Permutation tests for the correlation among genetic distances and measures of heterosis

Abstract

It is often found that heterosis tends to increase with genetic distance of the parents, though the correlation is not usually very close. It is therefore important to test the null hypothesis that the correlation is zero. The present work shows that standard procedures tend to yield too liberal tests, owing to the lack of independence among genetic distances and among heterosis estimates. A valid alternative is to use a permutation test, which was first suggested by Mantel [(1967) Cancer Res 27: 209–220). This test is well-known among plant breeders and geneticists, who often use it to test the correlation among two distance matrices. Its use is not restricted to the comparison of distance matrices. This is demonstrated in the present work, using two published datasets on marker-based genetic distances of maize inbreds or populations and heterosis of their crosses. It is shown that the test is also applicable in the presence of missing data.

Appendix

Using results on the distribution of linear functions of random variables, it is straightforward to see that heterosis estimates involving the same parent are stochastically dependent. For example, consider two mid-parant heterosis estimates h _ij =z _ij –(y _i +y _j )/2 (parents i and j) and h_ik=z_ik–(y _i +y _k )/2 (parents i and k), where z _ij , z _ik , y _i , y _j , and y _k are independent normal deviates with constant variance. The correlation of h _ij and h _ik is 1/6, so the two estimates are stochastically dependent.

Here is a simple example demonstrating that genetic distances involving the same parent are stochastically dependent. Assume that three lines i, j, and k are selected at random from a population of inbred lines, and consider a single locus with alleles a₁ und a₂. Let p be the allele frequency of a₁ and q the allele frequency of a₂ and define a dummy variable, w, indexed by parents. For example, w _i =0 when parent i is homozygous for allele a₁ and w _i =1 when parent i is homozygous for allele a₂. The Euclidean distances of the three parents at the locus are d _ij =|w _i -w _j |, d _ik =|w _i -w _k |, and d _jk =|w _j -w_k|. Table 4 gives the joint distribution of the three pairwise distances. To see that the distances are stochastically dependent, consider the conditional probability of the event d _ij =0, given that d _ik =1, P(d _ij =0|d _ik =1)=(p² q+pq²)/(p² q+pq²+p² q+pq²)=0.5, and the marginal probability of the event d _ij =0, P(d _ij =0)=p³+p² q+pq³+q³. Under stochastical independence, both probabilities must be equal. Obviously, P(d _ij =0|d _ik =1) ≠ P(d _ij =0) when p≠q, in which case the distances d _ij and d_ik are stochastically dependent. For example, when p=0.7, one finds P(d _ij =0)=0.58≠0.5=P(d _ij =0|d_ik=1).

Table 4 Possible realizations of random draws of z_i, z_j, and z_k with associated genetic distances d _ij =|z_i − z_j |, d _ik =|z _i − z _k |, and d_jk=|z_j − z _k | and joint probabilities P(d _ij , d _ik , d _jk )