Correction to: Improved precision in the analysis of randomized trials with survival outcomes, without assuming proportional hazards

1 Correction to: Lifetime Data Anal (2019) 25:439–468 https://doi.org/10.1007/s10985-018-9428-5

The original version of this article unfortunately contains mistakes. It has been corrected with this Correction.

2 Summary

The R code used for the data analysis and simulations in our manuscript (Díaz et al. 2018) had two errors, which we have corrected. All of the theoretical results in the paper are correct. It is our implementation of these in R code that had the errors, which impacts the data analysis and simulation results in Section 6. This erratum describes the errors and presents the updated results of the data analysis and simulations with the errors corrected.

The two errors were (i) incorrect coding of the auxiliary variable H used in the TMLE estimator, and (ii) incorrect coding of time t as numeric instead of as a factor in the adjusted estimators. These were corrected and the updated code is available at (Díaz 2018a). The updated results, given below, are qualitatively similar to the original results (i.e., there are no changes to our conclusions in the paper) except for the following: the updated TMLE () confidence interval coverage probabilities ranged from 93 to 95% (previously 94–95%); the updated bias of the TMLE was sometimes larger but still small (at most 3%) as a fraction of the treatment effect (14.9 days) and had negligible influence on the mean squared error; the adjusted inverse probability weighted estimator (which was not the focus of the paper) had larger variance in the updated results compared to the original results, leading to lower relative efficiency.

3 Corrections to R code

First, the R code for computing the auxiliary variable H(m, A, W) involved in the algorithm for our proposed estimator did not match the (correct) formula (17) from the paper. We have fixed this error in the R package (Díaz 2018a). Please refer to the GitHub commit given in that citation at the end of this erratum for the specific lines of code that were corrected.

Second, when computing all adjusted estimators, including and , the time t should have entered our R code (in the working models for $g_A, g_R$) as a variable of type factor rather than as numeric (see Díaz 2018b). This is needed in order to satisfy the requirement in Theorem 2 that dropout probabilities are estimated using models with saturated terms for time, treatment, and their interaction.

We have fixed the above two errors and reported the results of our revised analyses in Tables 1 through 3, whose numbering corresponds to the labels of tables in the original paper. Corresponding to our fix of the second error, the sentence “The model for $g_R$ for the adjusted estimators includes main terms for time (linear), treatment and their interaction, in addition to main terms for $W_1$ and $W_5$ and a treatment by $W_3$ interaction.” from Section 6.3 should now have “(linear)” replaced by “(factor)”.

4 CLEAR III trial data analysis

The results of analyzing the CLEAR III trial data were qualitatively similar to our initial findings. The Kaplan–Meier estimate of the RMST is 14.9 days (SE 5.3, 95% BCa CI 5.6–26.2). Our TMLE estimator has an estimated variance that is roughly 13% smaller than the (unadjusted) RMST difference based on the cMeier estimator (compared to 16% smaller as reported in the original manuscript).

Table 1 Estimated RMST (standard error) with 95% bias-corrected and accelerated (BCa) bootstrap confidence intervals based on the CLEAR III trial

Table 2 Revised simulation results for studies of size $n = 500$

Table 3 Revised simulation results for studies of size $n = 2000$

5 Simulation studies

Tables 2 and 3 display the new simulation results. The updated results were qualitatively similar to the original results, except for the following: the updated TMLE () confidence interval coverage probabilities ranged from 93 to 95% (previously 94–95%); the updated TMLE bias was sometime larger (with maximum absolute value 0.494 in Table 2, top half, positive treatment effect, scenario B), but still was small (at most 3%) as a fraction of the treatment effect (14.9 days) and had negligible influence on the mean squared error; the adjusted inverse probability weighted (adjusted IPW) estimator had larger variance.

The most important numerical differences in comparison with our initial results are highlighted in boldface, and all involve the adjusted IPW estimator. The only qualitative change to the conclusions of our initial study is as follows. Inclusion of time as a non-parametric term in our models increased variability in the inverse probability weights of . As a result, in most scenarios, this estimator was more variable. Most of the gains in MSE for the adjusted IPW compared to the Kaplan–Meier estimator that were reported in our original manuscript were not present after the errors were fixed.