The Design of Intervention Trials Involving Recurrent and Terminal Events

Clinical trials are often designed to assess the effect of therapeutic interventions on the incidence of recurrent events in the presence of a dependent terminal event such as death. Statistical methods based on multistate analysis have considerable appeal in this setting since they can incorporate changes in risk with each event occurrence, a dependence between the recurrent event and the terminal event, and event-dependent censoring. To date, however, there has been limited development of statistical methods for the design of trials involving recurrent and terminal events. Based on the asymptotic distribution of regression coefficients from a multiplicative intensity Markov regression model, we derive sample size formulas to address power requirements for both the recurrent and terminal event processes. We consider the design of trials for which separate marginal hypothesis tests are of interest for the recurrent and terminal event processes and deal with both superiority and non-inferiority tests. Simulation studies confirm that the designs satisfy the nominal power requirements in both settings, and an application to a trial evaluating the effect of a bisphosphonate on skeletal complications is given for illustration.     

Competitive or weak cooperative stochastic Lotka–Volterra systems conditioned on non-extinction

We are interested in the long time behavior of a two-type density-dependent biological population conditioned on non-extinction, in both cases of competition or weak cooperation between the two species. This population is described by a stochastic Lotka–Volterra system, obtained as limit of renormalized interacting birth and death processes. The weak cooperation assumption allows the system not to blow up. We study the existence and uniqueness of a quasi-stationary distribution, that is convergence to equilibrium conditioned on non-extinction. To this aim we generalize in two-dimensions spectral tools developed for one-dimensional generalized Feller diffusion processes. The existence proof of a quasi-stationary distribution is reduced to the one for a d-dimensional Kolmogorov diffusion process under a symmetry assumption. The symmetry we need is satisfied under a local balance condition relying the ecological rates. A novelty is the outlined relation between the uniqueness of the quasi-stationary distribution and the ultracontractivity of the killed semi-group. By a comparison between the killing rates for the populations of each type and the one of the global population, we show that the quasi-stationary distribution can be either supported by individuals of one (the strongest one) type or supported by individuals of the two types. We thus highlight two different long time behaviors depending on the parameters of the model: either the model exhibits an intermediary time scale for which only one type (the dominant trait) is surviving, or there is a positive probability to have coexistence of the two species.     

Cumulant dynamics of a population under multiplicative selection, mutation, and drift

We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation, and drift. The number of beneficial alleles in a multilocus system can be considered a trait under exponential selection. Equations of motion are derived for the cumulants of the trait distribution in the diffusion limit and under the assumption of linkage equilibrium. Because of the additive nature of cumulants, this reduces to the problem of determining equations of motion for the expected allele distribution cumulants at each locus. The cumulant equations form an infinite dimensional linear system and in an authored appendix Adam Prügel-Bennett provides a closed form expression for these equations. We derive approximate solutions which are shown to describe the dynamics well for a broad range of parameters. In particular, we introduce two approximate analytical solutions: (1) Perturbation theory is used to solve the dynamics for weak selection and arbitrary mutation rate. The resulting expansion for the system's eigenvalues reduces to the known diffusion theory results for the limiting cases with either mutation or selection absent. (2) For low mutation rates we observe a separation of time-scales between the slowest mode and the rest which allows us to develop an approximate analytical solution for the dominant slow mode. The solution is consistent with the perturbation theory result and provides a good approximation for much stronger selection intensities.     

On the fixation process of a beneficial mutation in a variable environment

A population that adapts to gradual environmental change will typically experience temporal variation in its population size and the selection pressure. On the basis of the mathematical theory of inhomogeneous branching processes, we present a framework to describe the fixation process of a single beneficial allele under these conditions. The approach allows for arbitrary time-dependence of the selection coefficient s(t) and the population size N(t), as may result from an underlying ecological model. We derive compact analytical approximations for the fixation probability and the distribution of passage times for the beneficial allele to reach a given intermediate frequency. We apply the formalism to several biologically relevant scenarios, such as linear or cyclic changes in the selection coefficient, and logistic population growth. Comparison with computer simulations shows that the analytical results are accurate for a large parameter range, as long as selection is not very weak.     

Statistical methods for panel data from a semi-Markov process, with application to HPV

Continuous-time, multistate processes can be used to represent a variety of biological processes in the public health sciences; yet the analysis of such processes is complex when they are observed only at a limited number of time points. Inference methods for such panel data have been developed for time homogeneous Markov models, but there has been little research done for other classes of processes. We develop likelihood-based methods for panel data from a semi-Markov process, where transition intensities depend on the duration of time in the current state. The proposed methods account for possible misclassification of states. To illustrate the methods, we investigate a three- and a four-state models in detail and apply the results to model the natural history of oncogenic genital human papillomavirus infections in women.     

The stationary distribution of allele frequencies when selection acts at unlinked loci

We consider population genetics models where selection acts at a set of unlinked loci. It is known that if the fitness of an individual is multiplicative across loci, then these loci are independent. We consider general selection models, but assume parent-independent mutation at each locus. For such a model, the joint stationary distribution of allele frequencies is proportional to the stationary distribution under neutrality multiplied by a known function of the mean fitness of the population. We further show how knowledge of this stationary distribution enables direct simulation of the genealogy of a sample at a single-locus. For a specific selection model appropriate for complex disease genes, we use simulation to determine what features of the genealogy differ between our general selection model and a multiplicative model.     

Bistability versus bimodal distributions in gene regulatory processes from population balance

In recent times, stochastic treatments of gene regulatory processes have appeared in the literature in which a cell exposed to a signaling molecule in its environment triggers the synthesis of a specific protein through a network of intracellular reactions. The stochastic nature of this process leads to a distribution of protein levels in a population of cells as determined by a Fokker-Planck equation. Often instability occurs as a consequence of two (stable) steady state protein levels, one at the low end representing the "off" state, and the other at the high end representing the "on" state for a given concentration of the signaling molecule within a suitable range. A consequence of such bistability has been the appearance of bimodal distributions indicating two different populations, one in the "off" state and the other in the "on" state. The bimodal distribution can come about from stochastic analysis of a single cell. However, the concerted action of the population altering the extracellular concentration in the environment of individual cells and hence their behavior can only be accomplished by an appropriate population balance model which accounts for the reciprocal effects of interaction between the population and its environment. In this study, we show how to formulate a population balance model in which stochastic gene expression in individual cells is incorporated. Interestingly, the simulation of the model shows that bistability is neither sufficient nor necessary for bimodal distributions in a population. The original notion of linking bistability with bimodal distribution from single cell stochastic model is therefore only a special consequence of a population balance model.     

Known unknowns in an imperfect world: incorporating uncertainty in recruitment estimates using multi‐event capture–recapture models

Studying the demography of wild animals remains challenging as several of the critical parts of their life history may be difficult to observe in the field. In particular, determining with certainty when an individual breeds for the first time is not always obvious. This can be problematic because uncertainty about the transition from a prebreeder to a breeder state – recruitment – leads to uncertainty in vital rate estimates and in turn in population projection models. To avoid this issue, the common practice is to discard imperfect data from the analyses. However, this practice can generate a bias in vital rate estimates if uncertainty is related to a specific component of the population and reduces the sample size of the dataset and consequently the statistical power to detect effects of biological interest. Here, we compared the demographic parameters assessed from a standard multistate capture–recapture approach to the estimates obtained from the newly developed multi‐event framework that specifically accounts for uncertainty in state assessment. Using a comprehensive longitudinal dataset on southern elephant seals, we demonstrated that the multi‐event model enabled us to use all the data collected (6639 capture–recapture histories vs. 4179 with the multistate model) by accounting for uncertainty in breeding states, thereby increasing the precision and accuracy of the demographic parameter estimates. The multi‐event model allowed us to incorporate imperfect data into demographic analyses. The gain in precision obtained has important implications in the conservation and management of species because limiting uncertainty around vital rates will permit predicting population viability with greater accuracy.     

Ergodicity-breaking reveals time optimal decision making in humans

Ergodicity describes an equivalence between the expectation value and the time average of observables. Applied to human behaviour, ergodic theories of decision-making reveal how individuals should tolerate risk in different environments. To optimize wealth over time, agents should adapt their utility function according to the dynamical setting they face. Linear utility is optimal for additive dynamics, whereas logarithmic utility is optimal for multiplicative dynamics. Whether humans approximate time optimal behavior across different dynamics is unknown. Here we compare the effects of additive versus multiplicative gamble dynamics on risky choice. We show that utility functions are modulated by gamble dynamics in ways not explained by prevailing decision theories. Instead, as predicted by time optimality, risk aversion increases under multiplicative dynamics, distributing close to the values that maximize the time average growth of in-game wealth. We suggest that our findings motivate a need for explicitly grounding theories of decision-making on ergodic considerations.     

Role of demographic dynamics and conflict in the population-area relationship for human languages

Many patterns displayed by the distribution of human linguistic groups are similar to the ecological organization described for biological species. It remains a challenge to identify simple and meaningful processes that describe these patterns. The population size distribution of human linguistic groups, for example, is well fitted by a log-normal distribution that may arise from stochastic demographic processes. As we show in this contribution, the distribution of the area size of home ranges of those groups also agrees with a log-normal function. Further, size and area are significantly correlated: the number of speakers p and the area a spanned by linguistic groups follow the allometric relation a proportional to p2, with an exponent z varying accross different world regions. The empirical evidence presented leads to the hypothesis that the distributions of p and a, and their mutual dependence, rely on demographic dynamics and on the result of conflicts over territory due to group growth. To substantiate this point, we introduce a two-variable stochastic multiplicative model whose analytical solution recovers the empirical observations. Applied to different world regions, the model reveals that the retreat in home range is sublinear with respect to the decrease in population size, and that the population-area exponent z grows with the typical strength of conflicts. While the shape of the population size and area distributions, and their allometric relation, seem unavoidable outcomes of demography and inter-group contact, the precise value of z could give insight on the cultural organization of those human groups in the last thousand years.     

Persistence in fluctuating environments

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations’ invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka–Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.

One day is fine, the next is black.—The Clash     

Population processes under the influence of disasters occurring independently of population size

Markov branching processes and in particular birth-and-death processes are considered under the influence of disasters that arrive independently of the present population size. For these processes we derive an integral equation involving a shifted and rescaled argument. The main emphasis, however, is on the (random) probability of extinction. Its distribution density satisfies an equation which can be solved numerically at least up to a multiplicative constant. In an example it is also found by simulation.     

Attributable risk estimation for adjusted disability multistate models: Application to nosocomial infections

Attributable risk has become an important concept in clinical epidemiology. In this paper, we suggest to estimate the attributable risk of nosocomial infections using a multistate approach. Recently, a multistate model (called progressive disability model in the literature) has been developed in order to take into consideration both the time‐dependency of the risk factor (e.g., nosocomial infections) and the presence of competing risks (e.g., death and discharge) at each time point. However, this approach does not take into account the possible heterogeneity of the study population. In this paper, we investigate an extension of this model and suggest an adjusted disability multistate model including covariates in each transition. This new multistate model has led us to define the concepts of overall and profiled attributable risk. We use a classical semiparametric approach to estimate the model and the new attributable risk. A simulation study is investigated and we show, in particular, that neglecting the presence of covariates when estimating the model can lead to an important bias. The methodology developed in this paper is applied to data on ventilator‐associated pneumonia in 12 French intensive care units.     

Ancestral processes in population genetics-the coalescent

A special stochastic process, called the coalescent, is of fundamental interest in population genetics. For a large class of population models this process is the appropriate tool to analyse the ancestral structure of a sample of n individuals or genes, if the total number of individuals in the population is sufficiently large. A corresponding convergence theorem was first proved by Kingman in 1982 for the Wright-Fisher model and the Moran model. Generalizations to a large class of exchangeable population models and to models with overlying mutation processes followed shortly later. One speaks of the "robustness of the coalescent, as this process appears in many models as the total population size tends to infinity. This publication can be considered as an introduction to the theory of the coalescent as well as a review of the most important "convergence-to-the-coalescent-theorems. Convergence theorems are not only presented for the classical exchangeable haploid case but also for larger classes of population models, for example for diploid, two-sex or non-exchangeable models. A review-like summary of further examples and applications of convergence to the coalescent is given including the most important biological forces like mutation, recombination and selection. The general coalescent process allows for simultaneous multiple mergers of ancestral lines.     

Convergence of the age structure: Applications of the projective metric

This paper states necessary and sufficient conditions for the convergence of the age structure (in a discrete time, one-sex model of population growth); it also contains a new and simple proof of the weak ergodic theorem of stable population theory. The main tool used to attain these results is Hilbert's notion of the projective metric. This metric provides a way of defining the distance between positive vectors in $\text{R}$ⁿ which has two important features: First, the distance between any two positive vectors depends only on the rays on which the vectors lie; and, second, positive matrices act as contractions in this metric.     

On the 'rate of aging' in heterogeneous populations

It is well known that the rate of aging is constant for populations described by the Gompertz law of mortality. However, this is true only when a population is homogeneous. In this note, we consider the multiplicative frailty model with the baseline distribution that follows the Gompertz law and study the impact of heterogeneity on the rate of aging in this population. We show that the rate of aging in this case is a function of age and that it increases in (calendar) time when the baseline mortality rate decreases.     

Convergence of one-dimensional diffusion processes to a jump process related to population genetics

A conjecture on the convergence of diffusion models in population genetics to a simple Markov chain model is proved. The notion of bi-generalized diffusion processes and their limit theorems are used systematically to prove the conjecture. Three limits; strong selection-weak mutation limit, moderate selection-weak mutation limit, weak selection-weak mutation limit are considered for typical diffusion models in population genetics.Supported in part by Research Grant from the Ministry of Education, Science and Culture of JapanSupported by the Air Force Office of Scientific Research Contract No. F49620 85C 0144     

Measures of success in a class of evolutionary models with fixed population size and structure

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth–death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.     

A bayesian approach to the multistate Jolly-Seber capture-recapture model

This article considers a Bayesian approach to the multistate extension of the Jolly-Seber model commonly used to estimate population abundance in capture-recapture studies. It extends the work of George and Robert (1992, Biometrika79, 677-683), which dealt with the Bayesian estimation of a closed population with only a single state for all animals. A super-population is introduced to model new entrants in the population. Bayesian estimates of abundance are obtained by implementing a Gibbs sampling algorithm based on data augmentation of the missing data in the capture histories when the state of the animal is unknown. Moreover, a partitioning of the missing data is adopted to ensure the convergence of the Gibbs sampling algorithm even in the presence of impossible transitions between some states. Lastly, we apply our methodology to a population of fish to estimate abundance and movement.