Nelson-Aalen Estimator Research Articles

Helicopter blade reliability: Statistical data analysis and modeling

The concept of reliability has been attracting attentions in mechanical engineering following the developments of the aerospace industries. Limited failure data and statistical analyses of helicopter components reliability exist in the technical literature. For filling this gap, a nonparametric analysis is conducted on the performance of the 338 blades of some Iranian helicopters, which were in service between 1974 and 2012. These blades have 41 different failure modes. In this paper, statistical reliability analysis is conducted based on two strategies: In strategy I, general failure is defined as scrapping or retirement of the blade. In strategy II, the blade is assumed to be subjected to different modes of failure and the cumulative mode-specific functions are derived for each failure modes using Nelson–Aalen estimator. The Kaplan–Meier estimator is used for calculating the nonparametric reliability functions. Confidence intervals are derived for the reliability results and parametric fits are conducted using the maximum likelihood estimation. An important result from parametric analysis is that the blade reliability has a 3-parameter Weibull distribution and so the blades exhibit an increasing failure rate. Finally, considering the mode-specific hazard functions, the failure mode 1, i.e., excessive vibration is observed to have major contribution to the blade failures.

Improved versions of greenwood estimators under koziol-green model

ABSTRACTIt is well known that the Greenwood estimators underestimate the variances of the Nelson-Aalen estimator and the Kaplan-Meier estimator. In this article, we reveal some “improved” versions of the Greenwood estimators under the Koziol-Green model.

Bayesian Competing Risks Model: An Application to Breast Cancer Clinical Trial with Incomplete Observations

Presences of Lost to follow up observations of patient in any clinical trial is natural and difficult to handle. Conclusion about the effective therapeutic region becomes conflicted in presence of lost-to follow up observations particularly in death-event data. It becomes more tedious to find out the best therapy under consideration. To make certain about compatible treatment, it is important to handle the data into proper and suitable manner. Using data from randomized clinical trial for patients with node negative breast cancer, we illustrated the approach to handle lost to follow up observation. The primary events are considered as death/alive with equal priority to the censored observations. Nelson-Aalen estimator and Bayesian approach are considered to handle the lost to follow up in the data set. Therapeutic differences on survival event were observed among the treatment under consideration. Competing risks model is found suitable choice to deal with death-event data in presences of lost-to follow up observations. Bayesian with non-informative priors and Nelson-Aalen estimator as distribution free approach are compatible to deal with death-event data.

Information bounds for nonparametric estimators of L-functionals and survival functionals under censored data

In the present paper we derive lower asymptotic information bounds of Cramer-Rao type for estimators of nonparametric statistical functionals. The results are based on dense differentiability and dense regularity concepts which lead to weak assumptions. As explicit examples L-estimators are treated. In addition a new rapid method for the treatment of survival functionals under randomly right censored data is presented. For instance, for the famous Kaplan-Meier and Nelson-Aalen estimators, our information bound is just the lower bound obtained earlier in the literature.

Estimating excess hazard ratios and net survival when covariate data are missing: strategies for multiple imputation.

Net survival is the survival probability we would observe if the disease under study were the only cause of death. When estimated from routinely collected population-based cancer registry data, this indicator is a key metric for cancer control. Unfortunately, such data typically contain a non-negligible proportion of missing values on important prognostic factors (eg, tumor stage). We carried out an empirical study to compare the performance of complete records analysis and several multiple imputation strategies when net survival is estimated via a flexible parametric proportional hazards model that includes stage, a partially observed categorical covariate. Starting from fully observed cancer registry data, we induced missingness on stage under three scenarios. For each of these scenarios, we simulated 100 incomplete datasets and evaluated the performance of the different strategies. Ordinal logistic models are not suitable for the imputation of tumor stage. Complete records analysis may lead to grossly misleading estimates of net survival, even when the missing data mechanism is conditionally independent of survival time given the covariates and the bias on the excess hazard ratios estimates is negligible. As key covariates are unlikely missing completely at random, studies estimating net survival should not use complete records. When the missingness can be inferred from available data, appropriate multiple imputation should be performed. In the context of flexible parametric proportional hazards models with a partially observed stage covariate, a multinomial logistic imputation model for stage should be used and should include the Nelson-Aalen cumulative hazard estimate and the event indicator.

On nonparametric hazard estimation.

The Nelson-Aalen estimator provides the basis for the ubiquitous Kaplan-Meier estimator, and therefore is an essential tool for nonparametric survival analysis. This article reviews martingale theory and its role in demonstrating that the Nelson-Aalen estimator is uniformly consistent for estimating the cumulative hazard function for right-censored continuous time-to-failure data.

Relation of Major Depression to Survival After Coronary Artery Bypass Grafting

The primary aim was to study the association between preoperative depression and long-term survival after coronary artery bypass grafting (CABG). Our secondary objective was to analyze the association between depression and cardiovascular events or all-cause mortality. In a nationwide, population-based, cohort study, all patients who underwent CABG in Sweden from 1997 to 2008 were included from the Swedish Web-system for Enhancement and Development of Evidence-based care in Heart disease Evaluated According to Recommended Therapies registry. Individual-level data were cross-linked from other national Swedish registers. Depression status and outcomes were obtained from the National Patient Register. The study population was 56,064 patients who underwent primary, isolated, nonemergent CABG. We identified 324 patients (0.6%) with depression before CABG. During a mean follow-up of 7.5years, 114 patients (35%) with depression died, compared with 13,767 patients (25%) in the control group. Depression was significantly associated with increased mortality and the combined end point of death or rehospitalization for myocardial infarction, heart failure, or stroke (multivariate-adjusted hazard ratios [95% confidence intervals] 1.65 [1.37 to 1.99] and 1.61 [1.38 to 1.89], respectively). In conclusion, we found a strong and significant association between depression and long-term survival in patients with established ischemic heart disease who underwent CABG. Depression was also associated with an increased risk for a combination of death or rehospitalization for heart failure, myocardial infarction, or stroke.

Logrank-type tests with presmoothing.

The logrank test is the most popular choice for testing the equality of two survival distributions with time-to-event data. It is based on the comparison of the Nelson-Aalen estimates of the corresponding cumulative hazard functions. The improvements of the logrank test in the literature have been accomplished using weighted logrank tests, with the weight chosen appropriately to maximize power. Notwithstanding, power also depends on the efficiency of the estimation of the cumulative hazard function. The presmoothed estimator has been shown to be more efficient than the Nelson-Aalen estimator under some general assumptions. We introduce a new logrank-type test that, instead of the Nelson-Aalen estimator, is based on its presmoothed counterpart. An extensive simulation study has been conducted to compare the performance of this new test with the classical one. This study shows that the new test has the proper size under the null hypothesis, while improving the power over a wide range of alternatives. The new test is illustrated with several real data examples.

Robust methods to improve efficiency and reduce bias in estimating survival curves in randomized clinical trials

In randomized clinical trials, improving efficiency and reducing bias due to chance imbalance in covariates among groups are always of considerable interest. The two purposes are often achieved by some type of covariate adjustment. In trials involving time-to-an-event, Kaplan-Meier and Nelson-Aalen estimators are the most popular nonparametric estimation of survival curves. However, these methods do not permit direct covariate adjustment, missing the important chance of improving efficiency and reducing bias. In this article, we propose robust, covariate adjusted analogues of the Nelson-Aalen and Kaplan-Meier estimators. The method is robust in that it does not require any additional modeling assumptions and hence the resulting estimators are again nonparametric. The robustness is achieved by taking advantage of the study design, i.e., treatments are randomized. Large-sample properties of the proposed estimators are developed, which show that the improvement in efficiency is guaranteed asymptotically. Simulation studies using reasonably small sample sizes further demonstrate the efficiency gain and the ability to reduce or remove bias resulted from chance imbalance to a large degree, e.g., more than 10-fold reduction in bias is achieved. Efficiency improvement and bias reduction are also illustrated by application to a cancer clinical trial. The proposed methods may help to resolve the tension between the need to make best use of data and the unwillingness to make additional assumptions in analyzing data from clinical trials.

Clinical state transitions during advanced life support (ALS) in in-hospital cardiac arrest

BackgroundWhen providing advanced life support (ALS) in cardiac arrest, the patient may alternate between four clinical states: ventricular fibrillation/tachycardia (VF/VT), pulseless electrical activity (PEA), asystole, and return of spontaneous circulation (ROSC). At the end of the resuscitation efforts, either death has been declared or sustained ROSC has been obtained. The aim of this study was to describe and analyze the clinical state transitions during ALS among patients experiencing in-hospital cardiac arrest. Methods and resultsThe defibrillator files from 311 in-hospital cardiac arrests at the University of Chicago Hospital (IL, USA) and St. Olav University Hospital (Trondheim, Norway) were analyzed (clinicaltrials.gov: NCT00920244). The transitions between clinical states were annotated along the time axis and visualized as plots of the state prevalence according to time. The cumulative intensity of the state transitions was estimated by the Nelson–Aalen estimator for each type of state transition, and for the intensities of overall state transitions. Between 70% and 90% of patients who eventually obtained sustained ROSC had progressed to ROSC by approximately 15–20min of ALS, depending on the initial rhythm. Patients behaving unstably after this time period, i.e., alternating between ROSC, VF/VT and PEA, had a high risk of ultimately being declared dead. ConclusionsWe provide an overall picture of the intensities and patterns of clinical state transitions during in-hospital ALS. The majority of patients who obtained sustained ROSC obtained this state and stabilized within the first 15–20min of ALS. Those who continued to behave unstably after this time point had a high risk of ultimately being declared dead.

A nonparametric approach to medical survival data: Uncertainty in the context of risk in mortality analysis

Medical survival right-censored data of about 850 patients are evaluated to analyze the uncertainty related to the risk of mortality on one hand and compare two basic surgery techniques in the context of risk of mortality on the other hand. Colorectal data come from patients who underwent colectomy in the University Hospital of Ostrava. Two basic surgery operating techniques are used for the colectomy: either traditional (open) or minimally invasive (laparoscopic). Basic question arising at the colectomy operation is, which type of operation to choose to guarantee longer overall survival time.Two non-parametric approaches have been used to quantify probability of mortality with uncertainties. In fact, complement of the probability to one, i.e. survival function with corresponding confidence levels is calculated and evaluated. First approach considers standard nonparametric estimators resulting from both the Kaplan–Meier estimator of survival function in connection with Greenwood's formula and the Nelson–Aalen estimator of cumulative hazard function including confidence interval for survival function as well. The second innovative approach, represented by Nonparametric Predictive Inference (NPI), uses lower and upper probabilities for quantifying uncertainty and provides a model of predictive survival function instead of the population survival function.The traditional log-rank test on one hand and the nonparametric predictive comparison of two groups of lifetime data on the other hand have been compared to evaluate risk of mortality in the context of mentioned surgery techniques. The size of the difference between two groups of lifetime data has been considered and analyzed as well.Both nonparametric approaches led to the same conclusion, that the minimally invasive operating technique guarantees the patient significantly longer survival time in comparison with the traditional operating technique.

Matching methods for obtaining survival functions to estimate the effect of a time-dependent treatment.

In observational studies of survival time featuring a binary time-dependent treatment, the hazard ratio (an instantaneous measure) is often used to represent the treatment effect. However, investigators are often more interested in the difference in survival functions. We propose semiparametric methods to estimate the causal effect of treatment among the treated with respect to survival probability. The objective is to compare post-treatment survival with the survival function that would have been observed in the absence of treatment. For each patient, we compute a prognostic score (based on the pre-treatment death hazard) and a propensity score (based on the treatment hazard). Each treated patient is then matched with an alive, uncensored and not-yet-treated patient with similar prognostic and/or propensity scores. The experience of each treated and matched patient is weighted using a variant of Inverse Probability of Censoring Weighting to account for the impact of censoring. We propose estimators of the treatment-specific survival functions (and their difference), computed through weighted Nelson-Aalen estimators. Closed-form variance estimators are proposed which take into consideration the potential replication of subjects across matched sets. The proposed methods are evaluated through simulation, then applied to estimate the effect of kidney transplantation on survival among end-stage renal disease patients using data from a national organ failure registry.

996 NATION-WIDE ANALYSIS OF SURVIVAL AFTER RADICAL PROSTATECTOMY FOR CLINICALLY LOCALIZED PROSTATE CANCER IN DENMARK 1995-2011

INTRODUCTION AND OBJECTIVES: PSA-screening for prostate cancer (PCa) has never been recommended in Denmark. Radical prostatectomy (RP) for clinically localized PCa was introduced in 1995. The objective of study was to describe survival after RP in a nationwide cohort of Danish PCa patients. METHODS: Prospective observational study of 6489 men with localized PCa treated with RP at 6 different hospitals in Denmark between 1995 and 2011. Survival was described using Kaplan-Meier estimates. Causes of death were obtained in national registry and cross-checked with patient files. Cumulative incidence of death; any cause and PCa-specific was described using Nelson-Aalen estimates. Risk for PCa death was analyzed in a Cox multivariate regression model using covariates age, cT-category, PSA and biopsy Gleason score (GS). RESULTS: Median follow-up was 4 years. During follow-up, 328 patients died, 109 (33.2%) of PCa and 219 (66.8%) of other causes. Six patients (0.09%) died within 30 days of surgery. In multivariate analysis, cT-category was a predictor of PCa death (p .001). Compared to T1 disease, both cT2c (HR 2.2) and cT3 (HR 7.2) significantly increased the risk of PCa death. For every doubling of PSA the risk of PCa death was increased by 34.8% (p 0.001). Biopsy GS 4 3 and 8 were associated with an increased risk of PCa death compared to biopsy GS 6 of 2.3 and 2.7 (p 0.003), respectively. The cumulative incidence of any-cause mortality after 10 years was 15.4% (95% CI:13.2-17.7) and 6.6% (95% CI:4.9-8.2) for PCa mortality. CONCLUSIONS: Our results seem comparable to high-volume, single institution, single surgeon series u even in a country where PCa are primarily clinically detected and patients are managed by small volume, multi-surgeon institutions. The true benefit of RP in localized PCa compared other treatment strategies remains controversial.

Survival after radical prostatectomy for clinically localised prostate cancer: a population‐based study

To describe survival and cause of death in a nationwide cohort of Danish patients with prostate cancer undergoing radical prostatectomy (RP). To describe risk factors associated with prostate cancer mortality. Observational study of 6489 men with localised prostate cancer treated with RP at six different hospitals in Denmark between 1995 and 2011. Survival was described using Kaplan-Meier estimates. Causes of death were obtained from the national registry and cross-checked with patient files. Cumulative incidence of death, any cause and prostate cancer-specific, was described using Nelson-Aalen estimates. Risk for prostate cancer death was analysed in a Cox multivariate regression model using the covariates: age, cT-category, PSA level and biopsy Gleason score. The median follow-up was 4 years. During follow-up, 328 patients died, 109 (33.2%) from prostate cancer and 219 (66.8%) from other causes. Six patients (0.09%) died ≤30 days of RP. In multivariate analysis, cT-category was a predictor of prostate cancer death (P < 0.001). Compared with T1 disease, both cT2c (hazard ratio [HR] 2.2) and cT3 (HR 7.2) significantly increased the risk of prostate cancer death. For every doubling of PSA level the risk of prostate cancer death was increased by 34.8% (P < 0.001). Biopsy Gleason score 4 + 3 and ≥8 were associated with an increased risk of prostate cancer death compared with biopsy Gleason score ≤ 6 of 2.3 and 2.7 (P = 0.003), respectively. The cumulative hazard of all-cause and prostate cancer-specific mortality after 10 years was 15.4% (95% confide3nce interval [CI] 13.2-17.7) and 6.6% (95% CI 4.9-8.2) respectively. We present the first survival analysis of a complete, nationwide cohort of men undergoing RP for localised prostate cancer. The main limitation of the study was the relatively short follow-up. Interestingly, our national results are comparable to high-volume, single institution, single surgeon series.

Estimating the Number of Double-Strand Breaks Formed During Meiosis from Partial Observation

Analyzing the basic mechanism of DNA double-strand breaks (DSB) formation during meiosis is important for understanding sexual reproduction and genetic diversity. The location and amount of meiotic DSBs can be examined by using a common molecular biological technique called Southern blotting, but only a subset of the total DSBs can be observed; only DSB fragments still carrying the region recognized by a Southern blot probe are detected. With the assumption that DSB formation follows a nonhomogeneous Poisson process, we propose two estimators of the total number of DSBs on a chromosome: (1) an estimator based on the Nelson-Aalen estimator, and (2) an estimator based on a record value process. Further, we compared their asymptotic accuracy.

Non-Parametric Survival Analysis of Infectious Disease Data

SummaryThe paper develops non-parametric methods based on contact intervals for the analysis of infectious disease data. The contact interval from person i to person j is the time between the onset of infectiousness in i and infectious contact from i to j, where we define infectious contact as a contact sufficient to infect a susceptible individual. The hazard function of the contact interval distribution equals the hazard of infectious contact from i to j, so it provides a summary of the evolution of infectiousness over time. When who infects whom is observed, the Nelson–Aalen estimator produces an unbiased estimate of the cumulative hazard function of the contact interval distribution. When who infects whom is not observed, we use an expectation–maximization algorithm to average the Nelson–Aalen estimates from all possible combinations of who infected whom consistent with the observed data. This converges to a non-parametric maximum likelihood estimate of the cumulative hazard function that we call the marginal Nelson–Aalen estimate. We study the behaviour of these methods in simulations and use them to analyse household surveillance data from the 2009 influenza A(H1N1) pandemic.

Estimation of a decreasing hazard of patients with acute coronary syndrome

The Kaplan-Meier, Nelson-Aalen and Breslow estimators are widely used in the analysis of right-censored time to event data in medical applications. These methods are fully non-parametric and do not put any restriction on the shape of the hazard curve. In some applications, this leads to implausible estimates of the hazard course over time. With non-parametric shape-constrained estimation techniques, one can facilitate an increasing or decreasing hazard and thus generate estimators that better match the biological reasoning, without being as restrictive as parametric methods. We illustrate the advantage of such techniques in the analysis of a large clinical trial in cardiology. Simulation results show that in case the true hazard is monotone, the non-parametric shape-constrained estimators are more accurate than the traditional estimators on the hazard level. On the (cumulative) distribution function level, the shape-constrained estimators show similar performance as the traditional ones.

Review of Case-Mix Corrected Survival Curves

Survival is an end point of immense interest in cardiothoracic research. In observational studies, the comparison of survival between groups of patients is usually accomplished using a toolbox that includes Kaplan-Meier survival curves and the Cox model. The Cox model yields comparisons between groups adjusted for case-mix differences, whereas the Kaplan-Meier is a plot of survival over time without adjustment. During the past decade, new methods have emerged for case-mix adjustment of survival curves and are increasingly being used in cardiothoracic research. The purpose of this report is to describe, illustrate, and review several approaches to case-mix adjusted survival (or event-free) curves.

Adjusted Hazard Rate Estimator Based on a Known Censoring Probability

In most reliability studies involving censoring, one assumes that censoring probabilities are unknown. We derive a nonparametric estimator for the survival function when information regarding censoring frequency is available. The estimator is constructed by adjusting the Nelson–Aalen estimator to incorporate censoring information. Our results indicate significant improvements can be achieved if available information regarding censoring is used. We compare this model to the Koziol–Green model, which is also based on a form of proportional hazards for the lifetime and censoring distributions. Two examples of survival data help to illustrate the differences in the estimation techniques.

On the errors committed by sequences of estimator functionals

Consider a sequence of estimators $\hat \theta _n $ which converges almost surely to θ 0 as the sample size n tends to infinity. Under weak smoothness conditions, we identify the asymptotic limit of the last time $\hat \theta _n $ is further than ɛ away from θ 0 when ɛ → 0+. These limits lead to the construction of sequentially fixed width confidence regions for which we find analytic approximations. The smoothness conditions we impose is that $\hat \theta _n $ is to be close to a Hadamard-differentiable functional of the empirical distribution, an assumption valid for a large class of widely used statistical estimators. Similar results were derived in Hjort and Fenstad (1992) for the case of Euclidean parameter spaces; part of the present contribution is to lift these results to situations involving parameter functionals. The apparatus we develop is also used to derive appropriate limit distributions of other quantities related to the far tail of an almost surely convergent sequence of estimators, like the number of times the estimator is more than ɛ away from its target. We illustrate our results by giving a new sequential simultaneous confidence set for the cumulative hazard function based on the Nelson-Aalen estimator and investigate a problem in stochastic programming related to computational complexity.