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Correcting for sampling variability in maximum likelihood-based one-sample log-rank tests

Moritz Fabian Danzer, Rene Schmidt

TL;DR

This work analyzes estimation uncertainty for the common situation that the reference curve is estimated parametrically using the maximum likelihood method, and indicates how the variance estimation of the one-sample log-rank test can be adapted in order to take this variability into account.

Abstract

Single-arm studies in the early development phases of new treatments are not uncommon in the context of rare diseases or in paediatrics. If an assessment of efficacy is to be made at the end of such a study, the observed endpoints can be compared with reference values that can be derived from historical data. For a time-to-event endpoint, a statistical comparison with a reference curve can be made using the one-sample log-rank test. In order to ensure the interpretability of the results of this test, the role of the reference curve is crucial. This quantity is often estimated from a historical control group using a parametric procedure. Hence, it should be noted that it is subject to estimation uncertainty. However, this aspect is not taken into account in the one-sample log-rank test statistic. We analyse this estimation uncertainty for the common situation that the reference curve is estimated parametrically using the maximum likelihood method, and indicate how the variance estimation of the one-sample log-rank test can be adapted in order to take this variability into account. The resulting test procedures are illustrated using a data example and analysed in more detail using simulations, particularly in comparison with established two-sample methods.

Correcting for sampling variability in maximum likelihood-based one-sample log-rank tests

TL;DR

This work analyzes estimation uncertainty for the common situation that the reference curve is estimated parametrically using the maximum likelihood method, and indicates how the variance estimation of the one-sample log-rank test can be adapted in order to take this variability into account.

Abstract

Single-arm studies in the early development phases of new treatments are not uncommon in the context of rare diseases or in paediatrics. If an assessment of efficacy is to be made at the end of such a study, the observed endpoints can be compared with reference values that can be derived from historical data. For a time-to-event endpoint, a statistical comparison with a reference curve can be made using the one-sample log-rank test. In order to ensure the interpretability of the results of this test, the role of the reference curve is crucial. This quantity is often estimated from a historical control group using a parametric procedure. Hence, it should be noted that it is subject to estimation uncertainty. However, this aspect is not taken into account in the one-sample log-rank test statistic. We analyse this estimation uncertainty for the common situation that the reference curve is estimated parametrically using the maximum likelihood method, and indicate how the variance estimation of the one-sample log-rank test can be adapted in order to take this variability into account. The resulting test procedures are illustrated using a data example and analysed in more detail using simulations, particularly in comparison with established two-sample methods.

Paper Structure

This paper contains 16 sections, 7 theorems, 49 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Parametric estimation
  4. Testing procedures
  5. Application example
  6. Simulation study
  7. Data generation
  8. Results
  9. Type I error rate
  10. Power
  11. Discussion
  12. Derivation of the asymptotic distribution
  13. Simulation results
  14. Type I error rate
  15. Type I error rate (Exponential and Weibull MLEs)
  16. ...and 1 more sections

Key Result

Lemma 1

Let $(X_n)_{n\in\mathbb{N}}$ and $(Y_n)_{n\in\mathbb{N}}$ two sequences of $\mathbb{R}$-valued random variables where $X_n$ and $Y_n$ are independent for any $n \in \mathbb{N}$. If $X_n \overset{\mathcal{D}}{\to} X$ and $Y_n \overset{\mathcal{D}}{\to} Y$ as $n \to \infty$, then $X_n + Y_n \overset{\

Figures (11)

  • Figure 1: Kaplan-Meier estimates from simulated data inspired by reconstructed data and the survival curve of a log-logistic distribution fitted to the historic control group data.
  • Figure 2: Empirical two-sided type I error rates for seven different testing procedures with fixed sample size in the experimental group and varying allocation ratios (left) or varying sample size in the experimental group with fixed allocation ratio (right).
  • Figure 3: Empirical one-sided type I error rates for seven different testing procedures with fixed sample size in the experimental group and varying allocation ratios (left) or varying sample size in the experimental group with fixed allocation ratio (right).
  • Figure 4: Empirical power for five procedures that showed acceptable adherence to the nominal type I error rate with fixed sample size in the experimental group and varying allocation ratios (left) or varying sample size in the experimental group with fixed allocation ratio (right).
  • Figure S1: Two- and one-sided empirical rejection rates of the null hypothesis $H_0$ with Weibull-distributed data with shape parameter $\kappa = 0.5$ for four different sample sizes in the experimental cohort in dependence of the allocation ratio.
  • ...and 6 more figures

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • ...and 4 more