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.     

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 Cavalieri's principle is dealt with in some detail. The efficiency of the systematic fractionating of an object is also illustrated.     

Confidence intervals in Cavalieri sampling

The aim of this study is to derive a formula that allows the prediction, from a Cavalieri data sample, of an appropriate confidence interval for a parameter Q. Two different approaches are used to address the problem. The first approach is to investigate whether the asymptotic distribution of the Cavalieri estimator exists when the sampling period T tends to zero. In particular, the distribution of the standardized version of the Cavalieri estimator z_T is analysed for a measurement function f whose smoothness constant is an integer number m. The analysis reveals that when the first noncontinuous derivative of f, f^(m), exhibits a unique discontinuity, the asymptotic distribution of z_T exists and it has a bounded support. An analytical expression of the distribution is derived for the cases m = 0 and 1. However, when f^(m) has two or more discontinuities, the asymptotic distribution of z_T does not exist and its support may be unbounded. In the second approach, a generalized version of the refined Euler Mac‐Laurin summation formula, valid for measurement functions with a fractional, rather than just an integer, smoothness constant, is applied to the Cavalieri estimator. As a result, a formula that predicts a lower and upper bound for the true parameter is derived for small T. This bound prediction formula is applied to Cavalieri data samples of human cerebral cortex. In particular, for sample sizes n = 8, 12 and 16, the true volume of cerebral cortex is bounded by relative distances 8%, 4% and 2% of the Cavalieri estimate, respectively.     

New variance expressions for systematic sampling: the filtering approach

We present a collection of variance models for estimators obtained by geometric systematic sampling with test points, quadrats, and n‐boxes in general, on a bounded domain in n‐dimensional Euclidean space ?ⁿ, n = 1, 2, ... , and for systematic rays and sectors on the circle. The approach adopted ? termed the filtering approach ? is new and different from the current transitive approach. This report is only preliminary, however, because it includes only variance models in terms of the covariogram of the measurement function. The estimation step is in preparation.     

On error prediction in circular systematic sampling

An extended covariogram model is discussed for estimating the precision of circular systematic sampling. The extension is motivated by recent developments in shape analysis of featureless planar objects. Preliminary simulation results indicate that it is important to consider the extended covariogram model.     

On the precision of systematic sampling: a review of Matheron's transitive methods

G. Matheron's theory of regionalized variables provides a suitable basis for obtaining variance approximations for estimators of integrals from systematically sampled observations, with applications in geostatistics, image analysis, stereology and numerical quadrature techniques in general. The approximations are often fairly accurate for practical purposes. The methods are, however, not sufficiently widespread outside the field of geostatistics. The purpose of this paper is to present in an informal way the transitive part of the methods (relevant to the design-based approach) and a number of stereological applications.     

Evaluation of variance models for fractionator sampling of trees

We compared the performance of several models for predicting, from small samples, the precision of estimates of the total number of blossoms on fruit trees obtained using a three‐stage fractionator, in which the sampling units were defined by the tree structure: (1) primary branches and stem (2) secondary branches and shoots and (3) flowering buds. The models considered were the semiempirical models of Cruz‐Orive (1990, 2004 ) (CO), a random sample model (SR), a Poisson model (P), successive differences (D) and repeated systematic sampling (R). Procedures that relied upon a single sample and a model of the variance (SR, P, D) were not able to predict the estimator variance because the considered structures strongly violated model assumptions. The semiempirical CO model performed acceptably in some cases where model assumptions were violated to a moderate degree. The repeated systematic sampling procedure, which does not rely upon a model of the variance, usually provided very good predictions when the resampling terms were distributed appropriately across more than one sampling stage.     

Application of design‐based stereology for estimation of absolute volume and surface area of the articular and calcified cartilage compartments of undecalcified human femoral heads

Design‐based stereological methods using systematic uniform random sampling, the Cavalieri estimator and vertical sections are used to investigate undecalcified human femoral heads. Ten entire human femoral heads, obtained from normal women and normal men, were systematically sampled and thin undecalcified vertical sections were obtained. Absolute volumes and surface areas of the entire femoral head, the articular cartilage and the calcified cartilage compartments were estimated. In addition, the average thickness of the articular cartilage and the calcified cartilage were calculated. The stereological procedures applied to the human femoral heads resulted in average coefficient of errors, which were 0.03–0.06 for the volume estimates and 0.03–0.04 for the surface area estimates. We conclude that design‐based stereology using the Cavalieri estimator and vertical sections can successfully be used in large undecalcified tissue specimens, like the human femoral head, to estimate the absolute volume and surface area of macroscopic as well as of microscopic tissue compartments. The application of well‐known design‐based stereological methods carries potential advantage for investigating the pathology in inflammatory and degenerative joint diseases.     

Vertical LM sectioning and parallel CT scanning designs for stereology: application to human lung

Practical, unbiased stereological methods are described to estimate lung volume and external surface area, and total volume and surface area of relatively large and anisotropic structures (bronchi and arteries) inside the lung. The volume of each of five lung strata was estimated first by fluid displacement and then by computed tomography (CT) using Cavalieri's method; the reliability of CT was assessed through a calibration procedure, and image thresholding criteria for an accurate volume estimation using CT were established. The parallel, perfectly registered CT section images were also used to estimate the external surface area of each stratum by the spatial grid method. Unbiased estimation of internal surface areas in lung is a long-standing problem: since the structures are large and essentially void, large sections are needed; to facilitate identification, thin sections have to be used for light microscopy, and since such structures are anisotropic, the sections should be vertical. A practical stereological design is demonstrated here on an infant lung, which fulfils all these requirements. This study illustrates the potential of using unbiased stereology to characterize infant pulmonary hypoplasia.     

A simple and efficient alternative to implementing systematic random sampling in stereological designs without a motorized microscope stage

When properly applied, stereology is a very robust and efficient method to quantify a variety of parameters from biological material. A common sampling strategy in stereology is systematic random sampling, which involves choosing a random sampling relevant objects start point outside the structure of interest, and sampling at sites that are placed at pre‐determined, equidistant intervals. This has proven to be a very efficient sampling strategy, and is used widely in stereological designs. At the microscopic level, this is most often achieved through the use of a motorized stage that facilitates the systematic random stepping across the structure of interest. Here, we report a simple, precise and cost‐effective software‐based alternative to accomplishing systematic random sampling under the microscope. We believe that this approach will facilitate the use of stereological designs that employ systematic random sampling in laboratories that lack the resources to acquire costly, fully automated systems.     

Sampling theory and automated simulations for vertical sections,applied to human brain

In recent years, there have been substantial developments in both magnetic resonance imaging techniques and automatic image analysis software. The purpose of this paper is to develop stereological image sampling theory (i.e. unbiased sampling rules) that can be used by image analysts for estimating geometric quantities such as surface area and volume, and to illustrate its implementation. The methods will ideally be applied automatically on segmented, properly sampled 2D images – although convenient manual application is always an option – and they are of wide applicability in many disciplines. In particular, the vertical sections design to estimate surface area is described in detail and applied to estimate the area of the pial surface and of the boundary between cortex and underlying white matter (i.e. subcortical surface area). For completeness, cortical volume and mean cortical thickness are also estimated. The aforementioned surfaces were triangulated in 3D with the aid of FreeSurfer software, which provided accurate surface area measures that served as gold standards. Furthermore, a software was developed to produce digitized trace curves of the triangulated target surfaces automatically from virtual sections. From such traces, a new method (called the ‘lambda method’) is presented to estimate surface area automatically. In addition, with the new software, intersections could be counted automatically between the relevant surface traces and a cycloid test grid for the classical design. This capability, together with the aforementioned gold standard, enabled us to thoroughly check the performance and the variability of the different estimators by Monte Carlo simulations for studying the human brain. In particular, new methods are offered to split the total error variance into the orientations, sectioning and cycloid components. The latter prediction was hitherto unavailable – one is proposed here and checked by way of simulations on a given set of digitized vertical sections with automatically superimposed cycloid grids of three different sizes. Concrete and detailed recommendations are given to implement the methods.