A Unified Discussion on the Concept of Score Functions Used in the Context of Nonparametric Linkage Analysis

In this article we try to discuss nonparametric linkage (NPL) score functions within a broad and quite general framework. The main focus of the paper is the structure, derivation principles and interpretations of the score function entity itself. We define and discuss several families of one-locus score function definitions, i.e. the implicit, explicit and optimal ones. Some generalizations and comments to the two-locus, unconditional and conditional, cases are included as well. Although this article mainly aims at serving as an overview, where the concept of score functions are put into a covering context, we generalize the noncentrality parameter (NCP) optimal score functions in Angquist et al. (2007) to facilitate--through weighting--for incorporation of several plausible distinct genetic models. Since the genetic model itself most oftenly is to some extent unknown this facilitates weaker prior assumptions with respect to plausible true disease models without loosing the property of NCP-optimality.Moreover, we discuss general assumptions and properties of score functions in the above sense. For instance, the concept of identical by descent (IBD) sharing structures and score function equivalence are discussed in some detail.