MaxCombo Tests for Uncertain Effect Timing

Why Combine Several Weighted Logrank Tests?

A weighted logrank test can emphasize events at different parts of follow-up. An early-weighted test may be sensitive to an early treatment effect, while a late-weighted test may be more sensitive to a delayed effect. If the timing is uncertain, choosing one weight after looking at the observed treatment pattern risks overstating evidence. A MaxCombo test instead specifies several weights in advance and combines their test statistics by taking their maximum. Its calibration accounts for the fact that the statistics usually use the same patients and events, so they are correlated.

The procedure is aimed at a prespecified, one-sided question: is there evidence of a treatment benefit in the direction encoded by the score? It is not a license to try many weights and report whichever looks most favorable without adjusting for that search. Here, “maximum” means the largest of the signed, benefit-oriented standardized statistics—not the largest absolute statistic.

For clarity, consider two randomized groups and an event time \(t_i\). Let \(Y_1(t_i)\) and \(Y_0(t_i)\) be the numbers at risk in the treatment and control groups just before that time, \(Y(t_i)=Y_1(t_i)+Y_0(t_i)\), and \(d(t_i)\) the total number of events. If \(O_1(t_i)\) is the treatment-group event count, its null expected count is \(Y_1(t_i)d(t_i)/Y(t_i)\). A weighted score can be written as

Thus, fewer treatment events than expected contribute positively. The variance \(V_j\) is the usual variance for that weighted logrank score, accounting for the risk sets, weights, and ties as appropriate. In practice, the scores and their covariance are computed together; one should not assume the component statistics are independent merely because their weights differ.

Null Calibration: The Joint Distribution Matters

The usual large-sample justification is that, under the null hypothesis of no treatment difference in the event-time distributions, the vector of prespecified weighted logrank scores is approximately multivariate normal with mean zero. Let \(C_{jk}=\operatorname{Cov}(U_j,U_k)\), estimated under the null, and let \(R\) be the correlation matrix of the standardized statistics. Its entries are

Under this approximation, \((Z_1,\ldots,Z_K)\) has a multivariate normal null distribution with mean vector zero and correlation matrix \(R\). If the observed maximum is \(m\), the one-sided p-value is

The probability on the right is a joint multivariate normal probability. It can be evaluated with a multivariate normal calculation or approximated by simulation from the specified joint null distribution. The same principle gives a level-\(\alpha\) critical value \(c\): choose \(c\) such that \(P(\max_j Z_j\leq c\mid H_0)=1-\alpha\). This calibration controls the probability of rejecting \(H_0\) at the prespecified analysis under the null, subject to the approximation and assumptions.

What the Maximum Buys—and What It Costs

Taking the maximum allows the data to obtain evidence through whichever prespecified weight is most responsive to the actual effect pattern. If treatment benefit is delayed, for example, a late-emphasizing statistic might be larger than an ordinary logrank statistic. But maximizing over several statistics creates more opportunities for a large value under the null. The joint calibration raises the rejection threshold, or equivalently adjusts the p-value, to account for those opportunities.

The ordinary logrank test is a single component with a particular time weighting and can be efficient when its assumptions about the effect pattern are appropriate. A single prespecified weighted logrank test can be more focused when the relevant timing is well justified in advance. MaxCombo trades some of that focused power for protection against uncertainty across its chosen weight patterns. It does not guarantee greater power than every single test for every alternative.

For two standard normal components with correlation \(\rho\), the null probability is instead \(P(Z_1<m,Z_2<m)\) under that correlation. With more components, the full matrix \(R\) matters. A common practical approach is to simulate many vectors from the estimated joint null distribution, calculate the maximum for each vector, and compare the observed maximum with this simulated reference distribution. Simulation error should be small relative to the desired precision of the p-value or critical value.

Power When Effect Timing Is Uncertain

Under an alternative, the standardized vector is often approximated by a multivariate normal distribution with a nonzero mean vector \(\boldsymbol{\delta}\) and a covariance matrix that may be approximated by its null value for planning. The power of a test rejecting when \(M>c\) is then \(P(\max_j Z_j>c)\) under that alternative distribution. If the effect timing matches one of the prespecified weights, the corresponding component may have a relatively large mean. If it falls between the patterns represented by the weights, no component is guaranteed to be highly responsive.

Assumptions, Diagnostics, and Limitations

The large-sample null calibration relies on standard weighted logrank conditions. Treatment assignment should be randomized or otherwise support the intended group comparison; subjects should be independent; and censoring should be noninformative for the event process within the analysis groups or strata. The weights and the component family must be specified before examining the treatment contrasts. The score variances and cross-covariances must be calculated consistently with the chosen weights and design. A small number of events, sparse risk sets, or extreme weights can make a normal approximation less reliable.

Proportional hazards are not required for the test to be defined or for its null calibration. Non-proportional effects are part of the motivation for using multiple time weights. However, a single MaxCombo p-value does not summarize the shape, timing, or clinical size of an effect; effect estimates and survival summaries still need context. Nor does a significant result establish which component represents the “true” timing. The largest observed component is selected by construction and may be an unstable description of the effect pattern.

Useful checks include confirming that the intended weights were fixed in advance, verifying the signs and scales of all scores, inspecting the estimated correlation matrix for valid entries and plausible dependence, and checking that the computation uses the full joint distribution. If the correlation matrix is nearly singular, some components may be redundant; this is not automatically an error, but it can signal that the nominal collection of weights supplies little distinct information. If estimated correlations are invalid or numerical calculations are unstable, investigate the covariance calculation, ties, risk sets, and simulation procedure rather than silently substituting independence.

MaxCombo is also distinct from repeated interim testing. Its adjustment handles the maximum across the specified weighted statistics at the analysis being calibrated; it does not by itself account for repeatedly examining accumulating data at multiple analysis times. Any sequential monitoring plan requires an appropriate sequential design and calibration in addition to choosing the weighted statistics.

Common mistake: “The largest component is \(2.10\), so its ordinary one-sided normal p-value is the MaxCombo p-value.” That ignores the chance that any one of several statistics could be large under \(H_0\). The correction is to calculate \(P(\max_j Z_j\geq2.10\mid H_0)\) using the joint null distribution and the component correlations. A second error is to use the maximum absolute statistic for a benefit-only test; that changes the rejection region and must not be presented as the same one-sided procedure.