[R-sig-eco] Chi square distribution of Mahalanobis distance -how many degrees of freedom

[Jon Olav Skoien]

Dear list,

The squared Mahalanobis distance is defined as:
D2 = (X-m)' C (X-m)
When this is used in ecological modelling, X refers to the environmental 
variables at a location i, m and C are the mean and inverted covariance 
matrix of the same variables at a set of locations where a species has 
been observed. It is then usually assumed that this squared distance is 
Chi square distributed with n-1 degrees of freedom, n being the number 
of variables, as described in the original paper of Clark et al. (1993). 
The same assumption is used in the mahasuhab function of adehabitat. 
However, when reading non-ecological texts about linear models, the 
Mahalanobis distance seems to be a special case of quadratic forms, 
which for an invertible covariance matrix are described to have rank(C) 
= n degrees of freedom, see e.g. Christensen (2011, p9) and different 
hits with the search string:
mahalanobis chi square degrees of freedom
I cannot verify the reliability of all these sites, but it is striking 
that in most ecological links the authors seem to use n-1 degrees of 
freedom whereas in most non-ecological links the authors seem to use n 
degrees of freedom.

Clark et al. (1993) do not give a reference in the description of the 
Chi square distribution, but earlier in the theory section there are 
some references to a book of Morrison (1976). Could this be the source? 
Does anyone know why it appears to be a discrepancy in the  degrees of 
freedom? I have not found any discussions about this, is there anything 
I have missed?

Thanks in advance,
Jon


Christensen, R., 2011. Plane answers to complex questions: The theory of 
linear models, 4 ed. New York: Springer Science.

Clark, J.D., Dunn, J.E. & Smith, K.G., 1993. A multivariate model of 
female black bear habitat use for a geographical information system. 
Journal of Wildlife Management 57, 519-526.

Morrison, D. F., 1976. Multivariate statistical methods. McGraw-Hill 
Book Co., New York, N.Y., 415 pp.

-- 
Jon Olav Skøien
Joint Research Centre - European Commission
Institute for Environment and Sustainability (IES)
Land Resource Management Unit

Via Fermi 2749, TP 440,  I-21027 Ispra (VA), ITALY

jon.skoien at jrc.ec.europa.eu
Tel:  +39 0332 789206

Disclaimer: Views expressed in this email are those of the individual and do not necessarily represent official views of the European Commission.