Z-Statistic-Based Sample Size Adaptation

From an Interim Z Statistic to an Enrollment Rule

An interim Z statistic summarizes the treatment contrast observed so far relative to its standard error. A sample-size adaptation rule can use that statistic to decide whether the planned enrollment is sufficient under a prespecified effect assumption, or whether additional enrollment is allowed. The decision is not simply “increase when the interim result is weak”: a very weak result may be unable to attain the conditional-power target even at the maximum sample size, while a sufficiently favorable result may already meet the target at the original plan.

The key is to specify the entire mapping before examining interim results: the conditional-power model, the target, the permitted sample-size range, and what to do when the target is already met or cannot be met. The introductory material on conditional power and deterministic re-estimation rules supplies the interpretation of conditional power; here the emphasis is on expressing the rule in terms of the observed Z statistic and finding its thresholds.

Conditional Power as a Function of the Interim Z

Under the canonical model, the interim statistic has the form \(Z_1=(\theta I_1+\varepsilon_1)/\sqrt{I_1}\), where \(\varepsilon_1\) has variance \(I_1\). The future increment of the unstandardized score has mean \(\theta^*(I-I_1)\) under the prespecified effect and variance \(I-I_1\). Conditional on the observed interim statistic \(z\), the final Z statistic therefore has mean \((z\sqrt{I_1}+\theta^*(I-I_1))/\sqrt I\) and variance \((I-I_1)/I\). This uses independent increments: the future increment remains random, but the interim contribution is fixed once \(z\) is observed.

The conditional power at final information \(I\) is the probability of crossing the final critical value, conditional on the interim result and calculated under \(\theta^*\). Write \(\Phi\) for the standard normal cumulative distribution function.

A target conditional power \(q\), such as \(0.80\), corresponds to a target normal quantile \(z_q=\Phi^{-1}(q)\). For a fixed candidate \(I\), the interim Z value at which conditional power equals \(q\) is obtained by setting the argument of \(\Phi\) equal to \(z_q\) and solving for \(z\).

Thus \(CP(z,I;\theta^*)\geq q\) exactly when \(z\geq z_q(I)\), for fixed \(I>I_1\). This threshold is useful for defining regions of the interim Z scale. It does not imply that conditional power is monotone as a function of final information \(I\); the conditional mean and variance both change when \(I\) changes.

Turning the Calculation into a Prespecified Rule

A practical rule needs more than a target \(q\). It also needs an original final information level \(I_0\), a maximum allowed level \(I_{\max}\), and explicit actions at both extremes. One transparent rule is:

1. If \(CP(z,I_0;\theta^*)\geq q\), keep the original planned enrollment.
2. If \(CP(z,I_{\max};\theta^*)<q\), do not increase enrollment, because the target cannot be reached at the cap under this calculation.
3. Otherwise, choose the smallest permitted final information level above \(I_0\) for which \(CP(z,I;\theta^*)\geq q\).

The word “permitted” matters. A design may specify a set of allowed enrollment levels rather than permit every real-valued information value. For such a rule, evaluate conditional power at the allowed choices and select according to the prespecified criterion. Do not assume that a larger sample always raises conditional power at a fixed observed \(z\); verify the candidate values or establish the behavior for the design being used.

Why the Rule Needs Prespecification and Diagnostics

A Z-based rule is attractive because it converts an interim result into an auditable decision: given the prespecified inputs, the same \(z\) always leads to the same action. The thresholds also make the rule easy to communicate. For example, the maximum-size threshold derived above identifies which interim Z values can attain the target at the cap, while evaluating \(CP(z,I_0;\theta^*)\) identifies values for which no increase is needed.

The conditional-power calculation is only as credible as its assumptions. Check that the interim statistic uses the planned information scale; that future observations contribute the assumed independent information increment; that the variance model remains appropriate; and that the effect \(\theta^*\), final test, and critical value were specified consistently. If interim data are used to change these assumptions informally, the calculated rule is no longer the rule that was evaluated in advance.

Most importantly, a conditional-power rule alone does not establish that the final test has type I error \(\alpha\). Because enrollment depends on an interim statistic, the distribution of the final test statistic under \(H_0\) can differ from that of a fixed-sample design. The design must evaluate its operating characteristics under the null and use a testing procedure justified for the adaptation. Likewise, a target conditional power under \(\theta^*\) is not a guarantee of unconditional power across all possible interim results.

A fixed-sample design is simpler and avoids an interim enrollment decision, but it cannot respond to the observed interim result. A rule that always increases enrollment by a fixed amount is simpler than a Z-dependent rule, but does not tailor the increase to conditional power. A Z-threshold formulation instead makes the decision regions explicit. It differs from a rule written directly in terms of an estimated treatment effect: both may use related interim information, but their thresholds and implications must be derived under their own prespecified definitions.