Meta-analysis of median survival times with inverse-variance weighting

TL;DR

This paper tackles the challenge of meta-analyzing outcome measures based on median survival times when primary studies report medians with confidence intervals but not standard errors. It introduces a Wald-approximation-based inverse-variance approach that derives within-study SEs from published CIs, and extends this to meta-analyze the median, difference of medians, and ratio of medians, with a theoretical consistency result for Brookmeyer-Crowley intervals. Through extensive study- and meta-analysis–level simulations, the method shows near-benchmark performance for moderate-to-large effective sample sizes across common CI constructions, while highlighting reduced accuracy with small samples or heavy censoring. An NSCLC OS application demonstrates the method's practical utility, and the authors provide software implementations to facilitate adoption in practice.

Abstract

We consider the problem of meta-analyzing outcome measures based on median survival times. Primary studies with time-to-event outcomes often report estimates of median survival times and confidence intervals based on the Kaplan-Meier estimator. However, outcome measures based on median survival are rarely meta-analyzed, as standard inverse-variance weighted methods require within-study standard errors that are typically not reported. In this article, we consider an inverse-variance weighted approach to meta-analyze median survival times that estimates the within-study standard errors from the reported confidence intervals. We show that this method consistently estimates the standard error of median survival when applied to confidence intervals constructed by the Brookmeyer-Crowley method. We conduct a series of simulation studies evaluating the performance of this approach at the study level (i.e., for estimating the standard error of median survival) and the meta-analytic level (i.e., for estimating the pooled median, difference of medians, and ratio of medians) for commonly used confidence intervals for median survival, including the Brookmeyer-Crowley method and nonparametric bootstrap. We find that this approach often performs comparably to a benchmark approach that uses the true within-study standard errors for meta-analyzing median-based outcome measures when within-study sample sizes are moderately large (e.g., above 50). However, when the effective sample sizes are small, the method can yield biased estimates of within-study standard errors. We illustrate an application of this approach in a meta-analysis evaluating survival benefits of being assigned to experimental arms versus comparator arms in randomized trials for non-small cell lung cancer therapies.

Figures (6)

• Figure 1: Study-level simulation results for the scenarios with the exponential and Weibull event time distributions. The box plots illustrate the estimated standard errors (SEs) of the median survival time from the Wald approximation-based approach. The peach boxes correspond to when the Brookmeyer-Crowley method based on a log transformation was used to construct the 95% confidence interval; The green boxes correspond to the Brookmeyer-Crowley method based on a log-minus-log transformation; The purple boxes correspond to the nonparametric bootstrap method. The true standard errors are illustrated by red dots
• Figure 2: Study-level simulation results for the scenarios with small sample sizes and the exponential event time distribution. The box plots illustrate the estimated standard errors (SEs) of the median survival time from the Wald approximation-based approach. The peach boxes correspond to when the Brookmeyer-Crowley method based on a log transformation was used to construct the 95% confidence interval; The green boxes correspond to the Brookmeyer-Crowley method based on a log-minus-log transformation; The purple boxes correspond to the nonparametric bootstrap method. The true standard errors are illustrated by red dots
• Figure 3: Estimates of the pooled median (left panel) and the between-study variance (right panel). The blue boxes correspond to the Wald approximation-based approach, and the red boxes correspond to the benchmark approach. The red dots indicate the true values
• Figure 4: Estimates of the pooled difference of medians (left panel) and the between-study variance (right panel). The blue boxes correspond to the Wald approximation-based approach, and the red boxes correspond to the benchmark approach. The red dots indicate the true values
• Figure 5: Estimates of the pooled ratio of medians (left panel) and the between-study variance (right panel). The blue boxes correspond to the Wald approximation-based approach, and the red boxes correspond to the benchmark approach. The red dots indicate the true values

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Proposition 1
• Remark 3