Alan Welsh

Modelling the abundance of rare species: statistical models for counts with extra zeros

Abstract We consider several statistical models for the analysis of the abundance of a rare species and these are illustrated in detail with data for the abundance of Leadbeaters Possum in montane ash forests of south-eastern Australia. These data are characterised by a discrete distribution with the zero class inflated. In many statistical problems the parameters of this distribution depend on covariates, such as the number of hollow bearing trees. We advocate a conditional model which is simple to interpret and readily fitted. We show how to obtain standard errors for the parameter estimates. We also show how to estimate the mean abundance of animals at a site. The methods outlined in this paper offer a powerful framework for the study of problems having a discrete response (like abundance) with the zero class inflated.

Generalized additive modelling and zero inflated count data

This paper describes a flexible method for modelling zero inflated count data which are typically found when trying to model and predict species distributions. Zero inflated data are defined as data that has a larger proportion of zeros than expected from pure count (Poisson) data. The standard methodology is to model the data in two steps, first modelling the association between the presence and absence of a species and the available covariates and second, modelling the relationship between abundance and the covariates, conditional on the organism being present. The approach in this paper extends previous work to incorporate the use of Generalized Additive Models (GAM) in the modelling steps. The paper develops the link and variance functions needed for the use of GAM with zero inflated data. It then demonstrates the performance of the models using data on stem counts of Eucalyptus mannifera in a region of South East Australia.

Local Estimating Equations

Abstract Estimating equations have found wide popularity recently in parametric problems, yielding consistent estimators with asymptotically valid inferences obtained via the sandwich formula. Motivated by a problem in nutritional epidemiology, we use estimating equations to derive nonparametric estimators of a “parameter” depending on a predictor. The nonparametric component is estimated via local polynomials with loess or kernel weighting; asymptotic theory is derived for the latter. In keeping with the estimating equation paradigm, variances of the nonparametric function estimate are estimated using the sandwich method, in an automatic fashion, without the need (typical in the literature) to derive asymptotic formulas and plug-in an estimate of a density function. The same philosophy is used in estimating the bias of the nonparametric function; that is, an empirical method is used without deriving asymptotic theory on a case-by-case basis. The methods are applied to a series of examples. The applicatio...

Testing for Glaucoma with the spatial frequency doubling illusion

We examined the performance of tests for glaucoma based on the spatial frequency doubling (FD) illusion. Contrast thresholds for seeing the FD illusion in four large visual field regions were measured from 340 subjects who were tested up to seven times over 2 years. Median sensitivities of 91% at specificities of 95% were obtained. Test-retest variability for the worst hemifield thresholds averaged 2.22 db +/- 0.09 S.E. for all tested groups, and significant progression was observed for glaucoma suspects over the seven visits, indicating that tests based on the FD illusion can detect diffuse early glaucomatous loss.

Robust Restricted Maximum Likelihood in Mixed Linear Models

SUMMARY Definitions of robust maximum likelihood (robust ML) and robust restricted maximum likelihood (robust REML) are introduced, and the definitions are applied to data from biological and chemical experiments. A simulation study is undertaken to investigate the asymptotic properties of robust ML and robust REML in small samples and to examine the advantages of using robust methods. 1. Introduction and Definitions Linear models with multiple sources of error are widely used in designed experiments across many scientific fields. An example of such an experiment is described by Patterson and Nabugoomu (1992), from Patterson and Silvey (1980). Six varieties of wheat were grown at ten centres that formed a sample of the main types of growing area for wheat in Scotland, and the yields in tonnes/hectare were recorded. The experiment is unbalanced because, of 60 possible variety-centre combinations, only 46 were used. At seven centres, four varieties were grown and at the remaining three centres, all six varieties were grown. In this paper we fit the simplest mixed linear model proposed for this data by Patterson and Nabugoomu, namely:

Fitting and Interpreting Occupancy Models

We show that occupancy models are more difficult to fit than is generally appreciated because the estimating equations often have multiple solutions, including boundary estimates which produce fitted probabilities of zero or one. The estimates are unstable when the data are sparse, making them difficult to interpret, and, even in ideal situations, highly variable. As a consequence, making accurate inference is difficult. When abundance varies over sites (which is the general rule in ecology because we expect spatial variance in abundance) and detection depends on abundance, the standard analysis suffers bias (attenuation in detection, biased estimates of occupancy and potentially finding misleading relationships between occupancy and other covariates), asymmetric sampling distributions, and slow convergence of the sampling distributions to normality. The key result of this paper is that the biases are of similar magnitude to those obtained when we ignore non-detection entirely. The fact that abundance is subject to detection error and hence is not directly observable, means that we cannot tell when bias is present (or, equivalently, how large it is) and we cannot adjust for it. This implies that we cannot tell which fit is better: the fit from the occupancy model or the fit ignoring the possibility of detection error. Therefore trying to adjust occupancy models for non-detection can be as misleading as ignoring non-detection completely. Ignoring non-detection can actually be better than trying to adjust for it.

Marginal longitudinal nonparametric regression: Locality and efficiency of spline and kernel methods

We consider nonparametric regression in a longitudinal marginal model of generalized estimating equation (GEE) type with a time-varying covariate in the situation where the number of observations per subject is finite and the number of subjects is large. In such models, the basic shape of the regression function is affected only by the covariate values and not otherwise by the ordering of the observations. Two methods of estimating the nonparametric function can be considered: kernel methods and spline methods. Recently, surprising evidence has emerged suggesting that for kernel methods previously proposed in the literature, it is generally asymptotically preferable to ignore the correlation structure in our marginal model and instead assume that the data are independent, that is, working independence in the GEE jargon. As seen through equivalent kernel results, in univariate independent data problems splines and kernels have similar behavior; smoothing splines are equivalent to kernel regression with a specific higher-order kernel, and hence smoothing splines are local. This equivalence suggests that in our marginal model, working independence might be preferable for spline methods. Our results suggest the opposite; via theoretical and numerical calculations, we provide evidence suggesting that for our marginal model, marginal smoothing and penalized regression splines are not local in their behavior. In contrast to the kernel results, our evidence suggests that when using spline methods, it is worthwhile to account for the correlation structure. Our results also suggest that spline methods appear to be more efficient than the previously proposed kernel methods for our marginal model.

Nonparametric function estimation of the relationship between two repeatedly measured variables

We describe methods for estimating the regression function nonparametrically and for estimating the variance components in a simple variance component model which is sometimes used for repeated measures data or data with a simple clustered structure. We consider a number of different ways of estimating the regression function. The main results are that the simple pooled estimator which treats the data as independent performs very well asymptotically but that we can construct estimators which perform better asymptotically in some circumstances.

Modelling correlated zero-inflated count data

This paper extends the two-component approach to modelling count data with extra zeros, considered by Mullahy (1986), Heilbron (1994) and Welsh et al. (1996), to take account of possible serial dependence between repeated observations. Generalized estimating equations (Liang & Zeger, 1986) are constructed for each component of the model by incorporating correlation matrices into each of the maximum likelihood estimating equations. The proposed method is demonstrated on weekly counts of Noisy Friarbirds (Philemon cornic-ulatus), which were recorded by observers for the Canberra Garden Bird Survey (Hermes, 1981)

Contrasting mammal responses to vegetation type and fire

The response of terrestrial mammals and arboreal marsupials to past burning history as well as a year prior to, and then for 4 years after, a major wildfire in 2003 at Booderee National Park, Jervis Bay Territory was quantified. The present study encompassed extensive repeated surveys at a set of 109 replicated sites stratified by vegetation type and fire history. It was found that most species exhibited significant differences in presence and abundance between major vegetation types. Detections of long-nosed bandicoot (Perameles nasuta) increased significantly in all vegetation types surveyed, in both burnt and unburnt areas. Temporal patterns in captures of three species of small mammals (bush rat (Rattus fuscipes), swamp rat (Rattus lutreolus) and brown antechinus (Antechinus stuartii)) showed a trend for lower numbers of captures on burnt sites compared with unburnt sites. Three species of arboreal marsupials, common ringtail possum (Pseudocheirus peregrinus), greater glider (Petauroides volans) and common brushtail possum (Trichosurus vulpecula), were moderately common and all showed marked differences in abundance between vegetation types. Whereas P. peregrinus and P. volans exhibited a temporal decline between 2003 and 2006, T. vulpecula exhibited a general increase from 2003 levels. However, arboreal marsupial responses did not appear to be directly fire related.