Non-uniform systematic sampling in stereology

Non-uniform systematic sampling designs in stereology are studied. Various methods of constructing non-uniform systematic sampling points from prior knowledge of the measurement function are presented. As an example, we consider area estimation from lengths of linear intercepts. The efficiency of two area estimators, based on non-uniform sampling of parallel lines, is compared to that of the classical 2D Cavalieri estimator, based on uniform sampling, in a sample of planar profiles from transverse sections of 41 small myelinated axons. The comparison is based on simulations. It is concluded that for profiles of this type one of the non-uniform sampling schemes is more efficient than the traditional uniform sampling scheme. Other examples where non-uniform systematic sampling may be used are in area estimation from lines emanating from a fixed point, area estimation from concentric circles or spirals and curve length estimation from sweeping lines. It is shown that proportional-to-size sampling is a special case of non-uniform systematic sampling. Finally, the effect of noise in the observations is discussed.