Conditional Error Across Multiple Interim Looks

Why Conditional Error Must Be Updated Across Looks

After an interim analysis, the probability of a later rejection depends on what has already been observed. If the trial continues through several more looks, that probability includes all remaining chances to cross a rejection boundary—not just the chance of crossing at the final analysis. The conditional error is this remaining probability under \(H_0\), conditional on the interim data and on having reached the interim without rejection.

This calculation is useful when a design is being assessed or changed after interim data become available. It makes explicit the Type I error still available at each history. The introductory material on conditional error establishes the principle; here the focus is its recursive calculation across multiple future looks.

Canonical Model and Conditional Transitions

Assume a one-sided test with \(K\) planned looks at strictly increasing information times \(0<t_1<\cdots<t_K=1\). Under the null model, the standardized statistics \(Z_1,\ldots,Z_K\) have the canonical joint distribution: each has a standard normal marginal distribution, and \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\) for \(i<j\). The design rejects at look \(i\) if \(Z_i\ge c_i\), where \(c_i\) is the prespecified boundary. The information times and boundaries are fixed for this calculation.

This model is appropriate when the standardized score process has independent increments under \(H_0\), with information indexing its variance. It is also necessary that the stated information times, statistic, and boundaries describe the design being evaluated. The canonical joint distribution is not automatic for arbitrary statistics or data-dependent information; if its assumptions fail, the recursion below need not give valid conditional error.

For adjacent looks, the canonical model gives the transition \(Z_{i+1}\mid Z_i=z\sim N\!\left(\sqrt{t_i/t_{i+1}}\,z,\;1-t_i/t_{i+1}\right)\). Thus the transition density is normal with mean \(\sqrt{t_i/t_{i+1}}\,z\) and variance \(1-t_i/t_{i+1}\). This Markov property is what makes a backward recursion possible: once the current statistic is known, earlier statistics do not provide additional information about future statistics under this model.

Backward Recursion for Remaining Error

Write \(E_j(z)\) for the conditional probability of rejection at some later look, given that the trial has reached look \(j\), has not rejected, and \(Z_j=z\). Set \(E_K(z)=0\), because there are no looks after \(K\). For \(j<K\), let \(f_{j+1}(x\mid z)\) be the density of the transition distribution from look \(j\) to \(j+1\). The recursion is

The first integral is the probability of rejecting immediately at the next look. In the second, the next statistic falls below its boundary, so the trial continues; the remaining error from that point is \(E_{j+1}(x)\). These cases are disjoint and cover all possible outcomes at look \(j+1\). Starting at \(j=K-1\) and working backwards therefore accounts for rejection at every future look without counting any path more than once.

The design’s overall Type I error is different from a conditional error at a realized interim value. It is the probability, under \(H_0\), of crossing at any look from the start. Conditional error is calculated at a particular history, after conditioning on survival and the observed statistic. In an error-controlled redesign, the relevant conditional budget is the original design’s conditional error at that history; it is not automatically the overall \(\alpha\).

Calculation Checks, Alternatives, and Limitations

A practical calculation proceeds backwards from the last look. At each step, compute the probability of crossing the next boundary, then integrate the conditional error beyond that look over values that do not cross. Numerical integration can be used when the recursion has no closed form. The resulting conditional error is between 0 and 1, and it cannot be smaller than the probability of crossing at the immediate next look, because that probability is included in the total.

A direct alternative is to calculate a multivariate-normal probability for the union of all future crossing events, conditional on the interim data. It expresses the same probability but can be computationally demanding as the number of remaining looks grows. The recursion instead uses one-dimensional integrals and the Markov transition. For a single remaining look, both approaches reduce to a normal tail probability.

• Check the conditioning history. Use the observed statistic at the current look and condition on survival through that look. A rejection at an earlier look means the future recursion is irrelevant for that path.
• Check the transition variance. It must be \(1-t_i/t_{i+1}\), not \(t_i/t_{i+1}\). The transition variance is positive when information strictly increases.
• Check that every future look is included. A final-look tail alone is not the conditional error if there is an earlier future boundary.
• Check the model and rule. The displayed recursion assumes the canonical transitions and fixed upper boundaries. A different stopping rule, statistic, or conditional distribution requires a corresponding calculation.

A common mistake is to treat the conditional error after an interim look as the original \(\alpha\), or to add separate marginal crossing probabilities as if the future looks were independent. The remaining error is conditional on the observed history, and future statistics are correlated through the canonical process. The recursion respects both facts by using conditional transition distributions and by partitioning paths into crossing now versus continuing.

The method has limits. It does not establish that a chosen canonical model is appropriate, nor does it fix invalid information-time estimates or an incorrectly specified boundary. If the transition law does not depend on the observed history only through the current statistic, the scalar recursion is insufficient: the conditioning state must contain all relevant information. Numerical integration should also be checked for stability, especially when boundaries are extreme or many looks remain.