Testing goodness of fit of a uniform truncation model

Several goodness-of-fit tests of a lifetime distribution have been suggested in the literature; many take into account censoring and/or truncation of event times. In some contexts, a goodness-of-fit test for the truncation distribution is of interest. In particular, better estimates of the lifetime distribution can be obtained when knowledge of the truncation law is exploited. In cross-sectional sampling, for example, there are theoretical justifications for the assumption of a uniform truncation distribution, and several studies have used it to improve the efficiency of their survival estimates. The duality of lifetime and truncation in the absence of censoring enables methods for testing goodness of fit of the lifetime distribution to be used for testing goodness of fit of the truncation distribution. However, under random censoring, this duality does not hold and different tests are required. In this article, we introduce several goodness-of-fit tests for the truncation distribution and investigate their performance in the presence of censored event times using simulation. We demonstrate the use of our tests on two data sets.     

Estimating the distribution of a renewal process from times at which events from an independent process are detected

The analysis of length-biased data has been mostly limited to the interarrival interval of a renewal process covering a specific time point. Motivated by a surveillance problem, we consider a more general situation where this time point is random and related to a specific event, for example, status change or onset of a disease. We also consider the problem when additional information is available on whether the event intervals (interarrival intervals covering the random event) end within or after a random time period (which we call a window period) following the random event. Under the assumptions that the occurrence rate of the random event is low and the renewal process is independent of the random event, we provide formulae for the estimation of the distribution of interarrival times based on the observed event intervals. Procedures for testing the required assumptions are also furnished. We apply our results to human immunodeficiency virus (HIV) test data from public test sites in Seattle, Washington, where the random event is HIV infection and the window period is from the onset of HIV infection to the time at which a less sensitive HIV test becomes positive. Results show that the estimator of the intertest interval length distribution from event intervals ending within the window period is less biased than the estimator from all event intervals; the latter estimator is affected by right truncation. Finally, we discuss possible applications to estimating HIV incidence and analyzing length-biased samples with right or left truncated data.     

Estimating incident population distribution from prevalent data

A prevalent sample consists of individuals who have experienced disease incidence but not failure event at the sampling time. We discuss methods for estimating the distribution function of a random vector defined at baseline for an incident disease population when data are collected by prevalent sampling. Prevalent sampling design is often more focused and economical than incident study design for studying the survival distribution of a diseased population, but prevalent samples are biased by design. Subjects with longer survival time are more likely to be included in a prevalent cohort, and other baseline variables of interests that are correlated with survival time are also subject to sampling bias induced by the prevalent sampling scheme. Without recognition of the bias, applying empirical distribution function to estimate the population distribution of baseline variables can lead to serious bias. In this article, nonparametric and semiparametric methods are developed for distribution estimation of baseline variables using prevalent data.     

Nonparametric and semiparametric estimation with sequentially truncated survival data

In observational cohort studies with complex sampling schemes, truncation arises when the time to event of interest is observed only when it falls below or exceeds another random time, that is, the truncation time. In more complex settings, observation may require a particular ordering of event times; we refer to this as sequential truncation. Estimators of the event time distribution have been developed for simple left-truncated or right-truncated data. However, these estimators may be inconsistent under sequential truncation. We propose nonparametric and semiparametric maximum likelihood estimators for the distribution of the event time of interest in the presence of sequential truncation, under two truncation models. We show the equivalence of an inverse probability weighted estimator and a product limit estimator under one of these models. We study the large sample properties of the proposed estimators and derive their asymptotic variance estimators. We evaluate the proposed methods through simulation studies and apply the methods to an Alzheimer's disease study. We have developed an R package, seqTrun , for implementation of our method.     

Nonparametric estimation of the cumulative incidences of competing risks under double truncation

Registry data typically report incident cases within a certain calendar time interval. Such interval sampling induces double truncation on the incidence times, which may result in an observational bias. In this paper, we introduce nonparametric estimation for the cumulative incidences of competing risks when the incidence time is doubly truncated. Two different estimators are proposed depending on whether the truncation limits are independent of the competing events or not. The asymptotic properties of the estimators are established, and their finite sample performance is investigated through simulations. For illustration purposes, the estimators are applied to childhood cancer registry data, where the target population is peculiarly defined conditional on future cancer development. Then, in our application, the cumulative incidences inform on the distribution by age of the different types of cancer.     

A mathematical model for the population dynamics of army ants

A stochastic cellular automata model for the population dynamics of the army antEciton burchelli on Barro Colorado Island in Panama is set up. It is simulated on the computer and shown to give good agreement with biological data. It is analysed using two approximations akin to the mean field approximation in statistical mechanics, and good agreement with the simulations is obtained. Finally, the role of distance between successive statary phase bivouacs is discussed with regard to the rate of colony growth. There are two aspects of the biological system studied here that make it of general importance. First, the population is structured, since the size of each colony of army ants is crucial. Second, the spatial behaviour of the population, as in many others, is not diffusion-like, although it is random. This has implications for the kind of model that is chosen.     

A model of microbial contamination of a water reservoir

A three year record of daily fecal coliform counts in a Massachusetts water reservoir has the appearance of an irregular time series punctuated by outbursts of varying duration. The pattern is described in terms of a probabilistic model where the fluctuations in the ‘regular’ and ‘explosive’ regimes are governed by two sets of probabilities. It has been assumed that the random oscillations has a lognormal distribution, and that once an explosion threshold has been exceeded the increments or decrements in the population size have fixed probability distributions. The threshold for triggering an outburst was estimated by examining the randomness of the autocorrelation function of the record after it is filtered to eliminate peaks of progressively increasing magnitude. Once the threshold has been identified, the mean and standard deviation of the underlying lognormal distribution could be estimated directly from remains found in the record after all the peaks were removed. The probabilities of an increment and decrement during the outbursts and their relative magnitudes could also be estimated using simple formulas. These estimated parameter values were then used to generate realistic records with known threshold levels, which were subsequently used to assess the procedure’s feasibility and sensitivity.     

Probabilistic Firing of Neurons Considered as a First Passage Problem

A formalized neuron receiving unitary excitatory impulses at random is considered. Each impulse provokes an effect of equal magnitude and of a duration not constant for each impulse, but which varies according to an exponential distribution. The effects sum until a threshold is reached when a response occurs. The distribution of intervals between successive responses is computed and compared with those obtained from a model in which the effects decay exponentially with time. Upon introducing inhibitory impulses also, the theory is applied to data on discharge characteristics of driven and spontaneously active thalamic neurons reported in the literature.     

RESPONSE TO SELECTION IN PARTIALLY SELF-FERTILIZING POPULATIONS. I. SELECTION ON A SINGLE TRAIT

Self-fertilization is a common form of reproduction in plants and it has important implications for quantitative trait evolution. Here, I present a model of selection on quantitative traits that can accommodate any level of self-fertilization. The “structured linear model” (SLM) predicts the evolution of the mean phenotype as a function of three distinct quantities: the mean additive genetic value, the directional dominance, and the mean inbreeding coefficient. Stochastic simulations of truncation selection demonstrate the accuracy of the SLM in predicting changes in the mean and variance of a quantitative trait over the full range of selfing rates. They also illustrate how complex interactions between selection and mating system determine the population distribution of inbreeding coefficients and also the amount of linkage disequilibrium. Changes in the genetic variance due to linkage disequilibria, which are commonly referred to as the “Bulmer effect,” are greatly magnified by selfing. This complicates the relationship between selfing rate and response to selection. Like the random mating theory, the parameters of the SLM can be estimated from phenotypic data.     

On estimating the probability of aperiodic outbursts of microbial populations from their fluctuating counts

The irregular sequence of counts of a microbial population, in the absence of observable corresponding environmental changes (e.g., temperature), can be regarded as reflecting the interplay of several unknown or random factors that favor or inhibit growth. Since these factors tend to balance one another, the fluctuations usually remain within bounds, and only by a coincidence—when all or most act in unison—does an ‘outburst’ occur. This situation can be represented mathematically as a sequence of independent random variables governed by a probability distribution. The concept was applied to reported microbial counts of ground meat and wastewater. It is found that the lognormal distribution could serve as a model, and that simulations from this model are indistinguishable from actual records. The parameters of the lognormal (or other) distribution can then be used to estimate the probability of a population outburst, i.e., an increase above a given threshold. Direct estimation of the outburst probability based on frequency of occurrence is also possible, but in some situations requires an impractically large number of observations. We compare the efficiency of these two methods of estimation. Such methods enable translation of irregular records of microbial counts into actual probabilities of an outburst of a given magnitude. Thus, if the environment remains ’stable’ or in dynamic equilibrium, the fluctuations should not be regarded merely as noise, but as a source of information and an indicator of potential population outbursts even where obvious signs do not exist.     

Behavioural ecology of mass recruitment in the army ant Neivamyrmex nigrescens

The successful rearing of the army ant Neivamyrmex nigrescens in the laboratory has enabled us to demonstrate that the pheromone trail deposited by recruiting workers is qualitatively different from the ants' exploratory trail. The recruitment trail alone can initiate as strong a mass recruitment response as can a recruiting worker that physically interacts with nestmates. The rapidity with which workers are aroused is due to secondary recruitment. Army ants are able to assemble a critical striking force before food is located, as a result of mass recruitment to new terrain. In addition to feeding behaviour, mass recruitment occurs when army ants emigrate to new nests. In both behavioural contexts, primary and secondary recruiters run more rapidly than exploring ants, and with exaggerated vertical motor patterns.     

Spatial Variation in Army Ant Swarm Raiding and its Potential Effect on Biodiversity

Invertebrate communities of the tropical rain forest floor are highly diverse, characterized by patchy species distribution patterns and high variation in species density. Spatial variation in the foraging activity of swarm raiding army ants, prime invertebrate predators in tropical rain forests, is discussed as a mechanism contributing to these patterns, but highly resolved long‐term data on army ant raiding on the local and landscape scale are hitherto lacking. In this study, 196 positions in 11 study sites in a tropical rain forest in western Kenya were continuously monitored over ~4 mo for the occurrence of swarm raids of army ants. Using population simulation analyses, the consequences of army ant raiding for prey communities were assessed. We found an unexpectedly high variation in raid rates at the study site and landscape scale. The weekly chance of communities to become raided by army ants was on average 0.11, but ranged from 0 to 0.50 among the 196 positions. Simulating population developments of two Lotka–Volterra species—showing slight trade‐offs between competitive strength and resistance to army ant raids—in the real raiding landscapes showed that the observed spatial variation in raid rates may produce high prey diversity at larger spatial scales (due to high β‐diversity) and strong variation in species density. Our results indicate that high spatial variation in army ant swarm raiding is a mechanism capable of generating patchy species distribution patterns and maintaining the high biodiversity of invertebrate communities of the tropical rain forest floor.     

A first principles derivation of animal group size distributions

Several empirical studies have shown that the animal group size distribution of many species can be well fit by power laws with exponential truncation. A striking empirical result due to Niwa is that the exponent in these power laws is one and the truncation is determined by the average group size experienced by an individual. This distribution is known as the logarithmic distribution. In this paper we provide first principles derivation of the logarithmic distribution and other truncated power laws using a site-based merge and split framework. In particular, we investigate two such models. Firstly, we look at a model in which groups merge whenever they meet but split with a constant probability per time step. This generates a distribution similar, but not identical to the logarithmic distribution. Secondly, we propose a model, based on preferential attachment, that produces the logarithmic distribution exactly. Our derivation helps explain why logarithmic distributions are so widely observed in nature. The derivation also allows us to link splitting and joining behavior to the exponent and truncation parameters in power laws.     

演替顶极阶段森林群落优势树种分布的变动趋势研究

本项研究通过对典型顶极森林群落和群落演替过程分析，表明处于演替顶极阶段顶极群落发育过程中，优势树种总体分布有由集群向随机的扩散趋势，各优势树种在此过程也由集群分布减幅波动地扩散为随机分布，而且是朝着优势树种种间联结关系和相关关系减弱的方向随机扩散。发育成熟的顶极森林群落中，优势树种的总体呈随机分布格局，各优势树种也呈随机分布格局镶嵌于总体的随机格局中。这种镶嵌形式，可使绝大多数优势树种种间联结关系和线性相关关系均达不到显著程度。该分布格局反映在成熟的顶级群落中，绝大多数优势树种个体间的相互影响很小，对微环境选择的差异较大     

Merits of Modelling Multivariate Survival Data Using Random Effects Proportional Odds Model

Recently, there has been a great deal of interest in the analysis of multivariate survival data. In most epidemiological studies, survival times of the same cluster are related because of some unobserved risk factors such as the environmental or genetic factors. Therefore, modelling of dependence between events of correlated individuals is required to ensure a correct inference on the effects of treatments or covariates on the survival times. In the past decades, extension of proportional hazards model has been widely considered for modelling multivariate survival data by incorporating a random effect which acts multiplicatively on the hazard function. In this article, we consider the proportional odds model, which is an alternative to the proportional hazards model at which the hazard ratio between individuals converges to unity eventually. This is a reasonable property particularly when the treatment effect fades out gradually and the homogeneity of the population increases over time. The objective of this paper is to assess the influence of the random effect on the within‐subject correlation and the population heterogeneity. We are particularly interested in the properties of the proportional odds model with univariate random effect and correlated random effect. The correlations between survival times are derived explicitly for both choices of mixing distributions and are shown to be independent of the covariates. The time path of the odds function among the survivors are also examined to study the effect of the choice of mixing distribution. Modelling multivariate survival data using a univariate mixing distribution may be inadequate as the random effect not only characterises the dependence of the survival times, but also the conditional heterogeneity among the survivors. A robust estimate for the correlation of the logarithm of the survival times within a cluster is obtained disregarding the choice of the mixing distributions. The sensitivity of the estimate of the regression parameter under a misspecification of the mixing distribution is studied through simulation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-parametric estimation of the age-at-onset distribution from a cross-sectional sample

We propose and study a simple and innovative non-parametric approach to estimate the age-of-onset distribution for a disease from a cross-sectional sample of the population that includes individuals with prevalent disease. First, we estimate the joint distribution of two event times, the age of disease onset and the survival time after disease onset. We accommodate that individuals had to be alive at the time of the study by conditioning on their survival until the age at sampling. We propose a computationally efficient expectation–maximization (EM) algorithm and derive the asymptotic properties of the resulting estimates. From these joint probabilities we then obtain non-parametric estimates of the age-at-onset distribution by marginalizing over the survival time after disease onset to death. The method accommodates categorical covariates and can be used to obtain unbiased estimates of the covariate distribution in the source population. We show in simulations that our method performs well in finite samples even under large amounts of truncation for prevalent cases. We apply the proposed method to data from female participants in the Washington Ashkenazi Study to estimate the age-at-onset distribution of breast cancer associated with carrying BRCA1 or BRCA2 mutations.     

A study of the effects of truncation in bivariate normal distribution

The moments of bivariate normal distribution, which is truncated with respect to both the random variables, are obtained by using the orthogonal expansion of the distribution and the properties of HERMITE polynomials. In particular the correlation coefficient of the truncated distribution is derived in terms of the actual correlation coefficient. In order to study the effect of truncation tables have been prepared of this correlation coefficient for certain given values of the actual correlation coefficient and for a few selected values of the points of truncation. A listing of the computer program for this purpose is also given.     

Probability functions for components of theDendroctonus frontalis—Host tree population system and their potential use with population models

Detailed investigation of the within-tree population system of Dendroctonus frontalisZimmermann has resulted in a large data base consisting of abundance estimates for various life stages. This data base was used to construct histograms for transformed estimates and several life stage indices. Histograms were also constructed for transformed values of adult residence time, brood development time, and several host-tree characteristics. Probability and cumulative density functions of the Weibull distribution were fitted, in tandem, to the scaled frequencies and interval means for each histogram. The inverse cumulative function is known, and with a uniform random number generator, allows the selection of random deviates from each distribution. This technique can be used for generating initial (starting) values in D. frontalis population models.     

Constructing gene co-expression networks and predicting functions of unknown genes by random matrix theory

Background  

Large-scale sequencing of entire genomes has ushered in a new age in biology. One of the next grand challenges is to dissect the cellular networks consisting of many individual functional modules. Defining co-expression networks without ambiguity based on genome-wide microarray data is difficult and current methods are not robust and consistent with different data sets. This is particularly problematic for little understood organisms since not much existing biological knowledge can be exploited for determining the threshold to differentiate true correlation from random noise. Random matrix theory (RMT), which has been widely and successfully used in physics, is a powerful approach to distinguish system-specific, non-random properties embedded in complex systems from random noise. Here, we have hypothesized that the universal predictions of RMT are also applicable to biological systems and the correlation threshold can be determined by characterizing the correlation matrix of microarray profiles using random matrix theory.     

Quantile regression for longitudinal data using the asymmetric Laplace distribution

In longitudinal studies, measurements of the same individuals are taken repeatedly through time. Often, the primary goal is to characterize the change in response over time and the factors that influence change. Factors can affect not only the location but also more generally the shape of the distribution of the response over time. To make inference about the shape of a population distribution, the widely popular mixed-effects regression, for example, would be inadequate, if the distribution is not approximately Gaussian. We propose a novel linear model for quantile regression (QR) that includes random effects in order to account for the dependence between serial observations on the same subject. The notion of QR is synonymous with robust analysis of the conditional distribution of the response variable. We present a likelihood-based approach to the estimation of the regression quantiles that uses the asymmetric Laplace density. In a simulation study, the proposed method had an advantage in terms of mean squared error of the QR estimator, when compared with the approach that considers penalized fixed effects. Following our strategy, a nearly optimal degree of shrinkage of the individual effects is automatically selected by the data and their likelihood. Also, our model appears to be a robust alternative to the mean regression with random effects when the location parameter of the conditional distribution of the response is of interest. We apply our model to a real data set which consists of self-reported amount of labor pain measurements taken on women repeatedly over time, whose distribution is characterized by skewness, and the significance of the parameters is evaluated by the likelihood ratio statistic.