Monte Carlo estimation of stage structured development from cohort data.

Cohort data are frequently collected to study stage-structured development and mortalities of many organisms, particularly arthropods. Such data can provide information on mean stage durations, among-individual variation in stage durations, and on mortality rates. Current statistical methods for cohort data lack flexibility in the specification of stage duration distributions and mortality rates. In this paper, we present a new method for fitting models of stage-duration distributions and mortality to cohort data. The method is based on a Monte Carlo within MCMC algorithm and provides Bayesian estimates of parameters of stage-structured cohort models. The algorithm is computationally demanding but allows for flexible specifications of stage-duration distributions and mortality rates. We illustrate the algorithm with an application to data from a previously published experiment on the development of brine shrimp from Mono Lake, California, through nine successive stages. In the experiment, three different food supply and temperature combination treatments were studied. We compare the mean duration of the stages among the treatments while simultaneously estimating mortality rates and among-individual variance of stage durations. The method promises to enable more detailed studies of development of both natural and experimental cohorts. An R package implementing the method and which allows flexible specification of stage duration distributions is provided.

Estimation of stage duration distributions and mortality under repeated cohort censuses.

Many processes in nature can be viewed as arising from subjects progressing through sequential stages and may be described by multistage models. Examples include disease development and the physiological development of plants and animals. We develop a multistage model for sampling designs where a small set of subjects is followed and the number of subjects in each stage is assessed repeatedly for a sequence of time points, but for which the subjects cannot be identified. The motivating problem is the laboratory study of developing arthropods through stage frequency data. Our model assumes that the same individuals are censused at each time, introducing among sample dependencies. This type of data often occur in laboratory studies of small arthropods but their detailed analysis has received little attention. The likelihood of the model is derived from a stochastic model of the development and mortality of the individuals in the cohort. We present an MCMC scheme targeting the posterior distribution of the times of development and times of death of individuals. This is a novel type of MCMC that uses customized proposals to explore a posterior with disconnected support arising from the fact that individual identities are unknown. The MCMC algorithm may be used for inference about parameters governing stage duration distributions and mortality rates. The method is demonstrated by fitting the development model to stage frequency data of a mealybug cohort placed on a grape vine.

Using uncertainty estimates in analyses of population time series.

Recording and monitoring wildlife is crucial for the conservation of wild species and the protection of their environment. The most common type of information reported from a monitoring scheme is a time series of population abundance estimates, but the potential of such data for analyzing population dynamics is limited due to lack of information on sampling error. Recent work has shown that replicating the sampling process and analyzing replicates jointly in a dynamical model can considerably increase estimation efficiency compared to analyzing population estimates alone. This method requires that independent replicates are available, and model fitting can be complex in general. Often, however, population estimates are accompanied by standard errors, or standard errors may be estimated from raw data using a sampling model. We evaluate a method where standard errors are used in combination with population estimates to account for sampling variability in state-space models of population dynamics. The method is simple and lends itself readily to data derived from many sampling procedures but ignores uncertainty in the standard errors themselves. We simulate data from a Gaussian state-space model where several observations, which may come from different sites, are available for the population at each time. Fitting the simulated data, we show that the method yields similar or even better results than a method utilizing all observations, even when there are few observations at each time. This holds under a range of simulation settings involving heteroscedastic observation error, site effects, and correlation among observations. We illustrate the approach on real data from the North American Breeding Bird Survey and show that it performs well in comparison to a more difficult maximum-likelihood analysis of the full data under non-Gaussian sampling error.

Fitting complex population models by combining particle filters with Markov chain Monte Carlo.

We show how a recent framework combining Markov chain Monte Carlo (MCMC) with particle filters (PFMCMC) may be used to estimate population state-space models. With the purpose of utilizing the strengths of each method, PFMCMC explores hidden states by particle filters, while process and observation parameters are estimated using an MCMC algorithm. PFMCMC is exemplified by analyzing time series data on a red kangaroo (Macropus rufus) population in New South Wales, Australia, using MCMC over model parameters based on an adaptive Metropolis-Hastings algorithm. We fit three population models to these data; a density-dependent logistic diffusion model with environmental variance, an unregulated stochastic exponential growth model, and a random-walk model. Bayes factors and posterior model probabilities show that there is little support for density dependence and that the random-walk model is the most parsimonious model. The particle filter Metropolis-Hastings algorithm is a brute-force method that may be used to fit a range of complex population models. Implementation is straightforward and less involved than standard MCMC for many models, and marginal densities for model selection can be obtained with little additional effort. The cost is mainly computational, resulting in long running times that may be improved by parallelizing the algorithm.

Stochastic development in biologically structured population models.

Variation in organismal development is ubiquitous in nature but omitted from most age- and stage-structured population models. I give a general approach for formulating and analyzing its role in density-independent population models using the framework of integral projection models. The approach allows flexible assumptions, including correlated development times among multiple life stages. I give a new Monte Carlo numerical integration approach to calculate long-term growth rate, its sensitivities, stable age-stage distributions and reproductive value. This method requires only simulations of individual life schedules, rather than iteration of full population dynamics, and has practical and theoretical appeal because it ties easily implemented simulations to numerical solution of demographic equations. I show that stochastic development is demographically important using two examples. For a desert cactus, many stochastic development models, with independent or correlated stage durations, can generate the same stable stage distribution (SSD) as the real data, but stable age-within-stage distributions and sensitivities of growth rate to demographic rates differ greatly among stochastic development scenarios. For Mediterranean fruit flies, empirical variation in maturation time has a large impact on population growth. The systematic model formulation and analysis approach given here should make consideration of variable development models widely accessible and readily extendible.

A simulation-approximation approach to sample size planning for high-dimensional classification studies.

Classification studies with high-dimensional measurements and relatively small sample sizes are increasingly common. Prospective analysis of the role of sample sizes in the performance of such studies is important for study design and interpretation of results, but the complexity of typical pattern discovery methods makes this problem challenging. The approach developed here combines Monte Carlo methods and new approximations for linear discriminant analysis, assuming multivariate normal distributions. Monte Carlo methods are used to sample the distribution of which features are selected for a classifier and the mean and variance of features given that they are selected. Given selected features, the linear discriminant problem involves different distributions of training data and generalization data, for which 2 approximations are compared: one based on Taylor series approximation of the generalization error and the other on approximating the discriminant scores as normally distributed. Combining the Monte Carlo and approximation approaches to different aspects of the problem allows efficient estimation of expected generalization error without full simulations of the entire sampling and analysis process. To evaluate the method and investigate realistic study design questions, full simulations are used to ask how validation error rate depends on the strength and number of informative features, the number of noninformative features, the sample size, and the number of features allowed into the pattern. Both approximation methods perform well for most cases but only the normal discriminant score approximation performs well for cases of very many weakly informative or uninformative dimensions. The simulated cases show that many realistic study designs will typically estimate substantially suboptimal patterns and may have low probability of statistically significant validation results.

Field-scale roles of density, temperature, nitrogen, and predation on aphid population dynamics.

Robust analyses of noisy, stage-structured, irregularly spaced, field-scale data incorporating multiple sources of variability and nonlinear dynamics remain very limited, hindering understanding of how small-scale studies relate to large-scale population dynamics. We used a novel, complementary Bayesian and frequentist state-space model analysis to ask how density, temperature, plant nitrogen, and predators affect cotton aphid (Aphis gossypii) population dynamics in weekly data from 18 field-years and whether estimated effects are consistent with small-scale studies. We found clear roles of density and temperature but not of plant nitrogen or predators, for which Bayesian and frequentist evidence differed. However, overall predictability of field-scale dynamics remained low. This study demonstrates stage-structured state-space model analysis incorporating bottom-up, top-down, and density-dependent effects for within-season (nearly continuous time), nonlinear population dynamics. The analysis combines Bayesian posterior evidence with maximum-likelihood estimation and frequentist hypothesis testing using average one-step-ahead residuals.