The use of quadrats and test systems in stereology,including magnification corrections

At present a model-free, design-based theory of unbiased estimation, and a model-based one of linear unbiased estimation of minimum variance, are available for stereology. The main developments rest upon the nested scheme {section (quadrat)}, whence the raw data are expressible in terms of area, length and number. The main aim of this paper is to complete the available model-based theory by introducing the step in which sections are analysed by point-counting via coherent test systems (CTSs). Using this development, the stereologist should be able to handle raw point and intersection counts optimally, in order to find the best estimator of a ratio R in a given specimen in a wide range of circumstances. The latter include, for instance, the use of different CTSs on different sections and of double CTSs on each section, as well as the case—(not uncommon in electron microscopy)—in which different sections from the same sample are observed at slightly different magnifications but analysed by quadrats (via automatic or semi-automatic image analysers, for instance), or CTSs of fixed sizes. The main conclusion pertaining to the latter case is that the estimators obtained via section-wise magnification corrections are in general superior to those corrected via a global, average magnification. In order to illustrate the methodology, a synthetic numerical example, and a real one, are given.