A proportional hazards model for arbitrarily censored and truncated data

Turnbull (1976, Journal of Royal Statistical Society, Series B 38, 290-295) proposed a method for nonparametric estimation of the distribution function when the data are incomplete because of censoring and truncation. However, as noted by Frydman (1994, Journal of Royal Statistical society, Series B 56, 71-74), Turnbull's method has to be modified to accommodate both truncation and censoring. This paper presents a detailed correction of Turnbull's method and an extension to the regression analysis: a method of fitting the proportional hazards model for arbitrarily censored and truncated data is developed. The method allows partial testing for zero regression coefficients. The test can be performed using the likelihood ratio test or the Wald test. The methodology is applied to estimate the distribution of the induction time of patients diagnosed with transfusion-associated AIDS and to estimate the distribution of time from diabetes onset to development of diabetic nephropathy for insulin-dependent diabetics.     

Tests of Race Models for Reaction Time in Experiments with Asynchronous Redundant Signals

We propose an extended hazard regression model which allows the spread parameter to be dependent on covariates. This allows a broad class of models which includes the most common hazard models, such as the proportional hazards model, the accelerated failure time model and a proportional hazards/accelerated failure time hybrid model with constant spread parameter. Simulations based on sub-classes of this model suggest that maximum likelihood performs well even when only small or moderate-size data sets are available and the censoring pattern is heavy. The methodology provides a broad framework for analysis of reliability and survival data. Two numerical examples illustrate the results.     

Environmental influences in hand preference: an African point of view

In the analysis of survival data from clinical trials and other studies, the censoring generally precludes estimation of the mean survival time. To accommodate censoring, Irwin (1949) proposed, as an alternative, estimation of the mean lifetime restricted to a suitably chosen time T. In this article we consider the use of Irwin's restricted mean as an index for comparing survival in different groups, using as an example published data from a randomized clinical trial in patients with primary biliary cirrhosis. Irwin's method, originally based on the actuarial survival estimator, is extended to incorporate covariates into the analysis through the use of piecewise exponential models. For comparing two survival curves, the logrank test is known to be optimal under proportional hazards alternatives. However, comparison of restricted means may outperform the logrank test in situations involving nonproportional hazard rates. We examine the size and power of these two procedures under various proportional and nonproportional hazards alternatives, with and without covariate adjustment. For survival curves that separate early in time the censored data generalization of the Wilcoxon test is known to exhibit high power, and we examine how the comparison of restricted means performs relative to this procedure also.     

Urn sampling and the proportional hazard model

This paper investigates the urn sampling analogue for the score statistic relating survival to covariates assuming a proportional hazard model. The exact permutation distribution can be calculated as well as the exact low order moments for arbitrary censoring patterns. The asymptotic distribution of the score statistic is an easy consequence. The method is naturally extended to deal with the multivariate case, time varying covariates and interval censoring. Finally the relationship between the censoring process, the survival times and covariates are studied considering different reference sets for the distribution of the score statistic. Some assumptions about the censoring process are investigated and as a consequence the effect of censoring is clarified.     

Bayesian variable selection method for censored survival data

A Bayesian variable selection method for censored data is proposed in this paper. Based on the sufficiency and asymptotic normality of the maximum partial likelihood estimator, we approximate the posterior distribution of the parameters in a proportional hazards model. We consider a parsimonious model as the full model with some covariates unobserved and replaced by their conditional expected values. A loss function based on the posterior expected estimation error of the log-risk for the proportional hazards model is used to select a parsimonious model. We derive computational expressions for this loss function for both continuous and binary covariates. This approach provides an extension of Lindley's (1968, Journal of the Royal Statistical Society, Series B 30, 31-66) variable selection criterion for the linear case. Data from a randomized clinical trial of patients with primary biliary cirrhosis of the liver (PBC) (Fleming and Harrington, 1991, Counting Processes and Survival Analysis) is used to illustrate the proposed method and a simulation study compares it with the backward elimination procedure.     

An empirical comparison of statistical tests for assessing the proportional hazards assumption of Cox's model

In the analysis of survival data using the Cox proportional hazard (PH) model, it is important to verify that the explanatory variables analysed satisfy the proportional hazard assumption of the model. This paper presents results of a simulation study that compares five test statistics to check the proportional hazard assumption of Cox's model. The test statistics were evaluated under proportional hazards and the following types of departures from the proportional hazard assumption: increasing relative hazards; decreasing relative hazards; crossing hazards; diverging hazards, and non-monotonic hazards. The test statistics compared include those based on partitioning of failure time and those that do not require partitioning of failure time. The simulation results demonstrate that the time-dependent covariate test, the weighted residuals score test and the linear correlation test have equally good power for detection of non-proportionality in the varieties of non-proportional hazards studied. Using illustrative data from the literature, these test statistics performed similarly.     

Large sample Bayesian inference on the parameters of the proportional hazard models

This paper considers large sample Bayesian analysis of the proportional hazards model when interest is in inference on the parameters and estimation of the log relative risk for specified covariate vectors rather than on prediction of the survival function. We use a normal prior distribution for the parameters and make inferences based on the derived posterior distribution. The suggested approach is much simpler than alternative Bayesian analyses previously suggested for the proportional hazards models. Using simulated data we compare estimates obtained from the Bayesian analysis with those obtained from the full proportional hazards model and the reduced model after backwards elimination. We show that under a wider range of assumptions, the Bayesian analysis provides reduced estimation errors and improved rejection of noise variables. Finally, we illustrate the methodology using data from a large study of prognostic markers in breast cancer.     

Evaluation of confounding effects in ROC studies

In many clinical studies, it is clear that external forces can affect the performance of diagnostic tests, as these factors influence the distributions of separator variables. A new estimator for the receiver operating characteristic (ROC) function is proposed; this estimator converges to the ROC function uniformly on the interval [0,1]. Using this new estimator, the author proposes to use Cox's proportional hazards regression model for the evaluation of confounding effects in ROC studies. The method can be used even when concomitant information is only available for the cases, for example, disease severity. A textbook example on prostate cancer is described for illustration.     

Prognostic factors for mortality in nasopharyngeal cancer: accounting for time-dependence of relative risks

BACKGROUND: Few studies have assessed the role of prognostic factors for mortality from nasopharyngeal cancer and even fewer used multivariable methods. Most of these studies relied on the Cox model without testing the proportional hazards assumption. METHODS: A cohort of 76 cases of nasopharyngeal cancer recorded in the Rh?ne, France, between 1980 and 1985, was followed until 1995. Proportional hazards assumption was tested for each putative prognostic factor. Two multivariable models were built using forward selection of prognostic factors: the Cox model and a flexible model in which variables not meeting the proportional hazards assumption were represented by a time-varying hazard ratio. RESULTS: Only Epstein Barr Virus Nuclear Antigen (EBNA) serology, a marker of infection by the Epstein-Barr virus, and tumoral extent were selected in the analysis based on the Cox model. In contrast, four prognostic factors were significant at alpha = 0.05 level in the flexible model: initial EBNA serology, tumoral histology, age and tumoral extent, the last two not verifying the proportional hazards assumption. The relative risk of age increases with duration of follow-up whereas the effect of tumoral extent changes in a non-monotonic pattern. CONCLUSION: We showed the importance of taking into account the non-proportionality of hazards which can influence results and yield new insights about the role of prognostic factors in nasopharyngeal cancer. Because of the small size of our cohort, our results have to be confirmed in an independent study.     

Preoperative predictors of cost in Medicare-age patients undergoing coronary artery bypass grafting

BACKGROUND: Identification of preoperative factors that contribute to the cost of coronary artery bypass grafting could aid in predicting the procedure's expense. In this study, 30 sociodemographic and clinical preoperative factors were examined with "survival analysis" techniques to determine characteristics related to total hospital cost. METHODS: Characteristics of all patients age 65 or older undergoing isolated coronary artery bypass grafting from July 1993 to April 1995 (n = 757) were recorded. Software was developed within the hospital's Transitions Systems, Inc, database to calculate the outcome variable of total cost. Nonparametric methods were used for the univariate analysis of the data, and the Cox proportional hazards model was used for the multivariable analysis, censoring 25 patients who died in the hospital. RESULTS: Median hospital cost from the day of the operation until discharge was $15,198. Median length of stay after the operation was 6 days. Multivariable analysis revealed that age, preoperative renal failure, history of cerebrovascular accident, low ejection fraction, and surgical urgency were independent predictors of total cost. CONCLUSIONS: This study, using an accurate representation of true hospital cost and a modeling technique that accounts for the confounding effect of in-hospital death on cost, provides a template for analysis of cost in other patient groups.     

Assessing gamma frailty models for clustered failure time data

Proportional hazards frailty models use a random effect, so called frailty, to construct association for clustered failure time data. It is customary to assume that the random frailty follows a gamma distribution. In this paper, we propose a graphical method for assessing adequacy of the proportional hazards frailty models. In particular, we focus on the assessment of the gamma distribution assumption for the frailties. We calculate the average of the posterior expected frailties at several followup time points and compare it at these time points to 1, the known mean frailty. Large discrepancies indicate lack of fit. To aid in assessing the goodness of fit, we derive and estimate the standard error of the mean of the posterior expected frailties at each time point examined. We give an example to illustrate the proposed methodology and perform sensitivity analysis by simulations.     

Level A in vivo-in vitro correlation: nonlinear models and statistical methodology

Some new nonlinear models for the relationship between the fraction of drug dose dissolved (absorbed) in vivo and that dissolved in vitro are described. The models are empirical in nature and are generalizations of the linear model that, at present, is the most commonly used model. The modeling approach is based on considering the time at which a drug molecule goes into solution (in vitro or in vivo) to be a random variable and relating the distribution functions using proportional odds, proportional hazards, and proportional reversed hazards models. The models are further extended by allowing the parameter that relates in vivo and in vitro to be a function of time. A statistical model for the data is developed and used as the basis for a statistical methodology for fitting these models. The methods are shown to be generalized linear mixed effects model (GLMM) methods. The models are fitted to some data sets, and the results demonstrate that these models have potential.     

Analysis of failure times for multiple infections following bone marrow transplantation: an application of the multiple failure time proportional hazards model

We report an application of the proportional hazards model for multiple failure times in a study arising from the Bone Marrow Transplant Database at the University of Minnesota. The study compared the risk of infections after transplantation for patients who received allogeneic bone marrow transplants from unrelated donors (URD) versus related donors (RD). In 249 patients there was a total of 365 infections in 2.5 years of follow-up. The multiple failure time model uses all the data and provides empirical estimates of standard errors that incorporate the within-person dependencies in the data. The estimate of relative risk associated with URD was 1.4 (naive 95 per cent confidence interval 1.14 to 1.73, empirical 1.08 to 1.79), compared to the estimate 1.6 (naive or empirical, 1.1 to 2.1) from the proportional hazards model on 165 first infections only. The multivariate model gives great flexibility in modelling, for example, in accommodating a separate base hazard function for each type of failure and in allowing analysis of intervals between infections as an alternative to analysis of time from a marker event, here transplantation.     

A specification test for univariate and multivariate proportional hazards models

This paper applies White's (1982, Econometrica 50, 1-25) information matrix (IM) test for correct model specification to proportional hazards models of univariate and multivariate censored survival data. Several alternative estimators of the test statistic are presented and their size performance examined. White also suggested an estimator of the parameter covariance matrix that was robust to certain forms of model misspecification. This has been subsequently proposed by others (e.g., Royall, 1986, International Statistical Review 54, 221-226) and applied by Huster, Brookmeyer, and Self (1989, Biometrics 45, 145-156) as part of an independence working model (IWM) approach to multivariate censored survival data. We illustrate how the IM test can be used for both univariate data and as part of the IWM approach to multivariate data.     

Flexible Bayesian modelling for survival data

The analysis of failure time data often involves two strong assumptions. The proportional hazards assumption postulates that hazard rates corresponding to different levels of explanatory variables are proportional. The additive effects assumption specifies that the effect associated with a particular explanatory variable does not depend on the levels of other explanatory variables. A hierarchical Bayes model is presented, under which both assumptions are relaxed. In particular, time-dependent covariate effects are explicitly modelled, and the additivity of effects is relaxed through the use of a modified neural network structure. The hierarchical nature of the model is useful in that it parsimoniously penalizes violations of the two assumptions, with the strength of the penalty being determined by the data.     

Neural networks as a prognostic tool for patients with non-small cell carcinoma of the lung

Patients with non-small cell carcinoma of the lung (NSCLC) have a poor prognosis (64 and 41% survival rates in Stages I and II). It is currently not possible to predict which patients with Stage I or II NSCLC will survive the disease. Sixty-seven patients with NSCLC, including 49 patients with Stage I NSCLC and 18 with Stage II disease (11 with squamous cell carcinomas, 35 with adenocarcinomas, and 21 with large cell carcinomas) were treated with lobectomy and followed for a minimum of 5 years. The tumors were studied with DNA flow cytometry and quantitative immunocytochemical studies for proliferation cell nuclear antigen, p53 protein, and MIB-1. The data were analyzed with backpropagation neural networks, univariate analysis of variance, the Kaplan-Meier survival method, and Cox proportional hazards model. The dependent variables were "free of disease" and "recurrence or dead from disease." Twenty neural network models were trained, using all cases but one, after 1883 to 2000 training cycles. At 5 years, 30 patients were free of disease and 37 were dead or had recurrence. Proliferating cell nuclear antigen was the only statistically significant prognostic factor by univariate analysis of variance and Cox proportional hazards analysis. The S phase was statistically significant by univariate analysis of variance (P <.05). All of the 20 models classified the test cases correctly. Study with backpropagation neural networks using multiple prognostic features from patients with NSCLC suggests that this technology might be useful for prediction of survival. This preliminary study must be validated with data from a larger group of patients with NSCLC before its clinical adequacy is established.     

Analysis of behavioral durations in observational studies of social interaction.

Many social science disciplines investigate dyadic or triadic social interaction by recording the interactive behaviors as qualitative codes. This article introduces a method for analyzing and interpreting variation in the durations of coded behaviors obtained during such interactions. Using survival analysis and event history analysis as its basis, this article discusses the use of regression techniques that enable the researcher to assess how the distributions of behavioral durations depend on covariate observations. An analysis of conflicts between mothers and children illustrates the discussion, using the proportional hazards model. Behavioral and attitudinal differences among the mother–son dyad predicted variation in the durations of mothers' negative statements to their sons. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Increasing efficiency from censored survival data by using random effects to model longitudinal covariates

When estimating a survival time distribution, the loss of information due to right censoring results in a loss of efficiency in the estimator. In many circumstances, however, repeated measurements on a longitudinal process which is associated with survival time are made throughout the observation time, and these measurements may be used to recover information lost to censoring. For example, patients in an AIDS clinical trial may be measured at regular intervals on CD4 count and viral load. We describe a model for the joint distribution of a survival time and a repeated measures process. The joint distribution is specified by linking the survival time to subject-specific random effects characterizing the repeated measures, and is similar in form to the pattern mixture model for multivariate data with nonignorable nonresponse. We also describe an estimator of survival derived from this model. We apply the methods to a long-term AIDS clinical trial, and study properties of the survival estimator. Monte Carlo simulation is used to estimate gains in efficiency when the survival time is related to the location and scale of the random effects distribution. Under relatively light censoring (20%), the methods yield a modest gain in efficiency for estimating three-year survival in the AIDS clinical trial. Our simulation study, which mimics characteristics of the clinical trial, indicates that much larger gains in efficiency can be realized under heavier censoring or with studies designed for long term follow up on survival.     

Identification and characterization of a new splicing variant of vascular endothelial growth factor: VEGF183

The purpose of this study was to determine implant survival rates by means of life table analyses for a cohort of patients not part of a prospective efficacy trial and treated by practitioners at varying experience levels. Prognostic variables associated with implant failure were identified by means of proportional hazards models and advanced statistical methods that account for patient effects. Ninety-nine consecutive patients treated from 1987 to 1991 with follow-up to 1994 were included in this retrospective study. A total of 384 dental implants (79.7% Br?nemark, 19.3% IMZ plasma-sprayed, 1% IMZ hydroxyapatite-coated) were placed and subsequently supported 108 prostheses. Survival and proportional hazards modeling were used to generate Kaplan-Meier survival curves and to identify variables associated with implant failure. Survey data analysis was used to adjust for any patient effects for variables identified as significant through the proportional hazards models. Thirty-four implants failed over the follow-up period (median follow-up time 3.6 years), resulting in an overall failure rate of 8.9%. Seventeen of 99 patients experienced an implant failure. When prosthesis type was excluded from the modeling process, survey data analysis identified posterior location and an implant width of less than 4.0 mm as being associated with implant failure (all P < .05).     

低温精馏空分产品能耗分摊的确定与计算

建立了一种氧-氮-氩三元物系的物性计算模型和空分精馏系统计算模型，并应用所建立的物性与空分精馏系统计算模型，分别计算了生产氧或氮单高纯产品、生产氧、氮双高纯产品和生产氧、氮、氩三高纯产品空分装置的能耗。在此基础上，提出了一种新的考虑主导产品地位的空分产品能耗分摊比例计算方法，解决了国内外长期以来由于氧、氮、氩互为副产品而带来的单个产品能耗无法确定的问题。此方法消除了以往计算产品能耗时人为规定能耗分摊比例的缺点，为空分产品的能耗和成本确定与分析提供了客观的依据。