Resumen de An application of the isometric log-ratio transformation in relatedness research

Jan Graffelman , I. Galv´an

• Genetic marker data contains information on the degree of relatedness of a pair of individuals. Relatedness investigations are usually based on the extent to which alleles of a pair of individuals match over a set of markers for which their genotype has been determined. A distinction is usually drawn between alleles that are identical-by-state (IBS) and alleles that are identical-by-descent (IBD). Since any pair of individuals can only share 0, 1 or 2 alleles IBS or IBD for any marker, 3-way compositions can be computed that consist of the fractions of markers sharing 0, 1 or 2 alleles IBS (or IBD) for each pair. For any given standard relationship (e.g. parent-offspring, sister-brother, etc.) the probabilities k0, k1 and k2 of sharing 0, 1 or 2 IBD alleles are easily deduced and are usually referred to as Cotterman’s coefficients. Marker data can be used to estimate these coefficients by maximum likelihood (Thompson, 1975; 1991). This maximization problem has the 2-simplex as its domain. If there is no inbreeding, then the maximum must occur in a subset of the 2-simplex. The maximization problem is then subject to an additional non-linear constraint (k 2 1 ≥ 4k0k2). Special optimization routines are needed that do respect all constraints of the problem. A re-parametrization of the likelihood in terms of isometric log-ratio (ilr) coordinates greatly simplifies the maximization problem. In isometric log-ratio coordinates the domain turns out to be rectangular, and maximization can be carried out by standard general-purpose maximization routines. We illustrate this point with some examples using data from the HapMap project