Median-Unbiased Estimates and Confidence-Interval Duality

Why Invert the Stopped Sampling Law?

When a trial can stop early, the observed estimate is selected through a rule that depends on interim data. Its sampling distribution is therefore generally different from the distribution it would have had at a fixed sample size. An estimate that ignores this distinction can be biased, and a confidence interval based on the fixed-sample law can have incorrect coverage. The central idea here is to use the distribution generated by the actual stopping rule for both estimation and interval construction.

Let \(\theta\) denote a scalar treatment effect, and let \(D\) denote all data observed up to stopping, including the stopping stage. Choose a scalar statistic \(T=T(D)\), with larger values representing evidence for larger treatment effects. Write \(F_\theta(t)=P_\theta(T\le t)\) for its cumulative distribution function when the true effect is \(\theta\), with probabilities taken under the complete, pre-specified stopping rule. The observed value is \(t_{\mathrm{obs}}\). The distribution must describe the statistic as actually recorded, not as though observation had continued to a fixed maximum sample size.

A median-unbiased estimate is a statement about the repeated-sampling median, not the mean. It need not equal the ordinary observed treatment-effect estimate, and it need not be the midpoint of a confidence interval. Its interpretation also depends on which statistic and which stopping rule define \(F_\theta\).

Conditions for Inversion and Confidence-Interval Duality

The simple inversion formula relies on several conditions. The parameter is scalar and ordered. The statistic has a continuous distribution under each candidate value of \(\theta\). At the observed statistic, \(F_\theta(t_{\mathrm{obs}})\) varies continuously and strictly decreases as \(\theta\) increases. Finally, its limits as \(\theta\) ranges over the parameter space must reach the probabilities being inverted. These conditions give a unique solution for the median and for the two confidence limits. If the model has nuisance parameters, the probabilities must be calibrated in a way that accounts for them; plugging in an unaccounted-for nuisance estimate does not automatically preserve coverage.

For a central confidence interval with nominal level \(1-\alpha\), invert the two tail probabilities of the stopped law. At the lower endpoint, the observed statistic is at the upper \(1-\alpha/2\) quantile under that candidate parameter. At the upper endpoint, it is at the lower \(\alpha/2\) quantile. The endpoint equations and the interval set are:

Because \(F_\theta(t_{\mathrm{obs}})\) decreases with \(\theta\), the first equation gives the lower limit and the second gives the upper limit. For a fixed true \(\theta\), continuity of the distribution implies that \(F_\theta(T)\) is uniform on \([0,1]\). Thus the probability that the true value belongs to the inverted set is \(P_\theta(\alpha/2\le F_\theta(T)\le 1-\alpha/2)=1-\alpha\). This is unconditional coverage under the specified stopping rule: the probability averages over all possible stopping stages and outcomes generated by that rule.

Why the Duality Works

The confidence interval is obtained by accepting candidate parameter values whose observed stopped statistic is not too extreme in either tail. The same CDF used to define the median estimate supplies these tail probabilities. The connection can be checked directly. For any continuous \(F_\theta\), the probability integral transform gives \(F_\theta(T)\sim\mathrm{Uniform}(0,1)\) when \(\theta\) is true. Retaining candidate values for which this transformed statistic lies between \(\alpha/2\) and \(1-\alpha/2\) therefore retains the true value with probability \(1-\alpha\).

The median estimate is the value for which the transformed observed statistic equals \(1/2\). The interval limits instead correspond to the two tail cutoffs. They are symmetric in probability scale, but not necessarily symmetric in treatment-effect units. A skewed stopped-law distribution can give unequal distances from the median estimate to the two limits.

This is the confidence-interval duality: confidence limits are obtained by inverting tests of candidate parameter values. Here the tests use the stopped-law tail probabilities. The equal-tailed intervals from the stopped sampling law use the same principle; the additional point-estimation perspective identifies the parameter value that makes the observed statistic a median. Neither procedure licenses replacing the stopped law with a fixed-sample distribution.

Discrete Laws, Diagnostics, and Limits

If \(T\) is discrete, \(F_\theta(T)\) is not continuously uniform. Ties create jumps, so solving an equation such as \(F_\theta(t_{\mathrm{obs}})=1/2\) may have no solution or may produce more than one. A median can still be defined by the two inequalities \(P_\theta(T\le t_{\mathrm{obs}})\ge1/2\) and \(P_\theta(T\ge t_{\mathrm{obs}})\ge1/2\). The resulting set of candidate values can be an interval or a more complicated set. Similarly, nonrandomized tail inversion can yield coverage greater than the nominal level because attainable probabilities jump over the desired cutoffs. Exact nominal coverage in a discrete model may require randomization; a nonrandomized construction should report its actual coverage guarantee rather than claim exact equality.

Before trusting an inversion, check that candidate CDF values move in the required direction and that each root exists within the parameter space. A flat region, multiple roots, or a root outside the allowed parameter range signals nonuniqueness or a boundary limit, not permission to select a convenient answer without explanation. Also check that the computation includes the actual efficacy and futility rules, the maximum sample size, and the distributional assumptions for the endpoint. If the standard error is estimated or nuisance parameters are present, the stated coverage result requires a method that accounts for that uncertainty.

A frequent mistake is to calculate a fixed-sample confidence interval from the observed sample size and call it a stopped-law interval. That calculation uses the wrong distribution whenever the monitoring rule changes the sampling behavior. The correction is to evaluate candidate effects under the pre-specified full stopping rule and invert those probabilities. Another mistake is to assume that a median-unbiased estimate must be the interval midpoint. It is the solution to a 50% CDF equation; the interval endpoints are solutions to two different tail equations.

Median inversion is useful because it gives a clear central estimate aligned with the sampling distribution relevant to the stopped design, while its tail inversion provides a principled uncertainty interval. Its limitations are equally important: uniqueness depends on ordering and continuity, coverage depends on correct calibration of the stopped law, and a scalar statistic may not capture all features of the observed data. Different choices of statistic or conditioning target can produce different valid procedures.