Combining Symbolic Regression with the Cox Proportional Hazards Model Improves Prediction of Heart Failure Deaths

Cox Model ◽

Symbolic Regression ◽

Data Set ◽

Abstract Background: Heart failure is a clinical syndrome characterised by a reduced ability of the heart to pump blood. Patients with heart failure have a high mortality rate, and physicians need reliable prognostic predictions to make informed decisions about the appropriate application of devices, transplantation, medications, and palliative care. In this study, we demonstrate that combining symbolic regression with the Cox proportional hazards model improves the ability to predict death due to heart failure compared to using the Cox proportional hazards model alone. Methods: We used a newly invented symbolic regression method called the QLattice to analyse a data set of medical records for 299 Pakistani patients diagnosed with heart failure. The QLattice identified a minimal set of mathematical transformations of the available covariates, which we then used in a Cox model to predict survival.Results: An exponential function of age, the inverse of ejection fraction, and the inverse of serum creatinine were identified as the best risk factors for predicting heart failure deaths. A Cox model fitted on these transformed covariates had improved predictive performance compared with a Cox model on the same covariates without mathematical transformations. Conclusion: Symbolic regression is a way to find transformations of covariates from patients’ medical records which can improve the performance of survival regression models. At the same time, these simple functions are intuitive and easy to apply in clinical settings. The direct interpretability of the simple forms may help researchers gain new insights into the actual causal pathways leading to deaths.

Combining symbolic regression with the Cox proportional hazards model improves prediction of heart failure deaths

10.1101/2021.01.15.21249874 ◽

2021 ◽

Author(s):

Casper Wilstup ◽

Chris Cave

Keyword(s):

Heart Failure ◽

Medical Records ◽

Proportional Hazards ◽

Cox Model ◽

Symbolic Regression ◽

Data Set ◽

AbstractHeart failure is a clinical syndrome characterised by a reduced ability of the heart to pump blood. Patients with heart failure have a high mortality rate, and physicians need reliable prognostic predictions to make informed decisions about the appropriate application of devices, transplantation, medications, and palliative care. In this study, we demonstrate that combining symbolic regression with the Cox proportional hazards model improves the ability to predict death due to heart failure compared to using the Cox proportional hazards model alone.We used a newly invented symbolic regression method called the QLat-tice to analyse a data set of medical records for 299 Pakistani patients diagnosed with heart failure. The QLattice identified a minimal set of mathematical transformations of the available covariates, which we then used in a Cox model to predict survival.An exponential function of age, the inverse of ejection fraction, and the inverse of serum creatinine were identified as the best risk factors for predicting heart failure deaths. A Cox model fitted on these transformed covariates had improved predictive performance compared with a Cox model on the same covariates without mathematical transformations.Symbolic regression is a way to find transformations of covariates from patients’ medical records which can improve the performance of survival regression models. At the same time, these simple functions are intuitive and easy to apply in clinical settings. The direct interpretability of the simple forms may help researchers gain new insights into the actual causal pathways leading to deaths.

Proportionally Difficult: Testing for Nonproportional Hazards in Cox Models

Political Analysis ◽

10.1093/pan/mpp044 ◽

2010 ◽

Vol 18 (2) ◽

pp. 189-205 ◽

Cited By ~ 59

Author(s):

Luke Keele

Keyword(s):

Proportional Hazards ◽

Cox Model ◽

Standard Test ◽

Cox Models ◽

Nonproportional Hazards ◽

Specification Errors ◽

The Cox proportional hazards model is widely used to model durations in the social sciences. Although this model allows analysts to forgo choices about the form of the hazard, it demands careful attention to the proportional hazards assumption. To this end, a standard diagnostic method has been developed to test this assumption. I argue that the standard test for nonproportional hazards has been misunderstood in current practice. This test detects a variety of specification errors, and these specification errors must be corrected before one can correctly diagnose nonproportionality. In particular, unmodeled nonlinearity can appear as a violation of the proportional hazard assumption for the Cox model. Using both simulation and empirical examples, I demonstrate how an analyst might be led astray by incorrectly applying the nonproportionality test.

Impact of renal function on mid-term outcomes in heart failure patients treated with tolvaptan

Therapeutic Advances in Cardiovascular Disease ◽

10.1177/1753944718819064 ◽

2019 ◽

Vol 13 ◽

pp. 175394471881906

Author(s):

Kensuke Fujioka ◽

Sumio Mizuno ◽

Taro Ichise ◽

Takao Matsui ◽

Hiroaki Hirase ◽

...

Keyword(s):

Heart Failure ◽

Renal Function ◽

Proportional Hazards ◽

Free Water ◽

Cardiac Events ◽

Hazards Model ◽

Continuous Use

Background: Although tolvaptan, an electrolyte-free water diuretic for congestive heart failure (HF), is reported to have no effect on long-term mortality or HF-related morbidity, there may exist some subgroups of patients who may receive beneficial effect of tolvaptan. The purpose of this study was to identify clinical factors associated with mid-term effect of tolvaptan on clinical outcomes of patients who discharged after acute HF. Methods: We retrospectively analyzed 140 patients (88 male; mean age, 77.1 ± 11.0 years) with acute HF who received tolvaptan (initial dose 8.6 ± 3.6 mg/day) during their hospitalization. They were divided into two groups according to how the tolvaptan was used at discharge; 77 in the tolvaptan-continued group and 63 in the discontinued group. Results: The Cox proportional hazards model revealed that eGFR was the only independent predictor for the occurrence of mid-term cardiac events (composite of re-hospitalization due to HF and all-cause death; aHR = 0.9870, p = 0.02597). The Kaplan–Meier survival curves of the two groups demonstrated no difference in cumulative event-free rates. In the subgroup with preserved renal function at admission (eGFR ⩾ 30 ml/min/1.73 m2), the continuous use of tolvaptan increased composite events (aHR = 2.130, p = 0.02549). Conclusions: The continuous use of tolvaptan after discharge did not affect mid-term cardiac events of HF overall but may be associated with increased cardiac events in the subgroup with preserved renal function. These findings suggest that the tolvaptan administration might need to be limited to treatment of in-hospital acute HF.

Proceedings of the Institution of Mechanical Engineers Part O Journal of Risk and Reliability ◽

Methods for displaying and calibration of Cox proportional hazards models

10.1177/1748006x17742779 ◽

2017 ◽

Vol 232 (1) ◽

pp. 105-115

Author(s):

Chrianna I Bharat ◽

Kevin Murray ◽

Edward Cripps ◽

Melinda R Hodkiewicz

Keyword(s):

Proportional Hazards ◽

Explanatory Power ◽

Remaining Useful Life ◽

Data Set ◽

Proportional Hazards Models ◽

Hazards Model ◽

Cox Proportional Hazards Models

Cox proportional hazards modelling is a widely used technique for determining relationships between observed data and the risk of asset failure when model performance is satisfactory. Cox proportional hazards models possess good explanatory power and are used by asset managers to gain insight into factors influencing asset life. However, validation of Cox proportional hazards models is not straightforward and is seldom considered in the maintenance literature. A comprehensive validation process is a necessary foundation to build trust in the failure models that underpin remaining useful life prediction. This article describes data splitting, model discrimination, misspecification and fit methods necessary to build trust in the ability of a Cox proportional hazards model to predict failures on out-of-sample assets. Specifically, we consider (1) Prognostic Index comparison for training and test sets, (2) Kaplan–Meier curves for different risk bands, (3) hazard ratios across different risk bands and (4) calibration of predictions using cross-validation. A Cox proportional hazards model on an industry data set of water pipe assets is used for illustrative purposes. Furthermore, because we are dealing with a non-statistical managerial audience, we demonstrate how graphical techniques, such as forest plots and nomograms, can be used to present prediction results in an easy to interpret way.

Beyond the Hazard Ratio: Generating Expected Durations from the Cox Proportional Hazards Model

British Journal of Political Science ◽

10.1017/s000712341700045x ◽

2017 ◽

Vol 50 (1) ◽

pp. 303-320 ◽

Cited By ~ 2

Author(s):

Jonathan Kropko ◽

Jeffrey J. Harden

Keyword(s):

Political Science ◽

Hazard Function ◽

Proportional Hazards ◽

Cox Model ◽

Hazards Model ◽

Hazard Ratios ◽

Quantities Of Interest

The Cox proportional hazards model is a commonly used method for duration analysis in political science. Typical quantities of interest used to communicate results come from the hazard function (for example, hazard ratios or percentage changes in the hazard rate). These quantities are substantively vague, difficult for many audiences to understand and incongruent with researchers’ substantive focus on duration. We propose methods for computing expected durations and marginal changes in duration for a specified change in a covariate from the Cox model. These duration-based quantities closely match researchers’ theoretical interests and are easily understood by most readers. We demonstrate the substantive improvements in interpretation of Cox model results afforded by the methods with reanalyses of articles from three subfields of political science.

An Application of the Cox-Aalen Model for Breast Cancer Survival

Austrian Journal of Statistics ◽

10.17713/ajs.v35i1.350 ◽

2016 ◽

Vol 35 (1) ◽

Cited By ~ 2

Author(s):

Ileana Baldi ◽

Giovannino Ciccone ◽

Antonio Ponti ◽

Stefano Rosso ◽

Roberto Zanetti ◽

...

Keyword(s):

Cancer Survival ◽

Proportional Hazards ◽

Cox Model ◽

Tumour Size ◽

Aalen Model ◽

Cancer Registry Data ◽

Semiparametric hazard function regression models are among the well studied risk models in survival analysis. The Cox proportional hazards model has been a popular choice in modelling data from epidemiological settings. The Cox-Aalen model is one of the tools for handling the problem of non-proportional effects in the Cox model. We show an application on Piedmont cancer registry data. We initially fit standard Cox model and with the help of the score process we detect the violation of the proportionality assumption. Covariates and risk factors that, on the basis of clinical reasoning, best model baseline hazard are then moved into the additive part of the Cox-Aalen model. Multiplicative effects results are consistent with those of the Cox model whereas only the Cox-Aalen model fully represents the timevarying effect of tumour size.

Does the Anti-Mullerian Hormone Decline Rate Improve the Prediction of Age at Menopause?

Frontiers in Endocrinology ◽

10.3389/fendo.2021.727229 ◽

2021 ◽

Vol 12 ◽

Author(s):

Fahimeh Ramezani Tehrani ◽

Ali Sheidaei ◽

Faezeh Firouzi ◽

Maryam Tohidi ◽

Fereidoun Azizi ◽

...

Keyword(s):

Proportional Hazards ◽

Cox Model ◽

Time Dependent ◽

Age At Menopause ◽

C Statistic ◽

Hazards Model ◽

Decline Rate

ObjectivesThere are controversial studies investigating whether multiple anti-Mullerian hormone (AMH) measurements can improve the individualized prediction of age at menopause in the general population. This study aimed to reexplore the additive role of the AMH decline rate in single AMH measurement for improving the prediction of age at physiological menopause, based on two common statistical models for analysis of time-to-event data, including time-dependent Cox regression and Cox proportional-hazards regression models.MethodsA total of 901 eligible women, aged 18–50 years, were recruited from the Tehran Lipid and Glucose Study (TLGS) population and followed up every 3 years for 18 years. The serum AMH level was measured at the time of recruitment and twice after recruitment within 6-year intervals using the Gen II AMH assay. The added value of repeated AMH measurements for the prediction of age at menopause was explored using two different statistical approaches. In the first approach, a time-dependent Cox model was plotted, with all three AMH measurements as time-varying predictors and the baseline age and logarithm of annual AMH decline as time-invariant predictors. In the second approach, a Cox proportional-hazards model was fitted to the baseline data, and improvement of the complex model, which included repeated AMH measurements and the logarithm of the AMH annual decline rate, was assessed using the C-statistic.ResultsThe time-dependent Cox model showed that each unit increase in the AMH level could reduce the risk of menopause by 87%. The Cox proportional-hazards model also improved the prediction of age at menopause by 3%, according to the C-statistic. The subgroup analysis for the prediction of early menopause revealed that the risk of early menopause increased by 10.8 with each unit increase in the AMH annual decline rate.ConclusionThis study confirmed that multiple AMH measurements could improve the individual predictions of the risk of at physiological menopause compared to single AMH measurements. Different alternative statistical approaches can also offer the same interpretations if the essential assumptions are met.

Money transfer between banks

International Journal of Bank Marketing ◽

10.1108/ijbm-04-2019-0131 ◽

2019 ◽

Vol 38 (2) ◽

pp. 283-295

Author(s):

Andrea Lippi ◽

Laura Barbieri ◽

Federica Poli

Keyword(s):

Proportional Hazards ◽

Financial Advisors ◽

Data Set ◽

Financial Intermediary ◽

Content Type ◽

Financial Advisor ◽

Purpose The purpose of this paper is to examine which individual traits of financial advisors influence portfolio transfer speed when a financial advisor recommends investors to migrate to a new financial intermediary. Design/methodology/approach With reference to the years 2014–2016, one of the three leading Italian tied-agent banks provided the authors with an exclusive and unique data set containing information regarding the financial advisors who had become tied agents, transferring their existing portfolios from their previous banks (traditional or tied-agent banks). The authors observed the ability of the migrant financial advisor in successfully transferring the entire portfolio declared within 12 months of observation. To investigate empirically which personal traits of financial advisors determine their success in the rapid transfer of clients’ portfolios to a new financial intermediary, the authors applied a Cox proportional hazards model. Findings The authors find that factors such as age, type of bank of origin and size of the managed financial portfolio positively affect the speed transfer. Practical implications The obtained results may be interesting for guiding recruiting policies of financial intermediaries. Social implications Regulators should closely examine the phenomenon analyzed in this paper to avoid conflict of interests. Originality/value The literature on this topic is scarce, mainly due to the lack of available data. This paper represents an original contribution to open a new field of research.

Impact of digital multi-sided platforms on firm survival in retail

Voprosy Ekonomiki ◽

10.32609/0042-8736-2021-8-93-122 ◽

2021 ◽

pp. 93-122

Author(s):

E. S. Andronova ◽

A. I. Rey ◽

G. R. Akzhigitova

Keyword(s):

Proportional Hazards ◽

Positive Impact ◽

Survival Function ◽

Firm Survival ◽

Data Set ◽

Hazards Model ◽

Kaplan Meier

This paper explores firm survival in Russian retail industry in cases of digital multi-sided platforms penetration such as aggregator Yandex.Market, marketplace Wildberries, electronic store Ozon and classified-ad service Avito. The panel data set of 130 thousand firms was analyzed using two methods: non-parametric Kaplan—Meier estimator and semi-parametric Cox proportional hazards model with time dependent covariates. Kaplan—Meier estimator calculates the survival function for censored data. Cox proportional hazards model examines the effect of platform penetration on hazard rates of differently sized firms in various industry spheres. Platforms-aggregators Yandex.Market and Wildberries have a strong positive impact on firm survival while platformsdisruptors Ozon and Avito increase likelihood of firm failure. The main results of platform influence in various industry spheres are as follows: the aggregator of price offers has a more positive impact on segments with high information asymmetry; and firms specialized on Wildberries key product categories enjoy lower hazard ratios of bankruptcy or liquidation. These hypotheses are not supported for Ozon and Avito platforms.

Additive Hazard Regression Models: An Application to the Natural History of Human Papillomavirus

Computational and Mathematical Methods in Medicine ◽

10.1155/2013/796270 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 12

Author(s):

Xianhong Xie ◽

Howard D. Strickler ◽

Xiaonan Xue

Keyword(s):

Proportional Hazards ◽

Cox Model ◽

Additive Model ◽

Hazards Model ◽

History Of ◽

Oncogenic Hpv ◽

Hiv Negative

There are several statistical methods for time-to-event analysis, among which is the Cox proportional hazards model that is most commonly used. However, when the absolute change in risk, instead of the risk ratio, is of primary interest or when the proportional hazard assumption for the Cox proportional hazards model is violated, an additive hazard regression model may be more appropriate. In this paper, we give an overview of this approach and then apply a semiparametric as well as a nonparametric additive model to a data set from a study of the natural history of human papillomavirus (HPV) in HIV-positive and HIV-negative women. The results from the semiparametric model indicated on average an additional 14 oncogenic HPV infections per 100 woman-years related to CD4 count < 200 relative to HIV-negative women, and those from the nonparametric additive model showed an additional 40 oncogenic HPV infections per 100 women over 5 years of followup, while the estimated hazard ratio in the Cox model was 3.82. Although the Cox model can provide a better understanding of the exposure disease association, the additive model is often more useful for public health planning and intervention.