Martingale Derivation Of Sequential Logrank Increments

Why derive sequential increments?

At an interim analysis, the logrank score is cumulative: it collects observed-minus-expected event contributions over the risk sets encountered so far. For sequential use, it matters not only how that total is calculated, but also why the contribution accumulated after one look has mean zero under the null, conditional on the history available at that look. A counting-process martingale makes this property explicit and gives a principled account of how variance builds over time.

The derivation below concerns a two-arm trial with a single event type and no tied event times. It explains the score and its variance process; it does not provide a method for handling ties. Risk-set allocation and the ordinary logrank variance can also be described directly through event-level comparisons. The martingale representation shows how those comparisons fit into a process evolving over calendar or follow-up time.

Model, notation, and assumptions

Index participants by \(i=1,\ldots,n\). Let \(Z_i=1\) for a participant assigned to treatment and \(Z_i=0\) for a participant assigned to control. Let \(N_i(t)\) count that participant’s events by time \(t\), and let \(Y_i(t)\) indicate whether the participant is at risk just before \(t\). Write \(Y_i(t-)\) when the left limit is needed to emphasize that the risk status is known immediately before the event time. Define the treatment and control risk-set sizes by \(Y_1(t-)=\sum_iY_i(t-)Z_i\) and \(Y_0(t-)=\sum_iY_i(t-)(1-Z_i)\), with \(Y(t-)=Y_1(t-)+Y_0(t-)\).

Let \(\mathcal{F}_t\) denote the observed history through time \(t\), including who remains at risk and which events have occurred. Under the null hypothesis \(H_0\), assume the two arms have the same event hazard \(\lambda_0(t)\) among participants still at risk. More precisely, conditional on the observed history, participant \(i\)’s event-counting process has intensity \(Y_i(t-)\lambda_0(t)\). Assume the intensity is well defined and integrable over the follow-up interval, and that participants’ event processes have no simultaneous jumps. Censoring must be non-informative for this event-intensity model, so that the stated intensity remains appropriate given the observed history. These assumptions are used to construct the martingales and their variance; violations require a different model or analysis.

From the logrank score to a martingale integral

At each event time, the logrank contribution is the observed treatment event indicator minus the treatment fraction in that event’s risk set. In continuous time, define the predictable treatment fraction just before \(t\) as \(\bar Z(t-)=Y_1(t-)/Y(t-)\), whenever \(Y(t-)>0\). The cumulative score through time \(t\) is

This formula is the cumulative sum of the risk-set contributions: an event in treatment contributes \(1-\bar Z\), and an event in control contributes \(-\bar Z\). To see why it is a martingale under \(H_0\), replace each \(dN_i\) by its martingale part \(dM_i\) plus its compensator \(Y_i(u-)\lambda_0(u)\,du\). The compensator contribution to the score is zero at every time:

because \(\bar Z(u-)=Y_1(u-)/Y(u-)\). Thus

The integrands depend only on the risk set immediately before time \(u\), so they are predictable. A sum of these predictable integrals against martingales is itself a martingale, provided the integrals are well defined and have finite variance over the time interval. Consequently, for an interim time \(t_k\) after a previous look at \(t_{k-1}\),

This is the key sequential statement: under \(H_0\), the next score increment has conditional mean zero given the information at the previous look. It does not say that a particular observed increment must be near zero.

How the variance accumulates

For counting-process martingales with no simultaneous events, the different participants’ event martingales have zero cross-variation. The predictable quadratic variation of \(U(t)\), denoted \(\langle U\rangle(t)\), is therefore the sum of the integrated squared weights times the event intensities:

The sum inside the integral simplifies to a quantity determined by the two risk-set sizes. Since \(Z_i\) is binary, the treatment participants contribute \(Y_1(1-\bar Z)^2\) and the control participants contribute \(Y_0\bar Z^2\). Substituting \(\bar Z=Y_1/Y\) gives

Hence the predictable variance process is

This process increases as event intensity accumulates in risk sets containing participants in both arms. It contributes no variance when one arm has no one at risk, because then \(Y_1Y_0/Y=0\). The variance is predictable: at each instant, its rate depends on the risk set just before that instant and the null event intensity.

The usual event-based logrank variance is the corresponding observable risk-set calculation. In the no-ties setting, estimating the pooled cumulative hazard increment in a risk set by \(dN(u)/Y(u-)\), where \(dN(u)=\sum_i dN_i(u)\), yields the accumulated estimate

Thus each event adds the treatment risk-set fraction times the control risk-set fraction. This event-based \(V(t)\) estimates the predictable variance; it is not the same object as the continuously accumulated bracket \(\langle U\rangle(t)\). Under the martingale assumptions, \(E\{U(t)^2\}=E\{\langle U\rangle(t)\}\) when the score starts at zero and has finite second moment.

Worked sequential score and variance

For successive looks, each interval contributes the score and variance accumulated from the event risk sets in that interval. The score increments are martingale differences under \(H_0\), while the variance increments are nonnegative and add across intervals. The following event sequence illustrates the calculations. Assume no censoring between the listed events, no ties, and a common hazard under the null.

What the martingale property implies across looks

Martingale increments from non-overlapping intervals are uncorrelated under the stated assumptions. For two successive increments, with \(t_0<t_1<t_2\), the justification is

Here \(\Delta U_1\) is known by \(t_1\), and the conditional mean of the later increment is zero. This does not imply that the increments are independent: their distributions can depend on the risk sets and event intensities created by the trial’s evolving history. Nor does it imply that standardized statistics at different looks are independent; the cumulative score at a later look contains the earlier score.

A useful additional consequence is that the cumulative score has mean zero under \(H_0\). Its variance is determined by the expected predictable quadratic variation, and its variance process accumulates over time rather than resetting at each look. These properties motivate sequential analyses that retain the increments or their cumulative score and account for their shared history. A normal approximation may be useful when the information is sufficiently spread across many events, but the martingale mean-zero statement alone does not guarantee that an increment has a normal distribution.

Diagnostics, alternatives, and limitations

The derivation depends on the specified event intensity and risk-set history, so diagnostics should focus on whether those are credible and whether the calculations reflect the observed trial structure. Check that each event is assigned to the correct arm, that each risk set uses participants at risk immediately before the event, and that the score and variance contributions use the same risk set. Risk sets with only one arm present have zero variance contribution and cannot provide a two-arm comparison at that event.

Warning signs include unexpected simultaneous events when a no-ties derivation is being applied, missing or misclassified events, risk-set counts that do not reconcile with enrollment and censoring, or evidence that the common-hazard null intensity is an inadequate working model. Informative censoring can also invalidate the assumed intensity conditional on the observed history. These issues should be investigated rather than addressed by mechanically applying the event-level variance formula. The treatment of tied events requires a tie-aware calculation and is not derived here.

The counting-process route is especially useful for understanding conditional mean zero, variance accumulation, and why later increments are orthogonal to earlier ones. The risk-set allocation route provides a direct event-by-event calculation of observed-minus-expected contributions and their variance; it is often the more convenient computational description. Both connect to the same logrank score under the stated setting, but neither removes the need to check the risk sets and assumptions. The martingale derivation also has limits: it does not itself establish a normal approximation, calibrate a stopping boundary, or justify a particular censoring model.

• Mistake: “The null makes every observed increment equal zero.” Correction: The null makes the increment’s conditional expectation zero; a realized increment can be positive or negative.
• Mistake: “Zero covariance proves that increments are independent.” Correction: The martingale argument gives zero covariance for non-overlapping increments with finite second moments, not independence.
• Mistake: “The predictable variance and the event-based variance estimate are identical.” Correction: The predictable quadratic variation integrates the null intensity; the usual event-based risk-set sum estimates it.