Analysis of time‐to‐event data using Cox's proportional hazards (PH) model is ubiquitous in scientific research. A sample is taken from the population of interest and covariate information is collected on everyone. If the event of interest is rare and covariate information is difficult to collect, the nested case‐control (NCC) design reduces costs with minimal impact on inferential precision. Under PH, application of the Cox model to data from a NCC sample provides consistent estimation of the hazard ratio. However, under non‐PH, the finite‐sample estimates corresponding to the Cox estimator depend on the number of controls sampled and the censoring distribution. We propose two estimators based on a binary predictor of interest: one recovers the estimand corresponding to the Cox model under a simple random sample, while the other recovers an estimand that does not depend on the censoring distribution. We derive the asymptotic distribution and provide finite‐sample variance estimators.
Survival models are used to analyze time-to-event data in a variety of disciplines. Proportional hazard models provide interpretable parameter estimates, but proportional hazard assumptions are not always appropriate. Non-parametric models are more flexible but often lack a clear inferential framework. We propose a Bayesian treed hazards partition model that is both flexible and inferential. Inference is obtained through the posterior tree structure and flexibility is preserved by modeling the log-hazard function in each partition using a latent Gaussian process. An efficient reversible jump Markov chain Monte Carlo algorithm is accomplished by marginalizing the parameters in each partition element via a Laplace approximation. Consistency properties for the estimator are established. The method can be used to help determine subgroups as well as prognostic and/or predictive biomarkers in time-to-event data. The method is compared with some existing methods on simulated data and a liver cirrhosis dataset.
more » « less- NSF-PAR ID:
- 10491177
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Biometrics
- Volume:
- 80
- Issue:
- 1
- ISSN:
- 0006-341X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
Abstract -
Predicting the occurrence of a particular event of interest at future time points is the primary goal of survival analysis. The presence of incomplete observations due to time limitations or loss of data traces is known as censoring which brings unique challenges in this domain and differentiates survival analysis from other standard regression methods. The popularly used survival analysis methods such as Cox proportional hazard model and parametric survival regression suffer from some strict assumptions and hypotheses that are not realistic in most of the real-world applications. To overcome the weaknesses of these two types of methods, in this paper, we reformulate the survival analysis problem as a multi-task learning problem and propose a new multi-task learning based formulation to predict the survival time by estimating the survival status at each time interval during the study duration. We propose an indicator matrix to enable the multi-task learning algorithm to handle censored instances and incorporate some of the important characteristics of survival problems such as non-negative non-increasing list structure into our model through max-heap projection. We employ the L2,1-norm penalty which enables the model to learn a shared representation across related tasks and hence select important features and alleviate over-fitting in high-dimensional feature spaces; thus, reducing the prediction error of each task. To efficiently handle the two non-smooth constraints, in this paper, we propose an optimization method which employs Alternating Direction Method of Multipliers (ADMM) algorithm to solve the proposed multi-task learning problem. We demonstrate the performance of the proposed method using real-world microarray gene expression high-dimensional benchmark datasets and show that our method outperforms state-of-the-art methods.more » « less
-
more » « less
Summary The paper considers the problem of hypothesis testing and confidence intervals in high dimensional proportional hazards models. Motivated by a geometric projection principle, we propose a unified likelihood ratio inferential framework, including score, Wald and partial likelihood ratio statistics for hypothesis testing. Without assuming model selection consistency, we derive the asymptotic distributions of these test statistics, establish their semiparametric optimality and conduct power analysis under Pitman alternatives. We also develop new procedures to construct pointwise confidence intervals for the baseline hazard function and conditional hazard function. Simulation studies show that all tests proposed perform well in controlling type I errors. Moreover, the partial likelihood ratio test is empirically more powerful than the other tests. The methods proposed are illustrated by an example of a gene expression data set.
-
more » « less
Abstract We consider user retention analytics for online freemium role-playing games (RPGs). RPGs constitute a very popular genre of computer-based games that, along with a player’s gaming actions, focus on the development of the player’s in-game virtual character through a persistent exploration of the gaming environment. Most RPGs follow the freemium business model in which the gamers can play for free but they are charged for premium add-on amenities. As with other freemium products, RPGs suffer from the curse of high dropout rates. This makes retention analysis extremely important for successful operation and survival of their gaming portals. Here, we develop a disciplined statistical framework for retention analysis by modelling multiple in-game player characteristics along with the dropout probabilities. We capture players’ motivations through engagement times, collaboration and achievement score at each level of the game, and jointly model them using a generalized linear mixed model (glmm) framework that further includes a time-to-event variable corresponding to churn. We capture the interdependencies in a player’s level-wise engagement, collaboration, achievement with dropout through a shared parameter model. We illustrate interesting changes in player behaviours as the gaming level progresses. The parameters in our joint model were estimated by a Hamiltonian Monte Carlo algorithm which incorporated a divide-and-recombine approach for increased scalability in glmm estimation that was needed to accommodate our large longitudinal gaming data-set. By incorporating the level-wise changes in a player’s motivations and using them for dropout rate prediction, our method greatly improves on state-of-the-art retention models. Based on data from a popular action based RPG, we demonstrate the competitive optimality of our proposed joint modelling approach by exhibiting its improved predictive performance over competitors. In particular, we outperform aggregate statistics based methods that ignore level-wise progressions as well as progression tracking non-joint model such as the Cox proportional hazards model. We also display improved predictions of popular marketing retention statistics and discuss how they can be used in managerial decision making.
-
more » « less
Abstract We propose a constrained maximum partial likelihood estimator for dimension reduction in integrative (e.g., pan-cancer) survival analysis with high-dimensional predictors. We assume that for each population in the study, the hazard function follows a distinct Cox proportional hazards model. To borrow information across populations, we assume that each of the hazard functions depend only on a small number of linear combinations of the predictors (i.e., “factors”). We estimate these linear combinations using an algorithm based on “distance-to-set” penalties. This allows us to impose both low-rankness and sparsity on the regression coefficient matrix estimator. We derive asymptotic results that reveal that our estimator is more efficient than fitting a separate proportional hazards model for each population. Numerical experiments suggest that our method outperforms competitors under various data generating models. We use our method to perform a pan-cancer survival analysis relating protein expression to survival across 18 distinct cancer types. Our approach identifies six linear combinations, depending on only 20 proteins, which explain survival across the cancer types. Finally, to validate our fitted model, we show that our estimated factors can lead to better prediction than competitors on four external datasets.
