# Sampling and Stereology

# Programme

**Thursday 31 January 2013, 10:00-14:00**

Aarhus University, Department of Mathematics

**10:00-12:00 in Koll. D (room 211, building 1531)**

**13:00-14:00 in Aud. D1 (room 113, building 1531)**

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**10:00 Johanna F. Ziegel, University of Bern: Precision estimation for stereological volumes**

Abstract:

Volume estimators based on Cavalieri’s principle are widely used in the bio-sciences. For example in neuroscience, where volumetric measurements of brain structures are of interest, systematic samples of serial sections are obtained by magnetic resonance imaging or by a physical cutting procedure. The volume V is then estimated by v, which is the sum over the areas of the structure of interest in the section planes multiplied by the width of the sections, t > 0.

Assessing the precision of such volume estimates is a question of great practical importance, but statistically a challenging task due to the strong spatial dependence of the data and typically small sample sizes. In this talk an overview of classical and new approaches to this problem will be presented. A special focus will be given to some recent advances on distribution estimators and confidence intervals for v; see Hall and Ziegel (2011).

References

P. Hall and J. Ziegel. Distribution estimators and confidence intervals for stereological volumes. Biometrika, 98:417–431, 2011.

**11:00 Eva Vedel Jensen, Aarhus University: Introduction to stereology**

Abstract: Stereology is a collection of geometric and statistical methods by means of which quantitative properties of a spatial structure such as volume and surface area may be estimated from random geometric samples of the structure. Typical examples of samples are linear sections, planar sections, projections and intersections with point grids. In this talk, I will give an introduction to stereology with emphasis on the important role that sampling theory plays in the development of stereological methods.

**13:00 Yves Tillé, University of Neuchâtel: Doubly balanced spatial sampling with spreading and restitution of auxiliary totals**

Abstract:

Anton Grafström, Swedish University of Agricultural Sciences, Umeå, Sweden, and Yves Tillé, University of Neuchâtel, Switzerland

A new spatial sampling method is proposed in order to achieve a double property of balancing. The sample is spatially balanced or well spread so as to avoid selecting neighbouring units. Moreover, the method also enables to satisfy balancing equations on auxiliary variables available on all the sampling units because the Horvitz-Thompson estimator is almost equal to the population totals for these variables. The method works with any definition of distance in a multidimensional space and supports the use of unequal inclusion probabilities. The algorithm is simple and fast. Examples show that the method succeeds in using more information than the local pivotal method, the cube method and the Generalized Random-Tessellation Stratified sampling method, and thus performs better. An estimator of the variance for this sampling design is proposed in order to lead to an inference that takes the effect of the sampling design into account.

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