.96\sqrt{25})/\sqrt{15}) =\Phi((6-9.8)/3.8730)=\Phi(-0.9812)\approx0.1632\). The shortfall from 0.80 is \(0.80-0.1632=0.6368\), or about 63.7 percentage points. This value is conditional power under the assumed future effect and model.

3. The threshold is the minimum interim mean that makes the conditional power at the specified cap at least the target, given the stated boundary, variability, and assumed future effect. It does not establish a universal decision cutoff, determine the true treatment effect, or replace evaluation of the conditional power under a different model or boundary.

4. Neither interpretation is correct. The 0.55 is the probability of crossing the final efficacy boundary conditional on the observed interim data and the assumed model for future observations. It is not unconditional design power and is not a probability that \(H_0\) is true or false.