ABSTRACT
We propose vertical modelling as a natural approach to the problem of analysis of competing risks data when failure types are missing for some individuals. Under a natural missing-at-random assumption for these missing failure types, we use the observed data likelihood to estimate its parameters and show that the all-cause hazard and the relative hazards appearing in vertical modelling are indeed key quantities of this likelihood. This fact has practical implications in that it suggests vertical modelling as a simple and attractive method of analysis in competing risks with missing causes of failure; all individuals are used in estimating the all-cause hazard and only those with non-missing cause of failure for relative hazards. The relative hazards also appear in a multiple imputation approach to the same problem proposed by Lu and Tsiatis and in the EM algorithm. We compare the vertical modelling approach with the method of Goetghebeur and Ryan for a breast cancer data set, highlighting the different aspects they contribute to the data analysis.
Subject(s)
Models, Statistical , Risk Assessment/methods , Treatment Failure , Algorithms , Humans , Kaplan-Meier Estimate , Likelihood Functions , Proportional Hazards Models , Survival AnalysisABSTRACT
In this article, we propose a new approach to the problem of dynamic prediction of survival data in the presence of competing risks as an extension of the landmark model for ordinary survival data. The key feature of our method is the introduction of dynamic pseudo-observations constructed from the prediction probabilities at different landmark prediction times. They specifically address the issue of estimating covariate effects directly on the cumulative incidence scale in competing risks. A flexible generalized linear model based on these dynamic pseudo-observations and a generalized estimation equations approach to estimate the baseline and covariate effects will result in the desired dynamic predictions and robust standard errors. Our approach has a number of attractive features. It focuses directly on the prediction probabilities of interest, avoiding in this way complex modeling of cause-specific hazards or subdistribution hazards. As a result, it is robust against departures from these omnibus models. From a computational point of view an advantage of our approach is that it can be fitted with existing statistical software and that a variety of link functions and regression models can be considered, once the dynamic pseudo-observations have been estimated. We illustrate our approach on a real data set of chronic myeloid leukemia patients after bone marrow transplantation.
Subject(s)
Bone Marrow Transplantation/mortality , Data Interpretation, Statistical , Leukemia, Myelogenous, Chronic, BCR-ABL Positive/mortality , Leukemia, Myelogenous, Chronic, BCR-ABL Positive/surgery , Models, Statistical , Risk Assessment/methods , Survival Analysis , Computer Simulation , Humans , Incidence , PrognosisABSTRACT
We propose an extension of the landmark model for ordinary survival data as a new approach to the problem of dynamic prediction in competing risks with time-dependent covariates. We fix a set of landmark time points tLM within the follow-up interval. For each of these landmark time points tLM , we create a landmark data set by selecting individuals at risk at tLM ; we fix the value of the time-dependent covariate in each landmark data set at tLM . We assume Cox proportional hazard models for the cause-specific hazards and consider smoothing the (possibly) time-dependent effect of the covariate for the different landmark data sets. Fitting this model is possible within the standard statistical software. We illustrate the features of the landmark modelling on a real data set on bone marrow transplantation.
Subject(s)
Forecasting/methods , Proportional Hazards Models , Risk , Bone Marrow Transplantation/standards , Graft vs Host Disease/etiology , Humans , Leukemia, Myelogenous, Chronic, BCR-ABL Positive/therapy , Neoplasm Recurrence, LocalABSTRACT
We study an alternative approach for estimation in the competing risks framework, called vertical modeling. It is motivated by a decomposition of the joint distribution of time and cause of failure. The two elements of this decomposition are (1) the time of failure and (2) the cause of failure condition on time of failure. Both elements of the model are based on observable quantities, namely the total hazard and the relative cause-specific hazards. The model can be implemented using the standard software. The relative cause-specific hazards are flexibly estimated using multinomial logistic regression and smoothing splines. We show estimates of cumulative incidences from vertical modeling to be more efficient statistically than those obtained from the standard nonparametric model. We illustrate our methods using data of 8966 leukemia patients from the European Group for Blood and Marrow Transplantation.
