Why Stopping-Adjusted Estimates Differ
A group sequential design uses interim data both to decide whether to stop and to estimate the treatment effect. At an efficacy stop, the observed estimate is selected for being large enough to cross a boundary. The ordinary estimate can therefore be systematically high among trials that stop early, even when its sampling model would make it unbiased under fixed sampling. Adjustments address this selection in different ways, so they need not produce the same number.
The key comparison is not simply which estimate is “least biased.” One must specify the bias target: bias conditional on the observed stopping stage, bias averaged over all possible stopping paths, or a median target under the stopped sampling distribution. Variance and numerical stability also matter. An estimate that reduces one form of bias may have greater variability or be sensitive to the assumed stopping rule.
Model and Targets for Comparison
Use the canonical normal model with known variance and information fractions \(t_1<\cdots<t_K=1\). The parameter \(\theta\) is the effect on the standardized scale, and the cumulative standardized statistic at look \(k\) satisfies
The ordinary estimate at look \(k\) is \(\hat\theta_k=Z_k/\sqrt{t_k}\). This model assumes the normal sampling law is appropriate, the information fractions and stopping boundaries are specified correctly, and the variance used to standardize the statistic is known. The formulas below concern this model; estimated standard errors, model misspecification, or a different stopping rule can change the sampling law and the comparison.
Conditional and unconditional bias answer different questions. Conditional bias at look \(k\) concerns \(E_\theta(\hat\theta_k-\theta\mid T=k)\); unconditional bias concerns \(E_\theta(\hat\theta_T-\theta)\), averaging over all possible stopping looks. A conditional MLE adjusts the likelihood for the observed stopping event, but it does not generally make the estimate unbiased. A median-unbiased estimate targets a median, not a mean. Expectation correction targets the unconditional mean of the ordinary stopped estimate.
Worked Example: Bias and Variance at an Efficacy Stop
Suppose a standardized estimate \(Y\sim N(\theta,1)\) is observed at an interim look, and the design stops for efficacy when \(Y\ge 1\). Consider the conditional distribution among trials that stop at this look when \(\theta=-0.08\). The model is normal with unit variance, and the selection event is exactly the stated threshold.
For a normal variable truncated below at \(c\), its conditional mean is \(\theta+\lambda(a)\), where \(a=c-\theta\) and \(\lambda(a)=\phi(a)/[1-\Phi(a)]\). Here \(a=1-(-0.08)=1.08\). Thus
The ordinary estimate \(Y\), among trials stopping at this look, has conditional bias \(1.5096-(-0.08)=1.5896\) standardized effect units. The result is a conditional statement: it does not say that the ordinary estimate has this unconditional bias across all paths of a multi-look design.
The conditional variance of a lower-truncated unit-variance normal is \(1+a\lambda(a)-\lambda(a)^2\). Substitution gives
Selection has shifted the conditional mean substantially and reduced the conditional spread in this example. A smaller conditional variance does not mean that selection is harmless: the conditional mean is far from the true value.
Comparing Estimates at the Same Stopping Event
For a trial that stops at the first look, the conditional likelihood is the normal density divided by the probability of crossing the boundary. Its maximizer differs from the ordinary MLE because the denominator depends on \(\theta\). A stage-conditional median estimate instead solves a conditional probability equation. These adjustments are meaningful only if the selection event is correctly represented.
Worked Example: Ordinary, Conditional-Likelihood, and Median Estimates
In the same unit-variance model, suppose the efficacy boundary is \(c=1\) and the observed statistic is \(y=1.5\), so the trial stops at this look. The ordinary estimate is the normal MLE \(y\), namely \(1.500\) standardized effect units.
For the conditional MLE, the truncated-normal log likelihood, apart from constants, is \(-\tfrac12(y-\theta)^2-\log\{1-\Phi(c-\theta)\}\). Setting its derivative to zero gives
Substituting \(y=1.5\) and \(c=1\), solve \(1.5=\theta+\phi(1-\theta)/[1-\Phi(1-\theta)]\). Numerical solution gives approximately \(\theta=-0.131\): at this value \(1-\theta=1.131\), and the inverse Mills ratio is about \(1.631\), giving \(-0.131+1.631\approx1.500\). This conditional-likelihood estimate is much lower than the ordinary estimate because it accounts for the fact that observing a value above 1 was required for stopping.
The stage-conditional median-unbiased estimate solves
With \(y=1.5\) and \(c=1\), this is equivalent to \(\Phi(1.5-\theta)=\tfrac12[1+\Phi(1-\theta)]\). Its numerical root is approximately \(\theta=0.429\). At this value the two sides are both about \(0.858\) to three decimal places. The conditional median estimate is therefore about \(0.429\), different from both the ordinary estimate \(1.500\) and the conditional MLE \(-0.131\).
These are not competing calculations of one identical target. The conditional MLE maximizes a conditional likelihood, while the median estimate calibrates a conditional median. Neither calculation, by itself, establishes unconditional unbiasedness across a multi-look design.
Expectation Correction Across All Stopping Paths
An expectation-corrected estimate uses the unconditional stopped-sampling expectation. For a fixed design, define \(g(\theta)=E_\theta(\hat\theta_T)\). Given the observed ordinary stopped estimate \(y\), the expectation-corrected estimate solves \(g(\theta)=y\). If a unique solution exists, the resulting estimate is calibrated so that the expected ordinary stopped estimate at that parameter value equals the observed value. This is a mean-calibration property, not a guarantee of low variance or small error in every trial.
Worked Example: Deriving the Two-Look Correction Map
Consider two looks with \(t_1=0.5\) and \(t_2=1\). Stop at look 1 if \(Z_1\ge2\); otherwise continue to the maximum look 2. At look 2, the trial stops regardless of whether \(Z_2\) exceeds any efficacy threshold. The ordinary stopped estimate is \(Z_1/\sqrt{0.5}\) if \(Z_1\ge2\), and \(Z_2\) if \(Z_1<2\). The canonical model applies, with \(\operatorname{Corr}(Z_1,Z_2)=\sqrt{0.5}\).
The stopping events are \(A_1=\{Z_1\ge2\}\) and \(A_2=\{Z_1<2\}\). In particular, \(A_2\) includes every value of \(Z_2\) once the trial reaches look 2. Therefore
Let \(a=2-\theta/\sqrt{0.5}\). Since \(Z_1\sim N(\theta\sqrt{0.5},1)\), the truncated-normal identity gives \(E_\theta[Z_1 1(Z_1\ge2)]=\theta\sqrt{0.5}\{1-\Phi(a)\}+\phi(a)\). Also, the canonical joint normal model implies \(E_\theta(Z_2\mid Z_1)=\theta+\sqrt{0.5}(Z_1-\theta\sqrt{0.5})\). Taking its expectation over \(Z_1<2\) yields \(E_\theta[Z_2 1(Z_1<2)]=\theta\Phi(a)-\sqrt{0.5}\phi(a)\). Combining both contributions, including the factor \(1/\sqrt{0.5}\) for the first-look estimate, gives
At \(\theta=0\), \(a=2\), so \(g(0)=\phi(2)/\sqrt{2}\approx0.05399/1.4142=0.03818\) standardized effect units. Thus the ordinary stopped estimate has expectation about \(0.03818\), not zero, at \(\theta=0\) under this design. This calculation accounts for all paths: early stops contribute the first term and every trial reaching look 2 contributes the second. An expectation-corrected estimate for an observed value \(y\) solves \(y=\theta+\phi(2-\theta/\sqrt{0.5})/\sqrt{2}\), using the specified design.
How to Judge the Trade-Offs
There is no universal ranking by bias, variance, or stability. The ordinary MLE is simple and often stable to calculate, but can be appreciably biased among early efficacy stops. The conditional MLE directly conditions on the observed stopping event; its estimate can be sensitive to the boundary and can behave poorly near the edge of the possible truncated data. Median-unbiased estimation has a clear median calibration, but its numerical value need not be close to a mean-corrected estimate. Expectation correction targets the unconditional mean, yet requires integrating over every stopping path and solving a nonlinear equation.
A useful comparison reports the target and the sampling variation together. Conditional bias and variance should be evaluated among trials with the specified stopping event; unconditional bias and variance must average over the complete stopping rule. Do not compare a conditional variance for one stage with an unconditional variance as if they described the same population of trial outcomes.
Diagnostics should check whether the implemented probability calculation matches the actual design. Verify the information fractions, correlation structure, boundaries, and definition of the terminal look. For expectation correction, check that the event sets partition all possible stopping paths, that numerical integration includes the full continuation region, and that the correction equation has a unique, well-conditioned solution in the relevant range. Sensitivity to small changes in boundaries or observed statistics is a warning that the estimate may be unstable.
A common mistake is to define the maximum-look event as “reached look 2 and did not cross its efficacy boundary.” That incorrectly drops trials that reach the maximum and then cross the boundary: they still have \(T=2\), and their stopped estimates contribute to the unconditional expectation. Another mistake is to describe a median-unbiased estimate as unbiased. Median calibration concerns the middle of a sampling distribution, not its mean.
The introductory material on bias correction in multi-look designs motivates adjusting for stopping, but estimator choice still depends on the target and the design. These methods do not repair a misspecified normal model, an incorrectly encoded stopping rule, or unaccounted changes to the analysis plan. They are estimation procedures, not substitutes for clearly defining the estimand and reporting the stopping rule.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For \(Y\sim N(\theta,1)\) observed only when \(Y\ge c\), what quantity does the conditional MLE use that the ordinary MLE omits?
- In the two-look example, why must the second-look contribution include all outcomes with \(Z_1<2\), including those with large \(Z_2\)?
- At \(\theta=0\) in that design, calculate the expectation of the ordinary stopped estimate from the derived correction map.
- Does a median-unbiased estimate necessarily have zero mean bias? Explain.
Solutions. (1) It uses the likelihood conditional on the selection event \(Y\ge c\), including the parameter-dependent denominator \(1-\Phi(c-\theta)\). (2) Look 2 is the maximum look, so every trial not stopped at look 1 reaches it; the stopped estimate is \(Z_2\) regardless of its boundary status. (3) The map gives \(g(0)=\phi(2)/\sqrt{2}\approx0.05399/1.4142=0.03818\) standardized effect units. (4) No. Median-unbiasedness calibrates the median of the relevant stopped distribution to the parameter; it does not require the distribution’s mean to equal that parameter.