1. Imputation method to correct overestimated shape parameter of
Weibull hazard function in time-to-event modeling
Dong-Seok Yim1 , Sung Min Park1, Kyungmee Choi2, Seunghoon Han1
1Department of Clinical Pharmacology and Therapeutics, Seoul St. Mary’s Hospital, PIPET (Pharmacometrics Institute for Practical Education and Training), South Korea
2Division of Mathematics, College of Science and Technology, Hongik University, Sejong, South Korea
BACKGROUND
Hyper-acute, mass occurrence of events (e.g., at day 0~1) is not rare in biomedical data.
But, the overestimation of maximum likelihood estimator (MLE) of the Weibull shape parameter at survival analysis of such data has been neglected so far (e.g., Fig. 1).
This biased result is observed regardless of statistical software (R, SAS, NONMEM…)
Without correcting this bias, robust models for time-to-event data with hyper-acutely occurring character may not be developed.
Thus, we performed this simulation study and present a useful correction method that is applicable to time-to-event modeling in pharmacometrics as well as conventional survival analysis.
RESULTS
Iterative imputation worked like this…
Iteration
Stopped
(p>0.999)
Figure 4 Improvement of biases by the iterative imputation process (in the case of
Weibull-based imputation) –an example case
After repeated Kolmogorov-Smirnov test, when the p-value was at last > 0.999, we concluded that the K-M plots of reference dataset and the hybrid dataset (iteratively-renewed) were identical and stopped the iterative imputation.
It took 3~20 iterative imputation steps before attaining the p-value > 0.99 for each of reference datasets with differing Weibull shape parameters of 0.2 ~ 2)
Figure 1 An example of biased Weibull estimates for hyper-acutely occurring events in biomedical studies.
(A part of this topic has been presented in ACoP 2015.)
Question: When do we get biased maximum likelihood estimates?
Answer: Bias was obvious when the shape parameter β was < 0.5 .
(because information loss at day1 worsens.)
METHOD
Raw dataset simulation
Survival datasets (100 subjects / dataset) per shape parameters(β) ranging from 0.2 to 2 were simulated using the rweibull (30, β) function in the R Statistical Software.
The scale parameter α was fixed to 30.
“Reference datasets” by modification of the raw datasets
Discretization: the raw dataset event times (t) were discretized by rounding up.
i.e., when 0 < t ≤ 1, t’s were rounded up to 1 (day) to mimic the survival data recorded by the unit of day.
Censoring: The raw dataset event times (t) > 28 (days) were discarded.
Estimation of α and β of reference datasets using survreg ( ) fuction of R: How big is the bias?
The biases from the conventional statistical method were first estimated.
Kaplan-Meier (K-M) curves and Bias were presented as results (Fig. 5).
Use of hybrid dataset to correct the bias (To impute the lost information at day1)
We replaced the day1 data with imputed (simulated) event time data (hybrid dataset):
Detailed methods are summarized in Fig. 2.
BIASED!!!
Figure 5 Comparison of K-M curves from simulated data and its estimates.
Black: reference datasets (simulated datasets)
Blue: MLE (and 95% P.I.’s in green shade) from conventional statistical analysis
See Fig. 3
See Fig. 4
Question: Which distribution to use for imputation, Weibull or Uniform?
Answer: Just Use Weibull.
Figure 6 Comparison of K-M curves of reference datasets and corresponding
hybrid datasets imputed from Weibull or Uniform distribution.
-Black: reference dataset
- Red: using uniform distribution (bias remained despite iteration: never passed K-M test.)
- Blue: using Weibull distribution
Table 1 Biases after applying imputation methods.
Figure 2 Workflow diagram of simulation study (K-S: Kolmogorov-Smirnov).
True value of
scale parameter
shape parameter
Bias
(Uniform)
(Weibull) 30
0.2
0.20
0.033
0.4
0.11
0.031
0.6
0.06
0.034
0.8
0.02
0.028
CONCLUSION
Sometimes, we come across biomedical data with hyper-acute, mass events at the very early observation interval. It is not a rare situation.
We confirmed that the conventional survival analysis was not appropriate to estimate time-to-event data when its shape parameter was ≤ 0.5.
We suggest the use of the iterative-imputation method presented herein when biased estimates are observed in the analysis of such dataset.
Figure 3 Schema of the iterative imputation process (simulation with Weibull parameters)