Sensitivity Analysis for Unknown Effect Onset Time

Why Vary the Onset Time?

A weighted logrank test can be powerful when its time weights emphasize the period in which a treatment effect occurs. But if the treatment effect is expected to emerge only after some delay, and that delay is uncertain, a power calculation at just one assumed onset can give a misleading impression of how dependable the design is. Sensitivity analysis makes that uncertainty visible: calculate power under a range of plausible onset times while keeping the planned design and test fixed.

The goal is not to identify the true onset time from planning assumptions. It is to show how conclusions about power and test choice change when the assumed onset moves. This is especially useful when the biological or clinical rationale supports delayed benefit but does not pin down its timing.

Model and Power Approximation

Suppose participants are randomized between treatment and control, and the planned primary test is a fixed weighted logrank test. Let \(\tau\) denote the treatment-effect onset time. For a simple delayed-effect model, assume the treatment and control hazards are equal before \(\tau\), and the treatment hazard is a constant fraction \(\theta\) of the control hazard after \(\tau\), where \(0<\theta<1\). Thus the hazard ratio is 1 before onset and \(\theta\) afterward. The hazard ratio is defined as the treatment hazard divided by the control hazard.

The approximation below uses fixed time bands. In band \(j\), let \(I_j\) be the planned logrank information, and let \(w_j\) be the prespecified test weight. Information is the variance contribution to the unweighted logrank score under the null; in a randomized trial it depends on the risk sets and allocation at event times. For a band wholly before onset, the assumed treatment effect contributes no signal. For a band wholly after onset, the effect contributes approximately \(-\log(\theta)\) per unit of information. The approximation treats information as fixed across the onset scenarios; substantial changes in survival or censoring that alter risk sets call for recalculating it under each scenario.

Here \(\mu(\tau)\) is the approximate mean of the standardized test statistic under the alternative, with its sign chosen so that treatment benefit is positive. Under the null, the standardized statistic is approximately standard normal. Under the alternative, it is approximated by a normal distribution with mean \(\mu(\tau)\) and variance 1. For a two-sided test at significance level \(\alpha\), approximate power is

where \(\Phi\) is the standard normal cumulative distribution function and \(z_{1-\alpha/2}\) is its \(1-\alpha/2\) quantile. This is a planning approximation, not an exact finite-sample calculation. It is most defensible when the score’s normal approximation is adequate, the event-information assumptions are credible, and the model for the time-varying hazard ratio is reasonable.

Deriving the Approximation

A weighted logrank score adds event-time contributions after multiplying each by its fixed weight. Under the null, the expected score is zero, and its variance is approximately the sum of the squared weights times the corresponding information contributions. Under a local alternative, the expected unweighted score contribution in a band is approximately its information multiplied by the log hazard contrast, with the sign reversed here so that benefit is positive. Adding weighted contributions across bands gives the numerator in the expression for \(\mu(\tau)\). Dividing this expected score by its null standard deviation gives the standardized mean:

This justification is an information-based, local-alternative approximation. It does not assert that the logrank score has exactly this mean under every fixed alternative. In a full design calculation, risk-set evolution, censoring, accrual, and the time-varying hazards can be incorporated directly, often by numerical integration or simulation. The sensitivity-analysis principle is unchanged: repeat the calculation under each onset scenario using the same prespecified test and consistent assumptions.

Worked Power Grid

Summarizing Uncertainty and Selecting a Test

A sensitivity table is useful even when no single scenario is considered most likely. It can show (1) power across the full plausible range, (2) the minimum power over a range regarded as clinically plausible, or (3) a scenario-weighted average when planning stakeholders have specified relative plausibility weights. These summaries answer different questions. Minimum power is a robustness summary; average power depends on the chosen scenario weights. Neither replaces reporting the individual scenario results.

For example, using the first worked example, suppose a planning group assigns scenario weights 0.20, 0.50, and 0.30 to immediate onset, onset after band 1, and onset after band 2. The weighted average ordinary-test power is \(0.20(0.935)+0.50(0.672)+0.30(0.188)=0.579\). For the late-weighted test it is \(0.20(0.898)+0.50(0.781)+0.30(0.320)=0.666\). These averages summarize the stated planning scenarios; the weights are not probabilities learned from trial data and do not alter the test’s type I error.

The principal alternatives have distinct trade-offs. The ordinary logrank test is familiar and performs well for proportional hazards, but may lose power when its time-constant weighting does not align with a delayed effect. A fixed late-weighted test can improve power under a planned delay pattern, but may lose power for early effects and is not automatically optimal for every delayed pattern. A MaxCombo test can protect against uncertainty across a prespecified collection of weights, but its critical value must account for selecting the largest statistic; its power cannot be calculated as though the best component had been chosen in advance. These options should be compared at the same overall significance level and under the same onset scenarios. An estimand-based alternative may be considered when a single hazard contrast is not an adequate summary, but its interpretation and analysis are separate design choices.

Assumptions, Diagnostics, and Limits

A credible grid analysis requires more than a list of onset values. State the hazard model before calculating power, including the post-onset hazard ratio and whether the effect is assumed to remain constant after onset. Specify the accrual and follow-up plans, censoring assumptions, allocation ratio, information or event-time model, test weights, sidedness, and significance level. If the onset falls inside a time band, split the band or calculate the active fraction using a finer time grid rather than assigning the entire band to before or after onset.

Check the analysis for signs that its assumptions are driving the answer. Useful checks include plotting power against onset time, confirming that each scenario uses the same test and alpha, and verifying that the information contributions reflect risk sets under that scenario. A sharp or unexpected change as onset moves slightly may indicate coarse time bands, an implementation error, or a genuine change in the fraction of information exposed to treatment benefit. If the treatment effect changes survival enough to alter later risk sets, recompute information rather than mechanically reusing the same \(I_j\). Simulation can assess the approximation when the normal model or fixed-information assumption is doubtful.

Sensitivity analysis has important limitations. A finite grid cannot guarantee performance at every possible onset time; include enough values to reveal important changes and refine the grid near steep transitions. A grid of onset times does not account for uncertainty in effect size, censoring, or other model features unless those are varied too. Finally, selecting the test after examining the observed treatment-effect timing can inflate type I error. Test weights, any combination procedure, the onset scenarios, and the selection criterion should be prespecified. The earlier material on weighted logrank tests provides the test-specific foundation; the present analysis concerns how uncertainty about onset affects the design comparison.

Common Pitfall

• Wrong reasoning: “The late-weighted test has the greatest power at the most likely delay, so I can use that power as the design’s power.”
• Correction: A single favored scenario hides sensitivity to onset uncertainty. Show the full scenario results and state whether selection is based on minimum power, a prespecified scenario-weighted average, or another explicit criterion. If the test is selected or adapted using trial outcomes, account for that selection in the inferential procedure.