Figure 1

Overview of approaches to estimate evolutionary distances and their covariances. A set of n sequences can be aligned jointly to obtain an MSA or in a pairwise optimal manner resulting in $\left(\begin{array}{c}n\\ 2\end{array}\right)$ optimal pairwise alignments (OPAs). Given a hypothesis of character homology, distance estimation per ML can essentially be done in two ways: jointly on a tree or pairwise. In the first case a tree's branch-lengths are estimated simultaneously. This requires an MSA. In the second case pairwise distances are estimated either from MSA induced pairwise alignments (IPAs) or from the OPAs. The distance estimators are afflicted with an error expressed by their variances and covariances. In all cases, the covariances can be modeled as a function of shared branch lengths, but this requires a phylogenetic tree. When distances are estimated based on an MSA, the variances and covariances can be obtained from ML theory or by bootstrapping over the MSA's columns. In the case of OPAs, these techniques cannot be directly applied (see Methods). We have previously presented a covariance estimator for the case where the two OPAs in question share a sequence (i.e. for triplets). In this paper, we introduce an estimator for the general case.