Proportionally Difficult: Testing for Nonproportional Hazards in Cox Models

2010 ◽  
Vol 18 (2) ◽  
pp. 189-205 ◽  
Author(s):  
Luke Keele
Keyword(s):  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Cox Model ◽  
Standard Test ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards Model ◽  
Cox Models ◽  
Nonproportional Hazards ◽  
Specification Errors ◽  
Hazards Model

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.

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

2021 ◽  
Author(s):  
Casper Wilstrup ◽  
Chris Cave
Keyword(s):  
Heart Failure ◽  
Medical Records ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Cox Model ◽  
Symbolic Regression ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards Model ◽  
Data Set ◽  
Hazards Model

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.

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

2016 ◽  
Vol 35 (1) ◽  
Author(s):  
Ileana Baldi ◽  
Giovannino Ciccone ◽  
Antonio Ponti ◽  
Stefano Rosso ◽  
Roberto Zanetti ◽  
...  
Keyword(s):  
Cancer Survival ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Cox Model ◽  
Tumour Size ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards Model ◽  
Aalen Model ◽  
Cancer Registry Data ◽  
Hazards Model

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.

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

2021 ◽  
Vol 12 ◽  
Author(s):  
Fahimeh Ramezani Tehrani ◽  
Ali Sheidaei ◽  
Faezeh Firouzi ◽  
Maryam Tohidi ◽  
Fereidoun Azizi ◽  
...  
Keyword(s):  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Cox Model ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards 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.

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

2017 ◽  
Vol 50 (1) ◽  
pp. 303-320 ◽  
Author(s):  
Jonathan Kropko ◽  
Jeffrey J. Harden
Keyword(s):  
Political Science ◽  
Hazard Function ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Cox Model ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards 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.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xianhong Xie ◽  
Howard D. Strickler ◽  
Xiaonan Xue
Keyword(s):  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Cox Model ◽  
Additive Model ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards 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.

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

2021 ◽  
Author(s):  
Casper Wilstup ◽  
Chris Cave
Keyword(s):  
Heart Failure ◽  
Medical Records ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Cox Model ◽  
Symbolic Regression ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards Model ◽  
Data Set ◽  
Hazards Model

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.

The Comparison of the Estimator of the Variance in the Cox Proportional Hazards Model

2014 ◽  
Vol 1049-1050 ◽  
pp. 1613-1616
Author(s):  
Yong Li
Keyword(s):  
Survival Data ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Cox Model ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards Model ◽  
Hazards Model

The Cox model is commonly used to model survival data as a function of covariates. In this paper we compare the three methods to estimate the variance of the parameters in Cox model and presents the simulation result.

scholarly journals Risk factors associated with major adverse cardiac and cerebrovascular events following percutaneous coronary intervention: a 10-year follow-up comparing random survival forest and Cox proportional-hazards model

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Maryam Farhadian ◽  
Sahar Dehdar Karsidani ◽  
Azadeh Mozayanimonfared ◽  
Hossein Mahjub
Keyword(s):  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Coronary Intervention ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards Model ◽  
Hazards Model ◽  
Cerebrovascular Events ◽  
Long Term Follow Up ◽  
Stent Length

Abstract Background Due to the limited number of studies with long term follow-up of patients undergoing Percutaneous Coronary Intervention (PCI), we investigated the occurrence of Major Adverse Cardiac and Cerebrovascular Events (MACCE) during 10 years of follow-up after coronary angioplasty using Random Survival Forest (RSF) and Cox proportional hazards models. Methods The current retrospective cohort study was performed on 220 patients (69 women and 151 men) undergoing coronary angioplasty from March 2009 to March 2012 in Farchshian Medical Center in Hamadan city, Iran. Survival time (month) as the response variable was considered from the date of angioplasty to the main endpoint or the end of the follow-up period (September 2019). To identify the factors influencing the occurrence of MACCE, the performance of Cox and RSF models were investigated in terms of C index, Integrated Brier Score (IBS) and prediction error criteria. Results Ninety-six patients (43.7%) experienced MACCE by the end of the follow-up period, and the median survival time was estimated to be 98 months. Survival decreased from 99% during the first year to 39% at 10 years' follow-up. By applying the Cox model, the predictors were identified as follows: age (HR = 1.03, 95% CI 1.01–1.05), diabetes (HR = 2.17, 95% CI 1.29–3.66), smoking (HR = 2.41, 95% CI 1.46–3.98), and stent length (HR = 1.74, 95% CI 1.11–2.75). The predictive performance was slightly better by the RSF model (IBS of 0.124 vs. 0.135, C index of 0.648 vs. 0.626 and out-of-bag error rate of 0.352 vs. 0.374 for RSF). In addition to age, diabetes, smoking, and stent length, RSF also included coronary artery disease (acute or chronic) and hyperlipidemia as the most important variables. Conclusion Machine-learning prediction models such as RSF showed better performance than the Cox proportional hazards model for the prediction of MACCE during long-term follow-up after PCI.

scholarly journals Elevated plasma growth and differentiation factor 15 predicts incident anemia in older adults aged 60 years and older

2020 ◽  
Author(s):  
Yuko Yamaguchi ◽  
Marta Zampino ◽  
Toshiko Tanaka ◽  
Stefania Bandinelli ◽  
Yusuke Osawa ◽  
...  
Keyword(s):  
Older Adults ◽  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
Cox Proportional Hazards ◽  
Cox Proportional Hazards Model ◽  
Differentiation Factor ◽  
Hazards Model ◽  
Increased Risk ◽  
The Relationship

Abstract Background Anemia is common in older adults and associated with greater morbidity and mortality. The causes of anemia in older adults have not been completely characterized. Although elevated circulating growth and differentiation factor 15 (GDF-15) has been associated with anemia in older adults, it is not known whether elevated GDF-15 predicts the development of anemia. Methods We examined the relationship between plasma GDF-15 concentrations at baseline in 708 non-anemic adults, aged 60 years and older, with incident anemia during 15 years of follow-up among participants in the Invecchiare in Chianti (InCHIANTI) Study. Results During follow-up, 179 (25.3%) participants developed anemia. The proportion of participants who developed anemia from the lowest to highest quartile of plasma GDF-15 was 12.9%, 20.1%, 21.2%, and 45.8%, respectively. Adults in the highest quartile of plasma GDF-15 had an increased risk of developing anemia (Hazards Ratio 1.15, 95% Confidence Interval 1.09, 1.21, P&lt;.0001) compared to those in the lower three quartiles in a multivariable Cox proportional hazards model adjusting for age, sex, serum iron, soluble transferrin receptor, ferritin, vitamin B12, congestive heart failure, diabetes mellitus, and cancer. Conclusions Circulating GDF-15 is an independent predictor for the development of anemia in older adults.

scholarly journals Dementia Adjudication Triggers Associated With Increased Mortality for Older Australians: Evidence From ASPREE

2020 ◽  
Vol 4 (Supplement_1) ◽  
pp. 161-161
Author(s):  
Jane Banaszak-Holl ◽  
Xiaoping Lin ◽  
Jing Xie ◽  
Stephanie Ward ◽  
Henry Brodaty ◽  
...  
Keyword(s):  
Proportional Hazards ◽  
Proportional Hazards Model ◽  
The Elderly ◽  
Cox Proportional Hazards ◽  
Service Planning ◽  
Cox Proportional Hazards Model ◽  
Cognitive Test ◽  
Risk Of Death ◽  
Hazards Model ◽  
Using Data

Abstract Research Aims: This study seeks to understand whether those with dementia experience higher risk of death, using data from the ASPREE (ASPirin in Reducing Events in the Elderly) clinical trial study. Methods: ASPREE was a primary intervention trial of low-dose aspirin among healthy older people. The Australian cohort included 16,703 dementia-free participants aged 70 years and over at enrolment. Participants were triggered for dementia adjudication if cognitive test results were poorer than expected, self-reporting dementia diagnosis or memory problems, or dementia medications were detected. Incidental dementia was adjudicated by an international adjudication committee using the Diagnostic and Statistical Manual for Mental Disorders (DSM-IV) criteria and results of a neuropsychological battery and functional measures with medical record substantiation. Statistical analyses used a cox proportional hazards model. Results: As previously reported, 1052 participants (5.5%) died during a median of 4.7 years of follow-up and 964 participants had a dementia trigger, of whom, 575 (60%) were adjucated as having dementia. Preliminary analyses has shown that the mortality rate was higher among participants with a dementia trigger, regardless of dementia adjudication outcome, than those without (15% vs 5%, Χ2 = 205, p &lt;.001). Conclusion: This study will provide important analyses of differences in the hazard ratio for mortality and causes of death among people with and without cognitive impairment and has important implications on service planning.

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