The Estimand After a Data-Dependent Stopping Time

The Target Does Not Move When the Trial Stops

A group sequential trial may stop after an early interim analysis or continue to its planned final analysis. The estimate reported at the end therefore comes from a data-dependent look. That fact changes the repeated-sampling distribution of the reported estimate, but it does not automatically change the treatment effect the trial was designed to estimate. Keeping these two ideas separate is essential: the estimand is the scientific target, while the stopped estimate is a random quantity whose distribution reflects both the data and the stopping rule.

For example, a trial might target the difference in mean change in a clinical outcome between randomized treatment groups at a fixed follow-up time in a defined population. If the trial stops early, the planned follow-up time and population do not become different merely because fewer participants were enrolled. The estimate may be less precise and its sampling distribution may differ from that of an estimate at the planned final look.

The definition presumes that each look’s estimate targets the same \(\theta\). This requires consistency of the population, outcome definition, treatment contrast, and follow-up interpretation across looks. If the protocol changes one of these, the target may change for a scientific reason; the change is not caused by the stopping time itself. The earlier material on stopping-induced bias concerns one important consequence of selection by a stopping rule. Here the focus is the more basic object: the entire repeated-sampling distribution of the estimate that the rule selects.

Representing the Stopped Estimate

Suppose there are \(K\) prespecified analyses. Let \(\hat{\theta}_k\) be the estimate at look \(k\), and let the rule specify when to stop using the interim data available at each look. Define \(A_k=\{T=k\}\), the event that the trial stops at look \(k\). The stopped estimate equals \(\hat{\theta}_k\) on \(A_k\). Since the events \(A_1,\ldots,A_K\) partition all possible trial outcomes, the unconditional distribution is

This is a mixture over stopping looks, but the components are joint probabilities: each estimate is considered together with the event that the stopping rule selects that look. When \(P(T=k)>0\), the same distribution can be written using conditional distributions:

The conditional distribution at look \(k\) is generally not the same as the marginal distribution of \(\hat{\theta}_k\). The event \(T=k\) is determined using observed data and can restrict which estimates occur at that look. Simply mixing the fixed-look marginal distributions with weights \(P(T=k)\) will therefore generally give the wrong answer.

The same decomposition gives the moments, when they exist:

where \(I(A_k)\) is 1 when \(A_k\) occurs and 0 otherwise. Then \(\operatorname{Var}(\hat{\theta}_T)=E(\hat{\theta}_T^2)-[E(\hat{\theta}_T)]^2\). These expressions do not assume that the stopped estimate is unbiased or normally distributed. They make explicit what must be averaged: the estimate on the event that the rule actually selects it.

A Canonical Repeated-Sampling Model

A useful model for a continuous treatment-effect estimate assumes prespecified, fixed information levels \(I_1<\cdots<I_K\) and approximately normal estimates. In the canonical model, the joint distribution is represented by Brownian motion \(W(I)\):

Here \(W(I)\) has mean zero, independent increments, and variance \(I\); consequently \(Z_k\) has variance 1 and mean \(\theta\sqrt{I_k}\). This model describes dependence across looks, not just the marginal standard error at each look. For \(j\leq k\),

The canonical approximation is appropriate when the information levels and analysis schedule are prespecified, the estimates target the same parameter, and the joint score or estimate process is adequately approximated by normal independent increments. It is not automatic for every endpoint or analysis. Material departures from the assumed covariance, information timing, or outcome model can invalidate numerical calculations based on this representation.

For a fixed stopping rule, the distribution of \(\hat{\theta}_T\) follows by applying the rule to the joint process. A short justification of the mixture formula is to partition the event \(\{\hat{\theta}_T\leq x\}\) according to the mutually exclusive stopping events:

Because the union is disjoint, probabilities add, giving the distribution formula above. The events \(A_k\) must be evaluated under the joint distribution of all looks involved in the rule; treating looks as independent when their estimates are correlated changes those probabilities and the resulting stopped distribution.

Numerical Illustration Under the Canonical Model

A continuous example shows why the joint model matters even when the estimate at each fixed look is normal. Suppose there are two looks with information \(I_1=50\) and \(I_2=100\), and let the true effect be \(\theta=0\) in the estimate’s measurement units. At the first look, stop if \(Z_1=\sqrt{50}\hat{\theta}_1>2\); otherwise continue to the final look. For this illustration, assume the canonical normal model, fixed information, and independent increments. The first-look estimate \(X_1=\hat{\theta}_1\) and the second-stage increment estimate \(X_2\) are then independent \(N(0,1/50)\) variables, and the final estimate is \((X_1+X_2)/2\).

The boundary is equivalent to \(X_1>2/\sqrt{50}\). Since \(X_1\) has standard deviation \(\sigma=1/\sqrt{50}=0.1414\), the stopping probability is \(P(Z>2)=0.02275\), where \(Z\sim N(0,1)\). Write \(\phi\) and \(\Phi\) for the standard normal density and distribution function. The truncated-normal identity \(E[X_1 I(X_1>2\sigma)]=\sigma\phi(2)\) gives the mean of the stopped estimate:

For the second term, independence and \(E(X_2)=0\) give \(E[X_2 I(X_1\leq2\sigma)]=0\), while \(E[X_1 I(X_1\leq2\sigma)]=-\sigma\phi(2)\). The stopped estimate’s mean is therefore slightly above the true value 0 in this illustration; this is a property of this particular rule and model, not a general numerical correction.

Its second moment can also be calculated from the same partition. For a standard normal variable, \(E[Z^2 I(Z>2)]=1-\Phi(2)+2\phi(2)\) and \(E[Z^2 I(Z\leq2)]=1-[1-\Phi(2)+2\phi(2)]\). Substitution, including the independent second-stage variance, yields

Subtracting the squared mean gives \(\operatorname{Var}(\hat{\theta}_T)\approx0.01183\), or a standard deviation of approximately \(0.1088\) in the estimate’s units. By contrast, the fixed final-look estimate has variance \(1/I_2=0.01000\). These calculations describe the unconditional distribution over repeated trials using this rule, including both possible stopping looks. They do not say that an individual reported estimate has a particular probability of being close to \(\theta\).

Diagnostics, Alternatives, and Limits

Before interpreting a stopped estimate’s distribution, verify that the analysis matches the design. Check that the look schedule and boundaries used in the calculation are the prespecified ones; that each \(\hat{\theta}_k\) estimates the same treatment contrast; that information and covariance across looks are represented correctly; and that the assumed normal or other sampling model is adequate. If the actual information timing differs, or if the analysis uses an outcome model with substantial small-sample or nonnormal behavior, a canonical normal calculation may be inaccurate. Simulation from a suitable trial model can be a useful diagnostic, provided the simulated stopping rule and data-generating assumptions are explicit.

Three distributions are easy to confuse. The fixed-look marginal distribution describes \(\hat{\theta}_k\) if the analysis were made at look \(k\) regardless of earlier results. The conditional distribution given \(T=k\) describes that estimate among trials that stop at \(k\). The unconditional stopped distribution describes \(\hat{\theta}_T\) across all repeated trials using the rule. The first is not a substitute for the third, and the second alone omits trials stopping at other looks.

The unconditional stopped distribution is the natural description when the question concerns repeated use of the entire prespecified design. A conditional distribution can instead be relevant when the target of inference explicitly conditions on the observed stopping look. Neither choice changes the scientific estimand by itself; it changes which repeated-sampling behavior is being described. A limitation of the canonical treatment is that it provides a model-based approximation, not a universal distribution for all treatment effects, endpoints, or adaptive designs.

• Mistake: “The trial stopped early, so the estimand is the effect in the smaller group enrolled.” Correction: The target population and outcome are defined by the estimand; early stopping changes observed information, not that definition.
• Mistake: “The stopped estimate is normal because every fixed-look estimate is normal.” Correction: Applying a data-dependent stopping rule can change the distribution; derive it from the joint estimates and stopping events.
• Mistake: “Weight the unrestricted look-specific distributions by stop probabilities.” Correction: Use distributions conditional on the stopping events, or equivalently the joint event probabilities.