Validation of discrete time-to-event prediction models in the presence of competing risks

Clinical prediction models play a key role in risk stratification, therapy assignment and many other fields of medical decision making. Before they can enter clinical practice, their usefulness has to be demonstrated using systematic validation. Methods to assess their predictive performance have been proposed for continuous, binary, and time-to-event outcomes, but the literature on validation methods for discrete time-to-event models with competing risks is sparse. The present paper tries to fill this gap and proposes new methodology to quantify discrimination, calibration, and prediction error (PE) for discrete time-to-event outcomes in the presence of competing risks. In our case study, the goal was to predict the risk of ventilator-associated pneumonia (VAP) attributed to Pseudomonas aeruginosa in intensive care units (ICUs). Competing events are extubation, death, and VAP due to other bacteria. The aim of this application is to validate complex prediction models developed in previous work on more recently available validation data.     

A Bayesian multi-risks survival (MRS) model in the presence of double censorings

Semi-competing risks data include the time to a nonterminating event and the time to a terminating event, while competing risks data include the time to more than one terminating event. Our work is motivated by a prostate cancer study, which has one nonterminating event and two terminating events with both semi-competing risks and competing risks present as well as two censoring times. In this paper, we propose a new multi-risks survival (MRS) model for this type of data. In addition, the proposed MRS model can accommodate noninformative right-censoring times for nonterminating and terminating events. Properties of the proposed MRS model are examined in detail. Theoretical and empirical results show that the estimates of the cumulative incidence function for a nonterminating event may be biased if the information on a terminating event is ignored. A Markov chain Monte Carlo sampling algorithm is also developed. Our methodology is further assessed using simulations and also an analysis of the real data from a prostate cancer study. As a result, a prostate-specific antigen velocity greater than 2.0 ng/mL per year and higher biopsy Gleason scores are positively associated with a shorter time to death due to prostate cancer.     

Estimating Abundance of an Unmarked,Low-Density Species using Cameras

Estimating abundance of wildlife populations can be challenging and costly, especially for species that are difficult to detect and that live at low densities, such as cougars (Puma concolor). Remote, motion-sensitive cameras are a relatively efficient monitoring tool, but most abundance estimation techniques using remote cameras rely on some or all of the population being uniquely identifiable. Recently developed methods estimate abundance from encounter rates with remote cameras and do not require identifiable individuals. We used 2 methods, the time-to-event and space-to-event models, to estimate the density of 2 cougar populations in Idaho, USA, over 3 winters from 2016–2019. We concurrently estimated cougar density using the random encounter model (REM), an existing camera-based method for unmarked populations, and genetic spatial capture recapture (SCR), an established method for monitoring cougar populations. In surveys for which we successfully estimated density using the SCR model, the time-to-event estimates were more precise and showed comparable variation between survey years. The space-to-event estimates were less precise than the SCR estimates and were more variable between survey years. Compared to REM, time-to-event was more precise and consistent, and space-to-event was less precise and consistent. Low sample sizes made the space-to-event and SCR models inconsistent from survey to survey, and non-random camera placement may have biased both of the camera-based estimators high. We show that camera-based estimators can perform comparably to existing methods for estimating abundance in unmarked species that live at low densities. With the time- and space-to-event models, managers could use remote cameras to monitor populations of multiple species at broader spatial and temporal scales than existing methods allow. © 2020 The Wildlife Society.     

Clustered restricted mean survival time regression

For multicenter randomized trials or multilevel observational studies, the Cox regression model has long been the primary approach to study the effects of covariates on time-to-event outcomes. A critical assumption of the Cox model is the proportionality of the hazard functions for modeled covariates, violations of which can result in ambiguous interpretations of the hazard ratio estimates. To address this issue, the restricted mean survival time (RMST), defined as the mean survival time up to a fixed time in a target population, has been recommended as a model-free target parameter. In this article, we generalize the RMST regression model to clustered data by directly modeling the RMST as a continuous function of restriction times with covariates while properly accounting for within-cluster correlations to achieve valid inference. The proposed method estimates regression coefficients via weighted generalized estimating equations, coupled with a cluster-robust sandwich variance estimator to achieve asymptotically valid inference with a sufficient number of clusters. In small-sample scenarios where a limited number of clusters are available, however, the proposed sandwich variance estimator can exhibit negative bias in capturing the variability of regression coefficient estimates. To overcome this limitation, we further propose and examine bias-corrected sandwich variance estimators to reduce the negative bias of the cluster-robust sandwich variance estimator. We study the finite-sample operating characteristics of proposed methods through simulations and reanalyze two multicenter randomized trials.     

White-Tailed Deer,Weather and Predation: a New Understanding of Winter Severity for Predicting Deer Mortality

Variation in white-tailed deer (Odocoileus virginianus) mortality during winter affects population growth in cold climates. Across the northern extent of their range, mortality increases with colder temperatures and snow. Few studies have examined the relationships between winter conditions and deer mortality, and no studies have concurrently studied this relationship for different ages of deer across multiple years and landscapes. We used recently developed cause-specific mortality models to evaluate temporal and age-class variation in deer mortality in farmland areas and compared to published results from forest areas in Wisconsin, USA, from 2011–2014. We then used temporally varying snow and temperature covariates to predict mortality trends using telemetry information from 860 deer. Cause-specific mortality in the farmland varied by age and year, similar to results from previous research in the forest. Human-related mortality was the leading cause of mortality in the farmland during most years and ranged from 4.3% to 10.3% for juveniles and 3.6% to 9.1% for adults from 2011–2014. Very little predation occurred in the farmland, and this differed from previous research in the forest where predation was the leading cause of mortality. During more severe winters (2013 and 2014), other mortality, usually associated with starvation, was the leading cause of mortality for juveniles in the farmland but not adults. In the forest, we found support for saturating effects of accumulated snow depth days >30.5 cm and accumulated temperature days >0°C on mortality. We also found support for the relationship of mortality with accumulated temperature days >0°C in the farmland but no relationship with snow depth. Deer tolerate sustained cold temperatures, but the timing of winter to spring transition is more important for deer survival in both forested and agricultural areas. In the absence of empirical survival information, managers can use our model to predict annual winter effects on deer survival, which can provide improved inference compared to traditional winter severity indices. Our results suggest changes in predator abundance may have minor influence on overwinter survival compared to winter weather. Based on mortality estimates from previous research, the highest predation rates on juvenile deer in the forest occurred when wolf (Canis lupus) counts were lowest and when wolf abundance was highest, juvenile deer predation rates were lowest. © 2021 The Wildlife Society.     

Joint modelling of repeated measurements and event time: Application to performance traits and survival of lambs bred in sub-humid tropics

We considered the analysis of a study for Dorper, Red Maasai and crossbred lambs born over a period of 6 years at the Diani Estate, Kenya. The study was designed to compare survival and performance traits of genotypes with differing susceptibilities to helminthiasis. The available data include information on time to death and repeated measurements of body weight, packed cell volume (PCV) and faecal egg count (FEC) of the animals. In the paper, we consider joint modelling of the survival time and the repeated measurements. Such an approach allows to account for the possible association between the survival and repeated measurement processes. The advantages and limitations of the joint modelling are discussed and illustrated using the Diani Estate study data.     

Gene-based association analysis for bivariate time-to-event data through functional regression with copula models

Several gene-based association tests for time-to-event traits have been proposed recently to detect whether a gene region (containing multiple variants), as a set, is associated with the survival outcome. However, for bivariate survival outcomes, to the best of our knowledge, there is no statistical method that can be directly applied for gene-based association analysis. Motivated by a genetic study to discover the gene regions associated with the progression of a bilateral eye disease, age-related macular degeneration (AMD), we implement a novel functional regression (FR) method under the copula framework. Specifically, the effects of variants within a gene region are modeled through a functional linear model, which then contributes to the marginal survival functions within the copula. Generalized score test statistics are derived to test for the association between bivariate survival traits and the genetic region. Extensive simulation studies are conducted to evaluate the type I error control and power performance of the proposed approach, with comparisons to several existing methods for a single survival trait, as well as the marginal Cox FR model using the robust sandwich estimator for bivariate survival traits. Finally, we apply our method to a large AMD study, the Age-related Eye Disease Study, and to identify the gene regions that are associated with AMD progression.     

Penalized estimation of frailty-based illness–death models for semi-competing risks

Semi-competing risks refer to the time-to-event analysis setting, where the occurrence of a non-terminal event is subject to whether a terminal event has occurred, but not vice versa. Semi-competing risks arise in a broad range of clinical contexts, including studies of preeclampsia, a condition that may arise during pregnancy and for which delivery is a terminal event. Models that acknowledge semi-competing risks enable investigation of relationships between covariates and the joint timing of the outcomes, but methods for model selection and prediction of semi-competing risks in high dimensions are lacking. Moreover, in such settings researchers commonly analyze only a single or composite outcome, losing valuable information and limiting clinical utility—in the obstetric setting, this means ignoring valuable insight into timing of delivery after preeclampsia has onset. To address this gap, we propose a novel penalized estimation framework for frailty-based illness–death multi-state modeling of semi-competing risks. Our approach combines non-convex and structured fusion penalization, inducing global sparsity as well as parsimony across submodels. We perform estimation and model selection via a pathwise routine for non-convex optimization, and prove statistical error rate results in this setting. We present a simulation study investigating estimation error and model selection performance, and a comprehensive application of the method to joint risk modeling of preeclampsia and timing of delivery using pregnancy data from an electronic health record.     

Improving sandwich variance estimation for marginal Cox analysis of cluster randomized trials

Cluster randomized trials (CRTs) frequently recruit a small number of clusters, therefore necessitating the application of small-sample corrections for valid inference. A recent systematic review indicated that CRTs reporting right-censored, time-to-event outcomes are not uncommon and that the marginal Cox proportional hazards model is one of the common approaches used for primary analysis. While small-sample corrections have been studied under marginal models with continuous, binary, and count outcomes, no prior research has been devoted to the development and evaluation of bias-corrected sandwich variance estimators when clustered time-to-event outcomes are analyzed by the marginal Cox model. To improve current practice, we propose nine bias-corrected sandwich variance estimators for the analysis of CRTs using the marginal Cox model and report on a simulation study to evaluate their small-sample properties. Our results indicate that the optimal choice of bias-corrected sandwich variance estimator for CRTs with survival outcomes can depend on the variability of cluster sizes and can also slightly differ whether it is evaluated according to relative bias or type I error rate. Finally, we illustrate the new variance estimators in a real-world CRT where the conclusion about intervention effectiveness differs depending on the use of small-sample bias corrections. The proposed sandwich variance estimators are implemented in an R package CoxBcv .