Event-Driven Selection Rules

Why Define a Look by Events?

In a time-to-event MAMS design, an intermediate event can provide information for deciding which experimental arms to retain. A selection look may therefore be triggered when a prespecified number of intermediate events has accrued, rather than at a prespecified calendar date. This makes the rule operationally clear, but the event count alone does not determine how precisely the arms can be compared: the information also depends on which participants remain at risk and on the treatment effects in the model.

Here the comparison is a Cox proportional-hazards comparison of each experimental arm with a shared control, using the intermediate time-to-event outcome. The selection rule can rank arms using their estimated intermediate log hazard ratios. It does not, by itself, establish that an arm improves the final endpoint or specify a calibrated threshold for retaining an arm.

Assume there are \(K\) experimental arms, indexed \(j=1,\ldots,K\), and a shared control indexed \(0\). Use treatment indicators \(x_j\), with the control represented by the zero vector and arm \(j\) represented by a vector whose \(j\)th entry is one. A Cox model specifies the hazard in arm \(j\) as the control hazard multiplied by \(\exp(\beta_j)\), where \(\beta_j\) is the log hazard ratio for the intermediate outcome. The principal assumptions here are proportional hazards over the analysis period, independent participants across individuals, non-informative censoring conditional on the model covariates, and correctly recorded event and risk-set times. For the derivation below, assume distinct event times; tied events require a stated tie method, such as the Breslow or Efron approximation.

Information at an Event

Immediately before event time \(t_i\), let \(Y_j(t_i)\) be the number at risk in arm \(j\). Conditional on one event occurring in that risk set, the Cox model assigns the event to an arm with probability proportional to the number at risk times its relative hazard. Thus the model-implied event probability is

with \(\beta_0=0\) for control. These \(q_{ij}\) are not generally the raw proportions at risk. The raw proportions \(Y_j(t_i)/\sum_hY_h(t_i)\) equal the event probabilities under the null model \(\boldsymbol{\beta}=0\), but away from the null the relative hazards weight the risk-set counts.

Let \(\mathbf{x}_i\) denote the treatment-indicator vector for the individual who experiences event \(i\). The event's partial log-likelihood contribution, up to terms not involving \(\boldsymbol{\beta}\), is the observed treatment indicator dotted with \(\boldsymbol{\beta}\), minus the log of the weighted risk-set total. Differentiating gives the score and its information contribution:

Here \(\mathbf{q}_i=(q_{i1},\ldots,q_{iK})^T\) contains the model-implied probabilities for experimental arms; the control probability is omitted from the vector because its treatment-indicator vector is zero. The matrix is the covariance matrix of the treatment-indicator vector under the model-implied event distribution in that risk set. The observed event identity contributes to the score, while the curvature of the partial likelihood is determined by the weighted risk set.

From Event Count to Contrast Precision

At a look, the observed information is the sum of the event contributions over all events observed by that look. In practice, it can be evaluated at the fitted Cox coefficients, or at a prespecified value for design calculations. The estimated covariance matrix of the log hazard-ratio estimates is approximately the inverse information matrix when the model is identifiable and the large-sample approximation is adequate. Shared control appears in the off-diagonal entries: estimates for two treatment-versus-control contrasts are generally correlated.

The expression is a local information calculation: for inference it is ordinarily evaluated at the fitted model, while a planning approximation may evaluate it under the null or another specified effect configuration. Under the null, the event probabilities reduce to raw risk-set proportions. If risk-set composition is approximately stable, with null proportions \(p_0,\ldots,p_K\), then \(D\) events give the approximation \(D\{\operatorname{diag}(p_1,\ldots,p_K)-\mathbf{p}\mathbf{p}^T\}\). It is an approximation, not a universal conversion from event count to information.

Worked Information Calculations

What the Event-Driven Rule Does—and Does Not—Guarantee

The information formula justifies why an event-driven rule can be useful: event occurrence contributes directly to the partial likelihood for the hazard contrasts. An event target also gives a reproducible trigger that does not depend on a fixed calendar date. However, an event count does not guarantee a particular amount of information. If one treatment arm has few people at risk, its event-indicator variation may be small; if an arm has no participants at risk, its contrast may not be estimable from that period. The shared-control covariance also means that the precision of the vector of comparisons is not captured by separate standard errors alone.

A practical selection rule should specify the event-count trigger and the comparison statistic in advance. For example, it may rank the fitted log hazard ratios, with more negative values favoring an arm when fewer intermediate events are desirable. The event-driven trigger determines when the statistic is evaluated; it does not determine the ranking direction, a threshold for dropping an arm, or the error properties of the overall selection procedure.

The information calculation follows from the curvature of the Cox partial likelihood. At an event, the negative second derivative is the covariance of the treatment indicators under the weighted risk-set distribution. Summing those curvatures gives the total information, which explains why changing risk-set composition changes information even when the event count is held fixed.

Diagnostics, Alternatives, and Limitations

At each look, inspect the risk-set sizes by arm, the fitted coefficients, and the eigenvalues or condition of the information matrix. Warning signs include an arm with very few participants at risk, a nearly singular information matrix, estimates with very large standard errors, or substantial changes in coefficients as events accrue. These features indicate that the nominal event target may not have delivered stable comparisons. The model also requires scrutiny: marked departures from proportional hazards or outcome-dependent censoring can undermine the Cox-based comparison, regardless of how many events have accrued.

Three timing choices answer related but distinct operational needs. A calendar-time look is scheduled at a date and may contain widely varying numbers of events. An event-driven look waits for an event target, making the endpoint count explicit while leaving information dependent on the risk sets. An information-driven look targets a prespecified information level, which more directly controls the intended precision but requires monitoring or forecasting information rather than only counting events. None automatically solves model misspecification or selection-threshold calibration.

Tied event times require a consistent partial-likelihood tie method; the distinct-event derivation should not be applied as if tied events were independent sequential failures without qualification. In addition, the inverse-information variance is an asymptotic approximation. Small event counts, sparse arms, heavy censoring, or unstable parameter estimates can make it unreliable. Finally, an event-driven selection rule concerns the intermediate outcome used for the arm decision. Its association with a final endpoint is a separate scientific question.