### ACCURACY OF ESTIMATES OF VOLUME FRACTION

#### Abstract

When estimating a volume fraction VV from point count fractions PP using Delesse's principle VV = PP, very little information on the accuracy of the estimator can be obtained from the basic theory of stereology. Existing methods for quantifying the variability of PP are mainly large-sample approximations such as Cochran's formula for the variance of a ratio. Cruz-Orive proposed an alternative method, but this requires statistical assumptions to be made on the point counts P, that do not hold in general. We introduce two alternative methods for quantifying the variability of PP, namely the bootstrap method and explicit statistical modelling of the bivariate distribution. The bootstrap method requires few statistical assumptions about the point counts but requires large sample size. The explicit statistical modelling method does make assumptions, but they can be checked directly from the data. To explore this approach, we propose a statistical model, the Type I Bivariate Binomial (BVB) distribution to model the pairs of count data (P, P). We show how to fit the BVB model to the data and how to assess the goodness-of-fit of this model. A formula for the variance of PP under the BVB model is also derived. The three approaches are compared in their application to nine example data sets taken from macroscopic sections of cerebral hemispheres of selected domesticated animals. The BVB model appears to be a good fit to these data sets. Implications for stereological estimation are discussed.

##### Keywords

bootstrap method; delta method; Monte Carlo; stereology; type I bivariate binomial distribution; volume fraction

DOI: 10.5566/ias.v19.p199-204