The importance of proper model specification in stereology

Abstract

It is desired to estimate, for a given phase Y of a 2-phase 3-dimensional opaque specimen X, its volume fraction V_V, surface fraction S_V and integral of mean curvature fraction K_V. Various estimates of these resulting from a random plane section (or sections) of X itself (the RESTRICTED case) are compared on the basis of bias, mean square error and likely feasibility in practice.

The same problem is considered in the case where the examinable specimen is but a random part of a much larger body Z, for which the corresponding stereological ratios are to be estimated (the EXTENDED case). It is seen that, although the same stereological formulae appertain, the sampling strategy is entirely different.

So far, the material of interest has been DETERMINISTIC. However, often in practice it is clear that it may be supposed RANDOM — the realization of a homogeneous spatial stochastic process. Then the deterministic extended case theory extends, with the great practical simplification that the specimen X need only be arbitrary relative to the material (not uniform random as before).

This theory seems relevant to the current controversy among stereologists as to the desirability of a completely standardized notation and nomenclature. It shows that V_V, S_V and K_V may have a number of different interpretations, so that further specification is essential. Thus, while admitting the advantages to be gained from such standardization, it does seem more important that stereological authors, early in their papers, clearly describe the type of model assumed, including a precise definition of the notation and a proper (probabilistic) specification of the sections (or projections) taken.

Financial support from the Roche Research Foundation, Basle, during the preparation of this paper is gratefully acknowledged.