# The Independence of Size and Shape before and after Scale Change

• James E. Mosimann
• James D. Malley
Conference paper
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 79)

## Summary

Let $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{X}$$ and $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{Q}$$ be k-dimensional positive random vectors related by an unequal linear scale change; that is Qi = ai Xi, ai > 0, i = 1,…,k, with some ai ≠ aj. In this paper we study the independence of shape and size (or size-ratios) before and after the scale change. If, before the change, shape is independent of size, G($$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{X}$$), then the new shape after the change is also independent of G($$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{X}$$). However, shape after the change is not independent of size after, G($$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{Q}$$), unless the ratio G($$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{X}$$)/G($$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{Q}$$) is degenerate. Similarly, if shape before the change is independent of a ratio of related size variables, then shape after the change cannot be independent of the same ratio applied to $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{Q}$$ unless again G($$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{X}$$)/G($$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{Q}$$) is degenerate. A consequence is that if proportions (shape) follow a generalized Dirichlet distribution before the change on $$\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle\thicksim}}}{X}$$ they cannot do so after the change.

## Key Words

size variables shape variables scale change proportions generalized Dirichlet distribution Dirichlet distribution isometry neutrality

## References

1. Connor, R. J. and Mosinann, J. E. (1969). Concepts of independence for proportions with a generalization of the Dirichlet distribution. Journal of the American Statistical Association, 64, 194–206.
2. Davis, M. B. (1963). On the theory of pollen analysis. American Journal of Science, 261, 897–912.
3. Davis, M. B. (1969). Climatic changes in southern Connecticut recorded by pollen deposition at Rogers Lake. Ecology, 50, 409–22.
4. Livingstone, D. A. (1968). Some interstadial and postglacial pollen diagrams from eastern Canada. Ecological Monographs, 38, 87–125.
5. Lochner, R. H. (1975). A generalized Dirichlet distribution in Bayesian life testing. Journal of the Royal Statistical Society, Series B, 37, 103–113.
6. Lukacs, E. (1970). Characteristic Functions (2nd edition). Hafner, New York.
7. Mosimann, J. E. (1970). Size allometry: size and shape variables with characterizations of the lognormal and generalized gamma distributions. Journal of the American Statistical Association, 65, 930–945.
8. Mosimann, J. E. (1975a). Statistical problems of size and shape. I. Biological applications and basic theorems. In Statistical Distributions in Scientific Work, Vol. 2, G. P. Patil, S. Kotz, and J. K. Ord, eds. Reidel, Dordrecht-Holland. Pages 187–217.Google Scholar
9. Mosimann, J. E. (1975b). Statistical problems of size and shape. II. Characterizations of the lognormal, gamma, and Dirichlet distributions. In Statistical Distributions in Scientific Work, Vol. 2, G. P. Patil, S. Kotz, and J. K. Ord, eds. Reidel, Dordrecht-Holland. Pages 219–239.Google Scholar
10. Mosimann, J. E. and Greenstreet, R. L. (1971). Representation-insensitive methods for paleoecological pollen studies. In Statistical Ecology, Vol. 1, G. P. Patil, E. C. Pielou, and W. E. Waters, eds. The Pennsylvania State University Press, University Park, Pennsylvania. Pages 23–58.Google Scholar
11. Okamoto, M. (1973). Distinctness of the eigenvalues of a quadratic form of a multivariate sample. Annals of Statistics, 1, 763–765.
12. Sprent, P. (1972). The mathematics of size and shape. Biometrics, 28, 23–38.