Bayesian semiparametric proportional odds models

Methodology for implementing the proportional odds regression model for survival data assuming a mixture of finite Polya trees (MPT) prior on baseline survival is presented. Extensions to frailties and generalized odds rates are discussed. Although all manner of censoring and truncation can be accommodated, we discuss model implementation, regression diagnostics, and model comparison for right-censored data. An advantage of the MPT model is the relative ease with which predictive densities, survival, and hazard curves are generated. Much discussion is devoted to practical implementation of the proposed models, and a novel MCMC algorithm based on an approximating parametric normal model is developed. A modest simulation study comparing the small sample behavior of the MPT model to a rank-based estimator and a real data example is presented.     

Kinetics and Mechanism of Bacterial Disinfection by Chlorine Dioxide

Survival data are presented for a fecal strain of Escherichia coli exposed to three concentrations of chlorine dioxide at four temperatures. Chick's first-order reaction equation is generalized to a pseudo nth-order model. Nonlinear least squares curve-fitting of the survival data to the nth order model was performed on an analogue computer. The data were observed to follow fractional order kinetics with respect to survival concentration, with an apparent activation energy of 12,000 cal/mole. Initial experiments support the thesis that the mechanism of chlorine dioxide kill occurs via disruption of protein synthesis.     

Joint models for multivariate longitudinal and multivariate survival data

Joint modeling of longitudinal and survival data is becoming increasingly essential in most cancer and AIDS clinical trials. We propose a likelihood approach to extend both longitudinal and survival components to be multidimensional. A multivariate mixed effects model is presented to explicitly capture two different sources of dependence among longitudinal measures over time as well as dependence between different variables. For the survival component of the joint model, we introduce a shared frailty, which is assumed to have a positive stable distribution, to induce correlation between failure times. The proposed marginal univariate survival model, which accommodates both zero and nonzero cure fractions for the time to event, is then applied to each marginal survival function. The proposed multivariate survival model has a proportional hazards structure for the population hazard, conditionally as well as marginally, when the baseline covariates are specified through a specific mechanism. In addition, the model is capable of dealing with survival functions with different cure rate structures. The methodology is specifically applied to the International Breast Cancer Study Group (IBCSG) trial to investigate the relationship between quality of life, disease-free survival, and overall survival.     

A Model for Censored Survival Data When There is an Intervening Event

A model is discussed for incorporating information from a time-dependent covariable (an intervening event) and covariables independent of time into the analysis of survival data. In the model, it is assumed that individuals are potentially subject to two paths to failure, one including the intervening event and the other not. Additional assumptions are that failure times associated with the two paths are independent and that the time to failure subsequent to the intervening event is dependent on the intervening event time. Allowing the underlying hazard rates for the model to follow a WEIBULL form, use of the model and methods for fitting and hypothesis testing are illustrated by application to a follow-up study involving industrial workers where disability retirement was the intervening event. Extensions of the model to accommodate grouped survival data are presented.     

Regression analysis of multivariate interval-censored failure time data with application to tumorigenicity experiments

This paper discusses multivariate interval-censored failure time data that occur when there exist several correlated survival times of interest and only interval-censored data are available for each survival time. Such data occur in many fields. One is tumorigenicity experiments, which usually concern different types of tumors, tumors occurring in different locations of animals, or together. For regression analysis of such data, we develop a marginal inference approach using the additive hazards model and apply it to a set of bivariate interval-censored data arising from a tumorigenicity experiment. Simulation studies are conducted for the evaluation of the presented approach and suggest that the approach performs well for practical situations.     

Comparison of methods for analysis of selective genotyping survival data

Survival traits and selective genotyping datasets are typically not normally distributed, thus common models used to identify QTL may not be statistically appropriate for their analysis. The objective of the present study was to compare models for identification of QTL associated with survival traits, in particular when combined with selective genotyping. Data were simulated to model the survival distribution of a population of chickens challenged with Marek disease virus. Cox proportional hazards (CPH), linear regression (LR), and Weibull models were compared for their appropriateness to analyze the data, ability to identify associations of marker alleles with survival, and estimation of effects when all individuals were genotyped (full genotyping) and when selective genotyping was used. Little difference in power was found between the CPH and the LR model for low censoring cases for both full and selective genotyping. The simulated data were not transformed to follow a Weibull distribution and, as a result, the Weibull model generally resulted in less power than the other two models and overestimated effects. Effect estimates from LR and CPH were unbiased when all individuals were genotyped, but overestimated when selective genotyping was used. Thus, LR is preferred for analyzing survival data when the amount of censoring is low because of ease of implementation and interpretation. Including phenotypic data of non-genotyped individuals in selective genotyping analysis increased power, but resulted in LR having an inflated false positive rate, and therefore the CPH model is preferred for this scenario, although transformation of the data may also make the Weibull model appropriate for this case. The results from the research presented herein are directly applicable to interval mapping analyses.     

Efficient analysis of Weibull survival data from experiments on heterogeneous patient populations

An efficient method is presented for analyses of death rated in one-way or cross-classified experiments where expected survival time for a patient at time of entry on trial is a function of observable covariates. The survival-time distribution used is a Weibull form of Cox's (1972) model. The analysis proceeds in two steps. In the first, goodness of fit of the model is checked, inefficient estimates of the parameters are obtained, and survival times adjusted for the entry covariates are calculated. In the second, efficient estimates and tests for the rate parameters are obtained. These can easily be calculated using hand or desk equipment. Reorganized data sets can be analyzed without repetition of step one, thereby reducing the computational load to hand level and facilitating exploratory data analysis.     

Bayesian approaches to joint cure-rate and longitudinal models with applications to cancer vaccine trials

Complex issues arise when investigating the association between longitudinal immunologic measures and time to an event, such as time to relapse, in cancer vaccine trials. Unlike many clinical trials, we may encounter patients who are cured and no longer susceptible to the time-to-event endpoint. If there are cured patients in the population, there is a plateau in the survival function, S(t), after sufficient follow-up. If we want to determine the association between the longitudinal measure and the time-to-event in the presence of cure, existing methods for jointly modeling longitudinal and survival data would be inappropriate, since they do not account for the plateau in the survival function. The nature of the longitudinal data in cancer vaccine trials is also unique, as many patients may not exhibit an immune response to vaccination at varying time points throughout the trial. We present a new joint model for longitudinal and survival data that accounts both for the possibility that a subject is cured and for the unique nature of the longitudinal data. An example is presented from a cancer vaccine clinical trial.     

Analysis of failure time data with multilevel clustering, with application to the child vitamin a intervention trial in Nepal

We consider the problem of estimating covariate effects in the marginal Cox proportional hazard model and multilevel associations for child mortality data collected from a vitamin A supplementation trial in Nepal, where the data are clustered within households and villages. For this purpose, a class of multivariate survival models that can be represented by a functional of marginal survival functions and accounts for hierarchical structure of clustering is exploited. Based on this class of models, an estimation strategy involving a within-cluster resampling procedure is proposed, and a model assessment approach is presented. The asymptotic theory for the proposed estimators and lack-of-fit test is established. The simulation study shows that the estimates are approximately unbiased, and the proposed test statistic is conservative under extremely heavy censoring but approaches the size otherwise. The analysis of the Nepal study data shows that the association of mortality is much greater within households than within villages.     

A formula for the mean lifetime of metapopulations in heterogeneous landscapes

We present a formula for the mean lifetime of metapopulations in heterogeneous landscapes. This formula provides new insights into the effect of the spatial structure of habitat networks on metapopulation survival, with consequences for modeling, landscape evaluation, and metapopulation management. In the whole study, the spatially realistic metapopulation model of Frank and Wissel is taken as a basis. First, as a key result on the way toward the desired formula, it is shown that a simple nonspatial (Levins-type) model is able to reproduce the behavior of the complex spatial model considered regarding the mean lifetime, provided its parameters appropriately summarize all the relevant details of spatial heterogeneity. Second, the formula presented reveals how data from species and landscape have to be combined to estimate the survival chance of a metapopulation without having to run any simulation or to solve numerically any model equation. Third, by taking the formula as a basis, landscape measures are derived that allow dissimilar habitat networks to be evaluated, compared, and ranked in terms of their effect on metapopulation survival. Fourth, a combination of analytical, nonlinear regression as well as aggregation techniques was used to deduce the formula presented. The potential of these techniques for simplifying (meta)population models that are complex due to spatial heterogeneity is discussed.     

A Bayesian semiparametric joint hierarchical model for longitudinal and survival data

This article proposes a new semiparametric Bayesian hierarchical model for the joint modeling of longitudinal and survival data. We relax the distributional assumptions for the longitudinal model using Dirichlet process priors on the parameters defining the longitudinal model. The resulting posterior distribution of the longitudinal parameters is free of parametric constraints, resulting in more robust estimates. This type of approach is becoming increasingly essential in many applications, such as HIV and cancer vaccine trials, where patients' responses are highly diverse and may not be easily modeled with known distributions. An example will be presented from a clinical trial of a cancer vaccine where the survival outcome is time to recurrence of a tumor. Immunologic measures believed to be predictive of tumor recurrence were taken repeatedly during follow-up. We will present an analysis of this data using our new semiparametric Bayesian hierarchical joint modeling methodology to determine the association of these longitudinal immunologic measures with time to tumor recurrence.     

Bayesian modeling of age-specific survival in bird nesting studies under irregular visits

In this article, a Bayesian model for age-specific nest survival rates is presented to handle the irregular visit case. Both informative priors and noninformative priors are investigated. The reference prior under this model is derived, and, therefore, the hyperparameter specification problem is solved to some extent. The Bayesian method provides a more accurate estimate of the total survival rate than the standard Mayfield method, if the age-specific hazard rates are not constant. The Bayesian method also lets the biologist look for high- and low-survival rates during the whole nesting period. In practice, it is common for data of several types to be collected in a single study. That is, some nests may be aged, others are not. Some nests are visited regularly; others are visited irregularly. The Bayesian method accommodates any mix of these sampling techniques by assuming that the aging and visiting activities have no effect on the survival rate. The methods are illustrated by an analysis of the Missouri northern bobwhite data set.     

Effect of covariate omission in Weibull accelerated failure time model: A caution

The accelerated failure time model is presented as an alternative to the proportional hazard model in the analysis of survival data. We investigate the effect of covariates omission in the case of applying a Weibull accelerated failure time model. In an uncensored setting, the asymptotic bias of the treatment effect is theoretically zero when important covariates are omitted; however, the asymptotic variance estimator of the treatment effect could be biased and then the size of the Wald test for the treatment effect is likely to exceed the nominal level. In some cases, the test size could be more than twice the nominal level. In a simulation study, in both censored and uncensored settings, Type I error for the test of the treatment effect was likely inflated when the prognostic covariates are omitted. This work remarks the careless use of the accelerated failure time model. We recommend the use of the robust sandwich variance estimator in order to avoid the inflation of the Type I error in the accelerated failure time model, although the robust variance is not commonly used in the survival data analyses.     

Modelling count and growth data with many zeros

We discuss the problem of modelling survival/mortality and growth data that are skewed with excess zeros. This type of data is a common occurrence in biological and environmental studies. The method presented here allows us to utilize both the survival/mortality and growth data when both data sets contain a large proportion of zeros. The method consists of four stages. Firstly the original data is divided into two sets; one contains all the surviving organisms and the other all of the mortalities. Secondly we calculate the actual growth of the surviving organisms and of the mortalities. Thirdly we count the number of surviving organisms for which growth has occurred and the number where no growth occurred, and the same count procedure is carried out on the mortalities. Next we model the survival/mortality data and growth/no growth data using logistic regression, and separately model the growth data using an ordinary regression. Finally we combine the three models to estimate the expected growth for a specific set of values of the explanatory variables. If we used another statistical method that did not involve the dead mussels or the ones with no growth, some of the information provided by these mussels would be lost. However, using the method we propose, all of the data collected are used to achieve an optimal estimation of the mussel growth. A case study of survival and growth of blue mussels (Mytilus galloprovincialis) and ribbed mussels (Aulacomya atra maoriana) trans-located from their natural distribution to different depths and sites along the axis of Doubtful Sound, New Zealand, is used for illustration.     

Analysis of mutant frequency curves and survival curves applied to the AL hybrid cell system

A model is presented for the statistical analysis of survival curves and mutant frequency curves for a hybrid cell system. The derivation of the model is given in the Appendix, and depends on simple assumptions about the distribution of insults, their repair, and the loss of a marker that is not rescued. A single formula (5) is found which relates a survival curve to the mutant frequency curve, i. e., the response curve for production of mutants per 10(5) survivors induced by a mutagen. The analysis is applied to loss of the a1 gene in AL-J1 hybrid cells submitted to Cesium gamma-rays. Previous experimental data using X-rays was reported by Waldren et al. (1986: Proc. Natl. Acad. Sci, USA 83, 4839.) Also, a derived formula (10), which predicts the probability that in a surviving cell a marker is lost and not rescued, will form the basis for testing the validity of the model in the future using new experimental data.     

Modeling survival of experimentally virus-infected laboratory animals--exploration of the survival diagram

In a recent paper (Sühnel & Veckenstedt, 1989, J. theor. Biol. 137, 27) we have proposed a new method of plotting survival data from experimentally virus-infected laboratory animals; the survival diagram. In this diagram two experiments, for which the mean number of virions inoculated is kept fixed but other parameters may vary, are compared. The variations in two basic quantities of survival analysis are simultaneously displayed: the standard mean survival time and the relative mean challenge virus dose, which is via a dose-response relation interrelated with the fraction of animals dying. It is analyzed in which manner variations in the kinetic parameters and the critical virus level necessary to produce a particular effect influence the location of the points of comparison in the survival diagram. The analysis presented is a prerequisite for further applications of this diagram and of the underlying mathematical model.     

Estimating survival of the shortfin mako Isurus oxyrinchus (Rafinesque) in the north-west Atlantic from tag-recapture data

Survival was estimated for shortfin mako Isurus oxyrinchus in the north-west Atlantic from tag-recapture data. The data used in this study were collected by the National Marine Fisheries Service Cooperative Shark Tagging Programme from 1962 to 2003. In total, 6309 shortfin mako sharks were tagged, of which 730 were recaptured. The high recapture rate of 11·6% for this species provided adequate recapture data to carry out survival analyses. Estimates of survival were generated with the computer software MARK, which provided a means for estimating parameters from tagged animals when they were recaptured. The results of several models are presented with various combinations of constant and time-specific survival and recovery rates. A parametric bootstrap and the median variance inflation factor ( ) approach were used to test the fit of the general model to the data. The estimated indicated a very good model fit. The models with time invariant survival rate had the most support from the data and no group or time period effects were found. Recovery rate ( f ) appeared to increase from 0·043 in the early years to 0·056 in the later years. The nominal survival rate of 0·59 year⁻¹ was adjusted with an estimated tag-shedding rate of 0·26 year⁻¹ to generate a final corrected annual survival estimate of 0·79 with a 95% CI of 0·71–0·87.     

On the use of multi-objective evolutionary algorithms for survival analysis

This paper proposes and evaluates a multi-objective evolutionary algorithm for survival analysis. One aim of survival analysis is the extraction of models from data that approximate lifetime/failure time distributions. These models can be used to estimate the time that an event takes to happen to an object. To use of multi-objective evolutionary algorithms for survival analysis has several advantages. They can cope with feature interactions, noisy data, and are capable of optimising several objectives. This is important, as model extraction is a multi-objective problem. It has at least two objectives, which are the extraction of accurate and simple models. Accurate models are required to achieve good predictions. Simple models are important to prevent overfitting, improve the transparency of the models, and to save computational resources. Although there is a plethora of evolutionary approaches to extract models for classification and regression, the presented approach is one of the first applied to survival analysis. The approach is evaluated on several artificial datasets and one medical dataset. It is shown that the approach is capable of producing accurate models, even for problems that violate some of the assumptions made by classical approaches.     

An effect-specific track-length model for radiations of intermediate to high LET

A model for the induction of transformation, mutation, and cell killing by radiations of intermediate to high linear energy transfer (LET) is presented. The mathematical formulation presupposes a constant probability per unit path length for damaging multiple subcellular targets by radiation of a fixed LET. The coupling between effects is accounted for through an explicit calculation of the probability that any specific combination of effects occurs in a given cell. This feature avoids the false assumption that cell killing and mutation (or transformation) are independent events. The resulting model then is applied to data on the in vitro survival, mutation, and transformation of cells by radiations of varying LET. A summary of estimated parameter values is provided and calculations of the effect of cellular flattening on transformation are presented.