Reference curve sampling variability in one–sample log–rank tests

Abstract

The one–sample log–rank test is the method of choice for single–arm Phase II trials with time–to–event endpoint. It allows to compare the survival of patients to a reference survival curve that typically represents the expected survival under standard of care. The one–sample log–rank test, however, assumes that the reference survival curve is known. This ignores that the reference curve is commonly estimated from historic data and thus prone to sampling error. Ignoring sampling variability of the reference curve results in type I error rate inflation. We study this inflation in type I error rate analytically and by simulation. Moreover we derive the actual distribution of the one–sample log–rank test statistic, when the sampling variability of the reference curve is taken into account. In particular, we provide a consistent estimate of the factor by which the true variance of the one-sample log–rank statistic is underestimated when reference curve sampling variability is ignored. Our results are further substantiated by a case study using a real world data example in which we demonstrate how to estimate the error rate inflation in the planning stage of a trial.

Two-sample survival tests based on control arm summary statistics

“…Notice that in theory one could use a corrected variance estimator b S 2 for Varð b M 0 ðsÞÞ (see Eq. ( 5) of [2]) instead of our approximation b S 2 OSLR ðs max Þ � ð1 þ pÞ. However, using the statistic Z π yields some advantages.…”

“…inflation depends in particular on the relative size of the control cohort compared to the intervention cohort [2].…”

“…In classical log-rank tests the prefixed reference curve is rather treated as deterministic. Doing so results in an inflation of the type I error rate, when in fact the prefixed reference curve resulted from an estimation process [2].…”

“…Then, application of the two-sample logrank test is not possible because the required complete historical survival data on individual patient level are not available. Conversely, it is also not permissible to use the published survival estimator as deterministic reference curve in a classical one-sample log-rank test, since this ignores the sampling error of the reference curve, leading to inflation of the type I error rate [2].…”

Untitled

“…In classical log-rank tests the prefixed reference curve is rather treated as deterministic. Doing so results in an inflation of the type I error rate, when in fact the prefixed reference curve resulted from an estimation process [2].…”