The precision of estimates in stereological analyses

When quantifying the total volume, number, length, or surface of morphological features in complex structures such as the brain, it is neither desirable nor feasible to make absolute determinations of these parameters. It would take a lifetime to count all of the neurons or measure the total length of dendrites in many parts of the brain. Estimates based on a relatively small number of samples of even the most complex structures will suffice, if the estimates are unbiased and have a sufficient amount of precision. It is important to have an understanding of what the precision of an estimate actually means and how it can be calculated and altered, to optimize studies with regard to the number of individuals, sections, and probes used in an analysis. An optimal sampling scheme is one that involves a minimum of sampling but still provides enough precision to draw a conclusion with a predetermined level of confidence.

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 general variance predictor for Cavalieri slices

A general variance predictor is presented for a Cavalieri design with slices of an arbitrary thickness t >or= 0. So far, prediction formulae have been available either for measurement functions with smoothness constant q = 0, 1, ... , and t >or= 0, or for fractional q in [0, 1] with t = 0. Because the possibility of using a fractional q adds flexibility to the variance prediction, we have extended the latter for any q in [0, 1] and t >or= 0. Empirical checks with previously published human brain data suggest an improved performance of the new prediction formula with respect to the hitherto available ones.

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.

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 R(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.