Robust joint modeling of longitudinal measurements and time to event data using normal/independent distributions: A Bayesian approach

Joint modeling of longitudinal data and survival data has been used widely for analyzing AIDS clinical trials, where a biological marker such as CD4 count measurement can be an important predictor of survival. In most of these studies, a normal distribution is used for modeling longitudinal responses, which leads to vulnerable inference in the presence of outliers in longitudinal measurements. Powerful distributions for robust analysis are normal/independent distributions, which include univariate and multivariate versions of the Student's t, the slash and the contaminated normal distributions in addition to the normal. In this paper, a linear‐mixed effects model with normal/independent distribution for both random effects and residuals and Cox's model for survival time are used. For estimation, a Bayesian approach using Markov Chain Monte Carlo is adopted. Some simulation studies are performed for illustration of the proposed method. Also, the method is illustrated on a real AIDS data set and the best model is selected using some criteria.     

Relating Cox's Proportional Hazard Model to the Exponential Model for Survival

An approximate representation is given for the partial likelihood estimate of the regression coefficient in Cox's proportional hazard model which indicates how it measures the association between survival time and covariate. The case of a single covariate is concentrated on. The representation is closely related to the first step of a Newton-Raphson iteration, i.e. to the score test. A similar representation for the Feigl-Zelen exponential model shows that a similar type of association is being measured, if observed lifetimes are interpreted as expected lifetimes of ordered exponentials. Necessary and sufficient conditions for the existence of Cox's estimate in the simple case are also written down.     

Exact Maximum Likelihood Estimates of Cox's Regression Parameters Based on Categorized Factorial Data

This paper describes how Cox's Proportional Hazards model may be used to analyze dichotomized factorial data obtained from a right-censored epidemiological study where time to response is of interest. Exact maximum likelihood estimates of the relative mortality rates are obtained for any number of prognostic factors, as well as their joint asymptotic sampling distribution. These rates represent excess mortality due to the various levels of the prognostic factors. The results are used to discuss the effect of the factors on the survival probability distribution of a cohort of industrial workers who have been exposed to a carcinogen. Kaplan-Meier estimates of the survival function of the internal control population are used to determine the expected number of deaths in the study population. This method differs from the usual lite-table procedure. Asymptotic tests are proposed for some simultaneous and conditional statistical hypotheses.     

Maximum Likelihood Estimation of Genetic Parameters in Cell Populations

In this paper we consider a cell population such as bacteria consisting of two types of cells, mutant and nonmutant. Under the mutation and homogeneous pure birth processes, this paper derives a maximum likelihood estimation procedure for estimating mutation rate and birth rate. The method is applied to Newcombe's data; further some Monte Carlo studies are generated. The numerical results indicate that the method is quite efficient for estimating genetic parameters in cell populations.     

On the empirical choice of the time window for restricted mean survival time

The t-year mean survival or restricted mean survival time (RMST) has been used as an appealing summary of the survival distribution within a time window [0, t]. RMST is the patient's life expectancy until time t and can be estimated nonparametrically by the area under the Kaplan-Meier curve up to t. In a comparative study, the difference or ratio of two RMSTs has been utilized to quantify the between-group-difference as a clinically interpretable alternative summary to the hazard ratio. The choice of the time window [0, t] may be prespecified at the design stage of the study based on clinical considerations. On the other hand, after the survival data have been collected, the choice of time point t could be data-dependent. The standard inferential procedures for the corresponding RMST, which is also data-dependent, ignore this subtle yet important issue. In this paper, we clarify how to make inference about a random “parameter.” Moreover, we demonstrate that under a rather mild condition on the censoring distribution, one can make inference about the RMST up to t, where t is less than or even equal to the largest follow-up time (either observed or censored) in the study. This finding reduces the subjectivity of the choice of t empirically. The proposal is illustrated with the survival data from a primary biliary cirrhosis study, and its finite sample properties are investigated via an extensive simulation study.     

Coital rates,sex‐selective infanticide,and sex ratios at birth: A reply to James

Abstract

The cumulative onset curves for smoking, drinking, and sexual intercourse have been tracked through adolescence with reasonable success by recursive equations positing an “epidemic” or contagious process. The gist of these models is that the likelihood of onset in the next time period is proportional to the prevalence of the behavior among an adolescent's peers in the current time period. The present paper extends this approach to official delinquency. The fits to the data (from the Philadelphia cohort studies) are extremely tight. Several conceptual mismatches between the theory underlying the model and the model itself are discussed.     

A spatial open-population capture-recapture model

A spatial open-population capture-recapture model is described that extends both the non-spatial open-population model of Schwarz and Arnason and the spatially explicit closed-population model of Borchers and Efford. The superpopulation of animals available for detection at some time during a study is conceived as a two-dimensional Poisson point process. Individual probabilities of birth and death follow the conventional open-population model. Movement between sampling times may be modeled with a dispersal kernel using a recursive Markovian algorithm. Observations arise from distance-dependent sampling at an array of detectors. As in the closed-population spatial model, the observed data likelihood relies on integration over the unknown animal locations; maximization of this likelihood yields estimates of the birth, death, movement, and detection parameters. The models were fitted to data from a live-trapping study of brushtail possums (Trichosurus vulpecula) in New Zealand. Simulations confirmed that spatial modeling can greatly reduce the bias of capture-recapture survival estimates and that there is a degree of robustness to misspecification of the dispersal kernel. An R package is available that includes various extensions.     

On the nature of errors involved in estimating stage-specific survival rates by Southwood's method for a population with overlapping stages

This paper has examined the effect of within-stage mortality on the estimation of stage-specific survival rates bySouthwood's (1978, p. 358) method. As pointed out bySouthwood , both the severity and timing of mortality affect the mean duration of a life stage, and consequently the estimate of the number of individuals entering that stage. Knowledge of the form of the survivorship curve permits correction of the estimate under certain circumstances. The use ofSouthwood's method with two overlapping stages having different rates and patterns of mortality leads to complex errors in the estimation of survival for the first stage. The nature of these errors is examined analytically and via a simulation model.Southwood's method is fairly robust, with moderate differences in mortality rates leading to acceptable errors in estimating survival for the first stage. When both the rate and pattern of mortality in both life stages are the same, then the survival estimate is made without error. Precise estimates of stage-specific survival will not usually be possible withSouthwood's method because of the errors introduced by the very parameters being measured. Direct measurement of mortality rates and survivorship patterns (seeSouthwood , 1978, p. 309) is strongly advised, at least in preliminary work.     

Survival and Recovery Rate of Canada Geese Staging in Interior Alaska

Abstract: Lesser Canada geese (Branta canadensis parvipes) are indistinguishable from other subspecies of small Canada geese on the wintering grounds using current survey methods. Consequently, managers are unable to adequately measure their abundance. Without direct estimates of abundance, researchers often use estimates of vital rates that influence abundance (e.g., annual survival) to monitor potential impact of harvest on the population. Based on capture and re-sighting data records of 567 geese marked from 1994 through 1998, we calculated annual survival and recovery rates for different age and sex classes of white-cheeked geese staging in interior Alaska. We compared those survival and recovery rates with those of other neck-collared white-cheeked geese. The best approximating model allowed survival to vary by age class while holding Seber's recovery probability (r̂) constant over sex, age class, and time. We estimated annual survival to be 0.49 (SE = 0.05) for hatch-year geese and 0.68 (SE = 0.03) for after-hatch-year geese based on the weighted average of all models with a change in Akaike's Information Criterion adjusted for small sample size and lack of fit < 4. Estimates of annual survival of white-cheeked geese in this study are among the lowest and recovery estimates are among the highest for migratory populations of neck-collared geese. Low survival estimates of Canada geese in our study suggest that harvest rates may be higher than in many other populations. Surveys to estimate abundance or other population parameters such as reproductive success and recruitment are necessary to determine whether this population is self-sustaining. Furthermore, we recommend monitoring abundance and harvest of small white-cheeked geese east and west of the Cascade Mountain Range separately to better determine harvest pressure on white-cheeked geese wintering east of the Cascades.     

Accounting for tagging‐to‐harvest mortality in a Brownie tag‐recovery model by incorporating radio‐telemetry data

The Brownie tag‐recovery model is useful for estimating harvest rates but assumes all tagged individuals survive to the first hunting season; otherwise, mortality between time of tagging and the hunting season will cause the Brownie estimator to be negatively biased. Alternatively, fitting animals with radio transmitters can be used to accurately estimate harvest rate but may be more costly. We developed a joint model to estimate harvest and annual survival rates that combines known‐fate data from animals fitted with transmitters to estimate the probability of surviving the period from capture to the first hunting season, and data from reward‐tagged animals in a Brownie tag‐recovery model. We evaluated bias and precision of the joint estimator, and how to optimally allocate effort between animals fitted with radio transmitters and inexpensive ear tags or leg bands. Tagging‐to‐harvest survival rates from >20 individuals with radio transmitters combined with 50–100 reward tags resulted in an unbiased and precise estimator of harvest rates. In addition, the joint model can test whether transmitters affect an individual's probability of being harvested. We illustrate application of the model using data from wild turkey, Meleagris gallapavo, to estimate harvest rates, and data from white‐tailed deer, Odocoileus virginianus, to evaluate whether the presence of a visible radio transmitter is related to the probability of a deer being harvested. The joint known‐fate tag‐recovery model eliminates the requirement to capture and mark animals immediately prior to the hunting season to obtain accurate and precise estimates of harvest rate. In addition, the joint model can assess whether marking animals with radio transmitters affects the individual's probability of being harvested, caused by hunter selectivity or changes in a marked animal's behavior.     

A case study of the influence of local weather on Aedes aegypti (L.) aging and mortality

The survival rate of mosquitoes is an important topic that affects many aspects of decision‐making in mosquito management. This study aims to estimate the variability in the survival rate of Ae. aegypti, and climate factors that are related to such variability. It is generally assumed that the daily probability of mosquito survival is independent of natural environment conditions and age. To test this assumption, a three‐year fieldwork (2005–2007) and experimental study was conducted at Fortaleza‐CE in Brazil with the aim of estimating daily survival rates of the dengue vector Aedes aegypti under natural conditions in an urban city. Survival rates of mosquitoes may be age‐dependent and statistical analysis is a sensitive approach for comparing patterns of mosquito survival. We studied whether weather conditions occurring on a particular day influence the mortality observed on that particular day. We therefore focused on the impact of daily meteorological fluctuations around a given climate average, rather than on the influence of climate itself. With regard to survival time, multivariate analyses using the stepwise logistic regression model, adjusted for daily temperature, relative humidity, and saturated vapor pressure deficit (SVPD), suggest that age, the seasonal factor, and the SVPD were the most dependent mortality factors. Similar results were obtained using the Cox proportional hazard model, which explores the relationships between the survival and explanatory variables.     

A nonparametric mixture model for cure rate estimation

Nonparametric methods have attracted less attention than their parametric counterparts for cure rate analysis. In this paper, we study a general nonparametric mixture model. The proportional hazards assumption is employed in modeling the effect of covariates on the failure time of patients who are not cured. The EM algorithm, the marginal likelihood approach, and multiple imputations are employed to estimate parameters of interest in the model. This model extends models and improves estimation methods proposed by other researchers. It also extends Cox's proportional hazards regression model by allowing a proportion of event-free patients and investigating covariate effects on that proportion. The model and its estimation method are investigated by simulations. An application to breast cancer data, including comparisons with previous analyses using a parametric model and an existing nonparametric model by other researchers, confirms the conclusions from the parametric model but not those from the existing nonparametric model.     

Using expert knowledge to incorporate uncertainty in cause‐of‐death assignments for modeling of cause‐specific mortality

Implicit and explicit use of expert knowledge to inform ecological analyses is becoming increasingly common because it often represents the sole source of information in many circumstances. Thus, there is a need to develop statistical methods that explicitly incorporate expert knowledge, and can successfully leverage this information while properly accounting for associated uncertainty during analysis. Studies of cause‐specific mortality provide an example of implicit use of expert knowledge when causes‐of‐death are uncertain and assigned based on the observer's knowledge of the most likely cause. To explicitly incorporate this use of expert knowledge and the associated uncertainty, we developed a statistical model for estimating cause‐specific mortality using a data augmentation approach within a Bayesian hierarchical framework. Specifically, for each mortality event, we elicited the observer's belief of cause‐of‐death by having them specify the probability that the death was due to each potential cause. These probabilities were then used as prior predictive values within our framework. This hierarchical framework permitted a simple and rigorous estimation method that was easily modified to include covariate effects and regularizing terms. Although applied to survival analysis, this method can be extended to any event‐time analysis with multiple event types, for which there is uncertainty regarding the true outcome. We conducted simulations to determine how our framework compared to traditional approaches that use expert knowledge implicitly and assume that cause‐of‐death is specified accurately. Simulation results supported the inclusion of observer uncertainty in cause‐of‐death assignment in modeling of cause‐specific mortality to improve model performance and inference. Finally, we applied the statistical model we developed and a traditional method to cause‐specific survival data for white‐tailed deer, and compared results. We demonstrate that model selection results changed between the two approaches, and incorporating observer knowledge in cause‐of‐death increased the variability associated with parameter estimates when compared to the traditional approach. These differences between the two approaches can impact reported results, and therefore, it is critical to explicitly incorporate expert knowledge in statistical methods to ensure rigorous inference.     

A joint model for longitudinal measurements and survival data in the presence of multiple failure types

Summary .   In this article we study a joint model for longitudinal measurements and competing risks survival data. Our joint model provides a flexible approach to handle possible nonignorable missing data in the longitudinal measurements due to dropout. It is also an extension of previous joint models with a single failure type, offering a possible way to model informatively censored events as a competing risk. Our model consists of a linear mixed effects submodel for the longitudinal outcome and a proportional cause-specific hazards frailty submodel ( Prentice et al., 1978 , Biometrics 34, 541–554) for the competing risks survival data, linked together by some latent random effects. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their standard errors using a profile likelihood method. The developed method works well in our simulation studies and is applied to a clinical trial for the scleroderma lung disease.     

Marginal hazards regression for retrospective studies within cohort with possibly correlated failure time data

Summary .  A retrospective dental study was conducted to evaluate the degree to which pulpal involvement affects tooth survival. Due to the clustering of teeth, the survival times within each subject could be correlated and thus the conventional method for the case–control studies cannot be directly applied. In this article, we propose a marginal model approach for this type of correlated case–control within cohort data. Weighted estimating equations are proposed for the estimation of the regression parameters. Different types of weights are also considered for improving the efficiency. Asymptotic properties of the proposed estimators are investigated and their finite sample properties are assessed via simulations studies. The proposed method is applied to the aforementioned dental study.     

A method for the estimation of parameters for natural stage-structured populations

A relatively simple method is proposed for the estimation of parameters of stage-structured populations from sample data for situation where (a) unit time survival rates may vary with time, and (b) the distribution of entry times to stage 1 is too complicated to be fitted with a simple parametric model such as a normal or gamma distribution. The key aspects of this model are that the entry time distribution is approximated by an exponential function withp parameters, the unit time survival rates in stages are approximated by anr parameter exponential polynomial in the stage number, and the durations of stages are assumed to be the same for all individuals. The new method is applied to four Zooplankton data sets, with parametric bootstrapping used to assess the bias and variation in estimates. It is concluded that good estimates of demographic parameters from stagefrequency data from natural populations will usually only be possible if extra information such as the durations of stages is known.     

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.     

Robust Joint Modeling of Longitudinal Measurements and Competing Risks Failure Time Data

Existing methods for joint modeling of longitudinal measurements and survival data can be highly influenced by outliers in the longitudinal outcome. We propose a joint model for analysis of longitudinal measurements and competing risks failure time data which is robust in the presence of outlying longitudinal observations during follow‐up. Our model consists of a linear mixed effects sub‐model for the longitudinal outcome and a proportional cause‐specific hazards frailty sub‐model for the competing risks data, linked together by latent random effects. Instead of the usual normality assumption for measurement errors in the linear mixed effects sub‐model, we adopt a t ‐distribution which has a longer tail and thus is more robust to outliers. We derive an EM algorithm for the maximum likelihood estimates of the parameters and estimate their standard errors using a profile likelihood method. The proposed method is evaluated by simulation studies and is applied to a scleroderma lung study (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Competing Risks Data Analysis with High-dimensional Covariates:An Application in Bladder Cancer

Analysis of microarray data is associated with the methodological problems of high dimension and small sample size. Various methods have been used for variable selection in highdimension and small sample size cases with a single survival endpoint. However, little effort has been directed toward addressing competing risks where there is more than one failure risks. This study compared three typical variable selection techniques including Lasso, elastic net, and likelihood-based boosting for high-dimensional time-to-event data with competing risks. The performance of these methods was evaluated via a simulation study by analyzing a real dataset related to bladder cancer patients using time-dependent receiver operator characteristic(ROC) curve and bootstrap.632+ prediction error curves. The elastic net penalization method was shown to outperform Lasso and boosting. Based on the elastic net, 33 genes out of 1381 genes related to bladder cancer were selected. By fitting to the Fine and Gray model, eight genes were highly significant(P 0.001). Among them, expression of RTN4, SON, IGF1 R, SNRPE, PTGR1, PLEK, and ETFDH was associated with a decrease in survival time, whereas SMARCAD1 expression was associated with an increase in survival time. This study indicates that the elastic net has a higher capacity than the Lasso and boosting for the prediction of survival time in bladder cancer patients.Moreover, genes selected by all methods improved the predictive power of the model based on only clinical variables, indicating the value of information contained in the microarray features.     

A generalized spatial niche model for aquatic macrophytes

The spatial niches of submerged macrophytes and some relevant statistics for niche extensions and boundaries are described. An interpretation of the potential niche as a probability measure of survival over an n-dimensional gradient space is given. To this end a set of probability measure of is introduced. It is shown that these measures closely relate to statistical concepts found in survival and failure time analysis. By this approach, potential niches are wholly defined as a statistical concept. Realized niches then arise as samples from the specified statistical model: hence appropriate niche location and size measures can be derived.The conceptualized model of the survival niche extends to a general model which describes the shape of realized spatial niches for aquatic macrophytes. This general model was derived from actual vegetation data sampled by diving in many Norwegian lakes. Spatial performance on the vertical gradient can be described by a product of doubly exponential functions. Each of these corresponds to a Gumbel EV1 distribution and expresses the probability distribution of the species' up- and downslope niche boundaries, respectively. Theoretically, both potential and realized niches separate into a persistent and a transient region, each related to ddifferent time scales. The deep-water patchiness, commonly observed for submerged macrophytes, might indicate the transient domain of a realized niche. Published data indicate that the proposed model had widespread applicability, and also relevance to, for example, periphyton.