Eva B. Vedel Jensen

The efficiency of systematic sampling in stereology and its prediction

The superior efficiency of systematic sampling at all levels in stereological studies is emphasized and various commonly used ways of implementing it are briefly described. Summarizing recent theoretical and experimental studies a set of very simple estimators of efficiency are presented and illustrated with a variety of biological examples. In particular, a nomogram for predicting the necessary number of points when performing point counting is provided. The very efficient and simple unbiased estimator of the volume of an arbitrary object based on Cavalieris principle is dealt with in some detail. The efficiency of the systematic fractionating of an object is also illustrated.

The efficiency of systematic sampling in stereology — reconsidered

In the present paper, we summarize and further develop recent research in the estimation of the variance of stereological estimators based on systematic sampling. In particular, it is emphasized that the relevant estimation procedure depends on the sampling density. The validity of the variance estimation is examined in a collection of data sets, obtained by systematic sampling. Practical recommendations are also provided in a separate section.

Stereological estimation of the volume‐weighted mean volume of arbitrary particles observed on random sections*

A stereological estimator of the weighted mean volume of particles of arbitrary shape is described. This unbiased estimator is based on simple point‐sampling of linear intercept lengths. The complete absence of shape assumptions effectively breaks the long‐standing ‘convexity‐barrier’: the only requirement here is that individual particles can be unambiguously identified by their profiles on random sections. Practical details of the simple estimation procedure and an example with very irregular particles are reported. Finally, an estimator of the variance of the weighted distribution of particle volume is discussed. This estimator is also valid for particles of arbitrary shape. For any mixture of ellipsoids (spheres, oblates, prolates and triaxial ellipsoids) the estimator is reduced to a simple function of measurements of diameters in the section plane.

Stereology for statisticians

Stereology for Statisticians sets out the principles of stereology from a statistical viewpoint, focusing on both basic theory and practical implications. This book discusses ways to effectively communicate statistical issues to clients, draws attention to common methodological errors, and provides references to essential literature. The first full text on design-based stereology, it opens with a review of classical and modern stereology, followed by a treatment of mathematical foundations. It then presents core techniques. The final chapters discuss implementing techniques in practical sampling designs, summarize understanding of the variance of stereological estimators, and describe open problems for further research.

Estimation of structural anisotropy based on volume orientation: a new concept

The quantification of anisotropy—its main direction and the degree of dispersion around it—is desirable in numerous research fields dealing with physical structures. Conventional methods are based on the orientation of interface elements. The results of these methods do not always agree with perceived anisotropy, and anisotropic structures do not necessarily turn out to be ‘anisotropic’ using these methods. In the present paper, we propose an alternative to curve and surface orientation, namely volume orientation. Using trabecular bone as an example of a two‐phase anisotropic structure, the new concept is studied in some detail. In particular, a parametric method of estimating volume orientation from sections is presented and discussed.

Particle sizes and their distributions estimated from line- and point-sampled intercepts. Including graphical unfolding

Information about particle size is currently obtained almost exclusively by the use of stereological methods which lead to estimates of the number distribution of linear particle size. The main point of this presentation is to stress the freedom to choose more appropriate parameters for size among a host of options, including particle surface area and volume. Moreover, particle size information may often be considered advantageously in terms of particle distributions based on structural characteristics rather than number distributions. Some of these other distribution types are correctly represented in samples of intercept lengths obtained by line‐ and point‐sampling, respectively. The known and quite simple theory of sampling intercepts is summarized and developed further in several different directions, including a derivation of the distribution of intercept length in ellipsoids, graphical unfolding procedures, and mean size estimators. The potential of the approach is illustrated—but not exhausted—by the existence of a general mean size estimator based on minimal assumptions regarding particle shape.

Stereological ratio estimation based on counts from integral test systems

The stereological practice of using integral test systems in the estimation of the fundamental stereological ratios is studied in the light of recent theoretical developments in sampling. The estimation of a ratio is based on counts only, obtained from two, in general different, test sets constituting the integral test system. The ordinary ratio‐of‐sums estimator based on counts from uniformly positioned integral test systems is compared with two estimators based on non‐uniform, weighted sampling. It is shown that the estimators based on weighted sampling are not, in general, unbiased. Furthermore, it is pointed out that the mean‐of‐ratios estimator based on replicated weighted sampling needs not have smaller MSE than the ordinary ratio‐of‐sums estimator based on replicated uniform sampling.

Lévy-based Cox point processes

In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

Inhomogeneous spatial point processes by location-dependent scaling

A new class of models for inhomogeneous spatial point processes is introduced. These locally scaled point processes are modifications of homogeneous template point processes, having the property that regions with different intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted Markov point processes are discussed in detail. It is shown that the locally scaled versions are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process. Approximations are suggested that simplify calculation of the density, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.

Fundamental stereological formulae based on isotropically orientated probes through fixed points with applications to particle analysis

A new set of fundamental stereological formulae based on isotropically orientated probes through fixed points is derived. Volume, surface area and integrals of curvature can be estimated by this method. Some of the estimators require, however, local 3‐D information. The estimators can be applied locally to each of a sample of particles whereby moments of particle volume, particle surface area and particle integral of curvature can be estimated. The results are derived from a generalized version of the so‐called Blaschke‐Petkantschin formula.