Parametric Proportional Hazards Model Research Articles

Bayesian Joint Modeling of Longitudinal Measurements and Time-to-Event Data Using Robust Distributions

Distributional assumptions of most of the existing methods for joint modeling of longitudinal measurements and time-to-event data cannot allow incorporation of outlier robustness. In this article, we develop and implement a joint modeling of longitudinal and time-to-event data using some powerful distributions for robust analyzing that are known as normal/independent distributions. These distributions include univariate and multivariate versions of the Student’s t, the slash, and the contaminated normal distributions. The proposed model implements a linear mixed effects model under a normal/independent distribution assumption for both random effects and residuals of the longitudinal process. For the time-to-event process a parametric proportional hazard model with a Weibull baseline hazard is used. Also, a Bayesian approach using the Markov-chain Monte Carlo method is adopted for parameter estimation. Some simulation studies are performed to investigate the performance of the proposed method under presence and absence of outliers. Also, the proposed methods are applied for analyzing a real AIDS clinical trial, with the aim of comparing the efficiency and safety of two antiretroviral drugs, where CD4 count measurements are gathered as longitudinal outcomes. In these data, time to death or dropout is considered as the interesting time-to-event outcome variable. Different model structures are developed for analyzing these data sets, where model selection is performed by the deviance information criterion (DIC), expected Akaike information criterion (EAIC), and expected Bayesian information criterion (EBIC).

Survival analysis of time‐to‐event data in respiratory health research studies

This article provides a review of techniques for the analysis of survival data arising from respiratory health studies. Popular techniques such as the Kaplan-Meier survival plot and the Cox proportional hazards model are presented and illustrated using data from a lung cancer study. Advanced issues are also discussed, including parametric proportional hazards models, accelerated failure time models, time-varying explanatory variables, simultaneous analysis of multiple types of outcome events and the restricted mean survival time, a novel measure of the effect of treatment.

Comparison of different parametric proportional hazards models for interval-censored data: A simulation study

Interval censoring occurs frequently in clinical trials, but is often simplified to a right censoring problem because statistical methods in this area are under developed. It is recognized that analyzing interval censored data as right-censored data can lead to biased results. Although statistical methods have been developed to estimate survival function and to test hypothesis, estimating hazard ratio (HR) in a proportional hazards (PH) model for interval censored data remains as a challenge. Semi-parametric PH model was developed but difficult to implement, and thus rarely used in practice. Parametric PH method can be easily implemented but received little attention in practice because the impact of mis-specifying baseline hazard function on HR estimate was not well understood. We examined the performance of parametric PH models, using 3 baseline hazard functions: exponential, Weibull, and a 10-piece exponential function, under different underlying data distributions and censoring schema, through an extensive simulation study. Data were generated from 6 different models representing a range of possible scenarios in clinical trials. The simulation study revealed that mis-specifying baseline hazard function had little impact on the HR estimates. Robust estimate of HR with little bias and small mean square errors (MSE) were obtained using a PH model with a Weibull or 10-piece exponential function approximating baseline hazard function. Bigger bias and MSE were observed when using an exponential function to approximate a complex baseline hazard function. Examples are included. Based on these findings, we advocate the use of parametric PH models for the analysis of interval censored data

UKPDS Outcomes Model 2: a new version of a model to simulate lifetime health outcomes of patients with type 2 diabetes mellitus using data from the 30 year United Kingdom Prospective Diabetes Study: UKPDS 82

The aim of this project was to build a new version of the United Kingdom Prospective Diabetes Study (UKPDS) Outcomes Model (UKPDS-OM1), a patient-level simulation tool for predicting lifetime health outcomes of people with type 2 diabetes mellitus. Data from 5,102 UKPDS patients from the 20 year trial and the 4,031 survivors entering the 10 year post-trial monitoring period were used to derive parametric proportional hazards models predicting absolute risk of diabetes complications and death. We re-estimated the seven original event equations and estimated new equations for diabetic ulcer and some second events. The additional data permitted inclusion of new risk factor predictors such as estimated GFR. We also developed four new equations for all-cause mortality. Internal validation of model predictions of cumulative incidence of all events and death was carried out and a contemporary patient-level dataset was used to compare 10 year predictions from the original and the new models. Model equations were based on a median 17.6 years of follow-up and up to 89,760 patient-years of data, providing double the number of events, greater precision and a larger number of significant covariates. The new model, UKPDS-OM2, is internally valid over 25 years and predicts event rates for complications, which are lower than those from the existing model. The new UKPDS-OM2 has significant advantages over the existing model, as it captures more outcomes, is based on longer follow-up data, and more comprehensively captures the progression of diabetes. Its use will permit detailed and reliable lifetime simulations of key health outcomes in people with type 2 diabetes mellitus.

Cholesteatoma in cleft lip and palate: A population‐based follow‐up study of children after ventilation tubes

To investigate the association of cleft conditions and the development of secondary cholesteatoma following middle ear ventilation tube insertion (MEVTI) in children. A retrospective cohort study of all children born after 1980 who underwent at least one MEVTI in a Western Australian hospital from 1980 to 2009 using administrative health data. The timing and number of MEVTIs, adenoidectomy, type of cleft conditions and cholesteatoma were identified along with demographic variables. Flexible parametric proportional hazards models and hazard functions using age as a time scale were used to estimate the relative rate of developing cholesteatoma. There were 56,949 children who underwent at least one MEVTI during the study period. There were 869 (1.5%) children who also had a diagnosis of a cleft condition. Overall, 594 (1.0%) children developed a secondary cholesteatoma. After taking length of follow-up into account, 6.9% (95% confidence interval [CI], 5.0-9.6) of children with cleft conditions developed cholesteatoma by 18 years of age compared to 1.5% (95% CI, 1.3-1.6) of children without cleft conditions. After adjusting for demographic and clinical variables, children with cleft conditions developed cholesteatoma 7.5 (95% CI, 3.8-18.2) times faster after first MEVTI compared to children without cleft conditions, although by the third MEVTI this difference was no longer statistically significant (P = .257). The rate of developing cholesteatoma in 2005 to 2009 was 0.5 (95% CI, 0.3-0.8) times that of 1980 to 1990. Children with cleft conditions are at increased risk of developing cholesteatoma compared to other children who had one or two MEVTIs, although the overall rate of cholesteatoma is declining.

Analysis of diabetic retinopathy data using shared inverse Gaussian frailty model

The dependence between individuals in a group is modeled by the group specific quantity, which can be interpreted as an unobserved covariates or “frailties” common to the individuals in the group and assumed to follow some distribution. We consider the shared frailty model in the frame work of parametric Cox proportional hazard model. The Cox regression model with the shared frailty factor allows for unobserved heterogeneity or for statistical dependence between the observed survival times. There are certain assumptions about the distribution of frailty and baseline distribution. The exponential distribution is the commonly used distribution for analyzing life time data. In this paper, we consider shared inverse Gaussian frailty model with bivariate exponential of Marshall-Olkin [20] distribution as baseline hazard for bivariate survival times. We fit the model to the real life bivariate survival data set of diabetic retinopathy data. Data are analyzed using Bayesian approach to the analysis of clustered survival data in which there is a dependence of failure time observations within the same group. The variance component estimation provides the estimated dispersion of the random effects. The problem of analyzing and estimating parameters of shared inverse Gaussian frailty for diabetic retinopathy data is the interest of this paper. Also, we carried out a test for independence using information criteria.

Flexible parametric modelling of cause-specific hazards to estimate cumulative incidence functions

BackgroundCompeting risks are a common occurrence in survival analysis. They arise when a patient is at risk of more than one mutually exclusive event, such as death from different causes, and the occurrence of one of these may prevent any other event from ever happening.MethodsThere are two main approaches to modelling competing risks: the first is to model the cause-specific hazards and transform these to the cumulative incidence function; the second is to model directly on a transformation of the cumulative incidence function. We focus on the first approach in this paper. This paper advocates the use of the flexible parametric survival model in this competing risk framework.ResultsAn illustrative example on the survival of breast cancer patients has shown that the flexible parametric proportional hazards model has almost perfect agreement with the Cox proportional hazards model. However, the large epidemiological data set used here shows clear evidence of non-proportional hazards. The flexible parametric model is able to adequately account for these through the incorporation of time-dependent effects.ConclusionA key advantage of using this approach is that smooth estimates of both the cause-specific hazard rates and the cumulative incidence functions can be obtained. It is also relatively easy to incorporate time-dependent effects which are commonly seen in epidemiological studies.

Parametric Cure Model versus Proportional Hazards Model in Survival Analysis of Breast Cancer and Other Malignancies

As cancer therapy has progressed dramatically, its goal has shifted toward cure of the disease (curative therapy) rather than prolongation of time to death (life-prolonging therapy). Consequently, the proportion of cured patients (c) has become an important measure of the long-term survival benefit derived from therapy. In 1949, Boag addressed this issue by developing the parametric log-normal cure model, which provides estimates of c and m where m is the mean of log times to death from cancer among uncured patients. Unfortunately, traditional methods based on the proportional hazards model like the Cox regression and log-rank tests cannot provide an estimate of either c or m. Rather, these methods estimate only the differences in hazard between two or more groups. In order to evaluate the long-term validity and usefulness of the parametric cure model compared with the proportional hazards model, we reappraised randomized controlled trials and simulation studies of breast cancer and other malignancies. The results reveal that: 1) the traditional methods fail to distinguish between curative and life-prolonging therapies; 2) in certain clinical settings, these methods may favor life-prolonging treatment over curative treatment, giving clinicians a false estimate of the best regimen; 3) although the Boag model is less sensitive to differences in failure time when follow-up is limited, it gains power as more failures occur. In conclusion, unless the disease is always fatal, the primary measure of survival benefit should be c rather than m or hazard ratio. Thus, the Boag lognormal cure model provides more accurate and more useful insight into the long-term benefit of cancer treatment than the traditional alternatives.

A Parametric Survival Model When a Covariate is Subject to Left-Censoring

Modeling survival data with a set of covariates usually assumes that the values of the covariates are fully observed. However, in a variety of applications, some values of a covariate may be left-censored due to inadequate instrument sensitivity to quantify the biospecimen. When data are left-censored, the true values are missing but are known to be smaller than the detection limit. The most commonly used ad-hoc method to deal with nondetect values is to substitute the nondetect values by the detection limit. Such ad-hoc analysis of survival data with an explanatory variable subject to left-censoring may provide biased and inefficient estimators of hazard ratios and survivor functions. We consider a parametric proportional hazards model to analyze time-to-event data. We propose a likelihood method for the estimation and inference of model parameters. In this likelihood approach, instead of replacing the nondetect values by the detection limit, we adopt a numerical integration technique to evaluate the observed data likelihood in the presence of a left-censored covariate. Monte Carlo simulations were used to demonstrate various properties of the proposed regression estimators including the consistency and efficiency. The simulation study shows that the proposed likelihood approach provides approximately unbiased estimators of the model parameters. The proposed method also provides estimators that are more efficient than those obtained under the ad-hoc method. Also, unlike the ad-hoc estimators, the coverage probabilities of the proposed estimators are at their nominal level. Analysis of a large cohort study, genetic and inflammatory marker of sepsis study, shows discernibly different results based on the proposed method. Naive use of detection limit in a parametric survival model may provide biased and inefficient estimators of hazard ratios and survivor functions. The proposed likelihood approach provides approximately unbiased and efficient estimators of hazard ratios and survivor functions.

Parametric proportional hazards model for mapping genomic imprinting of survival traits

A number of imprinted genes have been observed in plants, animals and humans. They not only control growth and developmental traits, but may also be responsible for survival traits. Based on the Cox proportional hazards (PH) model, we constructed a general parametric model for dissecting genomic imprinting, in which a baseline hazard function is selectable for fitting the effects of imprinted quantitative trait loci (iQTL) genotypes on the survival curve. The expectation-maximisation (EM) algorithm is derived for solving the maximum likelihood estimates of iQTL parameters. The imprinting patterns of the detected iQTL are statistically tested under a series of null hypotheses. The Bayesian information criterion (BIC) model selection criterion is employed to choose an optimal baseline hazard function with maximum likelihood and parsimonious parameterisation. We applied the proposed approach to analyse the published data in an F(2) population of mice and concluded that, among five commonly used survival distributions, the log-logistic distribution is the optimal baseline hazard function for the survival time of hyperoxic acute lung injury (HALI). Under this optimal model, five QTL were detected, among which four are imprinted in different imprinting patterns.

Fixed effects maximum likelihood estimation of a flexibly parametric proportional hazard model with an application to job exits

We extend the fixed effects maximum likelihood estimator to a proportional hazard model with a flexibly parametric baseline hazard. We use the method to estimate a job duration model for young men, and show that failure to account for unobserved fixed effects causes negative schooling and union effects to be downward biased.

Parametric Cox’s Model for Partly Interval-Censored Data with Application to AIDS Studies

 Abstract—The Parametric Cox's Proportional Hazard Model based on Expectation-Maximization (EM) algorithm for partly interval-censored data is studied. We mean by partly interval-censored data that the observed data include both the exact and interval-censored observations on the survival time of interest. Through the simulation data and real data, we demonstrate that the resulting estimate of regression coefficient and its associated standard error. Our proposal is easily implemented by using SAS software in the present of partly interval-censored data.

Life Expectancy of Colorectal Cancer Patients With Geriatric Syndromes

Colorectal cancer (CRC) is often diagnosed after age 70. Age and comorbidity reduce life expectancy independent of cancer stage and treatment. Geriatric syndromes (dementia, delirium, depression, falls, incontinence, pressure ulcers) are distinct morbidities that predict mortality in older adults independent of age and chronic disease; yet, we know little regarding their impact on life expectancy after CRC diagnosis. Aim: We assessed the prevalence of geriatric syndromes in a population-based sample of older CRC patients to quantify their impact onmedian life expectancy.Methods: This retrospective cohort study used the national administrative inpatient, outpatient and mortality data of veterans aged ≥ 60 years with CRC but no evidence of tumor spread beyond regional lymphatics. The prevalence of geriatric syndromes was assessed and a parametric proportional hazards model was used to examine their influence on mortality. Life expectancy was calculated using accelerated failure time models, stratified by advancing age, treatment and Deyo comorbidity score. Results: Of 60,265 veterans (98.4% male, 66.4% Caucasian, average age 74.5 [SD 7.8]), 29, 254 died during the period 10/01/03 to 10/01/08. Those who died were similar in age, gender, race, comorbidity, CRC site and receipt of therapy (chemotherapy, radiation, surgery) to those who lived. The prevalence of geriatric syndromes was 15%, with similar distributions of single versus multiple coexistent syndromes across cohorts. Falls were most frequent (61%), followed by dementia (12.8%), depression (11.7%) and incontinence (10.6%); however, pressure ulcers were most predictive of mortality (HR 2.5; 95% CI: 2.0-3.2). Life expectancy was strongly related to age and the presence of geriatric syndromes. For example, patients aged 60-69 with falls alone had an expected lifespan of 12.8 years, reduced to 6.9 years with coexistent psychiatric illness. Those 70-79 years with falls had a life expectancy of 7.1 years, shortened to 4.9 years with coexistent falls and psychiatric syndromes. In those ≥ 80 years, life expectancy was 4.8 years with falls, reduced by 40% with coexistent psychiatric illness, and limited to 1.4 years with falls, psychiatric illness and pressure ulcers. Chronic illness as identified by Deyo comorbidity score ≥ 1 further reduced life expectancy among patients of similar age and geriatric syndromes by an additional 16% to 50%. Conclusions: Geriatric syndromes are an important risk factor predicting mortality independent of age, treatment and comorbidity following CRC diagnosis. Gastroenterologists should identify them and consider their impact on life expectancy when discussing treatment options with older patients.

6. Effects of Exposure on Prevalence and Cumulative Relative Risk: Direct and Indirect Effects in a Recursive Hazard Model

This paper outlines decomposition methods for assessing how exposure affects prevalence and cumulative relative risk. Let x denote a vector of exogenous covariates and suppose that a single dimension of time t governs two event processes T(1) and T(2). If the occurrence of the event T(1) determines entry into the risk of the event T(2), then subgroup variation in T(1) will affect the prevalence T(2), even if subgroups in the population are otherwise identical. Although researchers often acknowledge this phenomenon, the literature has not provided procedures to assess the magnitude of an exposure effect of T(1) on the prevalence of T(2). We derive decompositions that assess how variation in exposure generated by direct and indirect effects of the covariates x affect measures of absolute and relative prevalence of T(2). We employ a parametric but highly flexible specification for baseline hazard for the T(1) and T(2) processes and use the resulting parametric proportional hazard model to illustrate the direct and indirect effects of family structure when T(1) is age at first sexual intercourse and T(2) is age at a premarital first birth for data on a cohort of nonhispanic white U.S. women.

Associations of serum uric acid with cardiovascular events and mortality in moderate chronic kidney disease.

It is unclear whether the presence of kidney disease modifies the associations of uric acid with cardiovascular events and death. In the limited access, public use Atherosclerosis Risk In Communities (ARIC) database, associations of serum uric acid levels with cardiovascular events and death were analysed using a parametric proportional hazards model and the modification of these associations by the presence of CKD was assessed using a likelihood ratio test. Of the 15 366 ARIC participants included in this analysis, 461 had CKD (eGFR <60 ml/min/1.73 m(2)). In both non-CKD and CKD sub-groups, participants with hyperuricaemia (> or = 7 mg/dl in men and > or = 6 mg/dl in women) compared to those with normal serum uric acid levels had higher waist circumference and fasting serum insulin levels. In the entire cohort, in a multivariate parametric proportional hazards model, each mg/dl increase in serum uric acid was associated with an increased hazard of cardiovascular events (HR 1.09, 95% CI 1.05-1.12) and death. A multiplicative interaction term of serum uric acid and CKD when added to the above models was significant (P < 0.001). The likelihood ratio test of the models with and without the interaction term was also significant (P < 0.001). In the non-CKD population, a multivariate analysis after adjusting for comorbidities and metabolic syndrome showed a significant association between hyperuricaemia and mortality (HR 1.18, 95% CI 1.04-1.33) but not for cardiovascular events (HR 1.07, 95% CI 0.96-1.19). In the CKD population, the association was not significant for both mortality and cardiovascular events. We conclude that hyperuricaemia is associated with insulin resistance and mortality in the non-CKD population. The presence of CKD attenuates the associations of uric acid with mortality. Interventional studies are warranted to establish the biological role of hyperuricaemia in mortality in non-CKD and CKD populations.

Models for recurring events with marginal proportional hazards

Semiparametric methods were proposed by Wei et al. (1989) to analyse recurring event-time data. They modelled the marginal distribution of each event time with a Cox proportional hazards model without imposing any constraint on the joint distribution of different event times. Therefore, it is unclear whether or not event times can simultaneously satisfy their respective marginal proportional hazards assumptions, while having continuous joint distribution. Often this leads to a difficulty of conducting simulation studies. In this note we construct parametric marginal proportional hazards models for recurring event times with proper joint density functions.

The methodology of Event History analysis and its application in the quality control of raw milk

Event-history analysis is among the most frequently adopted mathematical methods, mainly due to the numerous types of models, available in the analysis. This method is also appropriate for analysing risk. Specifically, it has been employed in medical and animal-breeding science to manage risks. In this article, I give a presentation of the basic concepts, the basic types of models and their theoretical basis. I deal with the nonparametric Kaplan Meier and the parametric Cox proportional hazard model. The case study is also analysed with these methods. The results shows, that the 16-stall caroussel is the most effective facility from the viewpoint of quality. Milking into assay crucible (sample- cup) is more effective than to the floor after 16 decades.

Mapping Quantitative Trait Loci With Censored Observations

The existing statistical methods for mapping quantitative trait loci (QTL) assume that the phenotype follows a normal distribution and is fully observed. These assumptions may not be satisfied when the phenotype pertains to the survival time or failure time, which has a skewed distribution and is usually subject to censoring due to random loss of follow-up or limited duration of the experiment. In this article, we propose an interval-mapping approach for censored failure time phenotypes. We formulate the effects of QTL on the failure time through parametric proportional hazards models and develop efficient likelihood-based inference procedures. In addition, we show how to assess genome-wide statistical significance. The performance of the proposed methods is evaluated through extensive simulation studies. An application to a mouse cross is provided.

A Prepayment Model for the Japanese Mortgage Loan Market: Prepayment-Type-Specific Parametric Model Approach

This paper proposes a framework for construction of a prepayment model suitedto the Japanese mortgage loan market and assesses the validity of thisframework based on an empirical analysis using data from Japan. In thisframework, a model is constructed for each of three prepayment types, namely,`full prepayment', `partial prepayment', and `subrogation', using a parametricproportional hazards model, which was also employed by Schwartz and Torous(1989). Combining these three types of models allows one to take into accountthe effects of partial prepayments, which are frequently used in the Japanesemortgage market, and to simultaneously construct a model for both prepaymentand default. Time-dependent (path-dependent) covariates are introduced intothe model, which are estimated by the maximum likelihood method based on thefull likelihood that takes into account the time-dependence of the covariates.Results of the empirical analysis indicate that the hazard functions differsubstantially depending on the prepayment type. In addition, results indicatethat the fit of the model can be improved by the distinction of prepaymenttypes and the introduction of the market interest rates as path-dependentcovariates.

A proportional hazard model with time-dependent parameters and covariates

In this paper we study a fully parametric proportional hazards model in which the covariates as well as the parameters of the base distribution can vary with time. We consider the application of this model to the case of extreme value base-line distribution with rightcensored data and obtain the equations for the estimate by maximum likelihood from the model parameters.