Dynamic clinical prediction models for discrete time‐to‐event data with competing risks—A case study on the OUTCOMEREA database

The development of clinical prediction models requires the selection of suitable predictor variables. Techniques to perform objective Bayesian variable selection in the linear model are well developed and have been extended to the generalized linear model setting as well as to the Cox proportional hazards model. Here, we consider discrete time‐to‐event data with competing risks and propose methodology to develop a clinical prediction model for the daily risk of acquiring a ventilator‐associated pneumonia (VAP) attributed to P. aeruginosa (PA) in intensive care units. The competing events for a PA VAP are extubation, death, and VAP due to other bacteria. Baseline variables are potentially important to predict the outcome at the start of ventilation, but may lose some of their predictive power after a certain time. Therefore, we use a landmark approach for dynamic Bayesian variable selection where the set of relevant predictors depends on the time already spent at risk. We finally determine the direct impact of a variable on each competing event through cause‐specific variable selection.     

A Bayesian Semiparametric Survival Model with Longitudinal Markers

Summary We consider inference for data from a clinical trial of treatments for metastatic prostate cancer. Patients joined the trial with diverse prior treatment histories. The resulting heterogeneous patient population gives rise to challenging statistical inference problems when trying to predict time to progression on different treatment arms. Inference is further complicated by the need to include a longitudinal marker as a covariate. To address these challenges, we develop a semiparametric model for joint inference of longitudinal data and an event time. The proposed approach includes the possibility of cure for some patients. The event time distribution is based on a nonparametric Pólya tree prior. For the longitudinal data we assume a mixed effects model. Incorporating a regression on covariates in a nonparametric event time model in general, and for a Pólya tree model in particular, is a challenging problem. We exploit the fact that the covariate itself is a random variable. We achieve an implementation of the desired regression by factoring the joint model for the event time and the longitudinal outcome into a marginal model for the event time and a regression of the longitudinal outcomes on the event time, i.e., we implicitly model the desired regression by modeling the reverse conditional distribution.     

Latent transition regression for mixed outcomes

Health status is a complex outcome, often characterized by multiple measures. When assessing changes in health status over time, multiple measures are typically collected longitudinally. Analytic challenges posed by these multivariate longitudinal data are further complicated when the outcomes are combinations of continuous, categorical, and count data. To address these challenges, we propose a fully Bayesian latent transition regression approach for jointly analyzing a mixture of longitudinal outcomes from any distribution. Health status is assumed to be a categorical latent variable, and the multiple outcomes are treated as surrogate measures of the latent health state, observed with error. Using this approach, both baseline latent health state prevalences and the probabilities of transitioning between the health states over time are modeled as functions of covariates. The observed outcomes are related to the latent health states through regression models that include subject-specific effects to account for residual correlation among repeated measures over time, and covariate effects to account for differential measurement of the latent health states. We illustrate our approach with data from a longitudinal study of back pain.     

Lagrangian molecular dynamics using selected conformational degrees of freedom,with application to the pseudorotation dynamics of furanose rings

Using internal conformational degrees of freedom for biopolymers as natural variables, and introducing a Lagrangian dynamics approach, one can simulate time-dependent processes over a much longer time scale than in classical Newtonian molecular dynamics (MD) techniques. Two factors contribute to this: a substantial reduction in the number of degrees of freedom and a very large increase in the size of the time step. We present the Lagrangian equations of motion for repuckering transitions in model furanose ( F ), ribose ( R ), and 2′-deoxyribose ( dR ) ring systems using the pseudorotation phase angle as the single dynamic variable. As in most Lagrangian analyses, the effective masses for the R and dR models are dependent on conformation, and we test the behavior of this variable mass (VM) model. Since the variation in effective mass is small, the VM model is compared with a simplified constant mass (CM) model, which is shown to be an excellent approximation. The equations of motion for the CM and VM models are integrated with the leapfrog and the iterative leapfrog algorithms, respectively. The Lagrangian dynamics approach reduces the number of degrees of freedom from about 40 to 1, and allows the use of time steps on the order of 20 fs, about an order of magnitude greater than is used in conventional MD simulations. © 1994 John Wiley & Sons, Inc.     

A weighted partial likelihood approach for zero‐truncated models

Zero‐truncated data arises in various disciplines where counts are observed but the zero count category cannot be observed during sampling. Maximum likelihood estimation can be used to model these data; however, due to its nonstandard form it cannot be easily implemented using well‐known software packages, and additional programming is often required. Motivated by the Rao–Blackwell theorem, we develop a weighted partial likelihood approach to estimate model parameters for zero‐truncated binomial and Poisson data. The resulting estimating function is equivalent to a weighted score function for standard count data models, and allows for applying readily available software. We evaluate the efficiency for this new approach and show that it performs almost as well as maximum likelihood estimation. The weighted partial likelihood approach is then extended to regression modelling and variable selection. We examine the performance of the proposed methods through simulation and present two case studies using real data.     

Applications of hidden hybrid Markov/semi‐Markov models: from stopover duration to breeding success dynamics

Usually in capture–recapture, a model parameter is time or time since first capture dependent. However, the case where the probability of staying in one state depends on the time spent in that particular state is not rare. Hidden Markov models are not appropriate to manage these situations. A more convenient approach would be to consider models that incorporate semi‐Markovian states which explicitly define the waiting time distribution and have been used in previous biologic studies as a convenient framework for modeling the time spent in a given physiological state. Here, we propose hidden Markovian models that combine several nonhomogeneous Markovian states with one semi‐Markovian state and which (i) are well adapted to imperfect and variable detection and (ii) allow us to consider time, time since first capture, and time spent in one state effects. Implementation details depending on the number of semi‐Markovian states are discussed. From a user's perspective, the present approach enhances the toolbox for analyzing capture–recapture data. We then show the potential of this framework by means of two ecological examples: (i) stopover duration and (ii) breeding success dynamics.     

Longitudinal Studies of Binary Response Data Following Case–Control and Stratified Case–Control Sampling: Design and Analysis

Summary We discuss design and analysis of longitudinal studies after case–control sampling, wherein interest is in the relationship between a longitudinal binary response that is related to the sampling (case–control) variable, and a set of covariates. We propose a semiparametric modeling framework based on a marginal longitudinal binary response model and an ancillary model for subjects' case–control status. In this approach, the analyst must posit the population prevalence of being a case, which is then used to compute an offset term in the ancillary model. Parameter estimates from this model are used to compute offsets for the longitudinal response model. Examining the impact of population prevalence and ancillary model misspecification, we show that time‐invariant covariate parameter estimates, other than the intercept, are reasonably robust, but intercept and time‐varying covariate parameter estimates can be sensitive to such misspecification. We study design and analysis issues impacting study efficiency, namely: choice of sampling variable and the strength of its relationship to the response, sample stratification, choice of working covariance weighting, and degree of flexibility of the ancillary model. The research is motivated by a longitudinal study following case–control sampling of the time course of attention deficit hyperactivity disorder (ADHD) symptoms.     

Towards a large-scale model of patient-specific epileptic spike-wave discharges

Clinical electroencephalographic (EEG) recordings of the transition into generalised epileptic seizures show a sudden onset of spike-wave dynamics from a low-amplitude irregular background. In addition, non-trivial and variable spatio-temporal dynamics are widely reported in combined EEG/fMRI studies on the scale of the whole cortex. It is unknown whether these characteristics can be accounted for in a large-scale mathematical model with fixed heterogeneous long-range connectivities. Here, we develop a modelling framework with which to investigate such EEG features. We show that a neural field model composed of a few coupled compartments can serve as a low-dimensional prototype for the transition between irregular background dynamics and spike-wave activity. This prototype then serves as a node in a large-scale network with long-range connectivities derived from human diffusion-tensor imaging data. We examine multivariate properties in 42 clinical EEG seizure recordings from 10 patients diagnosed with typical absence epilepsy and 50 simulated seizures from the large-scale model using 10 DTI connectivity sets from humans. The model can reproduce the clinical feature of stereotypy where seizures are more similar within a patient than between patients, essentially creating a patient-specific fingerprint. We propose the approach as a feasible technique for the investigation of patient-specific large-scale epileptic features in space and time.     

Survival analysis using auxiliary variables via multiple imputation, with application to AIDS clinical trial data

We develop an approach, based on multiple imputation, to using auxiliary variables to recover information from censored observations in survival analysis. We apply the approach to data from an AIDS clinical trial comparing ZDV and placebo, in which CD4 count is the time-dependent auxiliary variable. To facilitate imputation, a joint model is developed for the data, which includes a hierarchical change-point model for CD4 counts and a time-dependent proportional hazards model for the time to AIDS. Markov chain Monte Carlo methods are used to multiply impute event times for censored cases. The augmented data are then analyzed and the results combined using standard multiple-imputation techniques. A comparison of our multiple-imputation approach to simply analyzing the observed data indicates that multiple imputation leads to a small change in the estimated effect of ZDV and smaller estimated standard errors. A sensitivity analysis suggests that the qualitative findings are reproducible under a variety of imputation models. A simulation study indicates that improved efficiency over standard analyses and partial corrections for dependent censoring can result. An issue that arises with our approach, however, is whether the analysis of primary interest and the imputation model are compatible.     

Seasonal Invasion Dynamics in a Spatially Heterogeneous River with Fluctuating Flows

A key problem in environmental flow assessment is the explicit linking of the flow regime with ecological dynamics. We present a hybrid modeling approach to couple hydrodynamic and biological processes, focusing on the combined impact of spatial heterogeneity and temporal variability on population dynamics. Studying periodically alternating pool-riffle rivers that are subjected to seasonally varying flows, we obtain an invasion ratchet mechanism. We analyze the ratchet process for a caricature model and a hybrid physical–biological model. The water depth and current are derived from a hydrodynamic equation for variable stream bed water flows and these quantities feed into a reaction-diffusion-advection model that governs population dynamics of a river species. We establish the existence of spreading speeds and the invasion ratchet phenomenon, using a mixture of mathematical approximations and numerical computations. Finally, we illustrate the invasion ratchet phenomenon in a spatially two-dimensional hydraulic simulation model of a meandering river structure. Our hybrid modeling approach strengthens the ecological component of stream hydraulics and allows us to gain a mechanistic understanding as to how flow patterns affect population survival.     

Time‐Varying Latent Effect Model for Longitudinal Data with Informative Observation Times

Summary In analysis of longitudinal data, it is not uncommon that observation times of repeated measurements are subject‐specific and correlated with underlying longitudinal outcomes. Taking account of the dependence between observation times and longitudinal outcomes is critical under these situations to assure the validity of statistical inference. In this article, we propose a flexible joint model for longitudinal data analysis in the presence of informative observation times. In particular, the new procedure considers the shared random‐effect model and assumes a time‐varying coefficient for the latent variable, allowing a flexible way of modeling longitudinal outcomes while adjusting their association with observation times. Estimating equations are developed for parameter estimation. We show that the resulting estimators are consistent and asymptotically normal, with variance–covariance matrix that has a closed form and can be consistently estimated by the usual plug‐in method. One additional advantage of the procedure is that it provides a unified framework to test whether the effect of the latent variable is zero, constant, or time‐varying. Simulation studies show that the proposed approach is appropriate for practical use. An application to a bladder cancer data is also given to illustrate the methodology.     

Using "travel time" data to characterize the behavior of migrating animals

For migratory species, duration of migration, or "travel time," is often a critical variable in determining the cost of migration. Observed travel times are the result of both environmental factors such as air or water currents and the behavior of individuals. In an effort to distinguish among these components, I developed a migration model based on an advection-diffusion equation that characterizes population movements in terms of two biologically meaningful parameters: migration rate and rate of population spread. I applied the model to travel time data from juvenile chinook salmon (Onchorhynchus tshawytscha), which were tagged during their seaward migration. The tagged fish originated from three separate evolutionarily significant units (ESUs) as classified by the U. S. National Marine Fisheries Service. The model was expanded by allowing migration and diffusion rates to vary with fish length and river flow. Variability in travel times explained by these factors was strikingly similar from year to year within ESUs, and the migratory behavior revealed by the analysis was consistent with the life-history patterns that distinguish the ESUs. The approach presented here is easily adaptable to a wide range of migratory species and may be particularly useful for predicting how at-risk populations respond to variable conditions in regulated or otherwise disturbed migration habitats.     

Estimating environmental effects on population dynamics: consequences of observation error

Within the paradigm of population dynamics a central task is to identify environmental factors affecting population change and to estimate the strength of these effects. We here investigate the impact of observation errors in measurements of population densities on estimates of environmental effects. Adding observation errors may change the autocorrelation of a population time series with potential consequences for estimates of effects of autocorrelated environmental covariates. Using Monte Carlo simulations, we compare the performance of maximum likelihood estimates from three stochastic versions of the Gompertz model (log–linear first order autoregressive model), assuming 1) process error only, 2) observation error only, and 3) both process and observation error (the linear state–space model on log‐scale). We also simulated population dynamics using the Ricker model, and evaluated the corresponding maximum likelihood estimates for process error models. When there is observation error in the data and the considered environmental variable is strongly autocorrelated, its estimated effect is likely to be biased when using process error models. The environmental effect is overestimated when the sign of the autocorrelations of the intrinsic dynamics and the environment are the same and underestimated when the signs differ. With non‐autocorrelated environmental covariates, process error models produce fairly exact point estimates as well as reliable confidence intervals for environmental effects. In all scenarios, observation error models produce unbiased estimates with reasonable precision, but confidence intervals derived from the likelihood profiles are far too optimistic if there is process error present. The safest approach is to use state–space models in presence of observation error. These are factors worthwhile to consider when interpreting earlier empirical results on population time series, and in future studies, we recommend choosing carefully the modelling approach with respect to intrinsic population dynamics and covariate autocorrelation.     

A framework for inference about carnivore density from unstructured spatial sampling of scat using detector dogs

Wildlife management often hinges upon an accurate assessment of population density. Although undeniably useful, many of the traditional approaches to density estimation such as visual counts, livetrapping, or mark–recapture suffer from a suite of methodological and analytical weaknesses. Rare, secretive, or highly mobile species exacerbate these problems through the reality of small sample sizes and movement on and off study sites. In response to these difficulties, there is growing interest in the use of non-invasive survey techniques, which provide the opportunity to collect larger samples with minimal increases in effort, as well as the application of analytical frameworks that are not reliant on large sample size arguments. One promising survey technique, the use of scat detecting dogs, offers a greatly enhanced probability of detection while at the same time generating new difficulties with respect to non-standard survey routes, variable search intensity, and the lack of a fixed survey point for characterizing non-detection. In order to account for these issues, we modified an existing spatially explicit, capture–recapture model for camera trap data to account for variable search intensity and the lack of fixed, georeferenced trap locations. We applied this modified model to a fisher (Martes pennanti) dataset from the Sierra National Forest, California, and compared the results (12.3 fishers/100 km²) to more traditional density estimates. We then evaluated model performance using simulations at 3 levels of population density. Simulation results indicated that estimates based on the posterior mode were relatively unbiased. We believe that this approach provides a flexible analytical framework for reconciling the inconsistencies between detector dog survey data and density estimation procedures. © 2011 The Wildlife Society.     

An implicit approach to model plant infestation by insect pests

Various spatial approaches were developed to study the effect of spatial heterogeneities on population dynamics. We present in this paper a flux-based model to describe an aphid-parasitoid system in a closed and spatially structured environment, i.e. a greenhouse. Derived from previous work and adapted to host-parasitoid interactions, our model represents the level of plant infestation as a continuous variable corresponding to the number of plants bearing a given density of pests at a given time. The variation of this variable is described by a partial differential equation. It is coupled to an ordinary differential equation and a delay-differential equation that describe the parasitized host population and the parasitoid population, respectively. We have applied our approach to the pest Aphis gossypii and to one of its parasitoids, Lysiphlebus testaceipes, in a melon greenhouse. Numerical simulations showed that, regardless of the number and distribution of hosts in the greenhouse, the aphid population is slightly larger if parasitoids display a type III rather than a type II functional response. However, the population dynamics depend on the initial distribution of hosts and the initial density of parasitoids released, which is interesting for biological control strategies. Sensitivity analysis showed that the delay in the parasitoid equation and the growth rate of the pest population are crucial parameters for predicting the dynamics. We demonstrate here that such a flux-based approach generates relevant predictions with a more synthetic formalism than a common plant-by-plant model. We also explain how this approach can be better adapted to test different management strategies and to manage crops of several greenhouses.     

Coherent activation of a synthetic mammalian gene network

A quantitative analysis of naturally-occurring regulatory networks, especially those present in mammalian cells, is difficult due to their high complexity. Much simpler gene networks can be engineered in model organisms and analyzed as isolated regulatory modules. Recently, several synthetic networks have been constructed in mammalian systems. However, most of these engineered mammalian networks have been characterized using steady-state population level measurements. Here, we use an integrated experimental-computational approach to analyze the dynamical response of a synthetic positive feedback network in individual mammalian cells. We observe a switch-like activation of the network with variable delay times in individual cells. In agreement with a stochastic model of the network, we find that increasing the strength of the positive feedback results in a decrease in the mean delay time and a more coherent activation of individual cells. Our results are important for gaining insight into biological processes which rely on positive feedback regulation.     

A new approach for rapid detection of nearby thresholds in ecosystem time series

Massive changes to ecosystems sometimes cross thresholds from which recovery can be difficult, expensive and slow. These thresholds are usually discovered in post hoc analyses long after the event occurred. Anticipating these changes prior to their occurrence could give managers a chance to intervene. Here we present a novel approach for anticipating ecosystem thresholds that combines resilience indicators with Quickest detection of change points. Unlike existing methods, the Quickest detection method is updated every time a data point arrives, and minimizes the time to detect an approaching threshold given the users’ tolerance for false alarms. The procedure accurately detected an impending regime shift in an experimentally manipulated ecosystem. An ecosystem model was used to determine if the method can detect an approaching threshold soon enough to prevent a regime shift. When the monitored variable was directly involved in the interaction that caused the regime shift, detection was quick enough to avert collapse. When the monitored variable was only indirectly linked to the critical transition, detection came too late. The procedure is useful for assessing changes in resilience as ecosystems approach thresholds. However some thresholds cannot be detected in time to prevent regime shifts, and surprises will be inevitable in ecosystem management.     

Comparison of procedures to assess non-linear and time-varying effects in multivariable models for survival data

The focus of many medical applications is to model the impact of several factors on time to an event. A standard approach for such analyses is the Cox proportional hazards model. It assumes that the factors act linearly on the log hazard function (linearity assumption) and that their effects are constant over time (proportional hazards (PH) assumption). Variable selection is often required to specify a more parsimonious model aiming to include only variables with an influence on the outcome. As follow-up increases the effect of a variable often gets weaker, which means that it varies in time. However, spurious time-varying effects may also be introduced by mismodelling other parts of the multivariable model, such as omission of an important covariate or an incorrect functional form of a continuous covariate. These issues interact. To check whether the effect of a variable varies in time several tests for non-PH have been proposed. However, they are not sufficient to derive a model, as appropriate modelling of the shape of time-varying effects is required. In three examples we will compare five recently published strategies to assess whether and how the effects of covariates from a multivariable model vary in time. For practical use we will give some recommendations.     

Continuous time Markov models for binary longitudinal data

Longitudinal data usually consist of a number of short time series. A group of subjects or groups of subjects are followed over time and observations are often taken at unequally spaced time points, and may be at different times for different subjects. When the errors and random effects are Gaussian, the likelihood of these unbalanced linear mixed models can be directly calculated, and nonlinear optimization used to obtain maximum likelihood estimates of the fixed regression coefficients and parameters in the variance components. For binary longitudinal data, a two state, non-homogeneous continuous time Markov process approach is used to model serial correlation within subjects. Formulating the model as a continuous time Markov process allows the observations to be equally or unequally spaced. Fixed and time varying covariates can be included in the model, and the continuous time model allows the estimation of the odds ratio for an exposure variable based on the steady state distribution. Exact likelihoods can be calculated. The initial probability distribution on the first observation on each subject is estimated using logistic regression that can involve covariates, and this estimation is embedded in the overall estimation. These models are applied to an intervention study designed to reduce children's sun exposure.     

Regularized estimation in the accelerated failure time model with high-dimensional covariates

We consider two regularization approaches, the LASSO and the threshold-gradient-directed regularization, for estimation and variable selection in the accelerated failure time model with multiple covariates based on Stute's weighted least squares method. The Stute estimator uses Kaplan-Meier weights to account for censoring in the least squares criterion. The weighted least squares objective function makes the adaptation of this approach to multiple covariate settings computationally feasible. We use V-fold cross-validation and a modified Akaike's Information Criterion for tuning parameter selection, and a bootstrap approach for variance estimation. The proposed method is evaluated using simulations and demonstrated on a real data example.