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Marginal structural models have been increasingly used by analysts in recent years to account for confounding bias in studies with time‐varying treatments. The parameters of these models are often estimated using inverse probability of treatment weighting. To ensure that the estimated weights adequately control confounding, it is possible to check for residual imbalance between treatment groups in the weighted data. Several balance metrics have been developed and compared in the cross‐sectional case but have not yet been evaluated and compared in longitudinal studies with time‐varying treatment. We have first extended the definition of several balance metrics to the case of a time‐varying treatment, with or without censoring. We then compared the performance of these balance metrics in a simulation study by assessing the strength of the association between their estimated level of imbalance and bias. We found that the Mahalanobis balance performed best. Finally, the method was illustrated for estimating the cumulative effect of statins exposure over one year on the risk of cardiovascular disease or death in people aged 65 and over in population‐wide administrative data. This illustration confirms the feasibility of employing our proposed metrics in large databases with multiple time‐points. [ABSTRACT FROM AUTHOR]

An important task in survival analysis is choosing a structure for the relationship between covariates of interest and the time-to-event outcome. For example, the accelerated failure time (AFT) model structures each covariate effect as a constant multiplicative shift in the outcome distribution across all survival quantiles. Though parsimonious, this structure cannot detect or capture effects that differ across quantiles of the distribution, a limitation that is analogous to only permitting proportional hazards in the Cox model. To address this, we propose a general framework for quantile-varying multiplicative effects under the AFT model. Specifically, we embed flexible regression structures within the AFT model and derive a novel formula for interpretable effects on the quantile scale. A regression standardization scheme based on the g-formula is proposed to enable the estimation of both covariate-conditional and marginal effects for an exposure of interest. We implement a user-friendly Bayesian approach for the estimation and quantification of uncertainty while accounting for left truncation and complex censoring. We emphasize the intuitive interpretation of this model through numerical and graphical tools and illustrate its performance through simulation and application to a study of Alzheimer's disease and dementia. [ABSTRACT FROM AUTHOR]

This paper investigates the student dropout phenomenon in a technical Italian university from a time-to-event perspective. Shared frailty Cox time-dependent models are applied to analyse the careers of students enrolled in different engineering programs with the aim of identifying the determinants of student dropout through time, predicting the time to dropout as soon as possible and to observe how the dropout phenomenon varies across time and degree programs. The innovative contributions of this work are methodological and managerial. First, the adoption of shared frailty Cox models with time-varying covariates is relatively new to the student dropout literature and it allows to consider the student career evolution and the heterogeneity across degree programs. Second, understanding the dropout pattern over time and identifying the earliest moment for obtaining its accurate prediction allow policy makers to set timely interventions for students at risk of dropout. [ABSTRACT FROM AUTHOR]

Growth mixture modeling (GMM) is an analytical tool for identifying multiple unobserved sub-populations in longitudinal processes. In particular, it describes change patterns within each latent sub-population and investigates between-individual differences in within-individual change for each sub-group. A key research interest in using GMMs is examining how covariates influence the heterogeneity in change patterns. Liu & Perera (2022b) extended mixture-of-experts (MoE) models, which primarily focus on time-invariant covariates, to allow covariates to account for both within-group and between-group differences and investigate the heterogeneity in nonlinear trajectories. The present study further extends Liu & Perera, 2022b by examining the effects of time-varying covariates (TVCs) on trajectory heterogeneity. Specifically, we propose methods to decompose a TVC into an initial trait (the baseline value of the TVC) and a set of temporal states (interval-specific slopes or changes of the TVC). The initial trait is allowed to account for within-group differences in growth factors of trajectories (i.e., baseline effect), while the temporal states are allowed to impact observed values of a longitudinal process (i.e., temporal effects). We evaluate the proposed models using a simulation study and real-world data analysis. The simulation study demonstrates that the proposed models are capable of separating trajectories into several clusters and generally producing unbiased and accurate estimates with target coverage probabilities. The proposed models reveal the heterogeneity in initial trait and temporal states of reading ability across latent classes of students' mathematics performance. Additionally, the baseline and temporal effects on mathematics development of reading ability are also heterogeneous across the clusters of students. [ABSTRACT FROM AUTHOR]

Joint models for longitudinal and time‐to‐event data are often employed to calculate dynamic individualized predictions used in numerous applications of precision medicine. Two components of joint models that influence the accuracy of these predictions are the shape of the longitudinal trajectories and the functional form linking the longitudinal outcome history to the hazard of the event. Finding a single well‐specified model that produces accurate predictions for all subjects and follow‐up times can be challenging, especially when considering multiple longitudinal outcomes. In this work, we use the concept of super learning and avoid selecting a single model. In particular, we specify a weighted combination of the dynamic predictions calculated from a library of joint models with different specifications. The weights are selected to optimize a predictive accuracy metric using V‐fold cross‐validation. We use as predictive accuracy measures the expected quadratic prediction error and the expected predictive cross‐entropy. In a simulation study, we found that the super learning approach produces results very similar to the Oracle model, which was the model with the best performance in the test datasets. All proposed methodology is implemented in the freely available R package JMbayes2. [ABSTRACT FROM AUTHOR]

We evaluated the association between biomarkers and COVID-19 mortality. Baseline characteristics of 403 COVID-19 patients included sex and age; biomarkers, measured throughout the follow-up, included lymphocytes, neutrophils, ferritin, C-reactive protein, glucose, and LDH. Hazard ratios (HRs) and corresponding 95% credible intervals (CIs) were estimated through joint models (JMs) using a Bayesian approach. We fitted univariable (a single biomarker) and multivariable (all biomarkers) JMs. In univariable analyses, all biomarkers were significantly associated with COVID-19 mortality. In multivariable analysis, HRs were 1.78 (95% CI: 1.13–2.87) with a doubling of neutrophils levels, 1.49 (95% CI: 1.19–1.95) with a doubling of C-reactive protein levels, 2.66 (95% CI: 1.45–4.95) for an increase of 100 mg/dL of glucose, and 1.31 (95% CI: 1.12–1.55) for an increase of 100 U/L of LDH. No evidence of association was observed for lymphocytes and ferritin in multivariable analysis. Men had a higher COVID-19 mortality risk than women (HR = 1.75; 95% CI: 1.07–2.80) and age showed the strongest effect with a rapid increase from 60 years. These findings using JM confirm the usefulness of biomarkers in assessing COVID-19 severity and mortality. Monitoring trend patterns of such biomarkers can provide additional help in tailoring the appropriate care pathway. [ABSTRACT FROM AUTHOR]

Recurrent events such as hospitalisations are outcomes that can be used to monitor dialysis facilities' quality of care. However, current methods are not adequate to analyse data from many facilities with multiple hospitalisations, especially when adjustments are needed for multiple time scales. It is also controversial whether direct or indirect standardisation should be used in comparing facilities. This study is motivated by the need of the Centers for Medicare and Medicaid Services to evaluate US dialysis facilities using Medicare claims, which involve almost 8,000 facilities and over 500,000 dialysis patients. This scope is challenging for current statistical software's computational power. We propose a method that has a flexible baseline rate function and is computationally efficient. Additionally, the proposed method shares advantages of both indirect and direct standardisation. The method is evaluated under a range of simulation settings and demonstrates substantially improved computational efficiency over the existing R package survival. Finally, we illustrate the method with an important application to monitoring dialysis facilities in the U.S., while making time-dependent adjustments for the effects of COVID-19. [ABSTRACT FROM AUTHOR]

The accelerated failure time (AFT) model offers an important and useful alternative to the conventional Cox proportional hazards model, particularly when the proportional hazards assumption for a Cox model is violated. Since an AFT model is basically a log‐linear model, meaningful interpretations of covariate effects on failure times can be made directly. However, estimation of a semiparametric AFT model imposes computational challenges even when it only has time‐fixed covariates, and the situation becomes much more complicated when time‐varying covariates are included. In this paper, we propose a penalised likelihood approach to estimate the semiparametric AFT model with right‐censored failure time, where both time‐fixed and time‐varying covariates are permitted. We adopt the Gaussian basis functions to construct a smooth approximation to the nonparametric baseline hazard. This model fitting method requires a constrained optimisation approach. A comprehensive simulation study is conducted to demonstrate the performance of the proposed method. An application of our method to a motor neuron disease data set is provided. [ABSTRACT FROM AUTHOR]

Abstract Background Controlling cholesterol levels is one of the primary goals of preventing atherosclerotic plaque progression in patients undergoing coronary artery bypass graft (CABG) surgery. This study aimed to investigate the impact of serum cholesterol profile at multiple time points following isolated CABG surgery on long-term patient outcomes. Method This retrospective cohort study was conducted on the admission and follow-up data of isolated CABG patients from the Tehran Heart Center registry between 2009 and 2016. The association of low-density lipoprotein (LDL), high-density lipoprotein (HDL), and their ratio as an atherogenic index with major adverse cardiac and cerebrovascular events (MACCE) and all-cause mortality were evaluated using time-varying survival analysis methods. Result A total of 18657 patients were included in this analysis. After adjusting for known confounding factors, no significant difference in all-cause mortality and MACCE was observed at different LDL levels. The incidence of acute coronary syndrome (ACS) in patients with LDL > 100 mg/dl and LDL 100 mg/dl was also significantly higher compared to the control group (P -value = 0.033). Lower HDL levels were significantly associated with a higher MACCE (P -value

Abstract Objective This research note introduces a set of tools to conduct analysis of household structure and composition with either limited or comprehensive longitudinal data. The data used here are from Health and Demographic Surveillance Systems in Africa, but the methods can be adapted and applied to other longitudinal micro-data such as register data. A training manual describing the procedures for creating time-varying household measures step-by-step is supplied as supplementary material to this note. Code is provided in STATA but can easily be translated for other statistical software, and the logic for each step remains the same. Results The analysis of household structure demonstrates how with limited data (such as a household identifier), it is possible to construct time-varying measures of household membership, including household size or the number of members in specific age and sex groups. The analysis of household composition demonstrates how with expanded data (including links to parents in addition to residence status in the same household), it is possible to construct time-varying measures of household membership of specific kin, i.e. mother, sibling or grandparent. The results illustrated in this research note are a taste of what can be achieved by following the training manual in the supplementary material.

Objective: This research note introduces a set of tools to conduct analysis of household structure and composition with either limited or comprehensive longitudinal data. The data used here are from Health and Demographic Surveillance Systems in Africa, but the methods can be adapted and applied to other longitudinal micro-data such as register data. A training manual describing the procedures for creating time-varying household measures step-by-step is supplied as supplementary material to this note. Code is provided in STATA but can easily be translated for other statistical software, and the logic for each step remains the same. Results: The analysis of household structure demonstrates how with limited data (such as a household identifier), it is possible to construct time-varying measures of household membership, including household size or the number of members in specific age and sex groups. The analysis of household composition demonstrates how with expanded data (including links to parents in addition to residence status in the same household), it is possible to construct time-varying measures of household membership of specific kin, i.e. mother, sibling or grandparent. The results illustrated in this research note are a taste of what can be achieved by following the training manual in the supplementary material. [ABSTRACT FROM AUTHOR]

Background: Controlling cholesterol levels is one of the primary goals of preventing atherosclerotic plaque progression in patients undergoing coronary artery bypass graft (CABG) surgery. This study aimed to investigate the impact of serum cholesterol profile at multiple time points following isolated CABG surgery on long-term patient outcomes. Method: This retrospective cohort study was conducted on the admission and follow-up data of isolated CABG patients from the Tehran Heart Center registry between 2009 and 2016. The association of low-density lipoprotein (LDL), high-density lipoprotein (HDL), and their ratio as an atherogenic index with major adverse cardiac and cerebrovascular events (MACCE) and all-cause mortality were evaluated using time-varying survival analysis methods. Result: A total of 18657 patients were included in this analysis. After adjusting for known confounding factors, no significant difference in all-cause mortality and MACCE was observed at different LDL levels. The incidence of acute coronary syndrome (ACS) in patients with LDL > 100 mg/dl and LDL < 50 mg/dl was significantly higher than in the control group (P-value = 0.004 and 0.04, respectively). The incidence of cerebrovascular accidents (CVA) at LDL > 100 mg/dl was also significantly higher compared to the control group (P -value = 0.033). Lower HDL levels were significantly associated with a higher MACCE (P -value < 0.001), all-cause mortality (P -value < 0.001), ACS (P -value = 0.00), and CVA (P -value = 0.014). The atherogenic index was also directly related to MACCE and all its components (all P-values < 0.001). Conclusion: LDL/HDL ratio is suggested as a better marker for secondary prevention goals compared to LDL alone in patients undergoing CABG surgery. [ABSTRACT FROM AUTHOR]

Latent state-trait (LST) models are increasingly applied in psychology. Although existing LST models offer many possibilities for analyzing variability and change, they do not allow researchers to relate time-varying or time-invariant covariates, or a combination of both, to loading, intercept, and factor variance parameters in LST models. We present a general framework for the inclusion of nominal and/or continuous time-varying and time-invariant covariates in LST models. The new framework builds on modern LST theory and Bayesian moderated nonlinear factor analysis and is termed moderated nonlinear LST (MN-LST) framework. The MN-LST framework offers new modeling possibilities and allows for a fine-grained analysis of trait change, person-by-situation interaction effects, as well as inter- or intraindividual variability. The new MN-LST approach is compared to alternative modeling strategies. The advantages of the MN-LST approach are illustrated in an empirical application examining dyadic coping in romantic relationships. Finally, the advantages and limitations of the approach are discussed, and practical recommendations are provided. [ABSTRACT FROM AUTHOR]

We evaluated the association between biomarkers and COVID-19 mortality. Baseline characteristics of 403 COVID-19 patients included sex and age; biomarkers, measured throughout the follow-up, included lymphocytes, neutrophils, ferritin, C-reactive protein, glucose, and LDH. Hazard ratios (HRs) and corresponding 95% credible intervals (CIs) were estimated through joint models (JMs) using a Bayesian approach. We fitted univariable (a single biomarker) and multivariable (all biomarkers) JMs. In univariable analyses, all biomarkers were significantly associated with COVID-19 mortality. In multivariable analysis, HRs were 1.78 (95% CI: 1.13–2.87) with a doubling of neutrophils levels, 1.49 (95% CI: 1.19–1.95) with a doubling of C-reactive protein levels, 2.66 (95% CI: 1.45–4.95) for an increase of 100 mg/dL of glucose, and 1.31 (95% CI: 1.12–1.55) for an increase of 100 U/L of LDH. No evidence of association was observed for lymphocytes and ferritin in multivariable analysis. Men had a higher COVID-19 mortality risk than women (HR = 1.75; 95% CI: 1.07–2.80) and age showed the strongest effect with a rapid increase from 60 years. These findings using JM confirm the usefulness of biomarkers in assessing COVID-19 severity and mortality. Monitoring trend patterns of such biomarkers can provide additional help in tailoring the appropriate care pathway.

Abstract Background High bed-occupancy (capacity utilization) rates are commonly thought to increase in-hospital mortality; however, little evidence supports a causal relationship between the two. This observational study aimed to assess three time-varying covariates—capacity utilization, patient turnover and clinical complexity level— and to estimate causal effect of time-varying high capacity utilization on 14 day in-hospital mortality. Methods This retrospective population-based analysis was based on routine administrative data (n = 1,152,506 inpatient cases) of 102 Swiss general hospitals. Considering the longitudinal nature of the problem from available literature and expert knowledge, we represented the underlying data generating mechanism as a directed acyclic graph. To adjust for patient turnover and patient clinical complexity levels as time-varying confounders, we fitted a marginal structure model (MSM) that used inverse probability of treatment weights (IPTWs) for high and low capacity utilization. We also adjusted for patient age and sex, weekdays-vs-weekend, comorbidity weight, and hospital type. Results For each participating hospital, our analyses evaluated the ≥85th percentile as a threshold for high capacity utilization for the higher risk of mortality. The mean bed-occupancy threshold was 83.1% (SD 8.6) across hospitals and ranged from 42.1 to 95.9% between hospitals. For each additional day of exposure to high capacity utilization, our MSM incorporating IPTWs showed a 2% increase in the odds of 14-day in-hospital mortality (OR 1.02, 95% CI: 1.01 to 1.03). Conclusions Exposure to high capacity utilization increases the mortality risk of inpatients. Accurate monitoring of capacity utilization and flexible human resource planning are key strategies for hospitals to lower the exposure to high capacity utilization.

Time‐varying covariates can be important predictors when model based predictions are considered. A Cox model that includes time‐varying covariates is usually referred to as an extended Cox model. When only right censoring is presented in the observed survival times, the conventional partial likelihood method is still applicable to estimate the regression coefficients of an extended Cox model. However, if there are interval‐censored survival times, then the partial likelihood method is not directly available unless an imputation, such as the middle point imputation, is used to replaced the left‐ and interval‐censored data. However, such imputation methods are well known for causing biases. This paper considers fitting of the extended Cox models using the maximum penalised likelihood method allowing observed survival times to be partly interval censored, where a penalty function is used to regularise the baseline hazard estimate. We present simulation studies to demonstrate the performance of our proposed method, and illustrate our method with applications to two real datasets from medical research. [ABSTRACT FROM AUTHOR]

Time-varying covariates are of great interest in clinical research since they represent dynamic patterns which reflect disease progression. In cancer studies biomarkers values change as functions of time and chemotherapy treatment is modified by delaying a course or reducing the dose intensity, according to patient's toxicity levels. In this work, a Functional covariate Cox Model (FunCM) to study the association between time-varying processes and a time-to-event outcome is proposed. FunCM first exploits functional data analysis techniques to represent time-varying processes in terms of functional data. Then, information related to the evolution of the functions over time is incorporated into functional regression models for survival data through functional principal component analysis. FunCM is compared to a standard time-varying covariate Cox model, commonly used despite its limiting assumptions that covariate values are constant in time and measured without errors. Data from MRC BO06/EORTC 80931 randomised controlled trial for treatment of osteosarcoma are analysed. Time-varying covariates related to alkaline phosphatase levels, white blood cell counts and chemotherapy dose during treatment are investigated. The proposed method allows to detect differences between patients with different biomarkers and treatment evolutions, and to include this information in the survival model. These aspects are seldom addressed in the literature and could provide new insights into the clinical research. [ABSTRACT FROM AUTHOR]

Arctic-nesting geese face energetic challenges during spring migration, including ecological barriers and weather conditions (e.g., precipitation and temperature), which in long-lived species can lead to a trade-off to defer reproduction in favor of greater survival. We used GPS location and acceleration data collected from 35 greater white-fronted geese of the North American midcontinent and Greenland populations at spring migration stopovers, and novel applications of Bayesian dynamic linear models to test daily effects of minimum temperature and precipitation on energy expenditure (i.e., overall dynamic body acceleration, ODBA) and proportion of time spent feeding (PTF), then examined the daily and additive importance of ODBA and PTF on probability of breeding deferral using stochastic antecedent models. We expected distinct responses in behavior and probability of breeding deferral between and within populations due to differences in stopover area availability. Time-varying coefficients of weather conditions were variable between ODBA and PTF, and often did not show consistent patterns among birds, indicating plasticity in how individuals respond to conditions. An increase in antecedent ODBA was associated with a slightly increased probability of deferral in midcontinent geese but not Greenland geese. Probability of deferral decreased with increased PTF in both populations. We did not detect any differentially important time periods. These results suggest either that movements and behavior throughout spring migration do not explain breeding deferral or that ecological linkages between bird decisions during spring and subsequent breeding deferral were different between populations and across migration but occurred at different time scales than those we examined. [ABSTRACT FROM AUTHOR]

Background: Multivariate longitudinal data are under-utilized for survival analysis compared to cross-sectional data (CS - data collected once across cohort). Particularly in cardiovascular risk prediction, despite available methods of longitudinal data analysis, the value of longitudinal information has not been established in terms of improved predictive accuracy and clinical applicability. Methods: We investigated the value of longitudinal data over and above the use of cross-sectional data via 6 distinct modeling strategies from statistics, machine learning, and deep learning that incorporate repeated measures for survival analysis of the time-to-cardiovascular event in the Coronary Artery Risk Development in Young Adults (CARDIA) cohort. We then examined and compared the use of model-specific interpretability methods (Random Survival Forest Variable Importance) and model-agnostic methods (SHapley Additive exPlanation (SHAP) and Temporal Importance Model Explanation (TIME)) in cardiovascular risk prediction using the top-performing models. Results: In a cohort of 3539 participants, longitudinal information from 35 variables that were repeatedly collected in 6 exam visits over 15 years improved subsequent long-term (17 years after) risk prediction by up to 8.3% in C-index compared to using baseline data (0.78 vs. 0.72), and up to approximately 4% compared to using the last observed CS data (0.75). Time-varying AUC was also higher in models using longitudinal data (0.86–0.87 at 5 years, 0.79–0.81 at 10 years) than using baseline or last observed CS data (0.80–0.86 at 5 years, 0.73–0.77 at 10 years). Comparative model interpretability analysis revealed the impact of longitudinal variables on model prediction on both the individual and global scales among different modeling strategies, as well as identifying the best time windows and best timing within that window for event prediction. The best strategy to incorporate longitudinal data for accuracy was time series massive feature extraction, and the easiest interpretable strategy was trajectory clustering. Conclusion: Our analysis demonstrates the added value of longitudinal data in predictive accuracy and epidemiological utility in cardiovascular risk survival analysis in young adults via a unified, scalable framework that compares model performance and explainability. The framework can be extended to a larger number of variables and other longitudinal modeling methods. Trial registration: ClinicalTrials.gov Identifier: NCT00005130, Registration Date: 26/05/2000. [ABSTRACT FROM AUTHOR]

Most work in neural networks focuses on estimating the conditional mean of a continuous response variable given a set of covariates. In this article, we consider estimating the conditional distribution function using neural networks for both censored and uncensored data. The algorithm is built upon the data structure particularly constructed for the Cox regression with time-dependent covariates. Without imposing any model assumptions, we consider a loss function that is based on the full likelihood where the conditional hazard function is the only unknown nonparametric parameter, for which unconstrained optimization methods can be applied. Through simulation studies, we show that the proposed method possesses desirable performance, whereas the partial likelihood method and the traditional neural networks with L2 loss yields biased estimates when model assumptions are violated. We further illustrate the proposed method with several real-world data sets. The implementation of the proposed methods is made available at https://github.com/bingqing0729/NNCDE. [ABSTRACT FROM AUTHOR]

Background: High bed-occupancy (capacity utilization) rates are commonly thought to increase in-hospital mortality; however, little evidence supports a causal relationship between the two. This observational study aimed to assess three time-varying covariates—capacity utilization, patient turnover and clinical complexity level— and to estimate causal effect of time-varying high capacity utilization on 14 day in-hospital mortality. Methods: This retrospective population-based analysis was based on routine administrative data (n = 1,152,506 inpatient cases) of 102 Swiss general hospitals. Considering the longitudinal nature of the problem from available literature and expert knowledge, we represented the underlying data generating mechanism as a directed acyclic graph. To adjust for patient turnover and patient clinical complexity levels as time-varying confounders, we fitted a marginal structure model (MSM) that used inverse probability of treatment weights (IPTWs) for high and low capacity utilization. We also adjusted for patient age and sex, weekdays-vs-weekend, comorbidity weight, and hospital type. Results: For each participating hospital, our analyses evaluated the ≥85th percentile as a threshold for high capacity utilization for the higher risk of mortality. The mean bed-occupancy threshold was 83.1% (SD 8.6) across hospitals and ranged from 42.1 to 95.9% between hospitals. For each additional day of exposure to high capacity utilization, our MSM incorporating IPTWs showed a 2% increase in the odds of 14-day in-hospital mortality (OR 1.02, 95% CI: 1.01 to 1.03). Conclusions: Exposure to high capacity utilization increases the mortality risk of inpatients. Accurate monitoring of capacity utilization and flexible human resource planning are key strategies for hospitals to lower the exposure to high capacity utilization. [ABSTRACT FROM AUTHOR]

Heart failure (HF) is challenging public medical and healthcare systems. This study aimed to develop and validate a novel deep learning-based prognostic model to predict the risk of all-cause mortality for patients with HF. We also compared the performance of the proposed model with those of classical deep learning- and traditional statistical-based models. The present study enrolled 730 patients with HF hospitalized at Toho University Ohashi Medical Center between April 2016 and March 2020. A recurrent neural network-based model (RNNSurv) involving time-varying covariates was developed and validated. The proposed RNNSurv showed better prediction performance than those of a deep feed-forward neural network-based model (referred as "DeepSurv") and a multivariate Cox proportional hazard model in view of discrimination (C-index: 0.839 vs. 0.755 vs. 0.762, respectively), calibration (better fit with a 45-degree line), and ability of risk stratification, especially identifying patients with high risk of mortality. The proposed RNNSurv demonstrated an improved prediction performance in consideration of temporal information from time-varying covariates that could assist clinical decision-making. Additionally, this study found that significant risk and protective factors of mortality were specific to risk levels, highlighting the demand for an individual-specific clinical strategy instead of a uniform one for all patients. [ABSTRACT FROM AUTHOR]

Many clinical and epidemiological applications of survival analysis focus on interval‐censored events that can be ascertained only at discrete times of clinic visits. This implies that the values of time‐varying covariates are not correctly aligned with the true, unknown event times, inducing a bias in the estimated associations. To address this issue, we adapted the simulation‐extrapolation (SIMEX) methodology, based on assessing how the estimates change with the artificially increased time between clinic visits. We propose diagnostics to choose the extrapolating function. In simulations, the SIMEX‐corrected estimates reduced considerably the bias to the null and generally yielded a better bias/variance trade‐off than conventional estimates. In a real‐life pharmacoepidemiological application, the proposed method increased by 27% the excess hazard of the estimated association between a time‐varying exposure, representing the 2‐year cumulative duration of past use of a hypertensive medication, and the hazard of nonmelanoma skin cancer (interval‐censored events). These simulation‐based and real‐life results suggest that the proposed SIMEX‐based correction may help improve the accuracy of estimated associations between time‐varying exposures and the hazard of interval‐censored events in large cohort studies where the events are recorded only at relatively sparse times of clinic visits/assessments. However, these advantages may be less certain for smaller studies and/or weak associations. [ABSTRACT FROM AUTHOR]

Survival data with time-varying covariates are common in practice. If relevant, they can improve on the estimation of a survival function. However, the traditional survival forests—conditional inference forest, relative risk forest and random survival forest—have accommodated only time-invariant covariates. We generalize the conditional inference and relative risk forests to allow time-varying covariates. We also propose a general framework for estimation of a survival function in the presence of time-varying covariates. We compare their performance with that of the Cox model and transformation forest, adapted here to accommodate time-varying covariates, through a comprehensive simulation study in which the Kaplan-Meier estimate serves as a benchmark, and performance is compared using the integrated L 2 difference between the true and estimated survival functions. In general, the performance of the two proposed forests substantially improves over the Kaplan-Meier estimate. Taking into account all other factors, under the proportional hazard setting, the best method is always one of the two proposed forests, while under the non-proportional hazard setting, it is the adapted transformation forest. K -fold cross-validation is used as an effective tool to choose between the methods in practice. [ABSTRACT FROM AUTHOR]

Analyzing time to delisted status for listed firms with risk warnings in a stock market is important in risk management of the stock market. This analysis is entangled by the fact that not all listed firms with risk warnings will eventually be delisted, making a standard time-to-event analysis not suitable. The presence of time-varying factors that are related to the listed firms and the macro-economic environment adds another layer of challenge to the analysis. We propose to use a mixture cure model with time-varying covariates to analyze time to delisting in two Chinses stock markets. We identify issues in an existing method and propose a new method to better handle time-varying covariates in the mixture cure model. The model allows an exploration of the association between the probability that a listed firm will never be delisted and time-fixed covariates. The performance of the proposed estimation method is examined using simulation and compared with existing methods. The results of the data analysis reveal a few important time-varying covariates that have significant impacts on the time to delisted status. However, none of the measured time-fixed covariates is found to have a significant impact on the probability of never being delisted. [ABSTRACT FROM AUTHOR]

In modeling time-to-event data with long-term survivors, the proportional hazards model is widely used for its easy and direct interpretation as well as the flexibility to accommodate the past information and allow time-varying predictors. This becomes most relevant when the mortality of individuals converges with time, and the estimation and inference based upon the proportional odds model can often yield more accurate and reasonable results than the classical Cox's proportional hazards model. Along with the fast development of the data science technologies, computational challenges for survival data with increasing sample size and diverging parameter dimension exist. Currently, existing methods for analyzing such data are computationally inconvenient. In this paper, we propose efficient computational methods for analyzing survival data in the proportional odds model, where the nonparametric maximum likelihood approach is combined with the minorization-maximization (MM) algorithm and the regularization scheme to yield fast and accurate estimation and inferential procedures. The illustration of the methodology using extensive simulation studies and then the application to the Veterans' Administration lung cancer data is also given. [ABSTRACT FROM AUTHOR]

Abstract Objective To illustrate the challenges of estimating the effect of an exposure that is bounded by duration of follow-up on all-cause 28-day mortality, whilst simultaneously addressing missing data and time-varying covariates. Study design and methods BSI-FOO is a multicentre cohort study with the primary aim of quantifying the effect of modifiable risk factors, including time to initiation of therapy, on all-cause 28-day mortality in patients with bloodstream infection. The primary analysis involved two Cox proportional hazard models, first one for non-modifiable risk factors and second one for modifiable risk factors, with a risk score calculated from the first model included as a covariate in the second model. Modifiable risk factors considered in this study were recorded daily for a maximum of 28 days after infection. Follow-up was split at daily intervals from day 0 to 28 with values of daily collected data updated at each interval (i.e., one row per patient per day). Analytical challenges Estimating the effect of time to initiation of treatment on survival is analytically challenging since only those who survive to time t can wait until time t to start treatment, introducing immortal time bias. Time-varying covariates representing cumulative counts were used for variables bounded by survival time e.g. the cumulative count of days before first receipt of treatment. Multiple imputation using chained equations was used to impute missing data, using conditional imputation to avoid imputing non-applicable data e.g. ward data after discharge. Conclusion Using time-varying covariates represented by cumulative counts within a one row per day per patient framework can reduce the risk of bias in effect estimates. The approach followed uses established methodology and is easily implemented in standard statistical packages.

Time‐varying biomarkers reflect important information on disease progression over time. Dynamic prediction for event occurrence on a real‐time basis, utilizing time‐varying information, is crucial in making accurate clinical decisions. Functional principal component analysis (FPCA) has been widely adopted in the literature for extracting features from time‐varying biomarker trajectories. However, feature extraction via FPCA is conducted independent of the time‐to‐event response, which may not produce optimal results when the goal lies in prediction. With this consideration, we propose a novel supervised FPCA, where the functional principal components are determined to optimize the association between the time‐varying biomarker and time‐to‐event outcome. The proposed framework also accommodates irregularly spaced and sparse longitudinal data. Our method is empirically shown to retain better discrimination and calibration performance than the unsupervised FPCA method in simulation studies. Application of the proposed method is also illustrated in the Alzheimer's Disease Neuroimaging Initiative database. [ABSTRACT FROM AUTHOR]

Time‐varying covariates are often available in survival studies, and estimation of the hazard function needs to be updated as new information becomes available. In this article, we investigate several different easy‐to‐implement ways that random forests can be used for dynamic estimation of the survival or hazard function from discrete‐time survival data. Results from a simulation study indicate that all methods can perform well, and that none dominates the others. In general, situations that are more difficult from an estimation point of view (such as weaker signals and less data) favour a global fit, pooling over all time points, while situations that are easier from an estimation point of view (such as stronger signals and more data) favour local fits. [ABSTRACT FROM AUTHOR]

In this paper, we propose a semiparametric, tree-based joint latent class model for the joint behavior of longitudinal and time-to-event data. Existing joint latent class approaches are parametric and can suffer from high computational cost. The most common parametric approach, the joint latent class model, further restricts analysis to using time-invariant covariates in modeling survival risks and latent class memberships. The proposed tree method (joint latent class tree) is fast to fit, and permits time-varying covariates in all of its modeling components. We demonstrate the prognostic value of using time-varying covariates, and therefore the advantage of joint latent class tree over joint latent class model on simulated data. We apply joint latent class tree to a well-known data set (the PAQUID data set) and confirm its superior prediction performance and orders-of-magnitude speedup over joint latent class model. [ABSTRACT FROM AUTHOR]

Additive hazards model is often used to complement the proportional hazards model in the analysis of failure time data. Statistical inference of additive hazards model with time-dependent longitudinal covariates requires the availability of the whole trajectory of the longitudinal process, which is not realistic in practice. The commonly used last value carried forward approach for intermittently observed longitudinal covariates can induce biased parameter estimation. The more principled joint modeling of the longitudinal process and failure time data imposes strong modeling assumptions, which is difficult to verify. In this paper, we propose methods that weigh the distance between the observational time of longitudinal covariates and the failure time, resulting in unbiased regression coefficient estimation. We establish the consistency and asymptotic normality of the proposed estimators. Simulation studies provide numerical support for the theoretical findings. Data from an Alzheimer's study illustrate the practical utility of the methodology. [ABSTRACT FROM AUTHOR]

This study examines time-varying gender-responsive and gender-neutral predictors of recidivism over 3 years using baseline and quarterly follow-up interviews with 477 women released from a county jail. Of the 55 time-varying predictors tested in the longitudinal analysis, 39 were significant predictors of recidivism (new arrest or incarceration) even after controlling for baseline fixed predictors. Stepwise multivariate analysis simplified the model to 12 significant variables, including three time-varying variables associated with reduced risk of recidivism (custody of one’s children, self-help activity, environmental support), eight time-varying variables associated with increased risk of recidivism (illegal activity, type of crime, problems with probation/parole, days in jail/prison, number of sexual partners, past-year trauma, problem orientation, external pressure), and the composite measure of risk from baseline. These findings support the development of post-release re-entry services tailored for female offenders that address both gender-responsive and gender-neutral criminogenic risk factors.

In this article, we consider the problem of change‐point analysis for the count time series data through an integer‐valued autoregressive process of order 1 (INAR(1)) with time‐varying covariates. These types of features we observe in many real‐life scenarios especially in the COVID‐19 data sets, where the number of active cases over time starts falling and then again increases. In order to capture those features, we use Poisson INAR(1) process with a time‐varying smoothing covariate. By using such model, we can model both the components in the active cases at time‐point t namely, (i) number of nonrecovery cases from the previous time‐point and (ii) number of new cases at time‐point t. We study some theoretical properties of the proposed model along with forecasting. Some simulation studies are performed to study the effectiveness of the proposed method. Finally, we analyze two COVID‐19 data sets and compare our proposed model with another PINAR(1) process which has time‐varying covariate but no change‐point, to demonstrate the overall performance of our proposed model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Credit-granting institutions need to estimate the probability of loan default, which represents the chance a customer fails to make repayments as promised. Critically this estimation is intertwined with the competing risk a customer fully repays their loan while also having key predictive drivers with values that change over time. A conventional model in this setting is a competing risks Cox Model with time-varying covariates. However partial likelihood estimation of this model has two shortcomings: (1) the baseline hazard is not estimated, so calculating probabilities requires a further estimation step; and (2) a covariance matrix for both regression coefficients and the baseline hazard is not produced. This paper caters for these shortcomings by devising a maximum likelihood technique to jointly estimate regression coefficients and the cause-specific baseline hazards using constrained optimisation to ensure the latter's non-negativity. We show via simulation our technique produces regression coefficients estimates with lower bias in small samples with heavy censoring. When applied to a real-world credit risk dataset consisting of home loan data our Maximum Likelihood approach produces a smoother estimate of the cause-specific baseline hazards for default and redemption than those obtained using the Partial Likelihood and Breslow approach. This provides better clarity of the shape of these functions through both a less volatile central estimate as well as quantifying the error of this central estimate. We implement our method in R. [ABSTRACT FROM AUTHOR]

When constructing models to summarize clinical data to be used for simulations, it is good practice to evaluate the models for their capacity to reproduce the data. This can be done by means of Visual Predictive Checks (VPC), which consist of several reproductions of the original study by simulation from the model under evaluation, calculating estimates of interest for each simulated study and comparing the distribution of those estimates with the estimate from the original study. This procedure is a generic method that is straightforward to apply, in general. Here we consider the application of the method to time‐to‐event data and consider the special case when a time‐varying covariate is not known or cannot be approximated after event time. In this case, simulations cannot be conducted beyond the end of the follow‐up time (event or censoring time) in the original study. Thus, the simulations must be censored at the end of the follow‐up time. Since this censoring is not random, the standard KM estimates from the simulated studies and the resulting VPC will be biased. We propose to use inverse probability of censoring weighting (IPoC) method to correct the KM estimator for the simulated studies and obtain unbiased VPCs. For analyzing the Cantos study, the IPoC weighting as described here proved valuable and enabled the generation of VPCs to qualify PKPD models for simulations. Here, we use a generated data set, which allows illustration of the different situations and evaluation against the known truth. [ABSTRACT FROM AUTHOR]

Objective: To illustrate the challenges of estimating the effect of an exposure that is bounded by duration of follow-up on all-cause 28-day mortality, whilst simultaneously addressing missing data and time-varying covariates.Study Design and Methods: BSI-FOO is a multicentre cohort study with the primary aim of quantifying the effect of modifiable risk factors, including time to initiation of therapy, on all-cause 28-day mortality in patients with bloodstream infection. The primary analysis involved two Cox proportional hazard models, first one for non-modifiable risk factors and second one for modifiable risk factors, with a risk score calculated from the first model included as a covariate in the second model. Modifiable risk factors considered in this study were recorded daily for a maximum of 28 days after infection. Follow-up was split at daily intervals from day 0 to 28 with values of daily collected data updated at each interval (i.e., one row per patient per day).Analytical Challenges: Estimating the effect of time to initiation of treatment on survival is analytically challenging since only those who survive to time t can wait until time t to start treatment, introducing immortal time bias. Time-varying covariates representing cumulative counts were used for variables bounded by survival time e.g. the cumulative count of days before first receipt of treatment. Multiple imputation using chained equations was used to impute missing data, using conditional imputation to avoid imputing non-applicable data e.g. ward data after discharge.Conclusion: Using time-varying covariates represented by cumulative counts within a one row per day per patient framework can reduce the risk of bias in effect estimates. The approach followed uses established methodology and is easily implemented in standard statistical packages. [ABSTRACT FROM AUTHOR]

For several decades, cross-lagged panel models (CLPM) have been the dominant statistical model in relationship research for investigating reciprocal associations between two (or more) constructs over time. However, recent methodological research has questioned the frequent usage of the CLPM because, amongst other things, the model commingles within-person associations with between-person associations, while most developmental research questions pertain to within-person processes. Furthermore, the model presumes that there are no third variables that confound the relationships between the longitudinally assessed variables. Therefore, the usage of alternative models such as the Random-Intercept Cross-Lagged Panel Model (RI-CLPM) or the Latent Curve Model with Structured Residuals (LCM-SR) has been suggested. These models separate between-person from within-person variation and they also control for time constant covariates. However, there might also be third variables that are not stable but rather change across time and that can confound the relationships between the variables studied in these models. In the present article, we explain the differences between the two types of confounders and investigate how they affect the parameter estimates of within-person models such as the RI-CLPM and the LCM-SR. [ABSTRACT FROM AUTHOR]

For several decades, cross-lagged panel models (CLPM) have been the dominant statistical model in relationship research for investigating reciprocal associations between two (or more) constructs over time. However, recent methodological research has questioned the frequent usage of the CLPM because, amongst other things, the model commingles within-person associations with between-person associations, while most developmental research questions pertain to within-person processes. Furthermore, the model presumes that there are no third variables that confound the relationships between the longitudinally assessed variables. Therefore, the usage of alternative models such as the Random-Intercept Cross-Lagged Panel Model (RI-CLPM) or the Latent Curve Model with Structured Residuals (LCM-SR) has been suggested. These models separate between-person from within-person variation and they also control for time constant covariates. However, there might also be third variables that are not stable but rather change across time and that can confound the relationships between the variables studied in these models. In the present article, we explain the differences between the two types of confounders and investigate how they affect the parameter estimates of within-person models such as the RI-CLPM and the LCM-SR.

Heart failure (HF) is challenging public medical and healthcare systems. This study aimed to develop and validate a novel deep learning-based prognostic model to predict the risk of all-cause mortality for patients with HF. We also compared the performance of the proposed model with those of classical deep learning- and traditional statistical-based models. The present study enrolled 730 patients with HF hospitalized at Toho University Ohashi Medical Center between April 2016 and March 2020. A recurrent neural network-based model (RNNSurv) involving time-varying covariates was developed and validated. The proposed RNNSurv showed better prediction performance than those of a deep feed-forward neural network-based model (referred as “DeepSurv”) and a multivariate Cox proportional hazard model in view of discrimination (C-index: 0.839 vs. 0.755 vs. 0.762, respectively), calibration (better fit with a 45-degree line), and ability of risk stratification, especially identifying patients with high risk of mortality. The proposed RNNSurv demonstrated an improved prediction performance in consideration of temporal information from time-varying covariates that could assist clinical decision-making. Additionally, this study found that significant risk and protective factors of mortality were specific to risk levels, highlighting the demand for an individual-specific clinical strategy instead of a uniform one for all patients.

In modeling time-to-event data with long-term survivors, the proportional hazards model is widely used for its easy and direct interpretation as well as the flexibility to accommodate the past information and allow time-varying predictors. This becomes most relevant when the mortality of individuals converges with time, and the estimation and inference based upon the proportional odds model can often yield more accurate and reasonable results than the classical Cox’s proportional hazards model. Along with the fast development of the data science technologies, computational challenges for survival data with increasing sample size and diverging parameter dimension exist. Currently, existing methods for analyzing such data are computationally inconvenient. In this paper, we propose efficient computational methods for analyzing survival data in the proportional odds model, where the nonparametric maximum likelihood approach is combined with the minorization-maximization (MM) algorithm and the regularization scheme to yield fast and accurate estimation and inferential procedures. The illustration of the methodology using extensive simulation studies and then the application to the Veterans’ Administration lung cancer data is also given.

Extensions of the Kaplan‐Meier estimator have been developed to illustrate the relationship between a time‐varying covariate of interest and survival. In particular, Snapinn et al and Xu et al developed estimators to display survival for patients who always have a certain value of a time‐varying covariate. These estimators properly handle time‐varying covariates, but their clinical interpretation is limited. It is of greater clinical interest to display survival for patients whose covariates lie along certain defined paths. In this article, we propose extensions of Snapinn et al and Xu et al's estimators, providing crude and covariate‐adjusted estimates of the survival function for patients defined by covariate paths. We also derive analytical variance estimators. We demonstrate the utility of these estimators with medical examples and a simulation study. [ABSTRACT FROM AUTHOR]

Longitudinal biomarkers such as patient-reported outcomes (PROs) and quality of life (QOL) are routinely collected in cancer clinical trials or other studies. Joint modelling of PRO/QOL and survival data can provide a comparative assessment of patient-reported changes in specific symptoms or global measures that correspond to changes in survival. Motivated by a head and neck cancer clinical trial, we develop a class of trajectory-based models for longitudinal and survival data with disease progression. Specifically, we propose a class of mixed effects regression models for longitudinal measures, a cure rate model for the disease progression time (T P) and a Cox proportional hazards model with time-varying covariates for the overall survival time (T D) to account for T P and treatment switching. Under the semi-competing risks framework, the disease progression is the non-terminal event, the occurrence of which is subject to a terminal event of death. The properties of the proposed models are examined in detail. Within the Bayesian paradigm, we derive the decompositions of the deviance information criterion (DIC) and the logarithm of the pseudo-marginal likelihood (LPML) to assess the fit of the longitudinal component of the model and the fit of each survival component, separately. We further develop Δ DIC as well as Δ LPML to determine the importance and contribution of the longitudinal data to the model fit of the T P and T D data. [ABSTRACT FROM AUTHOR]

Standard partial likelihood methodology for the proportional hazards model with time-dependent covariates requires knowledge of the covariates at the observed failure times, which is not realistic in practice. A simple and commonly used estimator imputes the most recently observed covariate prior to each failure time, which is known to be biased. In this paper, we show that a weighted last observation carried forward approach may yield valid estimation. We establish the consistency and asymptotic normality of the weighted partial likelihood estimators and provide a closed form variance estimator for inference. The estimator may be conveniently implemented using standard software. Interestingly, the convergence rate of the estimator is slower than the parametric rate achieved with fully observed covariates but the same as that obtained with all lagged covariate values. Simulation studies provide numerical support for the theoretical findings. Data from an Alzheimer's study illustrate the practical utility of the methodology. [ABSTRACT FROM AUTHOR]

Adherence to medication is the process by which patients take their drugs as prescribed, and represents an issue in pharmacoepidemiological studies. Poor adherence is often associated with adverse health conditions and outcomes, especially in case of chronic diseases such as heart failure (HF). This turns out in an increased request for health care services, and in a greater burden for the health care system. In recent years, there has been a substantial growth in pharmacotherapy research, aimed at studying effects and consequences of proper/improper adherence to medication both for the increasing awareness of the problem and for the pervasiveness of poor adherence among patients. However, the way adherence is computed and accounted for into predictive models is far from being informative as it may be. In fact, it is usually analyzed as a fixed baseline covariate, without considering its time‐varying behavior. The purpose and novelty of this study is to define a new personalized monitoring tool exploiting time‐varying definition of adherence to medication, within a joint modeling approach. In doing so, we are able to capture and quantify the association between the longitudinal process of dynamic adherence to medication with the long‐term survival outcome. Another novelty of this approach consists of exploiting the potential of health care administrative databases in order to reconstruct the dynamics of drugs consumption through pharmaceutical administrative registries. In particular, we analyzed administrative data provided by Regione Lombardia ‐ Healthcare Division related to patients hospitalized for HF between 2000 and 2012. [ABSTRACT FROM AUTHOR]

With the exponential growth in data collection, multiple time‐varying biomarkers are commonly encountered in clinical studies, along with a rich set of baseline covariates. This paper is motivated by addressing a critical issue in the field of Alzheimer's disease (AD) in which we aim to predict the time for AD conversion in people with mild cognitive impairment to inform prevention and early treatment decisions. Conventional joint models of biomarker trajectory with time‐to‐event data rely heavily on model assumptions and may not be applicable when the number of covariates is large. This motivated us to consider a functional ensemble survival tree framework to characterize the joint effects of both functional and baseline covariates in predicting disease progression. The proposed framework incorporates multivariate functional principal component analysis to characterize the changing patterns of multiple time‐varying neurocognitive biomarker trajectories and then nest these features within an ensemble survival tree in predicting the progression of AD. We provide a fast implementation of the algorithm that accommodates personalized dynamic prediction that can be updated as new observations are gathered to reflect the patient's latest prognosis. The algorithm is empirically shown to perform well in simulation studies and is illustrated through the analysis of data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) (http://adni.loni.usc.edu/). We provide implementation of our proposed method in an R package funest. [ABSTRACT FROM AUTHOR]

People who are more socially integrated or have higher socio-economic status live longer. Recent studies in non-human primates show striking convergences with this human pattern: female primates with more social partners, stronger social bonds or higher dominance rank all lead longer lives. However, it remains unclear whether social environments also predict survival in male non-human primates, as it does in men. This gap persists because, in most primates, males disperse among social groups, resulting in many males who disappear with unknown fate and have unknown dates of birth. We present a Bayesian model to estimate the effects of time-varying social covariates on age-specific adult mortality in both sexes of wild baboons. We compare how the survival trajectories of both sexes are linked to social bonds and social status over the life. We find that, parallel to females, male baboons who are more strongly bonded to females have longer lifespans. However, males with higher dominance rank for their age appear to have shorter lifespans. This finding brings new understanding to the adaptive significance of heterosexual social bonds for male baboons: in addition to protecting the male's offspring from infanticide, these bonds may have direct benefits to males themselves. This article is part of the theme issue 'Evolution of the primate ageing process'. [ABSTRACT FROM AUTHOR]

Including time-varying covariates is a popular extension to the Cox model and a suitable approach for dealing with non-proportional hazards. However, partial likelihood (PL) estimation of this model has three shortcomings: (i) estimated regression coefficients can be less accurate in small samples with heavy censoring; (ii) the baseline hazard is not directly estimated and (iii) a covariance matrix for both the regression coefficients and the baseline hazard is not easily produced. We address these by developing a maximum likelihood (ML) approach to jointly estimate regression coefficients and baseline hazard using a constrained optimisation ensuring the latter's non-negativity. We demonstrate asymptotic properties of these estimates and show via simulation their increased accuracy compared to PL estimates in small samples and show our method produces smoother baseline hazard estimates than the Breslow estimator. Finally, we apply our method to two examples, including an important real-world financial example to estimate time to default for retail home loans. We demonstrate using our ML estimate for the baseline hazard can give much clearer corroboratory evidence of the 'humped hazard', whereby the risk of loan default rises to a peak and then later falls. [ABSTRACT FROM AUTHOR]

The Kaplan-Meier curve is a standard statistical tool that is used in cohort studies to illustrate how survival during follow-up depends on time-fixed covariates that are measured at baseline. For time-varying covariates, an extended Kaplan-Meier curve has been proposed that is constructed by letting subjects move across risk sets as their covariate levels change during follow-up. It has been claimed, but not proven, that, under a particular independence assumption, this extended Kaplan-Meier curve has a causal interpretation as representing a hypothetical cohort whose covariate values remain constant during follow-up. In this note, we show that, in the absence of confounding, this claim is indeed correct. However, we argue that the causal implications of this independence assumptions are highly unrealistic, and that a causal interpretation of the extended Kaplan-Meier curve is therefore typically unwarranted. [ABSTRACT FROM AUTHOR]