Worked Example: Three Arms With Interim Dropping

Why arm dropping affects the final estimate

In a multi-arm multi-stage trial, interim results determine which active arms continue. If an arm continues because its interim estimate was sufficiently favorable, an estimate that reuses those interim data will tend to be too favorable on average. This is selection bias: the estimate is being examined conditional on an event partly determined by that estimate.

With a shared control, the interim contrasts are also dependent. A control group with an unusually low mean, for example, makes all active-versus-control contrasts look more favorable. A correction must therefore account for both the selection rule and the joint distribution of the contrasts. The earlier material on parametric bootstrap for multi-stage selection provides the general simulation approach; here we apply it to three active arms and show the calculations.

Model, selection rule, and shared-control dependence

Assume independent, normally distributed patient outcomes within and across stages, a common known standard deviation \(\sigma\), and independent stage-one and stage-two patient samples. At stage one, all three active arms share one control group. An arm is retained if its one-sided interim statistic exceeds a prespecified threshold \(c\):

For this rule, the observed selection pattern is the joint event that each retained arm exceeds the threshold and each dropped arm does not. A bootstrap correction should condition on this exact pattern, not merely on the fact that some arms were retained.

Suppose each active arm has \(n_T\) patients and the shared control has \(n_C\) patients at stage one. The variance of each contrast is \(\sigma^2(1/n_T+1/n_C)\). For distinct arms \(j\) and \(k\), their covariance is \(\sigma^2/n_C\): their treatment groups are disjoint, but both contrasts subtract the same control mean. Thus their correlation is positive, not zero.

The ordinary final estimate for a retained arm combines its stage-one and stage-two contrasts. When the stage-specific variances are equal, this is their average. Selection affects that average because the stage-one part helped determine retention; the independent stage-two estimate alone is not selected by this interim rule.

Estimating and removing conditional selection bias

For an arm retained after stage one, write its ordinary all-stage estimate as \(\hat{\theta}_{j,\mathrm{all}}\). With equal stage-specific variances, it is \((\hat{\theta}_{j,1}+\hat{\theta}_{j,2})/2\). Its conditional selection bias at parameter value \(\boldsymbol{\theta}\) and selection pattern \(S\) is

The expectation is conditional on the selection event; it is not the unconditional bias of the estimator. Since the true effects are unknown, a practical parametric bootstrap substitutes fitted effects \(\tilde{\boldsymbol{\theta}}\), simulates trials under the assumed model, and keeps or generates replicates satisfying the observed selection pattern. The estimated conditional bias is the average simulated all-stage estimate minus the fitted effect for that arm. Subtracting this estimated bias from the observed all-stage estimate gives a plug-in bias-corrected estimate.

The bootstrap must reproduce the shared-control covariance, the interim threshold, the stage-specific sample sizes, and the analysis rule used to form the reported estimate. The adjustment is model-based and plug-in: it is not a guarantee of exact unbiasedness. It corrects the point estimate for estimated conditional bias; uncertainty in the correction and uncertainty in the treatment estimate still matter.

Why the correction has this form

The subtraction follows directly from the definition of conditional bias. If the conditional expectation of the ordinary estimate is \(\theta_j+B_j\), then subtracting \(B_j\) centers that conditional expectation on \(\theta_j\):

This identity justifies the direction of the adjustment. In practice \(B_j\) is unknown and replaced by \(\widehat{B}_j\), so the identity does not establish that the estimated correction is exactly unbiased. In particular, the fitted effects used in the bootstrap may themselves be affected by selection.

For equal stage variances, the stage-two component is independent of the selection event and has expectation \(\theta_j\). Consequently, the conditional bias in the all-stage average comes from the stage-one component and is half the conditional bias of the stage-one estimate. This is a useful check on a bootstrap implementation; it does not remove the need to simulate the joint selection event correctly.

Diagnostics, alternatives, and limitations

A correction is only as credible as its selection model and simulation. Check that simulated stage-one contrasts have the intended marginal standard errors and shared-control correlations, that the retention rule matches the protocol exactly, and that the retained-pattern frequency and conditional estimates are stable as the number of bootstrap replicates increases. Very few accepted replicates, large changes across reasonable fitted-effect choices, or estimates that change sharply with small changes in the assumed outcome distribution are warnings that the correction is unstable.

The parametric bootstrap is not the only approach. A stage-two-only estimate avoids bias from the interim selection when stage-two data are independent, but uses less information. A conditional likelihood or other selection-aware model can incorporate the selection event directly, but requires its own distributional assumptions and careful specification. Reporting the unadjusted all-stage estimate alongside a justified correction can aid transparency, provided the unadjusted value is not presented as if selection had no effect.

The bootstrap correction has important limitations. It relies on a correct outcome distribution, variance structure, allocation, selection rule, and treatment-effect parameterization. Plug-in effects may be poor near a selection boundary, and finite simulation adds Monte Carlo error. The point correction alone is not a confidence interval and does not account automatically for the uncertainty of estimating the bias. It also does not resolve multiplicity or make a causal claim beyond the randomized-trial estimand.