Statistics for Spatial Data.

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Table Of Integrals Series And Products

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 Cavalieri's 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 and its prediction

Use of high-resolution micro-computed tomography (microCT) imaging to assess trabecular and cortical bone morphology has grown immensely. There are several commercially available microCT systems, each with different approaches to image acquisition, evaluation, and reporting of outcomes. This lack of consistency makes it difficult to interpret reported results and to compare findings across different studies. This article addresses this critical need for standardized terminology and consistent reporting of parameters related to image acquisition and analysis, and key outcome assessments, particularly with respect to ex vivo analysis of rodent specimens. Thus the guidelines herein provide recommendations regarding (1) standardized terminology and units, (2) information to be included in describing the methods for a given experiment, and (3) a minimal set of outcome variables that should be reported. Whereas the specific research objective will determine the experimental design, these guidelines are intended to ensure accurate and consistent reporting of microCT-derived bone morphometry and density measurements. In particular, the methods section for papers that present microCT-based outcomes must include details of the following scan aspects: (1) image acquisition, including the scanning medium, X-ray tube potential, and voxel size, as well as clear descriptions of the size and location of the volume of interest and the method used to delineate trabecular and cortical bone regions, and (2) image processing, including the algorithms used for image filtration and the approach used for image segmentation. Morphometric analyses should be based on 3D algorithms that do not rely on assumptions about the underlying structure whenever possible. When reporting microCT results, the minimal set of variables that should be used to describe trabecular bone morphometry includes bone volume fraction and trabecular number, thickness, and separation. The minimal set of variables that should be used to describe cortical bone morphometry includes total cross-sectional area, cortical bone area, cortical bone area fraction, and cortical thickness. Other variables also may be appropriate depending on the research question and technical quality of the scan. Standard nomenclature, outlined in this article, should be followed for reporting of results.

https://asbmr.onlinelibrary.wiley.com/doi/pdf/10.1002/jbmr.141

Guidelines for assessment of bone microstructure in rodents using micro–computed tomography

A stereological method for obtaining estimates of the total number of neurons in five major subdivisions of the rat hippocampus is described. The new method, the optical fractionator, combines two recent developments in stereology: a three-dimensional probe for counting neuronal nuclei, the optical disector, and a systematic uniform sampling scheme, the fractionator. The optical disector results in unbiased estimates of neuron number, i.e., estimates that are free of assumptions about neuron size and shape, are unaffected by lost caps and over-projection, and approach the true number of neurons in an unlimited manner as the number of samples is increased. The fractionator involves sampling a known fraction of a structural component. In the case of neuron number, a zero dimensional quantity, it provides estimates that are unaffected by shrinkage before, during, and after processing of the tissue. Because the fractionator involves systematic sampling, it also results in highly efficient estimates. Typically only 100–200 neurons must be counted in an animal to obtain a precision that is compatible with experimental studies. The methodology is compared with those used in earlier works involving estimates of neuron number in the rat hippocampus and a number of new stereological methods that have particular relevance to the quantitative study of the structure of the nervous system are briefly described in an appendix.

Unbiased stereological estimation of the total number of neurons in the subdivisions of the rat hippocampus using the optical fractionator

In the present paper, we summarize and further develop recent reseach in the estimation of the variance of sterelogical 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.

The efficiency of systematic sampling in stereology--reconsidered.

SUMMARY

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.

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

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.

Stereology for Statisticians

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.

Eva B. Vedel Jensen

Papers

The efficiency of systematic sampling in stereology and its prediction

The efficiency of systematic sampling in stereology--reconsidered.

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

Stereology for Statisticians

Estimation of structural anisotropy based on volume orientation. A new concept.