Otso Ovaskainen, Hanna Rekola, Evgeniy Meyke, and Elja Arjas. 2008. Bayesian methods for analyzing movements in heterogeneous landscapes from mark–recapture data. Ecology 89:542–554.

Appendix A. A comparison between the maximum likelihood and Bayesian approaches.

Here we compare the Bayesian framework developed in this paper with the maximum likelihood (ML) method developed by Ovaskainen (2004). To do so, we reanalyzed the false heath fritillary butterfly (Melitaea diamina) considered by Ovaskainen (2004). The data consist of movements of 557 males and 285 females in a network of 14 habitat patches. While Ovaskainen (2004) considered only males, we analyze here both data sets.

We follow the Model A of Ovaskainen (2004), which classified the landscape simply to habitat patches and unsuitable matrix. We scale the habitat preference for the patches to 1, and denote the relative preference for the matrix by k. We assume that diffusivity (a) and mortality (c) do not depend on the habitat type. As the researchers spent more time per unit area in patch 1 than in the other patches, we estimate capture probability p separately for patch 1 and for the remaining 13 patches.

Figure A1 and Table A1 summarize the posterior distributions of the model parameters. Capture probability was much higher in patch 1 than in the other patches. In line with the result for the Glanville fritillary butterfly (main text), capture probability was higher for males than for females, P(p^M>p^F) = 0.997. In contrast to the Glanville fritillary data, mortality rate was higher for males than for females P(c^M>c^F) = 0.993. As expected, the parameters k and a are negatively correlated in the posterior distribution. While the marginal distributions for these parameters do not indicate large differences between the sexes, the joint distributions show no overlap. Males are more mobile in the sense that they move faster (high a) and leave the habitat patches more readily (high k).

Comparison between the estimates obtained by the Bayesian and ML methods is presented in Table A1. One should bear in mind, however, that the Bayesian credibility intervals and the confidence intervals derived in the case of ML estimation, here computed by applying bootstrapping, have completely different meanings in statistical inference. The Baysian credibility intervals provide a direct probabilistic evaluation of the "true" values of the model parameters, such as P(0.10<c^M<0.13) = 0.95. In contrast, the confidence intervals obtained by bootstrapping refer to sample variability in repeated sampling under similar circumstances. Thus the generally somewhat shorter confidence intervals, especially for the correlated parameters k and a, should not be taken as evidence of that the ML estimates are in some sense "more accurate".

A definite advantage of the Bayesian inferential methods used here is that they allow one to directly consider joint distributions of all model parameters of interest, thereby marginalizing, in each case, over the "nuisance parameters" that are left out. The comparisons of some parameter values for males and females that are reported above illustrate such possibilities. A further advantage of the Bayesian method is that it provides a direct possibility to assess uncertainty in joint distributions of several parameters (see, e.g., Fig. A1e).

The ML estimate roughly coincides with the median estimate of the Bayesian method, except for the parameters k and a, which show a substantial deviation. The deviation can be partly explained with the correlation structure (Fig. A1e), which makes the area of high likelihood appear as a narrow ridge extending over a large range of the paramaters a and k. As the straightforward maximization algorithm of Ovaskainen (2004) did not account for the correlation structure when locating the maximum likelihood, it is possible that the ML estimate includes more numerical error than the current estimates.

TABLE A1. Estimated parameter values obtained by the maximum-likelihood (ML) method and by the Bayesian method. In the ML method, the maximum likelihood and its 95% confidence interval based on bootstrapping is shown (from Ovaskainen 2004). In the Bayesian method, the median and 95% highest posterior density interval is shown.

† The parameters are the relative preference for the unsuitable matrix (k), diffusivity (a), mortality (c), and capture probability p. The capture probability was estimated separately for patch 1 in which the researchers spent more time per unit area (Ovaskainen 2004).

FIG. A1. Posterior densities for model parameters based on data on the false heath fritillary butterfly (Melitaea diamina). The panels a–d depict the marginal posterior distributions for the parameters measuring relative preference for the unsuitable matrix k (a), diffusivity a (b), mortality c (c), and capture probability p (d). Red lines correspond to females, blue lines to males, black dashed lines to prior distributions. In panel d, the colored dashed lines correspond to patch 1 for which capture probability was estimated separately. Panel e shows the joint posterior distributions of the parameters k and a. The two ellipses show the 50% and 95% quantiles of the prior.

LITERATURE CITED

Ovaskainen, O. 2004. Habitat-specific movement parameters estimated using mark–recapture data and a diffusion model. Ecology 85:242–257.