Appendix A. An extension of the eigenanalysis of selection ratios for design-III data.
We demonstrate here how the eigenanalysis of selection ratios may be extended to the analysis of design-III data. For design II, this analysis consists in the eigenanalysis of the triplet (W, P, D). It is straightforward to show that this method is equivalent to the eigenanalysis of the triplet (S, IJJ, D), where IJJ is the J×J identity matrix (square matrix with 1 on the diagonal and 0 elsewhere), and S (I rows × J columns) is the following matrix:
In designs III, availability is defined for each animal. Let pi/j be the proportion of habitat type i available to animal j so that
The selection ratios for designs III may be computed by
, for habitat type i and animal j (Manly et al. 2002).
Let T (I rows × J columns):
Then, the eigenanalysis of selection ratios for design-III data is the eigenanalysis of the triplet (T, IJJ, D). The origin of the column space, i.e., a row vector of 0 of length I, corresponds to a hypothetical habitat type that is used randomly by all animals. The origin of the row space, i.e., a column vector of 0 of length I, corresponds to a hypothetical habitat type that is used randomly by all animals. The case where a habitat type is not available to an animal, i.e. where pi/j = 0, is considered at the end of the appendix. The total inertia of this analysis is equal to
Thus the inertia of this eigenanalysis is again equal to the standard Chi-square statistic recommended by White and Garrott (1990) to test habitat selection with design-III data. This analysis is therefore an extension of the method proposed to analyze design-II data. Note that in some occasions, the availability pi/j may be equal to zero for some animals (e.g. when a given habitat type is absent for some animals whereas it is available for others). In this case, the selection ratio cannot be computed. When this situation occurs, two solutions can be considered.
One first option is to replace the missing values by the mean of the selection ratios for the considered habitat type. However, if the habitat selection varies from one animal to another, then the results may be misleading. For example, the habitat use by an animal may be the result of the habitat types available to it (functional responses, see Mysterud and Ims 1998). If the habitat has been present, the animal may have selected habitat differently. This solution is not recommended, when the aim of the analysis is to establish a typology of animals according to habitat selection.
One second, preferred, option is to replace the missing selection ratios by its expectation under random habitat use, i.e., by setting it equal to 1. Indeed, this analysis maximises the White and Garrott's measure of habitat selection. For a given animal, a habitat type with a selection ratio equal to 1 does not contributes at all to this measure. Therefore, by setting the missing selection ratios equal to 1, one ensures that the missing habitat types do not have any influence in the analysis. As noted in the paper, only the habitat types with large selection ratios for a given animal contribute to the first axes of the analysis.
Manly, B. F. J., L. L. McDonald, D. L. Thomas, T. L. MacDonald, and W. P. Erickson. 2002. Resource selection by animals. Statistical design and analysis for field studies. Second Edition. Kluwer Academic Publisher, London, UK.
Mysterud, A., and R. A. Ims. 1998. Functional responses in habitat use: availability influences relative use in trade-off situations. Ecology 79:14351441.
White, G. C., and R. A. Garrott. 1990. Analysis of wildlife radio-tracking data. Academic Press, London, UK.