Data generation for the Cox proportional hazards model with time-dependent covariates: a method for medical researchers

Time Line of Occlusion for Intracranial Aneurysms Treated with the Pipeline Embolization Device

BACKGROUND

Rates of aneurysm occlusion following treatment with flow-diverting stents have been quantified at predefined time points in clinical trials, but data characterizing the continuous temporal progression of aneurysm occlusion are lacking. This study used real-world variability in timing of angiographic follow-up to characterize the time line of aneurysm occlusion following treatment with the Pipeline embolization device (PED).

METHODS

All aneurysms treated with a PED at our institution between 2011 and 2020 were screened. Nonsaccular or ruptured aneurysms were excluded. Aneurysm occlusion status and time since treatment were recorded for each follow-up angiogram. Aneurysm occlusion was characterized using Kaplan-Meier and Cox proportional hazards analysis after censoring at last follow-up or subsequent treatment.

RESULTS

There were 290 aneurysms in 222 patients analyzed. The median time of observed aneurysm occlusion was 7.5 months, and overall rate of aneurysm occlusion was 77.9%. Larger aneurysms demonstrated a longer median time to occlusion and lower rate of aneurysm occlusion (P = 0.029). There were no observed differences in the time line of occlusion for aneurysms treated with a single PED or multiple PEDs (P = 0.889) or without or with adjunctive coiling (P = 0.771).

CONCLUSIONS

Aneurysms treated with a PED had a median time to observed occlusion of 7.5 months. Occlusion of larger aneurysms occurred more slowly than occlusion of smaller aneurysms following flow diversion. The number of PEDs deployed or the use of adjunctive coiling did not affect the time line or likelihood of aneurysm occlusion. These findings may guide optimal timing of follow-up after treatment with a PED.

Generating Survival Times Using Cox Proportional Hazards Models with Cyclic and Piecewise Time-Varying Covariates

Time-to-event outcomes with cyclic time-varying covariates are frequently encountered in biomedical studies that involve multiple or repeated administrations of an intervention. In this paper, we propose approaches to generating event times for Cox proportional hazards models with both time-invariant covariates and a continuous cyclic and piecewise time-varying covariate. Values of the latter covariate change over time through cycles of interventions and its relationship with hazard differs before and after a threshold within each cycle. The simulations of data are based on inverting the cumulative hazard function and a log link function for relating the hazard function to the covariates. We consider closed-form derivations with the baseline hazard following the exponential, Weibull, or Gompertz distribution. We propose two simulation approaches: one based on simulating survival data under a single-dose regimen first before data are aggregated over multiple-dosing cycles and another based on simulating survival data directly under a multiple-dose regimen. We consider both fixed intervals and varying intervals of the drug administration schedule. The method's validity is assessed in simulation experiments. The results indicate that the proposed procedures perform well in generating data that conform to their cyclic nature and assumptions of the Cox proportional hazards model.

Bootstrapping complex time-to-event data without individual patient data, with a view toward time-dependent exposures

We consider nonparametric and semiparametric resampling of multistate event histories by simulating multistate trajectories from an empirical multivariate hazard measure. One advantage of our approach is that it does not necessarily require individual patient data, but may be based on published information. This is also attractive for both study planning and simulating realistic real-world event history data in general. The concept extends to left-truncation and right-censoring mechanisms, nondegenerate initial distributions, and nonproportional as well as non-Markov settings. A special focus is on its connection to simulating survival data with time-dependent covariates. For the case of qualitative time-dependent exposures, we demonstrate that our proposal gives a more natural interpretation of how such data evolve over the course of time than many of the competing approaches. The multistate perspective avoids any latent failure time structure and sampling spaces impossible in real life, whereas its parsimony follows the principle of Occam's razor. We also suggest empirical simulation as a novel bootstrap procedure to assess estimation uncertainty in the absence of individual patient data. This is not possible for established procedures such as Efron's bootstrap. A simulation study investigating the effect of liver functionality on survival in patients with liver cirrhosis serves as a proof of concept. Example code is provided.

GENERATING SURVIVAL TIMES WITH TIME-VARYING COVARIATES USING THE LAMBERT W FUNCTION

Simulation studies provide an important statistical tool in evaluating survival methods, requiring an appropriate data-generating process to simulate data for an underlying statistical model. Many studies with time-to-event outcomes use the Cox proportional hazard model. While methods for simulating such data with time-invariant predictors have been described, methods for simulating data with time-varying covariates are sorely needed. Here, we describe an approach for generating data for the Cox proportional hazard model with time-varying covariates when event times follow an Exponential or Weibull distribution. For each distribution, we derive a closed-form expression to generate survival times and link the time-varying covariates with the hazard function. We consider a continuous time-varying covariate measured at regular intervals over time, as well as time-invariant covariates, in generating time-to-event data under a number of scenarios. Our results suggest this method can lead to simulation studies with reliable and robust estimation of the association parameter in Cox-Weibull and Cox-Exponential models.

Shared and unshared exposure measurement error in occupational cohort studies and their effects on statistical inference in proportional hazards models

Exposure measurement error represents one of the most important sources of uncertainty in epidemiology. When exposure uncertainty is not or only poorly accounted for, it can lead to biased risk estimates and a distortion of the shape of the exposure-response relationship. In occupational cohort studies, the time-dependent nature of exposure and changes in the method of exposure assessment may create complex error structures. When a method of group-level exposure assessment is used, individual worker practices and the imprecision of the instrument used to measure the average exposure for a group of workers may give rise to errors that are shared between workers, within workers or both. In contrast to unshared measurement error, the effects of shared errors remain largely unknown. Moreover, exposure uncertainty and magnitude of exposure are typically highest for the earliest years of exposure. We conduct a simulation study based on exposure data of the French cohort of uranium miners to compare the effects of shared and unshared exposure uncertainty on risk estimation and on the shape of the exposure-response curve in proportional hazards models. Our results indicate that uncertainty components shared within workers cause more bias in risk estimation and a more severe attenuation of the exposure-response relationship than unshared exposure uncertainty or exposure uncertainty shared between individuals. These findings underline the importance of careful characterisation and modeling of exposure uncertainty in observational studies.