A general class of Bayesian survival models with zero and nonzero cure fractions

We propose a new class of survival models which naturally links a family of proper and improper population survival functions. The models resulting in improper survival functions are often referred to as cure rate models. This class of regression models is formulated through the Box-Cox transformation on the population hazard function and a proper density function. By adding an extra transformation parameter into the cure rate model, we are able to generate models with a zero cure rate, thus leading to a proper population survival function. A graphical illustration of the behavior and the influence of the transformation parameter on the regression model is provided. We consider a Bayesian approach which is motivated by the complexity of the model. Prior specification needs to accommodate parameter constraints due to the non-negativity of the survival function. Moreover, the likelihood function involves a complicated integral on the survival function, which may not have an analytical closed form, and thus makes the implementation of Gibbs sampling more difficult. We propose an efficient Markov chain Monte Carlo computational scheme based on Gaussian quadrature. The proposed method is illustrated with an example involving a melanoma clinical trial.     

Two-component mixture cure rate model with spline estimated nonparametric components

Summary In some survival analysis of medical studies, there are often long-term survivors who can be considered as permanently cured. The goals in these studies are to estimate the noncured probability of the whole population and the hazard rate of the susceptible subpopulation. When covariates are present as often happens in practice, to understand covariate effects on the noncured probability and hazard rate is of equal importance. The existing methods are limited to parametric and semiparametric models. We propose a two-component mixture cure rate model with nonparametric forms for both the cure probability and the hazard rate function. Identifiability of the model is guaranteed by an additive assumption that allows no time-covariate interactions in the logarithm of hazard rate. Estimation is carried out by an expectation-maximization algorithm on maximizing a penalized likelihood. For inferential purpose, we apply the Louis formula to obtain point-wise confidence intervals for noncured probability and hazard rate. Asymptotic convergence rates of our function estimates are established. We then evaluate the proposed method by extensive simulations. We analyze the survival data from a melanoma study and find interesting patterns for this study.     

Rainfall and population dynamics of Grey Pileated Finch <Emphasis Type="Italic">Coryphospingus pileatus</Emphasis> (Aves: Passeriformes) in a Neotropical dry forest

In tropical dry environments rainfall periodicity may affect demographic parameters, resulting in fluctuations in bird abundance. We used capture–recapture data for the Grey Pileated Finch from a Neotropical dry forest to evaluate the hypothesis that intra- and inter-annual survival, individuals entrance and population abundance, are related to local rainfall. Sampling occurred across 3 years, with individuals captured, tagged and evaluated for age and presence of brood patch every 14 days. Using the POPAN formulation, we generated demographic models to evaluate study population temporal dynamics. Best-fit models indicated a low apparent annual survival in the first year (16%) compared to other years (between 47 and 62%), with this low value associated with an extreme drought. The abundance of juveniles at each capture occasion was significantly dependent on the accumulated precipitation in the previous 14 days, and the juvenile covariate was a strong predictor of the intra-annual entrance probability (natality). Individuals entrance during the reproductive period corresponded to 53, 52 and 75% of total ingress for each year, respectively. The trend in sampled population size indicated positive exponential growth (N_initial = 50, N_last = 600), with intra-annual fluctuations becoming progressively more intense. Low survival was relevant during population decline at study onset, while at study end intense Individuals entrance promoted rapid population growth. Thus, the indirect effects of rainfall and the combined effect of two demographic rates operated synergistically on the immediate population abundance of Grey Pileated Finch, an abundant bird in a Neotropical dry forest.     

The dynamics of hierarchical age-structured populations

An age-structured population is considered in which the birth and death rates of an individual of age a is a function of the density of individuals older and/or younger than a. An existence/uniqueness theorem is proved for the McKendrick equation that governs the dynamics of the age distribution function. This proof shows how a decoupled ordinary differential equation for the total population size can be derived. This result makes a study of the population's asymptotic dynamics (indeed, often its global asymptotic dynamics) mathematically tractable. Several applications to models for intra-specific competition and predation are given.     

A formulation of the lifetime distribution and the existence of its moments

Summary The lifetime distribution is formulated in terms of g(t), defined as the ratio of the hazard function to the tail probability function, to study the properties of the lifetime distribution. A criterion is provided for the asymptotic behaviour of g(t) and the hazard function. Criteria for the existence and non-existence of the moments of any probability distribution of a non-negative random variable are obtained in terms of the derivatives of g(t). Examples are given to illustrate the use of the criteria and applications made to stochastic models of population growth as well as other lifetime distributions.This research was supported under National Health Research and Development Project No. 605-7-434 (22)(48) of the Department of National Health and Welfare of Canada.     

Association Tests for a Censored Quantitative Trait and Candidate Genes in Structured Populations with Multilevel Genetic Relatedness

Summary Several statistical methods for detecting associations between quantitative traits and candidate genes in structured populations have been developed for fully observed phenotypes. However, many experiments are concerned with failure‐time phenotypes, which are usually subject to censoring. In this article, we propose statistical methods for detecting associations between a censored quantitative trait and candidate genes in structured populations with complex multiple levels of genetic relatedness among sampled individuals. The proposed methods correct for continuous population stratification using both population structure variables as covariates and the frailty terms attributable to kinship. The relationship between the time‐at‐onset data and genotypic scores at a candidate marker is modeled via a parametric Weibull frailty accelerated failure time (AFT) model as well as a semiparametric frailty AFT model, where the baseline survival function is flexibly modeled as a mixture of Polya trees centered around a family of Weibull distributions. For both parametric and semiparametric models, the frailties are modeled via an intrinsic Gaussian conditional autoregressive prior distribution with the kinship matrix being the adjacency matrix connecting subjects. Simulation studies and applications to the Arabidopsis thaliana line flowering time data sets demonstrated the advantage of the new proposals over existing approaches.     

Models for the spread of universally fatal diseases

In the formulation of models of S-I-R type for the spread of communicable diseases it is necessary to distinguish between diseases with recovery with full immunity and diseases with permanent removal by death. We consider models which include nonlinear population dynamics with permanent removal. The principal result is that the stability of endemic equilibrium may depend on the population dynamics and on the distribution of infective periods; sustained oscillations are possible in some cases.     

Generalized parametric cure models for relative survival

Cure models are used in time-to-event analysis when not all individuals are expected to experience the event of interest, or when the survival of the considered individuals reaches the same level as the general population. These scenarios correspond to a plateau in the survival and relative survival function, respectively. The main parameters of interest in cure models are the proportion of individuals who are cured, termed the cure proportion, and the survival function of the uncured individuals. Although numerous cure models have been proposed in the statistical literature, there is no consensus on how to formulate these. We introduce a general parametric formulation of mixture cure models and a new class of cure models, termed latent cure models, together with a general estimation framework and software, which enable fitting of a wide range of different models. Through simulations, we assess the statistical properties of the models with respect to the cure proportion and the survival of the uncured individuals. Finally, we illustrate the models using survival data on colon cancer, which typically display a plateau in the relative survival. As demonstrated in the simulations, mixture cure models which are not guaranteed to be constant after a finite time point, tend to produce accurate estimates of the cure proportion and the survival of the uncured. However, these models are very unstable in certain cases due to identifiability issues, whereas LC models generally provide stable results at the price of more biased estimates.     

Postfledging Survival of Laysan Ducks

ABSTRACT Precise and unbiased estimates of demographic parameters are necessary for effective population monitoring and to parameterize population models (e.g., population viability analyses). This is especially important for endangered species, where recovery planning and managers' decisions can influence species persistence. In this study, we used mark—recapture methods to estimate survival of fledged juveniles (hatch-yr [HY]) and adult (after-hatch-yr [AHY]) Laysan ducks (Anas laysanensis), an endangered anatid restricted to Laysan Island in the northwestern Hawaiian Islands. To better understand population dynamics, we examined how survival varied as a function of Laysan duck density during 1998–2004. Using random effects models, we also quantified process variation in survival, thereby quantifying the appropriate source of variation for future population models. The dataset supported variation in survival that was time (yr), age (AHY vs. HY), and sex specific. Due to small sample sizes, we did not examine time specificity in the survival of HY ducks. Survival of HY ducks was 0.832 (SE = 0.087) for females (n = 21) and 0.999 (SE < 0.001) for males (n = 15) during 1998–2001. Trends in time and density lacked support as sources of variation in the survival of AHY ducks during 1998–2004. After-hatch-year survival ranged from 0.792 (SE = 0.033) to 0.999 (SE < 0.001). Where we modeled survival as a random effect, annual survival for AHY females was 0.881 (SE = 0.017) and process variation (σ_s) was 0.034. For AHY males, annual survival (μ_s) was 0.906 (SE = 0.019) and process variation (σ_s) was 0.040. This information will improve existing population viability analysis models for Laysan ducks. We believe that monitoring the source and translocation populations will be paramount for increasing our understanding of Laysan duck dynamics, recovery planning, and population management.     

Dynamical consequences of harvest in discrete age-structured population models

The role of harvest in discrete age-structured one-population models has been explored. Considering a few age classes only, together with the overcompensatory Ricker recruitment function, we show that harvest acts as a weak destabilizing effect in case of small values of the year-to-year survival probability P and as a strong stabilizing effect whenever the survival probability approaches unity. In the latter case, assuming n=2 age classes, we find that harvest may transfer a population from the chaotic regime to a state where the equilibrium point (x₁*, x₂*) becomes stable. However, as the number of age classes increases (which acts as a stabilizing effect in non-exploited models), we find that harvest acts more and more destabilizing, in fact, when the number of age classes has been increased to n=10, our finding is that in case of large values of the survival probabilities, harvest may transfer a population from a state where the equilibrium is stable to the chaotic regime, thus exactly the opposite of what was found in case of n=2. On the other hand, if we replace the Ricker relation with the generalized Beverton and Holt recruitment function with abruptness parameter larger than 2, several of the conclusions derived above are changed. For example, when n is large and the survival probabilities exceed a certain threshold, the equilibrium will always be stable.Revised version: 18 September 2003     

Survival and hazard functions for progressive diseases using saddlepoint approximations

Saddlepoint approximations for the computation of survival and hazard functions are introduced in the context of parametric survival analysis. Although these approximations are computationally fast, accurate, and relatively straightforward to implement, their use in survival analysis has been lacking. We approximate survival functions using the Lugannani and Rice saddlepoint approximation to the distribution function or by numerically integrating the saddlepoint density approximation. The hazard function is approximated using the saddlepoint density and distribution functions. The approximations are especially useful for consideration of survival and hazard functions for waiting times in complicated models. Examples include total or partial waiting times for a disease that progresses through various stages (convolutions of distributions).     

Bayesian regularization for flexible baseline hazard functions in Cox survival models

Fully Bayesian methods for Cox models specify a model for the baseline hazard function. Parametric approaches generally provide monotone estimations. Semi‐parametric choices allow for more flexible patterns but they can suffer from overfitting and instability. Regularization methods through prior distributions with correlated structures usually give reasonable answers to these types of situations. We discuss Bayesian regularization for Cox survival models defined via flexible baseline hazards specified by a mixture of piecewise constant functions and by a cubic B‐spline function. For those “semi‐parametric” proposals, different prior scenarios ranging from prior independence to particular correlated structures are discussed in a real study with microvirulence data and in an extensive simulation scenario that includes different data sample and time axis partition sizes in order to capture risk variations. The posterior distribution of the parameters was approximated using Markov chain Monte Carlo methods. Model selection was performed in accordance with the deviance information criteria and the log pseudo‐marginal likelihood. The results obtained reveal that, in general, Cox models present great robustness in covariate effects and survival estimates independent of the baseline hazard specification. In relation to the “semi‐parametric” baseline hazard specification, the B‐splines hazard function is less dependent on the regularization process than the piecewise specification because it demands a smaller time axis partition to estimate a similar behavior of the risk.     

Gestational mutations and carcinogenesis

We present a mathematical formulation to evaluate the effects of gestational mutations on cancer risk. The hazard or incidence function of cancer is expressed in terms of the Probability Generating Function (PGF) of the number of normal and mutated cells at birth. Using Filtered Poisson Process Theory, we obtain the PGF for several models for the accumulation of gestational mutations. In particular, we develop expressions for the hazard function when one or two successive mutations could occur during gestation. We also calculate the hazard when the background gestational mutation rates are increased due to exposure to mutagens, such as prenatal radiation. To illustrate the use of our models, we apply them to colorectal cancer in the SEER database. We find that the proportion of cancer risk attributable to developmental mutations depends on age and that it could be quite significant when gestational mutation rates are high. The analysis of the SEER data also shows that gestational mutations could contribute to inter-individual variations in colorectal cancer risk.     

Dimension reduction for integrative survival analysis

We propose a constrained maximum partial likelihood estimator for dimension reduction in integrative (e.g., pan-cancer) survival analysis with high-dimensional predictors. We assume that for each population in the study, the hazard function follows a distinct Cox proportional hazards model. To borrow information across populations, we assume that each of the hazard functions depend only on a small number of linear combinations of the predictors (i.e., “factors”). We estimate these linear combinations using an algorithm based on “distance-to-set” penalties. This allows us to impose both low-rankness and sparsity on the regression coefficient matrix estimator. We derive asymptotic results that reveal that our estimator is more efficient than fitting a separate proportional hazards model for each population. Numerical experiments suggest that our method outperforms competitors under various data generating models. We use our method to perform a pan-cancer survival analysis relating protein expression to survival across 18 distinct cancer types. Our approach identifies six linear combinations, depending on only 20 proteins, which explain survival across the cancer types. Finally, to validate our fitted model, we show that our estimated factors can lead to better prediction than competitors on four external datasets.     

Survival benefit of sentinel lymph node biopsy in Asian melanoma patients

Sentinel lymph node biopsy (SLNB) provides important prognostic information for early-stage melanomas. However, statistics regarding the survival comparison between SLNB and nodal observation in Asia, where acral lentiginous melanoma (ALM) predominates, are limited. This study aimed to identify if SLNB offered survival benefits over nodal observation in early-stage melanomas in Taiwan. The retrospective study included 227 patients who met the SLNB criteria according to the National Comprehensive Cancer Network guidelines and were treated at National Taiwan University Hospital from June 1997 to June 2021. Survival analysis was performed using Kaplan–Meier curves and Cox proportional hazards regression models. Of the study population, ALM accounted for 73.1%; 161 patients (70.9%) underwent SLNB and 66 patients (29.1%) were under nodal observation. Multivariate analysis showed significantly improved melanoma-specific survival (hazard ratio [HR], 0.6; p = .02) in the SLNB group. Among those who underwent completion lymph node dissection (CLND), the non-sentinel node positivity rate was 44.4%. Immediate CLND resulted in significantly longer melanoma-specific survival and distant-metastasis-free survival (DMFS) compared to nodal observation. (HR, 0.2; p = .01 for melanoma-specific survival. HR, 0.3; p = .046 for DMFS). In conclusion, SLNB may provide survival benefits of cutaneous melanoma over nodal observation in the Taiwanese population.     

Modeling individual effects in the Cormack-Jolly-Seber model: a state-space formulation

Summary .   In population and evolutionary biology, there exists considerable interest in individual heterogeneity in parameters of demographic models for open populations. However, flexible and practical solutions to the development of such models have proven to be elusive. In this article, I provide a state-space formulation of open population capture–recapture models with individual effects. The state-space formulation provides a generic and flexible framework for modeling and inference in models with individual effects, and it yields a practical means of estimation in these complex problems via contemporary methods of Markov chain Monte Carlo. A straightforward implementation can be achieved in the software package WinBUGS . I provide an analysis of a simple model with constant parameter detection and survival probability parameters. A second example is based on data from a 7-year study of European dippers, in which a model with year and individual effects is fitted.     

珍稀濒危植物长苞铁杉种群生命表分析

以种群生命表及生存分析理论为基础,编制珍稀濒危植物长苞铁杉种群生命表,绘制死亡密度函数曲线、危险率函数曲线及生存曲线,分析种群动态趋势。结果表明,长苞铁杉种群有２个死亡高峰,存活曲线趋于Ｄｅｅｖｅｙ－Ⅱ型。     

Parametric proportional hazards model for mapping genomic imprinting of survival traits

A number of imprinted genes have been observed in plants, animals and humans. They not only control growth and developmental traits, but may also be responsible for survival traits. Based on the Cox proportional hazards (PH) model, we constructed a general parametric model for dissecting genomic imprinting, in which a baseline hazard function is selectable for fitting the effects of imprinted quantitative trait loci (iQTL) genotypes on the survival curve. The expectation–maximisation (EM) algorithm is derived for solving the maximum likelihood estimates of iQTL parameters. The imprinting patterns of the detected iQTL are statistically tested under a series of null hypotheses. The Bayesian information criterion (BIC) model selection criterion is employed to choose an optimal baseline hazard function with maximum likelihood and parsimonious parameterisation. We applied the proposed approach to analyse the published data in an F₂ population of mice and concluded that, among five commonly used survival distributions, the log-logistic distribution is the optimal baseline hazard function for the survival time of hyperoxic acute lung injury (HALI). Under this optimal model, five QTL were detected, among which four are imprinted in different imprinting patterns.     

Role of Ecological Modeling in Risk Assessment

Ecological models are useful tools for evaluating the ecological significance of observed or predicted effects of toxic chemicals on individual organisms. Current risk estimation approaches using hazard quotients for individual-level endpoints have limited utility for assessing risks at the population, ecosystem, and landscape levels, which are the most relevant indicators for environmental management. In this paper, we define different types of ecological models, summarize their input and output variables, and present examples of the role of some recommended models in chemical risk assessments. A variety of population and ecosystem models have been applied successfully to evaluate ecological risks, including population viability of endangered species, habitat fragmentation, and toxic chemical issues. In particular, population models are widely available, and their value in predicting dynamics of natural populations has been demonstrated. Although data are often limited on vital rates and doseresponse functions needed for ecological modeling, accurate prediction of ecological effects may not be needed for all assessments. Often, a comparative assessment of risk (e.g., relative to baseline or reference) is of primary interest. Ecological modeling is currently a valuable approach for addressing many chemical risk assessment issues, including screening-level evaluations.     

A stochastic computer model for simulating population growth

A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model's behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka's Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values or r_m and K. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetle Tribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.