The proportionator: Unbiased stereological estimation using biased automatic image analysis and non-uniform probability proportional to size sampling

Abstract

The proportionator is a novel and radically different approach to sampling with microscopes based on the well-known statistical theory (probability proportional to size-PPS sampling). It uses automatic image analysis, with a large range of options, to assign to every field of view in the section a weight proportional to some characteristic of the structure under study. A typical and very simple example, examined here, is the amount of color characteristic for the structure, marked with a stain with known properties. The color may be specific or not. In the recorded list of weights in all fields, the desired number of fields is sampled automatically with probability proportional to the weight and presented to the expert observer. Using any known stereological probe and estimator, the correct count in these fields leads to a simple, unbiased estimate of the total amount of structure in the sections examined, which in turn leads to any of the known stereological estimates including size distributions and spatial distributions. The unbiasedness is not a function of the assumed relation between the weight and the structure, which is in practice always a biased relation from a stereological (integral geometric) point of view. The efficiency of the proportionator depends, however, directly on this relation to be positive. The sampling and estimation procedure is simulated in sections with characteristics and various kinds of noises in possibly realistic ranges. In all cases examined, the proportionator is 2-15-fold more efficient than the common systematic, uniformly random sampling. The simulations also indicate that the lack of a simple predictor of the coefficient of error (CE) due to field-to-field variation is a more severe problem for uniform sampling strategies than anticipated. Because of its entirely different sampling strategy, based on known but non-uniform sampling probabilities, the proportionator for the first time allows the real CE at the section level to be automatically estimated (not just predicted), unbiased-for all estimators and at no extra cost to the user.

Getting Started in Stereology

“…For example, in Figure 2A, one would sample sections with a smaller intersectional spacing in the beginning of the series (left) as opposed to the later sections in the series (right), where little change occurs in the areas of the sectional profiles of the region of interest. The calculations of the efficiency of estimates made with proportional sampling are not the same as those described for uniform sampling (see Gardi et al 2008).…”

“…PROPORTIONATOR SAMPLING This strategy outlined in The Spacing Between Sections: Capturing the Peaks and Spacing the Probes applies to uniform random sampling, which is the most widely applicable approach to sampling. There is an alternative approach, proportionator sampling, which may be more efficient for distributions that are nonuniform (Dorph-Petersen et al 2000;Gardi et al 2008). Certain features might be clustered, for example, specific cell types in the striatum.…”

Alterations of Hematological Parameters, Hemoglobin and Hematocrit With Liver Enzymes, Aspartate Transaminase and Alanine Transaminase Among Patients With Chronic Kidney Disease Undergoing Hemodialysis in Aljouf Region, Saudi Arabia

“…In this study, a no-probability sample equation (n = N/1 + N (e) 2) was used to determine sample size [14], but for feasibility reasons, only 150 -170 samples for patients and 150 -180 for healthy individuals were chosen.…”

Design-based Stereology

“…The Proportionator is a recent estimator that combines the power of image analysis with stereology to greatly increase the efficiency of subsampling of histological sections (Gardi et al 2008a;Gardi et al 2008b). This method uses nonuniform random sampling (Dorph-Petersen et al 2000) guided by image analysis-detected features of interest in the stained sections.…”