Parameter Estimation for Partially Complete Time and Type of Failure Data

The theory of competing risks has been developed to asses a specific risk in presence of other risk factors. In this paper we consider the parametric estimation of different failure modes under partially complete time and type of failure data using latent failure times and cause specific hazard functions models. Uniformly minimum variance unbiased estimators and maximum likelihood estimators are obtained when latent failure times and cause specific hazard functions are exponentially distributed. We also consider the case when they follow Weibull distributions. One data set is used to illustrate the proposed techniques. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic prediction of cumulative incidence functions by direct binomial regression

In recent years there have been a series of advances in the field of dynamic prediction. Among those is the development of methods for dynamic prediction of the cumulative incidence function in a competing risk setting. These models enable the predictions to be updated as time progresses and more information becomes available, for example when a patient comes back for a follow‐up visit after completing a year of treatment, the risk of death, and adverse events may have changed since treatment initiation. One approach to model the cumulative incidence function in competing risks is by direct binomial regression, where right censoring of the event times is handled by inverse probability of censoring weights. We extend the approach by combining it with landmarking to enable dynamic prediction of the cumulative incidence function. The proposed models are very flexible, as they allow the covariates to have complex time‐varying effects, and we illustrate how to investigate possible time‐varying structures using Wald tests. The models are fitted using generalized estimating equations. The method is applied to bone marrow transplant data and the performance is investigated in a simulation study.     

Competing isogenic Campylobacter strains exhibit variable population structures in vivo

Consumption of poultry contaminated with Campylobacter jejuni is a risk factor for human gastrointestinal disease. The rational development of control strategies for Campylobacter within chickens requires an understanding of the colonization process at the molecular and population levels, both within and between hosts. Experiments employing competing strains of Campylobacter have been used to investigate colonization. Implicit in these studies is the assumption that the behavior of competing strains is reproducible between experiments. Variability in the recovery of mutants from the chicken gastrointestinal tract during signature-tagged mutagenesis studies demonstrated that this is not always the case. To further investigate this phenomenon in the absence of confounding factors due to phenotypic differences between mutants, we constructed individually identifiable wild-type isogenic tagged strains (WITS) that have indistinguishable phenotypes in pure culture. By using mixtures of WITS, it is possible to monitor the relative amounts of subpopulations of essentially wild-type bacteria. Using a 2-week-old chicken model of colonization, we observed unpredictable variations in population structure both within and between experiments, even in the simplest case of two competing strains. This variation occurred both when birds were simultaneously infected with two WITS and when birds inoculated with different WITS were cohoused. We present evidence for founder effects during initial colonization with subsequent bird-to-bird transmission. We suggest that these and phenotypic variation contribute to the observed variability. These factors render simple models of colonization which do not take them into account inappropriate for Campylobacter and impact the planning and interpretation of competition experiments using this organism.     

On maximum likelihood solutions for exponential survival models

"Stochastic survival models which adjust for covariate information have been developed by Beck (1979). These models can include one or two living states and several competing death states. The transitions between stages are assumed irreversible and the transition intensity functions are assumed to be independent of time but dependent upon the covariates." Explicit solutions of the maximum likelihood equations for such models when there are one or two dichotomous covariates are presented. Applications of these models to the case of heart transplants and lung cancer are discussed, and survival in two or four groups is compared. (summary in FRE)     

Validation of discrete time-to-event prediction models in the presence of competing risks

Clinical prediction models play a key role in risk stratification, therapy assignment and many other fields of medical decision making. Before they can enter clinical practice, their usefulness has to be demonstrated using systematic validation. Methods to assess their predictive performance have been proposed for continuous, binary, and time-to-event outcomes, but the literature on validation methods for discrete time-to-event models with competing risks is sparse. The present paper tries to fill this gap and proposes new methodology to quantify discrimination, calibration, and prediction error (PE) for discrete time-to-event outcomes in the presence of competing risks. In our case study, the goal was to predict the risk of ventilator-associated pneumonia (VAP) attributed to Pseudomonas aeruginosa in intensive care units (ICUs). Competing events are extubation, death, and VAP due to other bacteria. The aim of this application is to validate complex prediction models developed in previous work on more recently available validation data.     

Butterflies tailor their ejaculate in response to sperm competition risk and intensity

Males of many insects eclose with their entire lifetime sperm supply and have to allocate their ejaculates at mating prudently. In polyandrous species, ejaculates of rival males overlap, creating sperm competition. Recent models suggest that males should increase their ejaculate expenditure when experiencing a high risk of sperm competition. Ejaculate expenditure is also predicted to vary in relation to sperm competition intensity. During high intensity, where several ejaculates compete for fertilization of the female''s eggs, ejaculate expenditure is expected to be reduced. This is because there are diminishing returns of providing more sperm. Additionally, sperm numbers will depend on males'' ability to assess female mating status. We investigate ejaculate allocation in the polyandrous small white butterfly Pieris rapae (Lepidoptera). Males have previously been found to ejaculate more sperm on their second mating when experiencing increased risk of sperm competition. Here we show that males also adjust the number of sperm ejaculated in relation to direct sperm competition. Mated males provide more sperm to females previously mated with mated males (i.e. when competing with many sperm) than to females previously mated to virgin males (competing with few sperm). Virgin males, on the other hand, do not adjust their ejaculate in relation to female mating history, but provide heavier females with more sperm. Although virgin males induce longer non-receptive periods in females than mated males, heavier females remate sooner. Virgin males may be responding to the higher risk of sperm competition by providing more sperm to heavier females. It is clear from this study that males are sensitive to factors affecting sperm competition risk, tailoring their ejaculates as predicted by recent theoretical models.     

Mechanisms of sperm competition: testing the fair raffle

Sperm competition is a major force of sexual selection, but its implications for mating system and life-history evolution are just beginning to be understood. Of particular importance is understanding the mechanisms of sperm competition. Models have been developed to determine if sperm competition operates in a fair raffle process, whereby each sperm from competing males has an equal chance of fertilizing a female's ova, or if it operates in a loaded raffle process, whereby one male's sperm has a fertilization advantage. These models require data on relative sperm and offspring (paternity) numbers of competing males. Here we develop a model based on maximum-likelihood methods for differentiating between the fair and loaded raffle processes. The model calculates the relative competitiveness of two males' sperm (loadings) as well as the economy of scale (nonlinear returns to sperm number). Previous models implicitly assumed that there is no economy of scale, which may not be the case when there is cooperation or interference among sperm from a given male. We demonstrate that our model has superior power-in some instances more than double-than previous models. We apply our model to an example of sperm competition in the guppy (Poecilia reticulata) and show that the system may be characterized by a loaded raffle attributable to effects of second male precedence.     

Analysis of a lottery competition model with limited nutrient availability

How a plant species utilizes a limited nutrient is important for its survival. The purpose of this work is to examine how nutrient utilization mechanisms (for seed production) affect the coexistence of competing plant species. We construct a revised lottery model that uses one of three possible kinds of nutrient utilization functions. In all cases the models suggest that two species can coexist under certain circumstances, but that three species cannot coexist, at least when the nutrient utilization functions are continuous functions of nutrient uptake. However, in the discontinuous case three species can coexist in a state of sustained oscillations. The results suggest that one need pay close attention to the differences in the nutrient utilization mechanisms among competing plant species in order to ascertain the competitive outcome.     

Analysis of a lottery competition model with limited nutrient availability

How a plant species utilizes a limited nutrient is important for its survival. The purpose of this work is to examine how nutrient utilization mechanisms (for seed production) affect the coexistence of competing plant species. We construct a revised lottery model that uses one of three possible kinds of nutrient utilization functions. In all cases the models suggest that two species can coexist under certain circumstances, but that three species cannot coexist, at least when the nutrient utilization functions are continuous functions of nutrient uptake. However, in the discontinuous case three species can coexist in a state of sustained oscillations. The results suggest that one need pay close attention to the differences in the nutrient utilization mechanisms among competing plant species in order to ascertain the competitive outcome.     

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.     

Adversarial interspecies relationships facilitate population suppression by gene drive in spatially explicit models

Suppression gene drives bias their inheritance to spread through a population, potentially eliminating it when they reach high frequency. CRISPR homing suppression drives have already seen success in the laboratory, but several models predict that success may be elusive in population with realistic spatial structure due to extinction-recolonization cycles. Here, we extend our continuous space framework to include two competing species or predator–prey pairs. We find that in both general and mosquito-specific models, competing species or predators can facilitate drive-based suppression, albeit at the cost of an increased rate of drive loss outcomes. These results are robust in mosquito models with seasonal fluctuations. Our study illustrates the difficulty of predicting outcomes in complex ecosystems. However, our results are promising for the prospects of less powerful suppression gene drives to successfully eliminate target mosquito and other pest populations.     

Competing risk analysis for life table data with known observation times

"Several models have been proposed for the analysis of cohort mortality in the presence of competing risks.... This paper describes a maximum likelihood approach to the analysis of follow up data in life table format for the case of two competing risks--a specific cause and its competing complement. The model developed uses a robust survivorship assumption--the piecewise exponential--and takes into account information on time to death and time to withdrawal." (summary in GER)     

Persistence in models of three interacting predator-prey populations

This paper considers a class of deterministic models of three interacting populations with a view towards determining when all of the populations persist. In analytical terms persistence means that liminf_t→∞x(t)> 0 for each population x(t); in geometric terms, that each trajectory of the modeling system of differential equations is eventually bounded away from the coordinate planes. The class of systems considered allows three level food webs, two competing predators feeding on a single prey, or a single predator feeding on two competing prey populations. As a corollary to the last case it is shown that the addition of a predator can lead to persistence of a three population system where, without a predator, the two competing populations on the lower trophic level would have only one survivor. The basic models are of Kolmogorov type, and the results improve several previous theorems on persistence.     

Independent and competing disease risks: implications for host populations in variable environments

Disease models usually assume disease to act independently of other mortality- and morbidity-causing factors. Alternatively, disease may function as a competing risk factor, for example, killing already moribund hosts. Using tuberculosis (TB) in African buffalo as a model system, we explore consequences of competing or independent disease effects for host population dynamics. We include scenarios with density-dependent and density-independent effects of environmental variation, exemplified by variable food availability (driven by rainfall) and catastrophic droughts, respectively. Independent disease effects reduce population size linearly with prevalence, irrespective of the nature of environmental variation. Competing disease risks alter population size only if density-independent variation is present; then, disease reduces population size nonlinearly. Field data indicate that the net effect of TB on buffalo likely falls between the extremes of total independence and competition with other risk factors: TB increases mortality and decreases fecundity in some prime-aged buffalo, suggesting independent disease risks in these individuals, while similar disease effects in senescent buffalo may act as competing risks. Moreover, increased survival and fecundity of TB-negative buffalo may compensate for some disease-related losses. Model assumptions on independent or competing disease risks and environmental variability should be considered explicitly when assessing disease effects on wildlife populations.     

Competing Risk Analysis of Censored Survival Data

A competing risk model is developed to accommodate both planned Type I censoring and random withdrawals. MLE's, their properties, confidence regions for parameters and mean lifetimes are obtained for a model regarding random censoring as a competing risk and compared to those obtained for the model in which withdrawals are regarded as random censoring. Estimated net and crude probabilities are calculated and compared for the two models. The model is developed for two competing risks, one following a Weibull distribution and the other a Rayleigh distribution, and random withdrawals following a Weibull distribution.     

Explained variation and degrees of necessity and of sufficiency for competing risks survival data

In this contribution, the Schemper–Henderson measure of explained variation for survival outcomes is extended to accommodate competing events (CEs) in addition to events of interest. The extension is achieved by moving from the unconditional and conditional survival functions of the original measure to unconditional and conditional cumulative incidence functions, the latter obtained, for example, from Fine and Gray models. In the absence of CEs, the original measure is obtained as a special case. We define explained variation on the population level and provide two different types of estimates. Recently, the authors have achieved a multiplicative decomposition of explained variation into degrees of necessity and degrees of sufficiency. These measures are also extended to the case of competing risks survival data. A SAS macro and an R function are provided to facilitate application. Interesting empirical properties of the measures are explored on the population level and by an extensive simulation study. Advantages of the approach are exemplified by an Austrian study of breast cancer with a high proportion of CEs.     

A bias in ML estimates of branch lengths in the presence of multiple signals

Sequence data often have competing signals that are detected by network programs or Lento plots. Such data can be formed by generating sequences on more than one tree, and combining the results, a mixture model. We report that with such mixture models, the estimates of edge (branch) lengths from maximum likelihood (ML) methods that assume a single tree are biased. Based on the observed number of competing signals in real data, such a bias of ML is expected to occur frequently. Because network methods can recover competing signals more accurately, there is a need for ML methods allowing a network. A fundamental problem is that mixture models can have more parameters than can be recovered from the data, so that some mixtures are not, in principle, identifiable. We recommend that network programs be incorporated into best practice analysis, along with ML and Bayesian trees.     

Predation in multi-prey communities

We consider the effect of a top predator on the stability of a system of competing prey species. In the first instance, this is done in detail for two prey species where the predators either behave in a completely random way, interfere with each other or switch to the more abundant prey at any time. The analysis is then extended to the case of n similar prey species, either competing equally or competing with their two nearest neighbours in exploiting a one-dimensional resource spectrum. It is found that predator switching can produce local stability when the prey species overlap completely and even when the competition coefficients are greater than one. This, however, is more difficult to attain for nearest neighbour competition. In either case switching is advantageous to the predators, since it allows them to coexist successfully with their prey over a wider range of conditions.     

Regression modeling of competing risks data based on pseudovalues of the cumulative incidence function

Typically, regression models for competing risks outcomes are based on proportional hazards models for the crude hazard rates. These estimates often do not agree with impressions drawn from plots of cumulative incidence functions for each level of a risk factor. We present a technique which models the cumulative incidence functions directly. The method is based on the pseudovalues from a jackknife statistic constructed from the cumulative incidence curve. These pseudovalues are used in a generalized estimating equation to obtain estimates of model parameters. We study the properties of this estimator and apply the technique to a study of the effect of alternative donors on relapse for patients given a bone marrow transplant for leukemia.     

Analysis of cause of death: Competing risks or progressive illness‐death model?

The analysis of cause of death is increasingly becoming a topic in oncology. It is usually distinguished between disease‐related and disease‐unrelated death. A frequently used approach is to define death as disease‐related when a progression to advanced phases has occurred before, otherwise as disease‐unrelated. The data are often analyzed as competing risks, while a progressive illness‐death model might in fact describe the situation more precisely. In this study, we investigated under which circumstances this misspecification leads to biased estimations of the state occupation probabilities. We simulated data according to the progressive illness‐death model in various settings, analyzed them with a competing risks model and with a progressive illness‐death model and compared them to the true state occupation probabilities. Censoring was either added independently of the status or based on the patients' status. The simulations showed that the censoring mechanism was decisive for the bias while neither the progression hazard nor the Markov property was important. Further, we found a slightly increased standard deviation for the competing risk estimator when censoring was independent of the patients' status. For illustration, both methods were applied to two practical examples of chronic myeloid leukemia (CML): one randomized controlled trial and one registry data set. While in the first case both estimators yielded almost identical results, in the latter case, visible differences were found between both methods.