Survival Tree Method Research Articles

Optimal Survival Trees: A Dynamic Programming Approach

Survival analysis studies and predicts the time of death, or other singular unrepeated events, based on historical data, while the true time of death for some instances is unknown. Survival trees enable the discovery of complex nonlinear relations in a compact human comprehensible model, by recursively splitting the population and predicting a distinct survival distribution in each leaf node. We use dynamic programming to provide the first survival tree method with optimality guarantees, enabling the assessment of the optimality gap of heuristics. We improve the scalability of our method through a special algorithm for computing trees up to depth two. The experiments show that our method's run time even outperforms some heuristics for realistic cases while obtaining similar out-of-sample performance with the state-of-the-art.

Clinical-epidemiological characteristics and survival of cases of severe acute respiratory syndrome (SARS) due to COVID-19, according to the COVID-19 vaccination schedule in Brazil, 2021-2022: a prospective study.

To analyze the clinical and sociodemographic characteristics and survival of individuals with severe acute respiratory syndrome due to COVID-19 according to the COVID-19 vaccination schedule, Brazil, 2021-2022. This was a cohort study based on data from the Influenza Epidemiological Surveillance Information System; the Kaplan-Meier and Survival Tree methods were used to analyze survival. Among the 559,866 hospitalized cases, a higher proportion of vaccinated individuals was found among female (15.0%), elderly people aged ≥ 80 (34.5%), people from the Southeast region (15.7%), those who did not undergo respiratory support (21.2%) and those who did progress to death (15.2%); the survival curve showed that risk of death for unvaccinated individuals was higher in all age groups (p-value < 0.001); elderly people aged ≥ 80, who did not undergo mechanical ventilation and who had a booster dose had lower risk when compared to their peers who had two doses or were unvaccinated (hazard ratio = 0.64; 95%CI 0.62;0.67). Lowest risk of death was found in vaccinated individuals, especially those who had two doses or a booster dose as well. Prevalence was found to be high among unvaccinated individuals. Risk of death was lower among those vaccinated with a booster dose, compared to those not vaccinated, in all age groups analyzed. The number of hospitalizations of unvaccinated individuals with severe acute respiratory syndrome was high, which increases the demand for health services to care for these individuals. It is necessary to promote widespread vaccination of the entire population of Brazil, in addition to the regular provision of booster doses for the different population groups.

Tailored Risk Stratification in Severe Mitral Regurgitation and Heart Failure Using Supervised Learning Techniques

BackgroundSecondary mitral regurgitation (sMR) in the setting of heart failure (HF) has considerable impact on quality of life, HF rehospitalizations, and mortality. Identification of high-risk cohorts is essential to understand disease trajectories and for risk stratification. ObjectivesThis study aimed to provide a structured decision tree–like approach to risk stratification in patients with severe sMR and HF. MethodsThis observational study included 1,317 patients with severe sMR from the entire HF spectrum. Clinical, echocardiographic, and laboratory data were extracted for all patients. The primary end point was all-cause mortality. Survival tree analysis, a supervised learning technique, was applied to identify patient subgroups at risk of mortality and further stratified by HF subtype (preserved, mildly reduced, and reduced ejection fraction). ResultsUsing supervised learning (survival tree method), 8 distinct subgroups were identified that differed significantly in long-term survival. Subgroup 7, characterized by younger age (≤66 years), higher hemoglobin (>12.7 g/dL), and higher albumin levels (>40.6 g/L) had the best survival. In contrast, subgroup 5 displayed a 20-fold risk of mortality (hazard ratio: 20.38 [95% CI: 10.78-38.52]); P < 0.001 and had older age (>68 years), low serum albumin (≤40.6 g/L), and higher NT-proBNP levels (≥9,750 pg/mL). Unique subgroups were further identified for each type of HF subtypes. ConclusionsSupervised machine learning reveals heterogeneity in the sMR risk spectrum, highlighting the clinical variability in the population. A decision tree–like model can help identify differences in outcomes among subgroups and can help provide tailored risk stratification.

Optimal survival trees

Tree-based models are increasingly popular due to their ability to identify complex relationships that are beyond the scope of parametric models. Survival tree methods adapt these models to allow for the analysis of censored outcomes, which often appear in medical data. We present a new Optimal Survival Trees algorithm that leverages mixed-integer optimization (MIO) and local search techniques to generate globally optimized survival tree models. We demonstrate that the OST algorithm improves on the accuracy of existing survival tree methods, particularly in large datasets.

Myocardial Strain-Based on Regional Wall Motion Abnormalities Index : A Prognostic Marker of Incident Heart Failure. Cardiovascular MRI study from the Multi-Ethnic Study of Atherosclerosis (MESA)

Although several studies have investigated the prognostic value of the LV global circumferential strain for predicting the risk of heart failure (HF), very few studies have assessed the prognostic value of myocardial strain-based Regional Wall Motion Abnormalities (RWMA) index assessed by CMR tagging in asymptomatic individuals without a history of CVD. To assess the long-term prognostic value of myocardial strain and the RWMA index measured by tagged MRI for the prediction of CV events in the Multi-Ethnic Study of Atherosclerosis (MESA). Participants of the MESA who underwent a baseline CMR study including cardiac-tagged MRI using HARP were analysed. The AHA classification with 16-segments was used for strain data. RWMA score was defined by the number of segments with regiona strain > −16% (cut-off defined by survival tree method). Cox proportional hazard models were used to evaluate the association between RWMA index and incident HF. The survival tree method was used to determine the best cut-off to transform the RWMA score into a binary variable. Among the 1506 participants (age 63.3 ± 9.4 years, 54.6% men), 91 incident HFs were observed after a mean follow-up of 15.9 ± 3.1 years. RWMA score was associated with HF (HR 1.58, 95%CI [1.07-2.49], P = 0.002). Using an optimal cut-off to predict incident HF, RWMA > 66% of total myocardium was also associated with HF (HR 2.43, 95%CI [1.72-3.41], P < 0.001). After adjustment for traditional HF risk factors (age, male, ethnicity, BMI, hypertension, diabetes, dyslipidemia, smoking and hypercholesterolemia), higher RWMA score and RWMA > 66% were both independently associated with HF (HR 1.57, 95%CI [1.02-2.32], P = 0.038 and HR 1.67, 95%CI [1.03-2.94], P = 0.002; respectively) ( Table 1 ). In a multi-ethnic population, RWMA index assessed by cardiac-tagged MRI is independently associated with incident HF.

Evaluating Different Selection Criteria for Phase Type Survival Tree Construction

Due to its interpretability and intuitiveness, survival tree based analysis is a powerful Artificial Intelligence method for modelling longitudinal survival data, its relationship with covariates and the interrelationship between covariates. Furthermore, it is being increasingly used for a range of applications including clustering, prognostication and classification. Phase type survival tree methods have been demonstrated to have important applications, including clustering patients into clinically meaningful groups, patient pathway prognostication and forecasting bed requirements. In this article, we critically investigate and assess several selection information criteria with regards to their suitability and limitations when used as splitting criteria in phase type survival tree construction. As shown in Table 12, the results of this analysis are compared and discussed. Furthermore, a text mining approach is utilised to further assess correlations, which have been extracted from hospital data, between the three underlying diseases and the two different types of population groups, namely age and gender groups. Its aim is to provide further investigative tools. In fact, due to its ability to analyse large volumes of textual data, text mining can provide a useful approach to this research area.

An ensemble method for interval-censored time-to-event data.

Interval-censored data analysis is important in biomedical statistics for any type of time-to-event response where the time of response is not known exactly, but rather only known to occur between two assessment times. Many clinical trials and longitudinal studies generate interval-censored data; one common example occurs in medical studies that entail periodic follow-up. In this article, we propose a survival forest method for interval-censored data based on the conditional inference framework. We describe how this framework can be adapted to the situation of interval-censored data. We show that the tuning parameters have a non-negligible effect on the survival forest performance and guidance is provided on how to tune the parameters in a data-dependent way to improve the overall performance of the method. Using Monte Carlo simulations, we find that the proposed survival forest is at least as effective as a survival tree method when the underlying model has a tree structure, performs similarly to an interval-censored Cox proportional hazards model fit when the true relationship is linear, and outperforms the survival tree method and Cox model when the true relationship is nonlinear. We illustrate the application of the method on a tooth emergence data set.

Survival trees for interval-censored survival data.

Interval-censored data, in which the event time is only known to lie in some time interval, arise commonly in practice, for example, in a medical study in which patients visit clinics or hospitals at prescheduled times and the events of interest occur between visits. Such data are appropriately analyzed using methods that account for this uncertainty in event time measurement. In this paper, we propose a survival tree method for interval-censored data based on the conditional inference framework. Using Monte Carlo simulations, we find that the tree is effective in uncovering underlying tree structure, performs similarly to an interval-censored Cox proportional hazards model fit when the true relationship is linear, and performs at least as well as (and in the presence of right-censoring outperforms) the Cox model when the true relationship is not linear. Further, the interval-censored tree outperforms survival trees based on imputing the event time as an endpoint or the midpoint of the censoring interval. We illustrate the application of the method on tooth emergence data.

Survival trees for left-truncated and right-censored data, with application to time-varying covariate data.

Tree methods (recursive partitioning) are a popular class of nonparametric methods for analyzing data. One extension of the basic tree methodology is the survival tree, which applies recursive partitioning to censored survival data. There are several existing survival tree methods in the literature, which are mainly designed for right-censored data. We propose two new survival trees for left-truncated and right-censored (LTRC) data, which can be seen as a generalization of the traditional survival tree for right-censored data. Further, we show that such trees can be used to analyze survival data with time-varying covariates, essentially building a time-varying covariates survival tree. Implementation of the methods is easy, and simulations and real data analysis results show that the proposed methods work well for LTRC data and survival data with time-varying covariates, respectively.

A survival tree method for the analysis of discrete event times in clinical and epidemiological studies.

Survival trees are a popular alternative to parametric survival modeling when there are interactions between the predictor variables or when the aim is to stratify patients into prognostic subgroups. A limitation of classical survival tree methodology is that most algorithms for tree construction are designed for continuous outcome variables. Hence, classical methods might not be appropriate if failure time data are measured on a discrete time scale (as is often the case in longitudinal studies where data are collected, e.g., quarterly or yearly). To address this issue, we develop a method for discrete survival tree construction. The proposed technique is based on the result that the likelihood of a discrete survival model is equivalent to the likelihood of a regression model for binary outcome data. Hence, we modify tree construction methods for binary outcomes such that they result in optimized partitions for the estimation of discrete hazard functions. By applying the proposed method to data from a randomized trial in patients with filarial lymphedema, we demonstrate how discrete survival trees can be used to identify clinically relevant patient groups with similar survival behavior.

Bayesian network data imputation with application to survival tree analysis

Retrospective clinical datasets are often characterized by a relatively small sample size and many missing data. In this case, a common way for handling the missingness consists in discarding from the analysis patients with missing covariates, further reducing the sample size. Alternatively, if the mechanism that generated the missing allows, incomplete data can be imputed on the basis of the observed data, avoiding the reduction of the sample size and allowing methods to deal with complete data later on. Moreover, methodologies for data imputation might depend on the particular purpose and might achieve better results by considering specific characteristics of the domain. The problem of missing data treatment is studied in the context of survival tree analysis for the estimation of a prognostic patient stratification. Survival tree methods usually address this problem by using surrogate splits, that is, splitting rules that use other variables yielding similar results to the original ones. Instead, our methodology consists in modeling the dependencies among the clinical variables with a Bayesian network, which is then used to perform data imputation, thus allowing the survival tree to be applied on the completed dataset. The Bayesian network is directly learned from the incomplete data using a structural expectation–maximization (EM) procedure in which the maximization step is performed with an exact anytime method, so that the only source of approximation is due to the EM formulation itself. On both simulated and real data, our proposed methodology usually outperformed several existing methods for data imputation and the imputation so obtained improved the stratification estimated by the survival tree (especially with respect to using surrogate splits).

Application of Survival Tree Based on Texture Features Obtained through MRI of Patients with Brain Metastases from Breast Cancer

The information obtained by magnetic resonance imaging (MRI) is considered to possess great potential for providing the prognosis of cancer patients, although not been established. The goal of this study was to evaluate the covariates of the texture patterns obtained from MRI scans of patients with breast cancer brain metastases, which influence the survival time prognosis. The data of forty patients were analyzed using 29 covariates. Twenty-six covariates, which are focused on the texture patterns, were calculated from the gray-level co-occurrence matrix and wavelet coefficients obtained by transform of preoperative T1-weighted MRI scans. The remaining three covariates were age, Karnofsky Performance Scale, and the indicator of whether solitary or multiple metastases were present. These covariates are commonly used as the prognostic factors in medical research. The tree structure prognosis models were constructed by applying the survival tree method to these covariates. The obtained survival trees separated the patients into two or three groups between which there was a statistically significant distance. For the purpose of comparison, Cox regression analyses were performed to determine which covariates showed significant predictive values. All the covariates selected in the Cox analysis and survival tree method were texture features only. In particular, the energy of the gray-level co-occurrence matrix and wavelet coefficients showed a high performance in tree structure analysis. From these results, we conclude that the features obtained from simple medical images can be used to estimate the prognosis of brain metastases patients.

Rationale and Applications of Survival Tree and Survival Ensemble Methods.

Classification and Regression Trees (CART), and their successors-bagging and random forests, are statistical learning tools that are receiving increasing attention. However, due to characteristics of censored data collection, standard CART algorithms are not immediately transferable to the context of survival analysis. Questions about the occurrence and timing of events arise throughout psychological and behavioral sciences, especially in longitudinal studies. The prediction power and other key features of tree-based methods are promising in studies where an event occurrence is the outcome of interest. This article reviews existing tree algorithms designed specifically for censored responses as well as recently developed survival ensemble methods, and introduces available computer software. Through simulations and a practical example, merits and limitations of these methods are discussed. Suggestions are provided for practical use.

Bayesian survival trees for clustered observations, applied to tooth prognosis

AbstractTooth loss from periodontal disease or dental caries (decay) afflicts most adults over the course of their lives. Survival tree methods for correlated observations have shown potential for developing objective tooth prognosis systems; however, the current technology suffers either from prohibitive computational expense or unrealistic simplifying assumptions to overcome computational demands. In this article Bayesian tree methods are developed for correlated survival data, relying on a computationally feasible, yet flexible, frailty model with piecewise constant hazard function. Bayesian stochastic search methods, using a Laplace approximated marginal likelihood, are detailed for tree construction, and posterior ensemble averaged variable importance ranking and amalgamation procedures are developed. The proposed methods are used to assign each tooth from the Veteran Administration (VA) Dental Longitudinal Study to one of five prognosis categories and evaluate the effects of clinical factors and genetic polymorphisms in predicting tooth loss. The prognostic rules established may be used in clinical practice to optimize tooth retention and devise periodontal treatment plans.

Risk Factors for Recurrence After Lung Cancer Resection as Estimated Using the Survival Tree Method

Patients with lung cancer often present with recurrence, even after resection. The identification of risk factors for recurrence after resection is useful. Among 1,338 patients with lung cancer who underwent a complete resection, 277 developed recurrences post surgery. Data regarding the TNM factors, histologic subtype, and presence/absence of vessel invasion were analyzed retrospectively using the survival tree method to identify groups with a high risk of recurrence after resection. The results revealed that the T factor, the N factor, and lymphatic (ly) and blood (v) vessel invasion were related to the risk of recurrence, and six combinations of these factors were identified using the survival tree method: group A: v = 0, T ≤ 1b, ly = 0; group B: v = 0, T ≤ 1b, ly ≥ 1; group C: v = 0, T ≥ 2a; group D: v ≥ 1, N ≤ 1, T ≤ 2b; group E: v ≥ 1, N ≤ 1, T ≥ 3; and group F: v ≥ 1, N ≥ 2. The six groups were then further classified into three groups: a low-risk group (group A), a moderate-risk group (groups B, C, and D), and a high-risk group (groups E and F). The 5-year recurrence-free survival rate was approximately 98% for the low-risk group, 75% for the moderate-risk group, and 30% for the high-risk group. Combining the T, N, v, and ly factors allowed the precise identification of a group with a high risk of recurrence after resection.

Discrete-time survival trees and forests with time-varying covariates

The aim of this paper is to propose a new survival tree method for discrete-time survival data with time-varying covariates. This method can accommodate simultaneously time-varying covariates and time-varying effects. The method is then used for bankruptcy analysis of US firms that conducted an Initial Public Offerings between 1990 and 1999 using accounting and financial ratios.

Developing and comparing two different prognostic indexes for predicting disease-free survival of nonmetastatic breast cancer patients

To determine 2 different prognostic indexes (PI) for the differentiation of subgroups of nonmetastatic breast cancer patients with the Cox regression analysis and survival tree (ST) methods and the additional usage of the Kaplan-Meier estimates to investigate the predictive power of these methods. Materials and methods: Prognostic factors data were collected for 410 patients. The Cox regression analysis examines the relationship of the survival distribution and covariates. The ST method is a tree-structured survival analysis based on a recursive partitioning algorithm. In this study, Harrell's concordance indexes of models for training and test sets were computed. Furthermore, survival curves were estimated by the Kaplan-Meier method. Disease-free survival (DFS) was calculated from the time of initial diagnosis (initiation of the first treatment) to the first recurrence of disease. Results: After a median follow-up of 48 months, 100 (24.4%) patients have had at least 1 of the DFS events. In Cox regression analysis, we proposed the simple PI, which is a sum of axillary nodal and HER2/neu status. In the ST method, we identified 3 variables: HER2/neu, axillary nodal, and estrogen receptor status. The axillary nodal status was the most important determining factor for recurrence. Conclusion: We found that the PI of the ST and Cox regression methods had similar performance levels in predicting DFS, and the error rates of the models were close to each other in the training and test sets. Furthermore, we determined that the axillary nodal status and HER2/neu were the most important determining factors for prediction of DFS in breast cancer patients.

An empirical comparison of multivariable methods for estimating risk of death from coronary heart disease

Logistic regression and, more recently, Cox regression have been the predominant methods for identifying risk factors and developing risk estimation equations for coronary heart disease (CHD). Software for the regression tree method is now available for binary and survival outcomes and thus offers an alternative methodology. This paper compares these four methods for identifying significant risk factors from among a set of candidate factors and for estimating the risk of death from CHD using baseline and mortality follow-up data on 1,701 men participating in the Busselton Health Study. The candidate risk factors were age, body mass index, systolic and diastolic blood pressure, treatment for hypertension, cholesterol and smoking. Logistic regression, Cox proportional hazards regression, binary regression tree, and survival regression tree analyses have been applied to data obtained from the same cohort of men for CHD death risk estimation and prediction. The four methods are compared in terms of the variables selected, goodness-of-fit of models, similarity of cross-validated estimated risks for individuals, and ability to discriminate between those who died from CHD and those who did not die from CHD during the follow-up period, including the comparison of Receiver Operating Characteristic (ROC) curves. Although age and a blood pressure variable were selected by all four methods, body mass index was also selected by the regression tree methods and smoking was also selected by Cox regression. There was good, but not excellent, agreement between methods in estimates of risk for individuals, the areas under the ROC curves were 0.66 for the binary tree, 0.72 for logistic regression, 0.71 for the survival tree method and 0.78 for Cox regression. The average differences in estimated risk between those who died from CHD and those who did not die from CHD during the follow-up period were 0.051 for logistic regression, 0.070 for the binary tree method, 0.073 for the survival tree method and 0.088 for Cox regression. For a moderately sized cohort typical of many applications of these methods in the literature, the two methods which used the survival outcome performed better than the methods using a binary outcome. Despite selecting some different variables and showing moderate differences in risk estimates for individuals, the two binary approaches were similar in performance. Cox regression appeared to be superior to the survival tree method, but further larger studies of completely separate samples for model development and evaluation of prediction performance are required to confirm this finding.

An Empirical Comparison of Multivariable Methods for Estimating Risk of Death from Coronary Heart Disease

Background Logistic regression and, more recently, Cox regression have been the predominant methods for identifying risk factors and developing risk estimation equations for coronary heart disease (CHD). Software for the regression tree method is now available for binary and survival outcomes and thus offers an alternative methodology, This paper compares these four methods for identifying significant risk factors from among a set of candidate factors and for estimating the risk of death from CHD using baseline and mortality follow-up data on 1701 men participating in the Busselton Health Study. The candidate risk factors were age, body mass index, systolic and diastolic blood pressure, treatment for hypertension, cholesterol and smoking. Methods Logistic regression, Cox proportional hazards regression, binary regression tree, and survival regression tree analyses have been applied to data obtained from the same cohort of men for CHD death risk estimation and prediction. The four methods are compared in terms of the variables selected, goodness-of-fit of models, similarity of cross-validated estimated risks for individuals, and ability to discriminate between those who died from CHD and those who did not die from CHD during the follow-up period, including the comparison of Receiver Operating Characteristic (ROC) curves. Results Although age and a blood pressure variable were selected by all four methods, body mass index was also selected by the regression tree methods and smoking was also selected by Cox regression. There was good, but not excellent, agreement between methods in estimates of risk for individuals, the areas under the ROC curves were 0.66 for the binary tree, 0.72 for logistic regression, 0.71 for the survival tree method and 0.78 for Cox regression. The average differences in estimated risk between those who died from CHD and those who did not die from CHD during the follow-up period were 0.051 for logistic regression, 0.070 for the binary tree method, 0.073 for the survival tree method and 0.088 for Cox regression. Conclusions For a moderately sized cohort typical of many applications of these methods in the literature, the two methods which used the survival outcome performed better than the methods using a binary outcome. Despite selecting some different variables and showing moderate differences in risk estimates for individuals, the two binary approaches were similar in performance. Cox regression appeared to be superior to the survival tree method, but further larger studies of completely separate samples for model development and evaluation of prediction performance are required to confirm this finding.