MEASURES CALCULATED BY DifferInt

Dissimilarity

Elementary genic difference d between genetic types: Genic differences between two genetic types at the same level of integration are basically determined by the number of their individual genes that differ in type. If the numbers of copies of the i-th allele at the l-th gene locus are denoted by n_i;l and m_i;l, respectively, then the two genetic types differ by ∑_i,l|n_i;l - m_i;l| gene-type copies. This sum is maximal, equaling two times the total number K of individual genes represented in each object, if the objects share no gene-types (and thus differ completely). Since ∑_i,ln_i;l = ∑_i,lm_i;l = K holds, division of ∑_i,l |n_i;l - m_i;l| by 2·K yields a measure of genic difference that is bounded between zero and one. This measure of elementary genic difference is applicable to all levels of integration.

Pairwise genetic distance Δ between populations: The minimum degree to which the frequency distribution of the genetic types within one population must be transformed in order to make it match the distribution a second population, based on the elementary genic difference between genetic types [3,4]. For a specified set of marker loci, Δ equals the minimum extent to which the genetic types of individuals in one of the two populations must be altered in order to obtain the composition of genetic types in the other. Denote



where d(a,b) specifies the difference between genetic types a and b, and s(a,b) is a frequency shift. Frequency shifts are performed from types that are more frequent in the one population P than in the other Q to types that are less frequent in P than in Q. If the frequency p_a of type a in P exceeds the frequency q_a of this type in Q, then the excess p_a- q_a must be shifted to types deficient in P, such that ∑_bs(a, b) = p_a - q_a = p_a - min{p_a, q_a}. The shift process is continued for all types with a frequency excess in P until the frequencies of all types in P match those in Q. Since there may be more than one way of shifting, Δ is taken to be the minimum of the above sum over all admissible frequency shifts s, i.e.,



In [3,4] it is shown that finding a shift transformation s that minimizes Δ(s) is equivalent to solving the "Transportation Problem" [11] by linear programming methods. These methods are implemented in the computer program DeltaS [12].

Pairwise genetic distance d₀ between two populations: Neglecting genic differences between genotypes, the minimum degree to which the frequency distribution of the genetic types within one population must be transformed in order to make it match the distribution a second population, if the difference between two genetic types a and b simply equals 1 if a=b and 0 otherwise [3-7]. d₀ is the absolute genetic distance



between the frequencies p_a of the types a in population P and the frequencies q_a of the types a in population Q. It equals the reduction of the genetic distance Δ(P,Q) between populations P and Q.



Compositional genic differentiation

Complementary genic differentiation Δ_SD : The mean genetic distance Δ of each population to a complement population formed by pooling all other populations [2]. Δ_SD ranges from 0 and 1. Δ_SD equals 1 if all populations are genetically disjoint, i.e., no two populations share the same genetics types. Δ_SD equals 0 if all populations show identical frequency distributions for the genetic types.

Dispersive genic differentiation : The mean genetic distance Δ of each population to each of the other populations [1]. It holds that



Compositional genotypic differentiation

Complementary genotypic differentiation δ_SD : The mean genetic distance of each population to its complement population [8,9]. δ ranges between from 0 and 1. δ equals 1 if all populations are genetically disjoint, i.e., no two populations share the same genetics types. δ equals 0 if all populations show identical frequency distributions for the genetic types.

Dispersive genotypic differentiation : The mean genetic distance d₀ of each population to each of the other populations [1]. It holds that



Covariation C of differentiation between integration levels: The degree of correspondence between differentiation indices from two levels of integration can be determined by a measure of covariation. As was pointed out in [13], a suitable measure of covariation is



where the variables X_i and Y_i refer to genetic distances at two different levels of integration. In the case of the distances between a population and its complement, X_i and Y_i refer to Δ at two levels of integration. In the case of pairwise genetic distances between populations, X_i and Y_i refer to the i-th element of the distance matrix for each of the two levels of integration. C varies between -1 and +1 such that C = 1 for strictly positive and C = -1 for strictly negative covariation. It is undefined in the practically irrelevant case where a non-zero difference for one variable implies equality for the other.

Confidence intervals and P-values are estimated for all the observed measures of differentiation at each level of integration by permuting (a) the alleles among the individuals within each population< klink id="124874">[2] and (b) the individuals with their multilocus genotypes among the populations [1].

Main page | Previous: MEASURES | Next: PROPERTIES OF Δ