Sequential Inference for RMST Contrasts

Why Sequential Inference for an RMST Contrast Needs Its Own Boundaries

Restricted mean survival time (RMST) is useful when treatment effects vary over time, because it summarizes survival through a fixed horizon without requiring proportional hazards. But repeatedly testing the same RMST contrast creates a new problem: even if each look uses a valid test at level \(\alpha\), the chance of at least one false positive across several looks can exceed \(\alpha\). A sequential procedure therefore needs boundaries calibrated for the whole sequence of analyses, not just for one analysis in isolation.

This tutorial concerns a single treatment contrast and a horizon \(\tau\) chosen before examining interim results. At look \(k\), the estimate and its standard error provide a standardized statistic. Its distribution is related to the statistics at other looks because the data are cumulative: later analyses generally include many of the same participants as earlier ones. The boundary calculation must account for that dependence.

For a one-sided superiority test, the hypotheses are \(H_0:\Delta(\tau)\leq 0\) and \(H_a:\Delta(\tau)>0\). A positive contrast means greater average survival time through \(\tau\) in the treatment arm. The test is not a claim that treatment improves survival at every time point: it concerns the prespecified RMST contrast alone.

Assumptions for Repeated RMST Testing

The procedure begins with the estimand, horizon, analysis schedule, and error rate specified in advance. The horizon \(\tau\), treatment contrast, direction of testing, interim looks, and boundary-calibration method must not be chosen after inspecting the accumulating treatment differences. Otherwise the nominal error calculation does not describe the analysis actually performed.

Each look uses an RMST estimator appropriate to the observed follow-up and censoring. For the usual Kaplan–Meier-based estimator, the relevant conditions include independent treatment assignment and independent censoring within treatment arm (or conditional independent censoring under the model used). The estimator must be consistent, and its standard error must account for censoring and the estimation of the survival curve. The horizon must also be supported by follow-up: if little information is available near \(\tau\), the estimate may be imprecise or rely heavily on assumptions about censoring.

For boundary calibration, the standardized estimates are commonly approximated jointly by a multivariate normal distribution under the null. Write \(Z=(Z_1,\ldots,Z_K)\), with mean zero under the boundary null and correlation matrix \(R\). This is an approximation, not a guarantee of exact finite-sample normality. The correlations should reflect the repeated RMST estimators. A canonical group-sequential model sometimes uses \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\) for \(i<j\), where \(t_k\) is the information fraction at look \(k\). That relation should be used only when the design and statistic justify it. For RMST contrasts, the joint covariance can instead be estimated from the influence functions or obtained by suitable design-based simulation; one should not assume canonical correlations merely because looks are ordered in time.

How Alpha Spending Produces Sequential Boundaries

Let \(\alpha\) be the desired one-sided type I error rate and let \(g(t)\) be a nondecreasing spending function with \(g(0)=0\) and \(g(1)=1\). At look \(k\), the target cumulative probability of crossing an upper efficacy boundary under the null is \(\alpha g(t_k)\). If \(b_k\) is the boundary at look \(k\), it is chosen so that the probability of first crossing at that look equals the additional amount spent since the previous look:

For the first look, the probability on the left is simply \(P_0(Z_1\geq b_1)\). At each later look, the calculation accounts for the fact that trials crossing earlier boundaries would already have stopped. Summing the mutually exclusive first-crossing probabilities through look \(k\) gives \(\alpha g(t_k)\); when \(t_K=1\), the total is \(\alpha\) under the calibration model.

The boundary equation is generally solved numerically using the joint null distribution. A useful check is that the incremental crossing probability at look \(k\) cannot exceed the marginal probability \(P_0(Z_k\geq b_k)\). Therefore, if the desired increment is \(q\), the boundary must satisfy \(b_k\leq\Phi^{-1}(1-q)\) under a standard normal marginal. A proposed boundary above that limit cannot deliver the claimed increment, regardless of the correlation structure.

Worked Examples

Diagnostics, Alternatives, and Limitations

Before using a sequential RMST test, verify that the RMST horizon and contrast were prespecified, that each look uses a valid variance estimate, and that the covariance model describes the repeated estimates. Compare the assumed correlations with estimates from the design or from simulations under the null. Check that the computed boundaries reproduce the target cumulative crossing probabilities, and that the final cumulative probability is the planned \(\alpha\). If a boundary is reported, check that its marginal tail probability is at least as large as the incremental crossing probability it is supposed to produce.

A common mistake is to compare each interim \(Z_k\) with the ordinary single-analysis critical value and conclude that the nominal \(\alpha\) applies to the whole trial. That ignores repeated opportunities to reject. Another mistake is to add marginal tail probabilities as if they were the actual cumulative crossing probability. They give a union-bound upper limit, not generally the exact crossing probability, because the statistics are correlated. The correction is to use a joint model for exact alpha-spending calibration, or to describe the result honestly as a conservative bound when using Bonferroni thresholds.

Compared with a fixed-final-analysis test, sequential boundaries can permit an early efficacy conclusion while preserving a planned overall error rate. Compared with exact alpha-spending calibration, Bonferroni monitoring is simpler and less dependent on modeling the correlations, but may lose power because it can spend less than the available error. Exact calibration is only as reliable as its joint distribution approximation and covariance inputs. With small samples, heavy censoring near \(\tau\), unstable variance estimates, or changing analysis rules, nominal calibration may be inaccurate; design-based simulation can help assess operating characteristics.

For a two-sided test, the procedure must allocate the total error across both tails, commonly by calibrating lower and upper boundaries jointly with total null crossing probability \(\alpha\). The two one-sided tails should not each be assigned the full \(\alpha\). In all cases, crossing an efficacy boundary supports rejection of the specified null for the prespecified RMST contrast; it does not establish a benefit on every survival-time summary or justify causal interpretation beyond the randomized-trial estimand and its assumptions.