Estimation Of Survival Function Research Articles

Marginal regression approach for additive hazards models with clustered current status data

Current status data arise naturally from tumorigenicity experiments, epidemiology studies, biomedicine, econometrics and demographic and sociology studies. Moreover, clustered current status data may occur with animals from the same litter in tumorigenicity experiments or with subjects from the same family in epidemiology studies. Because the only information extracted from current status data is whether the survival times are before or after the monitoring or censoring times, the nonparametric maximum likelihood estimator of survival function converges at a rate of n(1/3) to a complicated limiting distribution. Hence, semiparametric regression models such as the additive hazards model have been extended for independent current status data to derive the test statistics, whose distributions converge at a rate of n(1/2) , for testing the regression parameters. However, a straightforward application of these statistical methods to clustered current status data is not appropriate because intracluster correlation needs to be taken into account. Therefore, this paper proposes two estimating functions for estimating the parameters in the additive hazards model for clustered current status data. The comparative results from simulation studies are presented, and the application of the proposed estimating functions to one real data set is illustrated.

Použití konečných směsí pravděpodobnostních rozdělení pro modelování rozdělení doby nezaměstnanosti v České republice

Unemployment belongs to the most serious economic and social problems of developed countries. Usually, the rate of unemployment is analysed. Another problem is the duration of unemployment and especially long-term unemployment. The unemployment duration in the Czech Republic in 2010 is analysed in the paper. The model uses data from the Labour Force Sample Survey, which is performed quarterly by the Czech Statistical Office. The probability distribution of unemployment duration is modelled with the use of finite mixtures of lognormal distributions with the observable components of membership, gender and education. The observations are right and interval-censored, exact values of the unemployment duration are not included in the data. Both parametric and non-parametric Kaplan-Meier methods are used to estimate the survival function. The estimated survival functions are compared graphically and medians are evaluated for each component. A positive effect of education on the duration of unemployment is found. Also, a greater median unemployment duration is found for women than for men. All the computations are made in the R software.

Density estimation on multivariate censored data with optional Polya tree

Analyzing the failure times of multiple events is of interest in many fields. Estimating the joint distribution of the failure times in a non-parametric way is not straightforward because some failure times are often right-censored and only known to be greater than observed follow-up times. Although it has been studied, there is no universally optimal solution for this problem. It is still challenging and important to provide alternatives that may be more suitable than existing ones in specific settings. Related problems of the existing methods are not only limited to infeasible computations, but also include the lack of optimality and possible non-monotonicity of the estimated survival function. In this paper, we proposed a non-parametric Bayesian approach for directly estimating the density function of multivariate survival times, where the prior is constructed based on the optional Pólya tree. We investigated several theoretical aspects of the procedure and derived an efficient iterative algorithm for implementing the Bayesian procedure. The empirical performance of the method was examined via extensive simulation studies. Finally, we presented a detailed analysis using the proposed method on the relationship among organ recovery times in severely injured patients. From the analysis, we suggested interesting medical information that can be further pursued in clinics.

An Application of Gamma Generalized Linear Model for Estimation of Survival Function of Diabetic Nephropathy Patients

Diabetic nephropathy (DN) is a generic term referring to deleterious effect on renal structure and/or function caused by diabetes mellitus. World Health Organization estimates that diabetes affects more than 170 million people worldwide and this number may rise to 370 million by 2030. The rate of rise in Serum Creatinine (SrCr) is a well-accepted marker for the progression of Diabetic Nephropathy (DN). In this paper, survival functions of type 2 diabetic patients with renal complication are estimated. Firstly, most appropriate distribution for duration of diabetes is selected through minimum Akaike Information Criterion value, Gamma distribution is found to be an appropriate model. Secondly, the parameters estimates of the selected distribution are obtained by fitting a Generalized Linear Model (GLM), with duration of diabetes as the response variable and predictors as SrCr and number of successes (number of times SrCr values exceed its normal range (1.4 mg/dl)). These covariates are linked with the response variable using two different link functions namely log and reciprocal links. Using the estimates of parameters obtained from generalized linear regression analysis, survival functions for different durations under both the links are estimated. Further we compared the estimated survival functions under both the links with Kaplan Meier (KM) estimates graphically. Findings suggested that the Kaplan Meier estimate and Gamma distribution under both links provided a close estimate of survival functions. Median survival time is 16.3 years and 16.8 years obtained from KM method and Gamma GLM respectively.

Nonparametric estimation of the conditional survival function for bivariate failure times

AbstractThis paper discusses the nonparametric estimation of the conditional survival function for bivariate failure time data in the presence of dependent censoring. We use local polynomial smoothing techniques to obtain a simple estimator of the conditional survival function and investigate its asymptotic properties. Several censoring structures are considered. The proposal is compared to an existing alternative by simulation and illustrated with two real data sets. The Canadian Journal of Statistics 41: 439–452; 2013 © 2013 Statistical Society of Canada

Estimating the survival functions in a censored semi-competing risks model

Rivest and Wells (2001) proposed estimators of the marginal survival functions in a right-censored model that assumes an Archimedean copula between the survival time and the censoring time. We study the extension of these estimators to the context of right-censored semi-competing risks data with an independent second level censoring time. We intensively use martingale techniques to derive their large sample properties under mild assumptions on the true distribution of the data. As compared to the simpler context of right-censored data, a primary difference is the need to enlarge the filtrations with respect to which we use the Doob-Meyer decompositions of counting processes.

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.

External validation of a Cox prognostic model: principles and methods

BackgroundA prognostic model should not enter clinical practice unless it has been demonstrated that it performs a useful role. External validation denotes evaluation of model performance in a sample independent of that used to develop the model. Unlike for logistic regression models, external validation of Cox models is sparsely treated in the literature. Successful validation of a model means achieving satisfactory discrimination and calibration (prediction accuracy) in the validation sample. Validating Cox models is not straightforward because event probabilities are estimated relative to an unspecified baseline function.MethodsWe describe statistical approaches to external validation of a published Cox model according to the level of published information, specifically (1) the prognostic index only, (2) the prognostic index together with Kaplan-Meier curves for risk groups, and (3) the first two plus the baseline survival curve (the estimated survival function at the mean prognostic index across the sample). The most challenging task, requiring level 3 information, is assessing calibration, for which we suggest a method of approximating the baseline survival function.ResultsWe apply the methods to two comparable datasets in primary breast cancer, treating one as derivation and the other as validation sample. Results are presented for discrimination and calibration. We demonstrate plots of survival probabilities that can assist model evaluation.ConclusionsOur validation methods are applicable to a wide range of prognostic studies and provide researchers with a toolkit for external validation of a published Cox model.

Continuation of the Etonogestrel Implant in Women Undergoing Immediate Postabortion Placement

Long-acting reversible contraceptive (LARC) methods can prevent repeat abortion. Two effective reversible LARC methods, the intrauterine device (IUD) and subdermal implant, have the potential to significantly reduce the rate of both unintended pregnancy and abortion. Some clinicians are concerned that women receiving an immediate postabortion etonogestrel implant may have higher rates of discontinuation compared with women who have the implant not in association with a recent pregnancy (interval insertion). Previous studies have not adequately investigated continuation rates for this implant following immediate postabortion insertion. The aim of this study was to compare 12-month continuation rates among women who received an immediate postabortion etonogestrel subdermal implant with those of women who had an interval insertion. The authors performed a secondary analysis of the Contraceptive CHOICE Project, a prospective cohort study designed to evaluate continuation and satisfaction for reversible LARC methods. Kaplan-Meier estimator of survival function and Cox proportional hazard models were used to compare continuation rates. Among 1178 women who received the implant during the study period, 141 received the implant immediately postabortion, and 935 had an interval insertion. Rates of 12-month continuation were similar in both groups: 81.5% for women who underwent immediate postabortion placement and 82.8% who had an interval placement (P = 0.54). Survival analysis showed that the likelihood of discontinuation among women who received an immediate postabortion implant was the same as those who had an interval placement; the adjusted hazard ratio was 1.04, with a 95% confidence interval of 0.68 to 1.60. In both groups, frequent or irregular bleeding was the most common reason given for discontinuation of the implant: 65.4% in the immediate postabortion group and 56.7% in the interval group (P = 0.29). These findings show that women who undergo immediate postabortion placement of the subdermal implant are no more likely to discontinue using it than women who have an interval placement. All women undergoing induced abortion should be offered immediate postabortion LARC to reduce the risk of unintended pregnancy and repeat abortion.

Canine Lymphomas

Canine lymphoma is the neoplasm most often treated by chemotherapy, yet there are few data to correlate response to therapy with its different subtypes. This study is based on biopsy specimens from 992 dogs for which lymphoma was the clinical diagnosis. All cases were phenotyped by immunohistochemistry for CD3 and CD79alpha. Cases with histiocytic proliferation were evaluated immunohistochemically for CD18. Clonality was verified in 12 cases by polymerase chain reaction (PCR). Survival (event time) data and complete survival information (cause of death or time to last follow-up) were available on 456 dogs. Additional covariate information when available included size, age, sex, phenotype, stage and grade of lymphoma, mitotic index, and treatment protocol. Because of the many subtypes of B- and T-cell lymphoma, the cases were grouped into 7 diagnostic categories: (1) benign hyperplasia; (2) low-grade B-cell; (3) high-grade B- and T-cell; (4) low-grade T-cell; (5) centroblastic large B-cell of all mitotic grades (subdivided by clinical stage); (6) immunoblastic large B-cell of all mitotic grades, and (7) high-grade peripheral T-cell. Grouping was determined by histological grade (based on mitotic rate/400× field, with low-grade 0-5, intermediate 6-10, and high-grade >10) and stage for survival function estimation. No association with survival was found for size (based on breed of dog) or sex. All diagnostic categories of indolent or low-grade type had low mitotic rates, whereas those with clinically high grades had high mitotic rates. The diagnostic category with the most cases was centroblastic large B-cell lymphoma. Compared with dogs in this largest represented group of lymphomas, dogs with high-grade lymphomas had significantly higher mortality rates, and dogs with low-grade T-cell lymphomas had significantly lower mortality rates. Treatments for high-, intermediate-, and low-grade lymphomas were divided into 4 groups: absence of treatment, chemotherapy with or without hydroxydaunorubicin, and only prednisone. Dogs with low-grade T-cell (T-zone) lymphomas had the longest median survival (622 days), whereas the shortest median survival was in dogs with T-cell high-grade (peripheral T-cell) subtype (162 days). The dogs with centroblastic large B-cell lymphomas had a median survival of 127 days with low stage, 221 days with intermediate stage, and 215 days with advanced stage. Dogs with T-zone lymphoma were probably diagnosed in later stages of disease because of the lack of signs associated with progression. As with human lymphomas, a histological diagnosis with immunophenotyping is a minimal requirement for diagnosis of a specific subtype.

Combining expert knowledge and local data for improved service life modeling of water supply networks

The presented approach aims to overcome the scarce data problem in service life modeling of water networks by combining subjective expert knowledge and local replacement data. A procedure to elicit imprecise quantile estimates of survival functions from experts, considering common cognitive biases, was developed and applied. The individual expert priors of the parameters of the service life distribution are obtained by regression over the stated distribution quantiles and aggregated into a single prior distribution. Furthermore, a likelihood function for the commonly encountered censored and truncated pipe replacement data is formulated. The suitability of the suggested Bayesian approach based on elicitation data from eight experts and real network data is demonstrated. Robust parameter estimates could be derived in data situations where frequentist maximum likelihood estimation is unsatisfactory, and to show how the consideration of imprecision and in-between-variance of experts improves posterior inference.

Large sample properties for a class of copulas in bivariate survival analysis

This work is concerned with asymptotic properties of the bivariate survival function estimator using the functional relationship between marginal survival functions and a class of copulas for the dependence structure. Specifically, we study consistency and weak convergence of the bivariate survival function estimator obtained considering a two-step procedure of estimation. The obtained results are found from a key decomposition of the bivariate survival function in quantities that can be studied separately. In particular, we use relating results to almost sure and weak convergence of estimators, almost sure convergence of uniformly equicontinuous functions, and the delta method for functionals.

Ophthalmic viscoelastic device injection for the treatment of flat anterior chamber after trabeculectomy: a case series study

PurposeFlat anterior chamber (FAC) in association with overfiltration is a complication after trabeculectomy. The aim of the current study was to investigate the efficacy of injection of an ophthalmic viscoelastic device (OVD) to treat FAC after trabeculectomy in a retrospective consecutive case series study.Materials and methodsTwenty-five consecutive patients who underwent injection of OVD for the treatment of FAC after trabeculectomy were analyzed. The primary outcome variable was the qualified success of trabeculectomy, which was defined at three different levels by the achievement of one of three IOP-related criteria with or without glaucoma medication. Kaplan–Meier plots were generated to estimate survival functions for trabeculectomy with OVD injection.ResultsTwenty-one patients (21/25 = 84.0%) recovered from FAC by one OVD injection. The postoperative intraocular pressure (IOP) measured at 12 months, 24 months, 36 months, 48 months, and 60 months was 14.7 ± 3.6 mmHg, 15.6 ± 7.0 mmHg, 14.2 ± 2.8 mmHg, 14.2 ± 3.8 mmHg, and 14.6 ± 2.7 mmHg, respectively. All postoperative IOP measurements were significantly lower than the preoperative IOP (P < 0.001). The probabilities of qualified success for 5 years after trabeculectomy with OVD injection were as follows: 24.4% (IOP ≦ 21 mmHg and ≧20% reduction of preoperative IOP), 16.5% (IOP ≦ 16 mmHg and ≧20% reduction) and 6.3% (IOP ≦ 12 mmHg and ≧30% reduction). One patient, who had undergone deep lamellar keratoplasty, experienced a complication associated with the injection of the viscoelastic material, Descemet’s membrane detachment.ConclusionInjection of OVD may be effective for the treatment of FAC; however, the long-term success rate was relatively low.

A goodness of fit test for the survival function under random right censoring

The present article contributes a goodness of fit test for the survival function under random right censoring. The test is based on a central limit theorem for the Integrated Square Error of an already existing in the literature kernel survival function estimate. Establishment of its asymptotic distribution yields the proposed test statistic for drawing decision on the null hypothesis of correctness of the assumed survival function. Numerical simulations quantify the empirical nominal level and power of the suggested test for various sample sizes and amounts of censoring and facilitate comparison with the power of the data driven Neyman goodness of fit test for censored samples.

단조 서포트벡터기계를 이용한 카플란-마이어 생존함수의 평활

서포트벡터 기계는 분류 및 비선형 함수추정에서 유용하게 사용되고 있는 통계적 기법이다. 본 논문에서는 두 개의 입력변수와 회귀함수의 단조 관계를 이용하여 단조 서포트벡터기계를 제안하고, Kaplan-Meier의 방법에 의해서 생존함수의 추정값이 주어진 경우 제안된 방법을 이용하여 생존 함수를 평활하는 방법 또한 제안한다. 모의실험에서는 실제 생존함수를 이용하여 Kaplan-Meier의 방법에 의한 생존함수의 추정값과의 성능을 비교함으로써 제안된 방법의 우수성을 보이기로 한다. Support vector machine is known to be the very useful statistical method in classification and nonlinear function estimation. In this paper we propose a monotone support vector regression (SVR) for the estimation of monotonically decreasing function. The proposed monotone SVR is applied to smooth the Kaplan-Meier estimate of survival function. Experimental results are then presented which indicate the performance of the proposed monotone SVR using survival functions obtained by exponential distribution.

Consistency of the Generalized MLE with Interval-Censored and Masked Competing Risks Data

We consider nonparametric estimation based on interval-censored competing risks data with masked failure cause. The generalized maximum likelihood estimator of the joint survival function of the failure time and the failure cause is studied under mixed case interval censorship and random partition masking. Strong consistency in the L 1(μ)-topology is established for some finite measure μ which is derived from the joint censoring and masking distribution. Under additional regularity assumptions we also establish the strong consistencies in the topologies of weak convergence, point-wise convergence, and uniform convergence.

Survival Function Estimates for Champions Tour Golfers

In this article, the authors present survival function estimates for elite professional golfers who have participated on the Champions professional golf tour. Kaplan–Meier survivor function estimates for these golfers show median survival of 89 years and mean life expectancy of 85 years. The authors find comparable results for winners of the U. S. Professional Golf Association (PGA) Senior Open championship, which is also restricted to players 50 years of age and over. Champions tour golfer life expectancies are high in comparison to the general male population of the United States and at the high end of the life expectancy advantages estimated for other elite athletes.

The Impact of Solitary and Multiple Positive Surgical Margins on Hard Clinical End Points in 1712 Adjuvant Treatment–Naive pT2–4 N0 Radical Prostatectomy Patients

BackgroundPositive surgical margins (PSMs) increase the risk of biochemical recurrence (BCR) after radical prostatectomy (RP), but their impact on hard clinical end points is a topic of ongoing discussion. ObjectiveTo evaluate the influence of solitary PSMs (sPSMs) and multiple PSMs (mPSMs) on important clinical end points. Design, setting, and participantsData from 1712 patients from the Centre Hospitalier Universitaire de Québec with pT2–4 N0 prostate cancer (PCa) and undetectable prostate-specific antigen after RP were analyzed. InterventionRP without neoadjuvant or adjuvant treatment. Outcome measurements and statistical analysisKaplan-Meier analysis estimated survival functions, and Cox proportional hazards models addressed predictors of clinical end points. Results and limitationsMedian follow-up was 74.9 mo. A total of 1121 patients (65.5%) were margin-negative, 281 patients (16.4%) had sPSMs, and 310 patients (18.1%) had mPSMs. A total of 280 patients (16.4%) experienced BCR, and 197 patients (11.5%) were treated with salvage radiotherapy (SRT). Sixty-eight patients (4.0%) received definitive androgen deprivation therapy, 19 patients (1.1%) developed metastatic disease, and 15 patients (0.9%) had castration-resistant PCa (CRPC). Thirteen patients (0.8%) died from PCa, and 194 patients (11.3%) died from other causes. Ten-year Kaplan-Meier estimates for BCR-free survival were 82% for margin-negative patients, 72% for patients with sPSMs, and 59% for patients with mPSMs (p<0.0001). Time to metastatic disease, CRPC, PCa-specific mortality (PCSM), or all-cause mortality did not differ significantly among the three groups (p=0.991, p=0.988, p=0.889, and p=0.218, respectively). On multivariable analysis, sPSMs and mPSMs were associated with BCR (hazard ratio [HR]: 1.711; p=0.001 and HR: 2.075; p<0.0001), but sPSMs and mPSMs could not predict metastatic disease (p=0.705 and p=0.242), CRPC (p=0.705 and p=0.224), PCSM (p=0.972 and p=0.260), or all-cause death (p=0.102 and p=0.067). The major limitation was the retrospective design. ConclusionsIn a cohort of patients who received early SRT in 70% of cases upon BCR, sPSMs and mPSMs predicted BCR but not long-term clinical end points. Adjuvant radiotherapy for margin-positive patients might not be justified, as only a minority of patients progressed to end points other than BCR. PCSM was exceeded 15-fold by competing risk mortality.

Estimating Propensity Scores and Causal Survival Functions Using Prevalent Survival Data

This article develops semiparametric approaches for estimation of propensity scores and causal survival functions from prevalent survival data. The analytical problem arises when the prevalent sampling is adopted for collecting failure times and, as a result, the covariates are incompletely observed due to their association with failure time. The proposed procedure for estimating propensity scores shares interesting features similar to the likelihood formulation in case-control study, but in our case it requires additional consideration in the intercept term. The result shows that the corrected propensity scores in logistic regression setting can be obtained through standard estimation procedure with specific adjustments on the intercept term. For causal estimation, two different types of missing sources are encountered in our model: one can be explained by potential outcome framework; the other is caused by the prevalent sampling scheme. Statistical analysis without adjusting bias from both sources of missingness will lead to biased results in causal inference. The proposed methods were partly motivated by and applied to the Surveillance, Epidemiology, and End Results (SEER)-Medicare linked data for women diagnosed with breast cancer.