Multinomial mixture model with heterogeneous classification probabilities

Royle and Link (Ecology 86(9):2505?C2512, 2005) proposed an analytical method that allowed estimation of multinomial distribution parameters and classification probabilities from categorical data measured with error. While useful, we demonstrate algebraically and by simulations that this method yields biased multinomial parameter estimates when the probabilities of correct category classifications vary among sampling units. We address this shortcoming by treating these probabilities as logit-normal random variables within a Bayesian framework. We use Markov chain Monte Carlo to compute Bayes estimates from a simulated sample from the posterior distribution. Based on simulations, this elaborated Royle-Link model yields nearly unbiased estimates of multinomial and correct classification probability estimates when classification probabilities are allowed to vary according to the normal distribution on the logit scale or according to the Beta distribution. The method is illustrated using categorical submersed aquatic vegetation data.     

An incidence-based richness estimator for quadrats sampled without replacement

Most richness estimators currently in use are derived from models that consider sampling with replacement or from the assumption of infinite populations. Neither of the assumptions is suitable for sampling sessile organisms such as plants where quadrats are often sampled without replacement and the area of study is always limited. In this paper, we propose an incidence-based parametric richness estimator that considers quadrat sampling without replacement in a fixed area. The estimator is derived from a zero-truncated binomial distribution for the number of quadrats containing a given species (e.g., species i) and a modified beta distribution for the probability of presence-absence of a species in a quadrat. The maximum likelihood estimate of richness is explicitly given and can be easily solved. The variance of the estimate is also obtained. The performance of the estimator is tested against nine other existing incidence-based estimators using two tree data sets where the true numbers of species are known. Results show that the new estimator is insensitive to sample size and outperforms the other methods as judged by the root mean squared errors. The superiority of the new method is particularly noticeable when large quadrat size is used, suggesting that a few large quadrats are preferred over many small ones when sampling diversity.     

Analyse „stetigkeitsbasierter“ Methoden zur Bestimmung von Stofffrachten in Fließgewässern

Aim and background

Increasing ecological awareness puts the load as a tool for water quality assessment in the foreground. Therefore it is very desirable to find optimal procedures for any given situation on basis of existing data. Criteria will be derived to fill that desire.

Method

The load for a water ingredient is an integrated entity of time-continuous loads (product of concentration and flow) for a given time, i.?e. a year (yearly load). Normally you can only observe a sample (in a statistical meaning) of these data. Therefore, load calculations vary with the concrete sample given. Each such calculation represents an estimation of the load, afflicted with some uncertainty. This uncertainty can be quantified, at least approximatively, by adding variance (scatter around “expected” mean) and the square of bias (divergence of “expected” mean from true mean). This quantity is called “mean squared error”. The methodological results will be illustrated by examples.

Focus

The main focus lies on the methodologies, which, besides data of concentrations observed discretely in time, additionally use quasi-continuously recorded data of other variates, namely the flow. For that situation, statistical models are developed and the load derived and estimated according to existing data. Hereby it is seen that all model based estimation formulae can be described as standard sampling formulae plus some additional terms which stem from comparison of some measurements (i.?e. mean) for quasi-continuous and discrete samples. This way a “continuity-correction” is performed on the standard formulae for the sampling case. This method is called “continuity-based” load determination.

Results

A surprising result is that no complicated mathematical-statistical models are necessary to determine the load. A simple linear concentration- or load-flow-model is sufficient. Additional variates lead, most of the time, to a larger mean squared error in load estimation, even when it is well known that they cannot detoriate the model fit. Only models which describe the concentration extraordinarily precise, near perfection, result in a smaller error.

Implications

It is enough to use one of the given formulae with some “continuity correction” to calculate a load. That can be done without using any software.     

Estimation of Normal Variance Based on Balanced and Unbalanced Ranked Set Samples

In this paper we address the problem of estimation of the variance of a normal population based on a balanced as well as an unbalanced ranked set sample (RSS), which is a modification of the original RSS of McIntyre (1952).We have proposed several methods of estimation of variance by combining different unbiased between and within estimators, and compared their performances     

Modelling species distributions with penalised logistic regressions: A comparison with maximum entropy models

An important aspect of species distribution modelling is the choice of the modelling method because a suboptimal method may have poor predictive performance. Previous comparisons have found that novel methods, such as Maxent models, outperform well-established modelling methods, such as the standard logistic regression. These comparisons used training samples with small numbers of occurrences per estimated model parameter, and this limited sample size may have caused poorer predictive performance due to overfitting. Our hypothesis is that Maxent models would outperform a standard logistic regression because Maxent models avoid overfitting by using regularisation techniques and a standard logistic regression does not. Regularisation can be applied to logistic regression models using penalised maximum likelihood estimation. This estimation procedure shrinks the regression coefficients towards zero, causing biased predictions if applied to the training sample but improving the accuracy of new predictions. We used Maxent and logistic regression (standard and penalised) to analyse presence/pseudo-absence data for 13 tree species and evaluated the predictive performance (discrimination) using presence-absence data. The penalised logistic regression outperformed standard logistic regression and equalled the performance of Maxent. The penalised logistic regression may be considered one of the best methods to develop species distribution models trained with presence/pseudo-absence data, as it is comparable to Maxent. Our results encourage further use of the penalised logistic regression for species distribution modelling, especially in those cases in which a complex model must be fitted to a sample with a limited size.     

Fourth-corner correlation is a score test statistic in a log-linear trait–environment model that is useful in permutation testing

Ecologists wish to understand the role of traits of species in determining where each species occurs in the environment. For this, they wish to detect associations between species traits and environmental variables from three data tables, species count data from sites with associated environmental data and species trait data from data bases. These three tables leave a missing part, the fourth-corner. The fourth-corner correlations between quantitative traits and environmental variables, heuristically proposed 20 years ago, fill this corner. Generalized linear (mixed) models have been proposed more recently as a model-based alternative. This paper shows that the squared fourth-corner correlation times the total count is precisely the score test statistic for testing the linear-by-linear interaction in a Poisson log-linear model that also contains species and sites as main effects. For multiple traits and environmental variables, the score test statistic is proportional to the total inertia of a doubly constrained correspondence analysis. When the count data are over-dispersed compared to the Poisson or when there are other deviations from the model such as unobserved traits or environmental variables that interact with the observed ones, the score test statistic does not have the usual chi-square distribution. For these types of deviations, row- and column-based permutation methods (and their sequential combination) are proposed to control the type I error without undue loss of power (unless no deviation is present), as illustrated in a small simulation study. The issues for valid statistical testing are illustrated using the well-known Dutch Dune Meadow data set.     

A capture-recapture model with heterogeneity and behavioural response

We develop the non-parametric maximum likelihood estimator (MLE) of the full M_bh capture-recapture model which utilizes both initial capture and recapture data and permits both heterogeneity (h) between animals and behavioural (b) response to capture. Our MLE procedure utilizes non-parametric maximum likelihood estimation of mixture distributions (Lindsay, 1983; Lindsay and Roeder, 1992) and the EM algorithm (Dempsteret al., 1977). Our MLE estimate provides the first non-parametric estimate of the bivariate capture-recapture distribution.Since non-parametric maximum likelihood estimation exists for submodels M_h (allowing heterogeneity only), Mb (allowing behavioural response only) and M₀ (allowing no changes), we develop maximum likelihood-based model selection, specifically the Akaike information criterion (AIC) (Akaike, 1973). The AIC procedure does well in detecting behavioural response but has difficulty in detecting heterogeneity.     

Density estimation using ranked-set sampling data

The ranked-set sampling (RSS) is applicable in practical problems where the variable of interest for an observed item is costly or time-consuming but the ranking of a set of items according to the variable can be easily done without actual measurement. In the context of RSS, the need for density estimation arises in certain statistical procedures. The density estimation also has its own interest. In this article, we develop a method for the density estimation using RSS data. We derive the properties of the resulted density estimate and compare it with its counterpart in simple random sampling (SRS). It is shown that the density estimate using RSS data provides a better estimate of the density than the usual density estimate using SRS data. The density estimate developed in this article can well serve various purposes in the context of RSS.     

Sampling variability and estimates of density dependence: a composite-likelihood approach

It is well known that sampling variability, if not properly taken into account, affects various ecologically important analyses. Statistical inference for stochastic population dynamics models is difficult when, in addition to the process error, there is also sampling error. The standard maximum-likelihood approach suffers from large computational burden. In this paper, I discuss an application of the composite-likelihood method for estimation of the parameters of the Gompertz model in the presence of sampling variability. The main advantage of the method of composite likelihood is that it reduces the computational burden substantially with little loss of statistical efficiency. Missing observations are a common problem with many ecological time series. The method of composite likelihood can accommodate missing observations in a straightforward fashion. Environmental conditions also affect the parameters of stochastic population dynamics models. This method is shown to handle such nonstationary population dynamics processes as well. Many ecological time series are short, and statistical inferences based on such short time series tend to be less precise. However, spatial replications of short time series provide an opportunity to increase the effective sample size. Application of likelihood-based methods for spatial time-series data for population dynamics models is computationally prohibitive. The method of composite likelihood is shown to have significantly less computational burden, making it possible to analyze large spatial time-series data. After discussing the methodology in general terms, I illustrate its use by analyzing a time series of counts of American Redstart (Setophaga ruticilla) from the Breeding Bird Survey data, San Joaquin kit fox (Vulpes macrotis mutica) population abundance data, and spatial time series of Bull trout (Salvelinus confluentus) redds count data.     

Using hidden Markov chains and empirical Bayes change-point estimation for transect data

Consider a lattice of locations in one dimension at which data are observed. We model the data as a random hierarchical process. The hidden process is assumed to have a (prior) distribution that is derived from a two-state Markov chain. The states correspond to the mean values (high and low) of the observed data. Conditional on the states, the observations are modelled, for example, as independent Gaussian random variables with identical variances. In this model, there are four free parameters: the Gaussian variance, the high and low mean values, and the transition probability in the Markov chain. A parametric empirical Bayes approach requires estimation of these four parameters from the marginal (unconditional) distribution of the data and we use the EM-algorithm to do this. From the posterior of the hidden process, we use simulated annealing to find the maximum a posteriori (MAP) estimate. Using a Gibbs sampler, we also obtain the maximum marginal posterior probability (MMPP) estimate of the hidden process. We use these methods to determine where change-points occur in spatial transects through grassland vegetation, a problem of considerable interest to plant ecologists.     

Design based estimation for ranked set sampling in finite populations

In this paper, we consider design-based estimation using ranked set sampling (RSS) in finite populations. We first derive the first and second-order inclusion probabilities for an RSS design and present two Horvitz–Thompson type estimators using these inclusion probabilities. We also develop an alternate Hansen–Hurwitz type estimator and investigate its properties. In particular, we show that this alternate estimator always outperforms the usual Hansen–Hurwitz type estimator in the simple random sampling with replacement design with comparable sample size. We also develop formulae for ratio estimator for all three developed estimators. The theoretical results are augmented by numerical and simulation studies as well as a case study using a well known data set. These show that RSS design can yield a substantial improvement in efficiency over the usual simple random sampling design in finite populations.     

Continuous dose-response modeling and risk analysis with the gamma and reciprocal gamma distributions

Kodell and West (1993) describe two methods for calculating pointwise upper confidence limits on the risk function with normally distributed responses and using a certain definition of adverse quantitative effect. But Banga et al. (2000) have shown that these normal theory methods break down when applied to skew data. We accordingly develop a risk analysis model and associated likelihood-based methodology when the response follows either a gamma or reciprocal gamma distribution. The model supposes that the shape (index) parameter k of the response distribution is held fixed while the logarithm of the scale parameter is a linear model in terms of the dose level. Existence and uniqueness of the maximum likelihood estimates is established. Asymptotic likelihood-based upper and lower confidence limits on the risk are solutions of the Lagrange equations associated with a constrained optimization problem. Starting values for an iterative solution are obtained by replacing the Lagrange equations by the lowest order terms in their asymptotic expansions. Three methods are then compared for calculating confidence limits on the risk: (i) the aforementioned starting values (LRAL method), (ii) full iterative solution of the Lagrange equations (LREL method), and (iii) bounds obtained using approximate normality of the maximum likelihood estimates with standard errors derived from the information matrix (MLE method). Simulation is used to assess coverage probabilities for the resulting upper confidence limits when the log of the scale parameter is quadratic in the dose level. Results indicate that coverage for the MLE method can be off by as much as 15% points and converges very slowly to nominal coverage levels as the sample size increases. Coverage for the LRAL and LREL methods, on the other hand, is close to nominal levels unless (a) the sample size is small, say N < 25, (b) the index parameter is small, say k 1, and (c) the direction of adversity is to the left for the gamma distribution or to the right for the reciprocal gamma distribution.     

On parametric estimation of population abundance for line transect sampling

Despite recent advances in nonparametric methods for estimating animal abundance, parametric methods are still used widely among biometricians due to their simplicity. In this paper, we propose an optimal shrinkage-type estimator and an empirical Bayes estimator for estimating animal density from line transect sampling data. The performances of the proposed estimators are compared with those of the maximum likelihood estimator and a bias-corrected maximum likelihood estimator both theoretically and numerically. Simulation results show that the optimal shrinkage-type estimator works the best if the detection function has a very thin tail (for example, the half normal detection function), while the maximum likelihood estimator is the best estimator if the detection function has relatively thick tail (for example, the polynomial detection function).     

Estimating survival rates with time series of standing age-structure data

It has long been recognized that age-structure data contain useful information for assessing the status and dynamics of wildlife populations. For example, age-specific survival rates can be estimated with just a single sample from the age distribution of a stable, stationary population. For a population that is not stable, age-specific survival rates can be estimated using techniques such as inverse methods that combine time series of age-structure data with other demographic data. However, estimation of survival rates using these methods typically requires numerical optimization, a relatively long time series of data, and smoothing or other constraints to provide useful estimates. We developed general models for possibly unstable populations that combine time series of age-structure data with other demographic data to provide explicit maximum likelihood estimators of age-specific survival rates with as few as two years of data. As an example, we applied these methods to estimate survival rates for female bison (Bison bison) in Yellowstone National Park, USA. This approach provides a simple tool for monitoring survival rates based on age-structure data.     

Methods for joint inference from panel survey and demographic data

A number of methods for joint inference from animal abundance and demographic data have been proposed in recent years, each with its own advantages. A new approach to analyzing panel survey and demographic data simultaneously is described. The approach fits population-dynamics models to the survey data, rather than to a single index of abundance derived from them and thus avoids disadvantages inherent in analyzing such an index. The methodology is developed and illustrated with British Lapwing data, and the results are compared with those obtained from existing approaches. The estimates of demographic parameters and population indices are similar for all methods. The results of a simulation study show that the new method performs well in terms of mean squared error.     

Nonparametric mean estimation using partially ordered sets

In ranked-set sampling (RSS), the ranker must give a complete ranking of the units in each set. In this paper, we consider a modification of RSS that allows the ranker to declare ties. Our sampling method is simply to break the ties at random so that we obtain a standard ranked-set sample, but also to record the tie structure for use in estimation. We propose several different nonparametric mean estimators that incorporate the tie information, and we show that the best of these estimators is substantially more efficient than estimators that ignore the ties. As part of our comparison of estimators, we develop new results about models for ties in rankings. We also show that there are settings where, to achieve more efficient estimation, ties should be declared not just when the ranker is actually unsure about how units rank, but also when the ranker is sure about the ranking, but believes that the units are close.     

Moment and maximum likelihood estimators for Weibull distributions under length- and area-biased sampling

Many of the most popular sampling schemes used in forestry are probability proportional to size methods. These methods are also referred to as size-biased because sampling is actually from a weighted form of the underlying population distribution. Length- and area-biased sampling are special cases of size-biased sampling where the probability weighting comes from a lineal or areal function of the random variable of interest, respectively. Often, interest is in estimating a parametric probability density of the data. In forestry, the Weibull function has been used extensively for such purposes. Estimating equations for method of moments and maximum likelihood for two- and three-parameter Weibull distributions are presented. Fitting is illustrated with an example from an area-biased angle-gauge sample of standing trees in a woodlot. Finally, some specific points concerning the form of the size-biased densities are reported.     

A generalized approach to modeling and estimating indirect effects in ecology

The need to model and test hypotheses about complex ecological systems has led to a steady increase in use of path analytical techniques, which allow the modeling of multiple multivariate dependencies reflecting hypothesized causation and mechanisms. The aim is to achieve the estimation of direct, indirect, and total effects of one variable on another and to assess the adequacy of whole models. Path analytical techniques based on maximum likelihood currently used in ecology are rarely adequate for ecological data, which are often sparse, multi-level, and may contain nonlinear relationships as well as nonnormal response data such as counts or proportion data. Here I introduce a more flexible approach in the form of the joint application of hierarchical Bayes, Markov chain Monte Carlo algorithms, Shipley's d-sep test, and the potential outcomes framework to fit path models as well as to decompose and estimate effects. An example based on the direct and indirect interactions between ants, two insect herbivores, and a plant species demonstrates the implementation of these techniques, using freely available software.     

Nonparametric MLE incorporation of heterogeneity and model testing into premarked cohort studies

We utilize mixture models and nonparametric maximum likelihood estimation to both develop a likelihood ratio test (lrt) for a common simplifying assumption and to allow heterogeneity within premarked cohort studies. Our methods allow estimation of the entire probability model and thus one can not only estimate many parameters of interest but one can also bootstrap from the estimated model to predict many things, including the standard deviations of estimators. Simulations suggest that our lrt has the appropriate protection for Type I error and often has good power. In practice, our lrt is important for determining the appropriateness of estimators and in examining if a simple design with only one capture period could be utilized for a future similar study.     

Weighted distributions and estimation of resource selection probability functions

Understanding how organisms selectively use resources is essential for designing wildlife management strategies. The probability that an individual uses a given resource, as characterized by environmental factors, can be quantified in terms of the resource selection probability function (RSPF). The present literature on the topic has claimed that, except when both used and unused sites are known, the RSPF is non-estimable and that only a function proportional to RSPF, namely, the resource selection function (RSF) can be estimated. This paper describes a close connection between the estimation of the RSPF and the estimation of the weight function in the theory of weighted distributions. This connection can be used to obtain fully efficient, maximum likelihood estimators of the resource selection probability function under commonly used survey designs in wildlife management. The method is illustrated using GPS collar data for mountain goats (Oreamnos americanus de Blainville 1816) in northwest British Columbia, Canada.