Blesson Chacko, Jonathan J Moss, Rupert Austin and Joachim Grevel
BAST Inc Ltd., Loughborough University Science and Enterprise Park, LE11 3AQ
Objectives: To compare the performance of various analysis methods to detect covariate effects on competing events. To explore the influence of sample size. To visualise model predictions versus observations.
Methods: 200 studies were repeatedly simulated while varying sample size (100 to 1000), intensity and direction of binary covariate influence on event of interest (response) and on competing event (dropout) with right censoring at a fixed cut-off time. The two competing events (response and dropout) were simulated using the same exponential hazard function.
The covariate effect on the response rate and on the individual response risk was investigated in all the simulated studies by the cause-specific hazard (CSH) and by the sub-distribution hazard (SDH), respectively. Covariate effects on the cumulative incidence of the events (cause-specific cumulative incidence function, CSCIF and sub-distribution cumulative incidence function, SDCIF) were visualised by indirectly calculating the CSCIF or by directly modelling the SDCIF.
Covariate effects on the CSH were analysed either by the semi-parametric Cox proportional hazard (PH) model or by parametric CSH modelling using exponential hazard functions. Covariate effects on the SDH were analysed with the semi-parametric Fine-Gray method [1] or by directly modelling the SDCIF with a modified three-parameter logistic hazard function and a generalised odds-rate link function under the constraint that the asymptotes of SDCIFs for the competing events must add up to one [2]. The utility of various link functions was explored.
Results: The type I and type II error (power) of detecting the true covariate effect on the response were calculated for various analysis methods. The performance of the Cox PH model and the parametric CSH model was similar and the ‘blind spot’ was found in the expected location where ‘no covariate effect on the response rate’ was true. The simultaneous covariate effect on the dropout rate was not tested with CSH. Statistical power increased, as expected, when the sample size was increased. Modelling the CSH alone did not identify a covariate effect on the individual response risk, and the covariate effect on the dropout remained unidentified due to the treatment of dropout as right-censoring. In contrast, the Fine-Gray model analysed the covariate effect on the individual risk of a response while taking the simultaneous risk of dropout into account. It provided a visualisation of the covariate effect on the SDCIF of the response. The ‘blind spot’ of this method lay where the covariate effect on the risk of response and dropout were equally strong. Similar performance was obtained by the direct parametric modelling of SDCIF with the generalised odds-rate link function fixed to mimic the proportional hazard (log-minus-log) model for regression (used also in the Fine-Gray model). Changing the parameter of the generalised odds-rate link function, however, showed that log-minus-log was not always the best model according to the Akaike information criterion (AIC). Therefore, the directly modelled SDCIF might fail to correctly estimate the covariate effect on the response risk when the SDCIF was poorly approximated. Visualisations of the SDCIF helped in model-building.
Conclusions: Covariate effects on the response rate, identified by the Cox PH model, describe the underlying aetiology and are not suited for predicting the covariate influence on future individual risks. No advantage is gained by parametric modelling of the CSH. This conclusion extends to most parametric time-to-event modelling presented at recent pharmacometric meetings. To identify individual risks in a situation of competing events (only two were simulated here, but the methods can handle any number) the SDH needs to be modelled. The semi-parametric Fine-Gray method is well established; a direct parametric modelling of the SDCIF qualifies the proportional hazard model or proposes useful alternatives. It also affords clear visualisation of the covariate effects in the presence of competing events similar to the standard visual predictive checks. Still, also the SDH models have their ‘blind spot’ where the covariate effect on the competing events is equal. For a comprehensive analysis of covariate effects on competing events, separate CSH models for each competing event and SDH models for the combined events are needed. With these models in place, future studies can be designed and powered adequately.
References:
[1] Fine, J. P., & Gray, R. J. (1999). A proportional hazards model for the subdistribution of a competing risk. Journal of the American statistical association, 94(446), 496-509.
[2] Shi, H., Cheng, Y., & Jeong, J. H. (2013). Constrained parametric model for simultaneous inference of two cumulative incidence functions. Biometrical Journal, 55(1), 82-96.
Reference: PAGE 27 (2018) Abstr 8424 [www.page-meeting.org/?abstract=8424]
Poster: Methodology - New Modelling Approaches