Prespecifying Weights Across Interim Looks

Why the Weighting Rule Must Be a Design Decision

A weighted logrank test can give more influence to events occurring at some parts of follow-up than to others. That can be useful when treatment effects are expected to change over time, such as when benefit is delayed. But a weight rule is part of the test itself: it determines how event information is combined. If the rule is chosen or altered after examining interim outcomes, the monitoring procedure is no longer simply the one originally planned.

Prespecification therefore means more than naming a weight family. The protocol needs to say how the weight at each event time is obtained, which data are used to estimate any survival quantities in the rule, which parameter values apply, how ties and risk sets are handled, and when analyses occur. A clearly specified rule can still produce different realized weights in different trials; what is fixed in advance is the algorithm that produces them.

Weighted Logrank Scores and Survival-Based Weights

Consider a randomized two-group time-to-event study. At distinct event time \(t_j\), let \(Y_{1j}\) and \(Y_{0j}\) be the numbers at risk in the treatment and control groups immediately before the event time, with \(Y_j=Y_{1j}+Y_{0j}\). Let \(d_{1j}\) and \(d_{0j}\) be the numbers of events at that time and \(d_j=d_{1j}+d_{0j}\). The observed-minus-expected treatment event contribution is

At a look that includes event times through \(t_L\), a weighted logrank score is the sum of these contributions multiplied by their weights. With \(w_j\) denoting the prespecified rule’s weight at \(t_j\),

Positive \(U_w\) means more treatment-group events than expected under equal event hazards, on the scale determined by the weights. For the examples below, define \(Z_w=-U_w/\sqrt{V_w}\), so a positive value points toward fewer weighted events in the treatment group. The sign convention must be stated, not inferred from the word “benefit.”

For tied event times, under the usual conditional allocation argument, the variance contribution is the weight squared times the hypergeometric variance:

This expression assumes \(Y_j>1\). When event times are distinct, \(d_j=1\), and the contribution reduces to \(w_j^2Y_{1j}Y_{0j}/Y_j^2\). The standardization is useful for describing the accumulated statistic; it does not, by itself, specify the sequential decision boundaries.

One common family uses the pooled Kaplan–Meier estimate. Write \(\hat S_{\mathrm{pool}}(t_j-)\) for the pooled estimate of survival immediately before \(t_j\), calculated using both randomized groups’ risk sets and events. A Fleming–Harrington-type rule is

The rule with \(\rho=\gamma=0\) gives weight one and is the ordinary logrank score. Positive \(\rho\) tends to retain more weight while estimated survival is high; positive \(\gamma\) gives more weight after the estimated cumulative event experience has grown. These are properties of the formula, not guarantees that a particular trial will have greater power.

What a Complete Prespecification Contains

Before the first interim analysis, specify a reproducible rule rather than leaving choices to the analyst at each look. In particular, record:

• The estimand and test direction: identify the event, treatment group, and which sign of \(U_w\) is favorable.
• The weight function and parameters: for a survival-based rule, state the exact formula and numerical \(\rho,\gamma\), rather than writing only “use a late-weighted test.”
• The survival estimator: state whether \(\hat S\) is pooled across randomized groups, how delayed entry is handled if applicable, and that the value used at an event time is the left limit, before that time’s events.
• Event and risk-set conventions: define the event-time ordering, the handling of tied times, and whether the variance uses the tied-event correction.
• The looks: state their planned timing rule, such as prespecified cumulative event targets or calendar dates, and how the analysis dataset is cut off at each look. Do not let an observed treatment contrast determine whether or when a scheduled look occurs unless that rule is itself part of the design.
• Operational details: identify data cut-off, censoring, and event adjudication conventions that affect which observations enter the calculation.

“Use pooled Kaplan–Meier weights with \(\rho=1\)” is incomplete if \(\gamma\), the time convention, or the look schedule is unspecified. Conversely, a protocol can use a deterministic calendar-time function \(w(t)\) rather than a survival estimate, provided its time origin, parameters, and evaluation convention are fixed. The key distinction is between a fixed rule and weights selected after observing which pattern of interim results appears most favorable.

Why the Left Limit and Pooled Estimate Matter

The value \(\hat S_{\mathrm{pool}}(t_j-)\) is computed from pooled event history strictly before \(t_j\). At a distinct event time, the Kaplan–Meier estimate changes only after processing that time’s event. Using the post-event estimate instead would assign a different weight to the event that caused the change. The difference may be small, but the analysis would no longer follow the stated convention.

Pooling both groups is a deliberate part of the rule. It provides one common weight for the event-time contribution rather than separate weights for treatment and control. The estimate is still calculated from observed data, so prespecifying its use does not mean the numerical weights are known before the trial. It means the source of the estimate, the formula, and its timing are known in advance.

A Short Justification for the Score and Variance

Under the null hypothesis of equal event hazards in the two randomized groups, condition on the risk set and on \(d_j\), the total number of events at a tied event time. The \(d_j\) event labels are then allocated between groups according to the risk-set proportions under the standard logrank conditional argument. Thus the treatment event count \(d_{1j}\) has conditional mean \(d_jY_{1j}/Y_j\), making the unweighted observed-minus-expected contribution have conditional mean zero.

Its conditional variance is the variance of a hypergeometric count: the number drawn is \(d_j\), the treatment fraction in the risk set is \(Y_{1j}/Y_j\), and the finite-population correction is \((Y_j-d_j)/(Y_j-1)\). Multiplication by a known weight \(w_j\) multiplies that contribution’s variance by \(w_j^2\). Summing the resulting contributions gives the displayed variance formula under the usual logrank variance calculation. This explains both why weights enter linearly in the score and why they are squared in the variance.

The argument also indicates what must be treated carefully in a sequential setting: the weight assigned to a contribution must follow the prespecified rule and be available from information defined before that event contribution is formed. Estimating weights from pooled prior survival history is one common convention. The full repeated-monitoring operating characteristics still depend on the specified design and its analysis; the score formula alone is not a guarantee of a particular error rate.

Timing the Looks and Comparing Alternatives

The weight rule and the look schedule are linked operationally. For an event-driven design, specify the event-count targets and the rule for handling a look that coincides with a tied event time. For calendar-driven monitoring, specify the cut-off and which events are included. Information time can describe the amount of accumulated score information, but a plan still needs a concrete procedure for identifying each analysis dataset. Planned looks should not be moved selectively because a particular observed treatment pattern seems promising.

An ordinary logrank test uses constant weights and is a straightforward alternative when a common hazard-ratio effect is a suitable working expectation. A prespecified weighted logrank test trades some emphasis across time to target a hypothesized non-proportional pattern; it can be less efficient if that pattern is wrong. Restricted mean survival time (RMST) contrasts offer a different summary, based on average survival time through a chosen horizon rather than a weighted event score. The choice concerns the scientific question and design, not simply which observed interim result looks larger.

Prespecified weights do not ensure robustness to an incorrect effect-onset assumption, nor do they replace a complete sequential design. They also cannot remove the consequences of sparse late risk sets: late-weighting may emphasize event contributions based on few participants, giving unstable information. Inspect the planned risk-set sizes and plausible event patterns during design, and report the actual weight rule and look timing transparently.

Common Mistakes and Diagnostics

• Writing only “use a late-weighted test.” That leaves the function, parameter values, and survival estimator open. Give the exact formula and every numerical parameter.
• Using the post-event survival estimate by accident. This changes the weight at the event that updates the Kaplan–Meier curve. State and use \(\hat S(t_j-)\) if the rule is based on pre-event survival.
• Changing the parameters after inspecting interim results. A new parameter choice is not the originally prespecified rule. Keep the chosen values fixed across looks.
• Failing to define which groups determine survival weights. Specify pooled or another explicitly justified estimator; do not leave the source to analyst discretion.
• Confusing a fixed rule with fixed numerical weights. A pooled-survival rule can yield different realized weights as data accumulate, even though its algorithm is unchanged.
• Ignoring late risk-set depletion. Check planned numbers at risk and event availability near later looks. A formula that emphasizes late events cannot create information when few participants remain under observation.

A useful design diagnostic is to reconstruct the weight sequence from the written specification using hypothetical event histories. If two analysts can follow the plan and obtain different weights because of an unstated pooling, tie, or timing choice, the prespecification is incomplete.