Allometric extension model for conditional distributions

When two groups are present, they are said to form an allometric model, if one group is the extension of the other group along the main axis of variation. The model is widely used in the context of principal component analysis, especially for the description of growth processes of creatures. In this paper, the notion of allometric extension model is applied to conditional distributions. More specifically, we derive a sufficient condition, for which the two conditional distributions given the sign of the first principal component form an allometric extension model.