Comparison of a traditional and a multilevel Cox Proportional Hazards model

Hierarchically clustered populations are often encountered in public health research, but the traditional methods used in analyzing this type of data are not always adequate. In the case of survival time data, more appropriate methods have only begun to surface in the last couple of decades. Such methods include multilevel statistical techniques which, although more complicated to implement than traditional methods, are more appropriate. ^ One population that is known to exhibit a hierarchical structure is that of patients who utilize the health care system of the Department of Veterans Affairs where patients are grouped not only by hospital, but also by geographic network (VISN). This project analyzes survival time data sets housed at the Houston Veterans Affairs Medical Center Research Department using two different Cox Proportional Hazards regression models, a traditional model and a multilevel model. VISNs that exhibit significantly higher or lower survival rates than the rest are identified separately for each model. ^ In this particular case, although there are differences in the results of the two models, it is not enough to warrant using the more complex multilevel technique. This is shown by the small estimates of variance associated with levels two and three in the multilevel Cox analysis. Much of the differences that are exhibited in identification of VISNs with high or low survival rates is attributable to computer hardware difficulties rather than to any significant improvements in the model. ^

Choosing between Cox proportional hazards and logistic models for interval-censored data via bootstrap

This work develops a new methodology in order to discriminate models for interval-censored data based on bootstrap residual simulation by observing the deviance difference from one model in relation to another, according to Hinde (1992). Generally, this sort of data can generate a large number of tied observations and, in this case, survival time can be regarded as discrete. Therefore, the Cox proportional hazards model for grouped data (Prentice & Gloeckler, 1978) and the logistic model (Lawless, 1982) can befitted by means of generalized linear models. Whitehead (1989) considered censoring to be an indicative variable with a binomial distribution and fitted the Cox proportional hazards model using complementary log-log as a link function. In addition, a logistic model can be fitted using logit as a link function. The proposed methodology arises as an alternative to the score tests developed by Colosimo et al. (2000), where such models can be obtained for discrete binary data as particular cases from the Aranda-Ordaz distribution asymmetric family. These tests are thus developed with a basis on link functions to generate such a fit. The example that motivates this study was the dataset from an experiment carried out on a flax cultivar planted on four substrata susceptible to the pathogen Fusarium oxysoprum. The response variable, which is the time until blighting, was observed in intervals during 52 days. The results were compared with the model fit and the AIC values.

Assessing time-by-covariate interactions in Cox proportional hazards regression models using cubic spline functions

This dissertation develops and explores the methodology for the use of cubic spline functions in assessing time-by-covariate interactions in Cox proportional hazards regression models. These interactions indicate violations of the proportional hazards assumption of the Cox model. Use of cubic spline functions allows for the investigation of the shape of a possible covariate time-dependence without having to specify a particular functional form. Cubic spline functions yield both a graphical method and a formal test for the proportional hazards assumption as well as a test of the nonlinearity of the time-by-covariate interaction. Five existing methods for assessing violations of the proportional hazards assumption are reviewed and applied along with cubic splines to three well known two-sample datasets. An additional dataset with three covariates is used to explore the use of cubic spline functions in a more general setting. ^

Concordance Probability and Discriminatory Power in Proportional Hazards Regression

The concordance probability is used to evaluate the discriminatory power and the predictive accuracy of nonlinear statistical models. We derive an analytic expression for the concordance probability in the Cox proportional hazards model. The proposed estimator is a function of the regression parameters and the covariate distribution only and does not use the observed event and censoring times. For this reason it is asymptotically unbiased, unlike Harrell's c-index based on informative pairs. The asymptotic distribution of the concordance probability estimate is derived using U-statistic theory and the methodology is applied to a predictive model in lung cancer.

Optimal and Robust Designs of Step-stress Accelerated Life Testing Experiments for Proportional Hazards Models

Accelerated life testing (ALT) is widely used to obtain reliability information about a product within a limited time frame. The Cox s proportional hazards (PH) model is often utilized for reliability prediction. My master thesis research focuses on designing accelerated life testing experiments for reliability estimation. We consider multiple step-stress ALT plans with censoring. The optimal stress levels and times of changing the stress levels are investigated. We discuss the optimal designs under three optimality criteria. They are D-, A- and Q-optimal designs. We note that the classical designs are optimal only if the model assumed is correct. Due to the nature of prediction made from ALT experimental data, attained under the stress levels higher than the normal condition, extrapolation is encountered. In such case, the assumed model cannot be tested. Therefore, for possible imprecision in the assumed PH model, the method of construction for robust designs is also explored.

Meta-analysis of individual patient data from randomized trials of chemotherapy plus cetuximab as first-line treatment for advanced non-small cell lung cancer

OBJECTIVES: Four randomized phase II/III trials investigated the addition of cetuximab to platinum-based, first-line chemotherapy in patients with advanced non-small cell lung cancer (NSCLC). A meta-analysis was performed to examine the benefit/risk ratio for the addition of cetuximab to chemotherapy. MATERIALS AND METHODS: The meta-analysis included individual patient efficacy data from 2018 patients and individual patient safety data from 1970 patients comprising respectively the combined intention-to-treat and safety populations of the four trials. The effect of adding cetuximab to chemotherapy was measured by hazard ratios (HRs) obtained using a Cox proportional hazards model and odds ratios calculated by logistic regression. Survival rates at 1 year were calculated. All applied models were stratified by trial. Tests on heterogeneity of treatment effects across the trials and sensitivity analyses were performed for all endpoints. RESULTS: The meta-analysis demonstrated that the addition of cetuximab to chemotherapy significantly improved overall survival (HR 0.88, p=0.009, median 10.3 vs 9.4 months), progression-free survival (HR 0.90, p=0.045, median 4.7 vs 4.5 months) and response (odds ratio 1.46, p<0.001, overall response rate 32.2% vs 24.4%) compared with chemotherapy alone. The safety profile of chemotherapy plus cetuximab in the meta-analysis population was confirmed as manageable. Neither trials nor patient subgroups defined by key baseline characteristics showed significant heterogeneity for any endpoint. CONCLUSION: The addition of cetuximab to platinum-based, first-line chemotherapy for advanced NSCLC significantly improved outcome for all efficacy endpoints with an acceptable safety profile, indicating a favorable benefit/risk ratio.

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.

The association of chronic kidney disease and dialysis treatment with foot ulceration and major amputation

Objective The objective of this study was to investigate the risk of chronic kidney disease (CKD) stage 4-5 and dialysis treatment on incidence of foot ulceration and major lower extremity amputation in comparison to CKD stage 3. Methods In this retrospective study, all individuals who visited our hospital between 2006 and 2012 because of CKD stages 3 to 5 or dialysis treatment were included. Medical records were reviewed for incidence of foot ulceration and major amputation. The time from CKD 3, CKD 4-5, and dialysis treatment until first foot ulceration and first major lower extremity amputation was calculated and analyzed by Kaplan-Meier curves and multivariate Cox proportional hazards model. Diabetes mellitus, peripheral arterial disease, peripheral neuropathy, and foot deformities were included for potential confounding. Results A total of 669 individuals were included: 539 in CKD 3, 540 in CKD 4-5, and 259 in dialysis treatment (individuals could progress from one group to the next). Unadjusted foot ulcer incidence rates per 1000 patients per year were 12 for CKD 3, 47 for CKD 4-5, and 104 for dialysis (P < .001). In multivariate analyses, the hazard ratio for incidence of foot ulceration was 4.0 (95% confidence interval [CI], 2.6-6.3) in CKD 4-5 and 7.6 (95% CI, 4.8-12.1) in dialysis treatment compared with CKD 3. Hazard ratios for incidence of major amputation were 9.5 (95% CI, 2.1-43.0) and 15 (95% CI, 3.3-71.0), respectively. Conclusions CKD 4-5 and dialysis treatment are independent risk factors for foot ulceration and major amputation compared with CKD 3. Maximum effort is needed in daily clinical practice to prevent foot ulcers and their devastating consequences in all individuals with CKD 4-5 or dialysis treatment.

Imputação múltipla de dados faltantes: exemplo de aplicação no Estudo Pró-Saúde

Dados faltantes são um problema comum em estudos epidemiológicos e, dependendo da forma como ocorrem, as estimativas dos parâmetros de interesse podem estar enviesadas. A literatura aponta algumas técnicas para se lidar com a questão, e, a imputação múltipla vem recebendo destaque nos últimos anos. Esta dissertação apresenta os resultados da utilização da imputação múltipla de dados no contexto do Estudo Pró-Saúde, um estudo longitudinal entre funcionários técnico-administrativos de uma universidade no Rio de Janeiro. No primeiro estudo, após simulação da ocorrência de dados faltantes, imputou-se a variável cor/raça das participantes, e aplicou-se um modelo de análise de sobrevivência previamente estabelecido, tendo como desfecho a história auto-relatada de miomas uterinos. Houve replicação do procedimento (100 vezes) para se determinar a distribuição dos coeficientes e erros-padrão das estimativas da variável de interesse. Apesar da natureza transversal dos dados aqui utilizados (informações da linha de base do Estudo Pró-Saúde, coletadas em 1999 e 2001), buscou-se resgatar a história do seguimento das participantes por meio de seus relatos, criando uma situação na qual a utilização do modelo de riscos proporcionais de Cox era possível. Nos cenários avaliados, a imputação demonstrou resultados satisfatórios, inclusive quando da avaliação de performance realizada. A técnica demonstrou um bom desempenho quando o mecanismo de ocorrência dos dados faltantes era do tipo MAR (Missing At Random) e o percentual de não-resposta era de 10%. Ao se imputar os dados e combinar as estimativas obtidas nos 10 bancos (m=10) gerados, o viés das estimativas era de 0,0011 para a categoria preta e 0,0015 para pardas, corroborando a eficiência da imputação neste cenário. Demais configurações também apresentaram resultados semelhantes. No segundo artigo, desenvolve-se um tutorial para aplicação da imputação múltipla em estudos epidemiológicos, que deverá facilitar a utilização da técnica por pesquisadores brasileiros ainda não familiarizados com o procedimento. São apresentados os passos básicos e decisões necessárias para se imputar um banco de dados, e um dos cenários utilizados no primeiro estudo é apresentado como exemplo de aplicação da técnica. Todas as análises foram conduzidas no programa estatístico R, versão 2.15 e os scripts utilizados são apresentados ao final do texto.

Assessing length of stay and associated characteristics of geriatric patients in Northern Ireland

The effective provision of care for the elderly is becoming increasingly more difficult. This is due to the rising proportion of elderly in the population, increasing demands placed on the health services and the financial strain placed on an already stretched economy. The research presented in this paper uses three different models to represent the length of stay distribution of geriatric patients admitted to one of the six key acute hospitals in Northern Ireland and various patient characteristics associated with their respective length of stay. The accurate modelling of bed usage within wards would enable hospital managers to prepare patient discharge packages and rehabilitation services in advance. The models presented within the paper include a Cox proportional hazards model, a Bayesian network with a discrete variable to represent length of stay and a special conditional phase-type model (C-Ph) with a connecting outcome node. This research demonstrates the new efficient fitting algorithm employed for Coxian phase-type distributions while updating C-Ph models for recent elderly patient data.

Incidence, determinants and outcome of chronic kidney disease after adult heart transplantation in the United Kingdom

Background: We investigated the incidence of chronic kidney disease (CKD) in the United Kingdom heart transplant population, identified risk factors for the development of CKD, and assessed the impact of CKD on subsequent survival.

Methods: Data from the UK Cardiothoracic Transplant Audit and UK Renal Registry were linked for 1732 adult heart transplantations, 1996 to 2007. Factors influencing time to CKD, defined as National Kidney Foundation CKD stage 4 or 5 or preemptive kidney transplantation, were identified using a Cox proportional hazards model. The effects of distinct CKD stages on survival were evaluated using time-dependent covariates.

Results: A total of 3% of patients had CKD at transplantation, 11% at 1-year and more than 15% at 6 years posttransplantation and beyond. Earlier transplantations, shorter ischemia times, female, older, hepatitis C virus positive, and diabetic recipients were at increased risk of developing CKD, along with those with impaired renal function pretransplantation or early posttransplantation. Significant differences between transplantation centers were also observed. The risk of death was significantly higher for patients at CKD stage 4, stage 5 (excluding dialysis), or on dialysis, compared with equivalent patients surviving to the same time point with CKD stage 3 or lower (hazard ratios of 1.66, 8.54, and 4.07, respectively).

Conclusions: CKD is a common complication of heart transplantation in the UK, and several risk factors identified in other studies are also relevant in this population. By linking national heart transplantation and renal data, we have determined the impact of CKD stage and dialysis treatment on subsequent survival in heart transplant recipients.

Influence of sex on the survival of patients with esophageal cancer

PURPOSE: The prognostic value of sex for esophageal cancer survival is currently unclear, and growing data suggest that hormonal influences may account for incidence disparities between men and women. Therefore, moving from the hypothesis that hormones could affect the prognosis of patients with esophageal cancer, we investigated the primary hypothesis that sex is associated with survival and the secondary hypotheses that the relationship between sex and survival depends, at least in part, on age, histology, and race/ethnicity.

PATIENTS AND METHODS: By using the SEER databases from 1973 to 2007, we identified 13,603 patients (34%) with metastatic esophageal cancer (MEC) and 26,848 patients (66%) with locoregional esophageal cancer (LEC). Cox proportional hazards model for competing risks were used for analyses.

RESULTS: In the multivariate analysis, women had longer esophageal cancer-specific survival (ECSS) than men in both MEC (hazard ratio [HR], 0.949; 95% CI, 0.905 to 0.995; P = .029) and LEC (HR, 0.920; 95% CI, 0.886 to 0.955; P < .001) cohorts. When age and histology were accounted for, there was no difference for ECSS between men and women with adenocarcinoma. In contrast, women younger than age 55 years (HR, 0.896; 95% CI, 0.792 to 1.014; P = .081) and those age 55 years or older (HR, 0.905; 95% CI, 0.862 to 0.950; P < .001) with squamous cell LEC had longer ECSS than men. In the squamous cell MEC cohort, only women younger than age 55 years had longer ECSS (HR, 0.823; 95% CI, 0.708 to 0.957; P = .011) than men.

CONCLUSION: Sex is an independent prognostic factor for patients with LEC or MEC. As secondary hypotheses, in comparison with men, women age 55 years or older with squamous cell LEC and women younger than age 55 years with squamous cell MEC have a significantly better outcome. These last two findings need further validation.

New statistical methods to assess the effect of time-dependent exposures in case-control studies

Contexte. Les études cas-témoins sont très fréquemment utilisées par les épidémiologistes pour évaluer l’impact de certaines expositions sur une maladie particulière. Ces expositions peuvent être représentées par plusieurs variables dépendant du temps, et de nouvelles méthodes sont nécessaires pour estimer de manière précise leurs effets. En effet, la régression logistique qui est la méthode conventionnelle pour analyser les données cas-témoins ne tient pas directement compte des changements de valeurs des covariables au cours du temps. Par opposition, les méthodes d’analyse des données de survie telles que le modèle de Cox à risques instantanés proportionnels peuvent directement incorporer des covariables dépendant du temps représentant les histoires individuelles d’exposition. Cependant, cela nécessite de manipuler les ensembles de sujets à risque avec précaution à cause du sur-échantillonnage des cas, en comparaison avec les témoins, dans les études cas-témoins. Comme montré dans une étude de simulation précédente, la définition optimale des ensembles de sujets à risque pour l’analyse des données cas-témoins reste encore à être élucidée, et à être étudiée dans le cas des variables dépendant du temps. Objectif: L’objectif général est de proposer et d’étudier de nouvelles versions du modèle de Cox pour estimer l’impact d’expositions variant dans le temps dans les études cas-témoins, et de les appliquer à des données réelles cas-témoins sur le cancer du poumon et le tabac. Méthodes. J’ai identifié de nouvelles définitions d’ensemble de sujets à risque, potentiellement optimales (le Weighted Cox model and le Simple weighted Cox model), dans lesquelles différentes pondérations ont été affectées aux cas et aux témoins, afin de refléter les proportions de cas et de non cas dans la population source. Les propriétés des estimateurs des effets d’exposition ont été étudiées par simulation. Différents aspects d’exposition ont été générés (intensité, durée, valeur cumulée d’exposition). Les données cas-témoins générées ont été ensuite analysées avec différentes versions du modèle de Cox, incluant les définitions anciennes et nouvelles des ensembles de sujets à risque, ainsi qu’avec la régression logistique conventionnelle, à des fins de comparaison. Les différents modèles de régression ont ensuite été appliqués sur des données réelles cas-témoins sur le cancer du poumon. Les estimations des effets de différentes variables de tabac, obtenues avec les différentes méthodes, ont été comparées entre elles, et comparées aux résultats des simulations. Résultats. Les résultats des simulations montrent que les estimations des nouveaux modèles de Cox pondérés proposés, surtout celles du Weighted Cox model, sont bien moins biaisées que les estimations des modèles de Cox existants qui incluent ou excluent simplement les futurs cas de chaque ensemble de sujets à risque. De plus, les estimations du Weighted Cox model étaient légèrement, mais systématiquement, moins biaisées que celles de la régression logistique. L’application aux données réelles montre de plus grandes différences entre les estimations de la régression logistique et des modèles de Cox pondérés, pour quelques variables de tabac dépendant du temps. Conclusions. Les résultats suggèrent que le nouveau modèle de Cox pondéré propose pourrait être une alternative intéressante au modèle de régression logistique, pour estimer les effets d’expositions dépendant du temps dans les études cas-témoins

Regression models for bivariate survival data

Multivariate lifetime data arise in various forms including recurrent event data when individuals are followed to observe the sequence of occurrences of a certain type of event; correlated lifetime when an individual is followed for the occurrence of two or more types of events, or when distinct individuals have dependent event times. In most studies there are covariates such as treatments, group indicators, individual characteristics, or environmental conditions, whose relationship to lifetime is of interest. This leads to a consideration of regression models.The well known Cox proportional hazards model and its variations, using the marginal hazard functions employed for the analysis of multivariate survival data in literature are not sufficient to explain the complete dependence structure of pair of lifetimes on the covariate vector. Motivated by this, in Chapter 2, we introduced a bivariate proportional hazards model using vector hazard function of Johnson and Kotz (1975), in which the covariates under study have different effect on two components of the vector hazard function. The proposed model is useful in real life situations to study the dependence structure of pair of lifetimes on the covariate vector . The well known partial likelihood approach is used for the estimation of parameter vectors. We then introduced a bivariate proportional hazards model for gap times of recurrent events in Chapter 3. The model incorporates both marginal and joint dependence of the distribution of gap times on the covariate vector . In many fields of application, mean residual life function is considered superior concept than the hazard function. Motivated by this, in Chapter 4, we considered a new semi-parametric model, bivariate proportional mean residual life time model, to assess the relationship between mean residual life and covariates for gap time of recurrent events. The counting process approach is used for the inference procedures of the gap time of recurrent events. In many survival studies, the distribution of lifetime may depend on the distribution of censoring time. In Chapter 5, we introduced a proportional hazards model for duration times and developed inference procedures under dependent (informative) censoring. In Chapter 6, we introduced a bivariate proportional hazards model for competing risks data under right censoring. The asymptotic properties of the estimators of the parameters of different models developed in previous chapters, were studied. The proposed models were applied to various real life situations.