Valence is a crucial concept in studying spatial voting and party competition. The widely adopted approach is to rely on intercepts of vote choice models and to infer, based on their size and direction, how valence affects party strategies in empirical settings. The approach suffers from fundamental statistical flaws. This contribution provides the statistical fundamentals to advance the empirical modeling of valence. It proposes an appropriate modeling approach to interpret intercepts as valences and alternate specifications to parameterize the effects of valence.

In stereology or applied geometric probability quantitative characterization of aggregates of particles from information on lower-dimensional sections plays a major role. Most stereological methods developed for particle aggregates are based on the assumption that the particles are of the same, known (simple) shape. Information on the volume-weighted distribution of particle size may, however, be obtained under fairly general assumptions about particle shape if particle volume is chosen as size parameter. In fact, there exists in this case an unbiased stereological estimator of the first moment under the sole assumption that the particles are convex. In the present paper, we consider a particle aggregate in ℝ and derive estimators of the q th moment of the volume-weighted distribution of particle volume, based on point-sampling of particles and measurements on q -flats through sampled particles. The estimators are valid for arbitrarily shaped particles but if the particles are non-convex it is necessary for the determination of the estimators to be able to identify the different separated parts on a q-flat through the particle aggregate which belong to the same particle. Explicit forms of the estimators are given for q = 1. For q = 2, an explicit form of one of the estimators is derived for an aggregate of triaxial ellipsoids in three-dimensional space.

Estimation of the properties of particles embedded in a solid medium via observations on random plane sections is studied. The particular properties considered are those associated with steel inclusions, for example, projected area fraction, total caliper length per unit volume and elongation factor. The approach is general, and with the minimum of assumptions about the particle shape, formulae are obtained for suitable estimators and their variances. Generalised moment relations are derived for the case of given, arbitrary shapes. Estimations of the density of particle centres based upon reciprocal observations are shown to lack robustness to small changes in shape and a more reliable density estimator is discussed. Consideration is given to the case of particles with elliptical profiles and to the use of aggregated results which arise when using automatic scanning equipment. Results are given for the case when there is length-biassing of the intercept distribution due to edge-effects for finite sample areas. Finally, the implication of a positive lower resolution point is discussed.