Modelling and analysis of recruitment distributions

Recruitment data for 18 marine fish stocks are smoothed using 10 parametric families of probability distributions. Comparative fit of the 10 families is assessed by means of the maximized log-likelihood. Results indicate that the gamma distribution provides an overall good fit in the right-hand tail of the data, but that some adjustment to the gamma distribution is called for in the left-hand tail. Weight functions and weighted distributions are suggested as one means of achieving the needed adjustment.     

Modelling skewed data with many zeros: A simple approach combining ordinary and logistic regression

We discuss a method for analyzing data that are positively skewed and contain a substantial proportion of zeros. Such data commonly arise in ecological applications, when the focus is on the abundance of a species. The form of the distribution is then due to the patchy nature of the environment and/or the inherent heterogeneity of the species. The method can be used whenever we wish to model the data as a response variable in terms of one or more explanatory variables. The analysis consists of three stages. The first involves creating two sets of data from the original: one shows whether or not the species is present; the other indicates the logarithm of the abundance when it is present. These are referred to as the presence data and the log-abundance data, respectively. The second stage involves modelling the presence data using logistic regression, and separately modelling the log-abundance data using ordinary regression. Finally, the third stage involves combining the two models in order to estimate the expected abundance for a specific set of values of the explanatory variables. A common approach to analyzing this sort of data is to use a ln (y+c) transformation, where c is some constant (usually one). The method we use here avoids the need for an arbitrary choice of the value of c, and allows the modelling to be carried out in a natural and straightforward manner, using well-known regression techniques. The approach we put forward is not original, having been used in both conservation biology and fisheries. Our objectives in this paper are to (a) promote the application of this approach in a wide range of settings and (b) suggest that parametric bootstrapping be used to provide confidence limits for the estimate of expected abundance.     

Using the bootstrap and fast Fourier transform to estimate confidence intervals of 2D kernel densities

Kernel density estimators are often used to estimate the utilization distributions (UDs) of animals. Kernel UD estimates have a strong theoretical basis and perform well, but are usually reported without estimates of error or uncertainty. It is intuitively and theoretically appealing to estimate the sampling error in kernel UD estimates using bootstrapping. However, standard equations for kernel density estimates are complicated and computationally expensive. Bootstrapping requires computing hundreds or thousands of probability densities and is impractical when the number of observations, or the area of interest is large. We used the fast Fourier transform (FFT) and discrete convolution theorem to create a bootstrapping algorithm fast enough to run on commonly available desktop or laptop computers. Application of the FFT method to a large (n>20,000) set of radio telemetry data would provide a 99.6% reduction in computation time (i.e., 1.6 as opposed to 444 hours) for 1000 bootstrap UD estimates. Bootstrap error contours were computed using data from a radio-collared polar bear (Ursus maritimus) in the Beaufort Sea north of Alaska.     

Maximum likelihood method for parameter estimation in non-linear models with below detection data

The maximum likelihood (ML) method for regression analyzes of censored data (below detection limit) for nonlinear models is presented. The proposed ML method has been translated into an equivalent least squares method (ML-LS). A two stage iterative algorithm is proposed to estimate statistical parameters from the derived least squares translation. The developed algorithm is applied to a nonlinear model for prediction of ambient air CO concentration in terms of concentrations of respirable particulate matter (RSPM) and NO₂. It has been shown that if censored data are ignored or estimated through simplifications such as (i) censored data are equal to detection limit, (ii) censored data are half of the difference between detection limit and lower limit (e.g., zero or background level) or (iii) censored data are equal to lower limit, this can cause significant bias in estimated parameters. The developed ML-LS method provided better estimates of parameters than any of the simplifications in censored data.