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.     

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.     

Evaluation of the precision of systematic sampling: Nugget effect and covariogram modelling

Systematic sampling designs are widely used in stereology. When an estimator of the total amount, Q, of the sampled variable is evaluated by such a procedure, the coefficient of error can be predicted by applying Matheron's theory of regionalized variables. To evaluate the accuracy of the estimate of Q, it is necessary to study the behaviour of the regionalized variable and to model its covariogram. Histological data with a low short-range variability and agronomic data with a pronounced nugget effect provided the biological material for extreme case studies. Results show that the short-range variability, if present, cannot be detected when only small samples are available. An underestimation of the coefficient of error is then to be expected. We propose several models of the covariogram, which can be used to test for the presence of a nugget effect. If a nugget effect is present, these models will provide better estimates of the coefficient of error. If there is no nugget effect a simplified method can be used and will provide reliable estimates of the coefficient of error.     

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.     

Prediction of the variance of stereological volume estimates from systematic sections using computer-intensive methods

The Cavalieri method is an unbiased estimator of the total volume of a body from its transectional areas on systematic sections. The coefficient of error (CE) of the Cavalieri estimator was predicted by a computer‐intensive method. The method is based on polynomial regression of area values on section number and simulation of systematic sectioning. The measurement function is modelled as a quadratic polynomial, with an error term superimposed. The relative influence of the trend and the error component is estimated by techniques of analysis of variance. This predictor was compared with two established short‐cut estimators of the CE based on transitive theory. First, all predictors were applied to data sets from six deterministic models with analytically known CE. For these models, the CE was best predicted by the older short‐cut estimator and by the computer‐intensive approach, if the measurement function had finite jumps. The best prediction was provided by the newer short‐cut estimator when the measurement function was continuous. The predictors were also applied to published empirical datasets. The first data set consisted of 10 series of areas of systematically sectioned rat hearts with 10–13 items, the second data set consisted of 13 series of systematically sampled transectional areas of various biological structures with 38–90 items. On the whole, similar mean values for the predicted CE were obtained with the older short‐cut estimator and the computer‐intensive method. These ranged in the same order of magnitude as resampling estimates of the CE from the empirical data sets, which were used as a cross‐check. The mean values according to the newer short‐cut CE estimator ranged distinctly lower than the resampling estimates. However, for individual data sets, it happened that the closest prediction as compared to the cross‐check value could be provided by any of the three methods. This finding is discussed in terms of the statistical variability of the resampling estimate itself.     

A note on the stereological implications of irregular spacing of sections

Stereological methods for serial sections traditionally assume that the sections are exactly equally spaced. In reality, the spacing and thickness of sections can be quite irregular. This may affect the validity and accuracy of stereological techniques, especially the Cavalieri estimator of volume. We present a new formula for the accuracy of the Cavalieri estimator that includes the effect of random variability in section spacing. A modest amount of variability in section spacing can cause a substantial increase in estimator variance.     

On the sampling of three-dimensional polycrystalline microstructures for distribution determination

How to sample three-dimensional microstructure and effectively reduce experimental error is a challenging problem. Taking seven single-phase polycrystalline structures generated by 400×400×400 Potts Monte Carlo simulation as the study object, effects of sampling strategy on the determination of various characteristic parameters of the grain size distribution and grain topology distribution are studied. The mean voxel value (or volume) of individual grains in the three-dimensional simulated microstructure varies from 4.56×10(4) to 1.0×10(3) , and the number of grains contained in the structure varies from 63 901 to 1403. The results show that, a minimum of 200 sampled grains can ensure the relative error to be less than 5% for determination of the mean grain volume, the mean grain face number and the coefficient of variance of the distribution of grain size and the grain face number. Whereas for the coefficient of the skewness and the kurtosis of grain size distribution or grain face number distribution, a minimum of 800 sampled grains are required for the same error level. However, if some exceptional big grains appear, e.g. a grain larger than with eight multiples of mean grain volume and/or three multiples of mean grain face number, abnormal values of the two parameters would be resulted, even the number of examined grains is over 1000.     

Unbiased estimation of human body composition by the Cavalieri method using magnetic resonance imaging

The classical methods for estimating the volume of human body compartments in vivo (e.g. skin-fold thickness for fat, radioisotope counting for different compartments, etc.) are generally indirect and rely on essentially empirical relationships — hence they are biased to unknown degrees. The advent of modern non-invasive scanning techniques, such as X-ray computed tomography (CT) and magnetic resonance imaging (MRI) is now widening the scope of volume quantification, especially in combination with stereological methods. Apart from its superior soft tissue contrast, MRI enjoys the distinct advantage of not using ionizing radiations. By a proper landmarking and control of the scanner couch, an adult male volunteer was scanned exhaustively into parallel systematic MR ‘sections’. Four compartments were defined, namely bone, muscle, organs and fat (which included the skin), and their corresponding volumes were easily and efficiently estimated by the Cavalieri method: the total section area of a compartment times the section interval estimates the volume of the compartment without bias. Formulae and nomograms are given to predict the errors and to optimize the design. To estimate an individual's muscle volume with a 5% coefficient of error, 10 sections and less than 10min point counting (to estimate the relevant section areas) are required. Bone and fat require about twice as much work. To estimate the mean muscle volume of a population with the same error contribution, from a random sample of six subjects, the workload per subject can be divided by √6, namely 4 min per subject. For a given number of sections planimetry would be as accurate but far more time consuming than point counting.     

气瓶水压试验方法的比较及应用分析

综合分析了气瓶在批量生产条件下,气瓶水压外测法和气瓶水压内测法的优缺点。通过改变气瓶水压内测法中受试瓶的试验状态,同时采用计算机控制技术,克服气瓶水压内测法试验受试验条件影响大的缺点,从而提高测量精度,以适合批量生产的条件。     

The saucor, a new stereological tool for analysing the spatial distributions of cells, exemplified by human neocortical neurons and glial cells

The 3D spatial arrangement of particles or cells, for example glial cells, with respect to other particles or cells, for example neurons, can be characterized by the radial number density function, which expresses the number density of so-called 'secondary' particles as a function of their distance to a 'primary' particle. The present paper introduces a new stereological method, the saucor, for estimating the radial number density using thick isotropic uniform random or vertical uniform random sections. In the first estimation step, primary particles are registered in a disector. Subsequently, smaller counting windows are drawn with random orientation around every primary particle, and the positions of all secondary particles within the windows are recorded. The shape of the counting windows is designed such that a large portion of the volume close to the primary particle is examined and a smaller portion of the volume as the distance to the primary object increases. The experimenter can determine the relation between these volumina as a function of the distance by adjusting the parameters of the window graph, and thus reach a good balance between workload and obtained information. Estimation formulae based on the Horvitz-Thompson theorem are derived for both isotropic uniform random and vertical uniform random designs. The method is illustrated with an example where the radial number density of neurons and glial cells around neurons in the human neocortex is estimated using thick vertical sections for light microscopy. The results indicate that the glial cells are clustered around the neurons and the neurons have a tendency towards repulsion from each other.     

Voltage dependence of the tunnelling current: an exact expression

An exact expression for the tunnelling current measured with a scanning tunnelling microscope (STM) is obtained. It clarifies the information deducible from the ‘spectroscopic mode’ of the STM and raises the question of the observability of surface states. The connection with the Transfer Hamiltonian approach is made, and the conditions of validity of the latter are analysed.