Non-proportional Hazards Model Research Articles

The changing educational and social class gradients in union dissolution: Evidence from a latecomer of the Second Demographic Transition

Most studies on the changing socioeconomic gradient of divorce have operationalized individuals’ socioeconomic status (SES) through education, often neglecting social class differences. Education may proxy cultural and cognitive skills, whereas social class could more accurately capture economic means. Additionally, existing research has predominantly focused on women and marital dissolutions. This study addresses these oversights by analyzing the educational and social class gradients of both marriage and cohabitation dissolutions among men and women in Italy—a latecomer to the Second Demographic Transition. We used non-proportional hazard models to estimate survival curves and union dissolution probabilities stratified by education, social class, and cohort. Our findings reveal a vanishing socioeconomic gradient of marital dissolution among women and a reversal from positive to negative among men across cohorts. These results challenge the conventional view that men’s higher SES always stabilizes unions and support Goode’s hypothesis on the reversal of the socioeconomic gradient of divorce for both genders. No clear SES gradient was found for cohabiting unions. Overall, the study demonstrates the significant predictive power of social class for marital dissolutions, even when controlling for education, emphasizing the need to consider both measures of SES to comprehensively account for different underlying mechanisms.

Non-proportional hazards model with a PVF frailty term: application with a melanoma dataset

Survival data analysis often uses the Cox proportional hazards (PH) model. This model is widely applied due to its straightforward interpretation of the hazard ratio under the assumption that the hazard rates for two subjects remain constant over time. However, in several randomized clinical trials with long-term survival data comparing two new treatments, it is frequently observed that Kaplan-Meier plots exhibit crossing survival curves. This violation of the PH assumption of the Cox PH model can not be applied to evaluate the treatment's effect on survival. This paper introduces a novel long-term survival model with non-PH that incorporates a frailty term into the hazard function. This model allows us to examine the effect of prognostic factors on survival and quantify the degree of unobservable heterogeneity. The model parameters are estimated using the maximum likelihood estimation procedure, and we evaluate the performance of the proposed models through simulation studies. Additionally, we demonstrate the applicability of our approach by fitting the models to a real skin cancer dataset.

Sequential monitoring of cancer immunotherapy trial with random delayed treatment effect

ABSTRACT Cancer immunotherapy trials are frequently characterized by a delayed treatment effect that violates the proportional hazards assumption. The log-rank test (LRT) suffers a substantial loss of statistical power under the nonproportional hazards model. Various group sequential designs using weighted LRTs (WLRTs) have been proposed under the fixed delayed treatment effect model. However, patients enrolled in immunotherapy trials are often heterogeneous, and the duration of the delayed treatment effect is a random variable. Therefore, we propose group sequential designs under the random delayed effect model using the random delayed distribution WLRT. The proposed group sequential designs are developed for monitoring the efficacy of the trial using the method of Lan-DeMets alpha-spending function with O’Brien-Fleming stopping boundaries or a gamma family alpha-spending function. The maximum sample size for the group sequential design is obtained by multiplying an inflation factor with the sample size for the fixed sample design. Simulations are conducted to study the operating characteristics of the proposed group sequential designs. The robustness of the proposed group sequential designs for misspecifying random delay time distribution and domain is studied via simulations.

Survextrap: a package for flexible and transparent survival extrapolation

BackgroundHealth policy decisions are often informed by estimates of long-term survival based primarily on short-term data. A range of methods are available to include longer-term information, but there has previously been no comprehensive and accessible tool for implementing these.ResultsThis paper introduces a novel model and software package for parametric survival modelling of individual-level, right-censored data, optionally combined with summary survival data on one or more time periods. It could be used to estimate long-term survival based on short-term data from a clinical trial, combined with longer-term disease registry or population data, or elicited judgements. All data sources are represented jointly in a Bayesian model. The hazard is modelled as an M-spline function, which can represent potential changes in the hazard trajectory at any time. Through Bayesian estimation, the model automatically adapts to fit the available data, and acknowledges uncertainty where the data are weak. Therefore long-term estimates are only confident if there are strong long-term data, and inferences do not rely on extrapolating parametric functions learned from short-term data. The effects of treatment or other explanatory variables can be estimated through proportional hazards or with a flexible non-proportional hazards model. Some commonly-used mechanisms for survival can also be assumed: cure models, additive hazards models with known background mortality, and models where the effect of a treatment wanes over time. All of these features are provided for the first time in an R package, survextrap, in which models can be fitted using standard R survival modelling syntax. This paper explains the model, and demonstrates the use of the package to fit a range of models to common forms of survival data used in health technology assessments.ConclusionsThis paper has provided a tool that makes comprehensive and principled methods for survival extrapolation easily usable.

Estimating age specific prevalence and force of infection in Zimbabwe using combined cross-sectional surveys from 2005 to 2015.

Age structured sexual mixing patterns have been noted to be associated with HIV prevalence and force of infection. Therefore, this study aimed to estimate the age dependent HIV force of infection using survey cross-sectional data from Zimbabwe. We fit generalized additive models namely; linear, semi-parametric, non-parametric and non-proportional hazards models. Using the 2005-06, 2010-11 and 2015 Zimbabwe Demographic Health Surveys data. The Akaike Information Criteria was used to select the best model. The best model was then used to estimate the age dependent HIV prevalence and force-of-infection. Based on birth year cohort-specific prevalence, the female HIV prevalence reaches the highest peak at around 29 years of age, then declines thereafter. Males have a lower cohort specific prevalence between 15 and 30 years than females. Male cohort-specific prevalence slightly decreases between the ages of 33 and 39, then peaks around the age of 40. The cohort-specific FOI is greater in females than in males throughout all age categories. In addition, the cohort-specific HIV FOI peaked at ages 22 and 40 for females and males, respectively. The observed 18-year age difference between the HIV FOI peaks of males and females. Our model was appealing because we did not assume that the FOI is stationary over time; however, we used serological survey data to distinguish the FOI's age-and-time effect. The cohort-specific FOI peaked 18 years earlier in females than males, indicative of age-mixing patterns. We recommend interventions that target younger females so as to reduce HIV transmission rates.

Cancer immunotherapy trial design with random delayed treatment effect and cure rate.

Immunotherapies are increasingly used for treating patients with advanced-stage cancers. However, cancer immunotherapy trials often present delayed treatment effects and long-term survivors which result nonproportional hazard models and challenge the immunotherapy trial designs. In this article, we proposed a general random delayed cure rate model for designing cancer immunotherapy trials. A sample size formula is derived for a weighted log-rank test. The accuracy of sample size estimation is assessed and compared with the existing methods via simulation studies. The sensitivities for misspecifying the random delay time are also studied through simulations.

Nonproportional hazards model with a frailty term for modeling subgroups with evidence of long-term survivors: Application to a lung cancer dataset.

With advancements in medical treatments for cancer, an increase in the life expectancy of patients undergoing new treatments is expected. Consequently, the field of statistics has evolved to present increasingly flexible models to explain such results better. In this paper, we present a lung cancer dataset with some covariates that exhibit nonproportional hazards (NPHs). Besides, the presence of long-term survivors is observed in subgroups. The proposed modeling is based on the generalized time-dependent logistic model with each subgroup's effect time and a random term effect (frailty). In practice, essential covariates are not observed for several reasons. In this context, frailty models are useful in modeling to quantify the amount of unobservable heterogeneity. The frailty distribution adopted was the weighted Lindley distribution, which has several interesting properties, such as the Laplace transform function on closed form, flexibility in the probability density function, among others. The proposed model allows for NPHs and long-term survivors in subgroups. Parameter estimation was performed using the maximum likelihood method, and Monte Carlo simulation studies were conducted to evaluate the estimators' performance. We exemplify this model's use by applying data of patients diagnosed with lung cancer in the state of São Paulo,Brazil.

Decline in Share Prices of Energy and Fuel Companies on the Warsaw Stock Exchange as a Reaction to the COVID-19 Pandemic

Many factors influence the prices of energy commodities and the value of energy and fuel companies. Among them there are the following factors: economic, social, environmental and political, and recently also the COVID-19 pandemic. The aim of the paper is to examine what the probability and intensity of a decrease in the prices of shares of energy and fuel companies listed on the Warsaw Stock Exchange (Poland) was during the first wave of the pandemic in the first quarter of 2020. The study used the survival analysis methods: the Kaplan-Meier estimator, the test of equality of duration curves and the Cox non-proportional hazards model. The analysis showed that the probability and intensity of price decline of energy and fuel companies in the initial period was the same as that of other companies. The differences become apparent only after 50 days from the established maximum of their value. The risk of price declines in energy and fuel companies increased significantly. This situation was related both to a temporary reduction in demand for energy and fuels, pandemic restrictions introduced in individual countries and the behaviour of stock market investors.

Unplanned Readmission Burden Impacts Survival in Patients with Contemporary Continuous-Flow Left Ventricular Assist Device: STS Intermacs Analysis

<h3>Purpose</h3> Survival on durable continuous-flow left ventricular assist device (CF-LVAD) now mirrors that of heart transplant patients. There is a paucity of data on how readmission burden impacts survival in such patients. This study aims to determine if readmission burden influences survival outcomes after contemporary CF-LVAD implant. <h3>Methods</h3> The STS Intermacs registry data were retrospectively evaluated for clinical outcomes in adults (≥18 years) with durable CF-LVAD implant between 1/2008-3/2020. Readmissions were defined as hemocompatibility-related adverse events (HRAE) vs non-HRAE (Table 1). A multiphase non-proportional hazard model with readmission as a time-varying covariate was used to analyze the impact of readmission on the risk of early (death predominant within first 6 months) and late hazard phases of death on CF-LVAD. <h3>Results</h3> 26,933 CF-LVAD adults (mean age 57 ± 13 years, 40% centrifugal, 49% destination therapy) had a median follow-up of 1.1 years [2.8 months, 3.4 years]. Overall 1 month and 5 year survival was 96% and 47%, respectively. 56,495 unplanned readmissions (23,617 HRAE and 32,057 non-HRAE) were observed in 17,221 patients. Most had one hospital readmission (31.5%). The cumulative number of overall, HRAE, and non-HRAE readmissions per patient were 0.83, 0.82 and 0.96 at 1 year and 5.7, 2.9 and 3.8 at 5 years, respectively. Unplanned readmission, along with frequency and length of stay (LOS) of non-HRAE was associated with a higher risk of late vs early death. Table 1 highlights the differential predictors associated with early and late death. <h3>Conclusion</h3> Most patients with contemporary CF-LVAD had at least one unplanned readmission. Unplanned readmission, along with predominantly frequency and LOS of non-HRAE, worsens the risk of late death. Future analyses should examine mitigation strategies to curb unplanned readmissions based on HRAE vs non-HRAE categorization.

Outcomes of mitral valve re-replacement for bioprosthetic structural valve deterioration

ObjectivesReoperation for structural valve deterioration (SVD) of bioprosthetic mitral valves carries a presumed high operative risk, and transcatheter mitral valve-in-valve implantation has emerged as an alternative. However, surgical risk and long-term outcome following mitral valve re-replacement in these patients remain ill-defined. Hence, we sought to evaluate outcomes and long-term survival following surgical mitral valve re-replacement and to identify risk factors for mortality. MethodsFrom January 1990 to January 2017, 525 patients underwent surgical mitral valve re-replacement at Cleveland Clinic for bioprosthetic SVD: 133 (25%) isolated operations and 392 (75%) with concomitant procedures. Surgical complications and modes of death were compiled, long-term mortality assessed, and risk factors identified using a multivariable nonproportional hazards model and random forest analysis. ResultsSVD was characterized by bioprosthetic regurgitation in 81% (425 out of 525) and stenosis in 44% (231 out of 525). One in-hospital death occurred after isolated valve re-replacement (0.75%) and 28 deaths occurred (7.1%; P = .003) after nonisolated re-replacement, 19 (68%) of which were from coagulopathy, vasoplegia, and multisystem organ failure. In the nonisolated group, incremental risk factors for time-related death after re-replacement included New York Heart Association functional class IV symptoms, concomitant coronary artery bypass grafting, prolonged cardiopulmonary bypass time, and transfusions. ConclusionsMitral valve re-replacement for bioprosthetic SVD was associated with low surgical risk and excellent long-term survival. Isolated mitral valve re-replacement for bioprosthetic SVD had near-zero surgical risk. Excessive cardiopulmonary bypass duration and multiple transfusions correlated with increased early mortality in nonisolated procedures, as did preoperative severe heart failure. Optimal surgical plan and timing of surgery are keys to success.

Extremes of Age Decrease Survival in Adults After Lung Transplant

Age has been implicated as a factor in the plateau of long-term survival after lung transplant. We used data from the Scientific Registry of Transplant Recipients to identify all recipients of lung transplant aged≥18 years of age between January 1, 2006, and February 19, 2015. A total of 14,253 patients were included in the analysis. Survival was estimated using a nonproportional hazard model and random-survival forest methodology was used to examine risk factors for death. Final selection of model variables was performed using bootstrap aggregation. Age was analyzed as both a continuous and categorical variable (age<30, 30-55, and >55 years). Risk factors for death were obtained for the entire cohort and additional age-specific risk factors were identified for each age category. The median age at transplant was 59 years. There were 1,098 (7.7%) recipients<30 years, 4,201 (29.5%) 30 to 55 years, and 8,954 (62.8%) >55 years of age. Age was the most significant risk factor for death at all time-points following transplant and its impact becomes more prominent as time from transplant increases. Risk factors for death for all patients included extremes of age, higher creatinine, single lung transplant, hospitalization before transplant, and increased bilirubin. Risk factors for death differed by age with social determinants of health disproportionately affecting survival for those in the youngest age category. The youngest and oldest adult recipients experienced the lowest posttransplant survival through divergent pathways that may present opportunities for intervention to improve survival after lung transplant.

A non-proportional hazards model with time-dependent variables for under-five mortality in Indonesia

Cox proportional hazards (PH) is known as the most popular model for the analysis of multivariate survival data. The main assumption of the model is that the hazard ratio among any two individuals in the population is constant over time. The violation of this assumption, however, may cause serious issues such as overestimation or underestimation of hazard risks and reducing the power of related statistical tests. The main objective of this research, therefore, is to extend the Cox PH model by adding time-dependent variables into the model to cope with the presence of non-PH. The proposed model, moreover, is applied to under-five mortality data based on Indonesian Demographic and Health Survey (IDHS) 2012. The results showed that the Akaike’s Information Criterion (AIC) and Bayesian Information Criterion (BIC) values of the extended Cox model with time-dependent variables are much lower than the Cox PH model. This means that the extended Cox model can increase significantly the goodness-of-fit of the Cox PH model to under-five mortality data in Indonesia.

Stratified-extended cox model in survival modeling of non-proportional hazard

Cox proportional hazard model is frequently used in survival analysis. Cox proportional hazard model is time independent covariate while many models involve time as a dependent covariate causing incomplete proportional hazard assumption, known as non-proportional hazard. The proposed model in this paper was a non-proportional hazard involving time-independent and time-dependent covariates. The approaching model was carried out by joining a stratified Cox and extended Cox model termed as Stratified-Extended Cox (SE Cox) model. The simulation of the SE Cox model resulted in small MSE for the parameter estimates. In addition, the goodness of value was more appropriate compared to the existing non-proportional hazard model. Hence, the SE Cox model was applied to evaluate student persistence in Universitas Terbuka, Indonesia.

Quantifying time-varying cause-specific hazard and subdistribution hazard ratios with competing risks data.

Various non-proportional hazard models have been developed in the literature for competing risks data. The regression coefficients under these models, however, typically cannot be compared directly. We propose new methods to quantify the average of the time-varying cause-specific hazard ratios and subdistribution hazard ratios through two general classes of transformations and weight functions that are chosen to reflect the relative importance of the hazard ratios in different time periods. We further propose an -norm type of test statistic that incorporates the test statistics for all possible pairs of the transformation function and weight function under consideration. Extensive simulations are conducted under various settings of the hazards and demonstrate that the proposed test performs well under all settings. An application to a clinical trial in follicular lymphoma is examined in detail.

Mortality in people with dementia, delirium, and unspecified cognitive impairment in the general hospital: prospective cohort study of 6,724 patients with 2 years follow-up.

PurposeCognitive impairment is common in older people admitted to hospital, but the outcomes are generally poorly understood, and previous research has shown inconsistent associations with mortality depending on the type of cognitive impairment examined and duration of follow-up. This study examines mortality in older people with any cognitive impairment during acute hospital admission.Patients and methodsProspective cohort of 6,724 people aged ≥65 years with a structured cognitive assessment on acute admission were included in this study. Cognitive spectrum disorder (CSD) was defined as delirium alone, known dementia alone, delirium superimposed on known dementia, or unspecified cognitive impairment. Mortality associated with different types of CSD was examined using a non-proportional hazards model with 2-year follow-up.ResultsOn admission, 35.4% of patients had CSD, of which 52.6% died within 2 years. After adjustment for demographics and comorbidity, delirium alone was associated with increased mortality in the 6 months post-admission (HR =1.45, 95% CI 1.28–1.65) and again after 1 year (HR =1.44, 95% CI 1.17–1.77). Patients with known dementia (alone or with superimposed delirium) had increased mortality only after 3 months from admission (HR =1.85, 95% CI 1.56–2.18 and HR =1.80, 95% CI 1.52–2.14) compared with patients with unspecified cognitive impairment after 6 months (HR =1.55, 95% CI 1.21–1.99). Similar but partially attenuated associations were seen after adjustment for functional ability.ConclusionMortality post-admission is high in older people with CSD. Immediate risk is highest in those with delirium, while dementia or unspecified cognitive impairment is associated with medium- to long-term risk. These findings suggest that individuals without dementia who develop delirium are more seriously ill (have required a larger acute insult in order to precipitate delirium) than those with pre-existing brain pathology (dementia). Further research to explain the mortality patterns observed is required in order to translate the findings into clinical care.

Survival affects decision making for fenestrated and branched endovascular aortic repair

ObjectiveRepair options for complex abdominal and thoracoabdominal aortic aneurysms (TAAAs) are evolving with increased experience and availability of less invasive endovascular techniques. Identifying risk factors for mortality after fenestrated and branched endovascular aortic repair (F/B-EVAR) could improve patient selection and facilitate decision making regarding who may benefit from prophylactic F/B-EVAR. MethodsWe evaluated 1091 patients in a prospective investigational device exemption trial who underwent F/B-EVAR from August 2001 to June 2015 for complex aortic aneurysms (CAAs). Multivariable analysis of risk factors for death was performed using a nonproportional hazards model and a nonparametric analysis using random survival forest technology. ResultsOperative mortality after F/B-EVAR was low (3.7%), with high CAA-related survival at 30 day and 5 years (96.8% and 94.0%, respectively). All-cause 5-year survival, however, was 46.2% and older age, heart failure, chronic obstructive pulmonary disease, renal disease, anemia, and coagulation disorders were risk factors. Risk was highest for those undergoing type I/II TAAA repairs and those with larger aneurysms. ConclusionsPatients with multiple comorbidities and those undergoing type I or II TAAA repair are at greatest risk of mortality; however, in this high-risk population, F/B-EVAR offers greater survival compared with that reported for the natural history of untreated aneurysms. Operative and early mortality is lower than the best-reported open repair outcomes, even in this high-risk population, suggesting a potential benefit in extending the use of F/B-EVAR to low-to-average risk CAA patients.

Comparison of methods for estimating the attributable risk in the context of survival analysis

BackgroundThe attributable risk (AR) measures the proportion of disease cases that can be attributed to an exposure in the population. Several definitions and estimation methods have been proposed for survival data.MethodsUsing simulations, we compared four methods for estimating AR defined in terms of survival functions: two nonparametric methods based on Kaplan-Meier’s estimator, one semiparametric based on Cox’s model, and one parametric based on the piecewise constant hazards model, as well as one simpler method based on estimated exposure prevalence at baseline and Cox’s model hazard ratio. We considered a fixed binary exposure with varying exposure probabilities and strengths of association, and generated event times from a proportional hazards model with constant or monotonic (decreasing or increasing) Weibull baseline hazard, as well as from a nonproportional hazards model. We simulated 1,000 independent samples of size 1,000 or 10,000. The methods were compared in terms of mean bias, mean estimated standard error, empirical standard deviation and 95% confidence interval coverage probability at four equally spaced time points.ResultsUnder proportional hazards, all five methods yielded unbiased results regardless of sample size. Nonparametric methods displayed greater variability than other approaches. All methods showed satisfactory coverage except for nonparametric methods at the end of follow-up for a sample size of 1,000 especially. With nonproportional hazards, nonparametric methods yielded similar results to those under proportional hazards, whereas semiparametric and parametric approaches that both relied on the proportional hazards assumption performed poorly. These methods were applied to estimate the AR of breast cancer due to menopausal hormone therapy in 38,359 women of the E3N cohort.ConclusionIn practice, our study suggests to use the semiparametric or parametric approaches to estimate AR as a function of time in cohort studies if the proportional hazards assumption appears appropriate.

C-reactive protein level on admission and time to and cause of death in patients hospitalized for acute heart failure.

We analysed the association between C-reactive protein (CRP) levels measured on admission and timing and cause of death among patients hospitalized for acute heart failure (AHF). The ATTEND study prospectively registered 4777 hospitalized AHF patients with data on CRP levels on admission. Mortality risks were assessed by univariable and multivariable Cox proportional and non-proportional hazards models. The overall median CRP level was 5.8 mg/L (intertertile range: 2.9-11.8 mg/L). There were significant increases in all-cause, cardiac, and non-cardiac mortalities from the lowest to highest CRP tertiles throughout the follow-up periods. Within 120 days after admission, CRP levels in the highest tertile (>11.8 mg/L) were independently associated with higher all-cause (hazard ratio [HR], 2.21; 95% confidence interval [CI], 1.69-2.88; P < 0.001), cardiac (HR, 1.88; 95% CI, 1.37-2.58; P < 0.001), and non-cardiac (HR, 3.21; 95% CI, 1.94-5.32; P < 0.001) deaths, while levels in the second tertile (2.9-11.8 mg/L) were not associated with poorer survival, compared with levels in the first tertile (<2.9 mg/L). However, in terms of cardiac death, the hazard ratios for patients in the third tertile decreased markedly with time and only CRP levels in second tertile were independently associated with poorer cardiac survival after the follow-up period of 120 days (HR, 1.44; 95% CI, 1.09-1.89; P = 0.011). Markedly elevated CRP levels at admission in patients with AHF may be associated with higher short-term cardiac and non-cardiac mortalities. In addition, modestly elevated CRP levels may be associated with higher mortality, especially cardiac mortality, after 120 days of long-term follow-up.

Semi-Parametric Non-Proportional Hazard Model With Time Varying Covariate

The application of survival analysis has extended the importance of statistical methods for time to event data that incorporate time dependent covariates. The Cox proportional hazards model is one such method that is widely used. An extension of the Cox model with time-dependent covariates was adopted when proportionality assumption are violated. The purpose of this study is to validate the model assumption when hazard rate varies with time. This approach is applied to model data on duration of infertility subject to time varying covariate. Validity is assessed by a set of simulation experiments and results indicate that a non proportional hazard model performs well in the phase of violated assumptions of the Cox proportional hazards.

State LGBT Rights Policy Outliers: Transsexual Birth Certificate Amendment Laws

This paper explores an anomaly in gay, lesbian, bisexual and transgender (LGBT) rights policy, laws allowing transsexual individuals to amend their birth certificates. Unlike most other LGBT rights policies, these statutes are often found in Southern and other conservative states. In fact, these laws are found in half of the Southern states. The array of states with these laws does not conform to the traditional pattern of morality politics laws that is commonly associated with LGBT rights. Using a Cox non-proportional hazards model, we find that the adoption of these laws was influenced by vertical diffusion of the Centers for Disease Control's model vital records recommendations. States with more professionalized bureaucracies, like Virginia and Georgia, were more likely to implement these recommended best practices. However, as transgender rights became more closely associated with the gay rights advocacy movement, this issue likely resembles morality policy. The result being that liberal and conservative elites respond to these policies in predictable manners. Notably, the political opportunity structure in Southern states has not allowed the passage of this type of statute since the incorporation of transgender rights into the LGBT social movement during the mid-1990s.