Bivariate linear regression (for two data columns), taking into account the expected phylogenetic covariation from a Brownian model for evolution (Grafen 1989; Pagel 1999; Symonds & Blomberg 2014).
The “tree collection” in Past must contain one or more trees with branch lengths (otherwise, branch lengths of 1 are assumed). The taxon names in the trees must match the row names in the spreadsheet. IMPORTANT: The first row in the data is not used, as it is assumed to refer to an outgroup. This (dummy) outgroup must be included in the tree – the branch length below the outgroup is not used. This is to ensure compatibility with the parsimony trees generated by Past, which are always rooted on the first row.
Standard errors on regression parameters are calculated according to Smaers & Rohlf (2016).
The PGLS var-covar matrix is transformed according to the parameter lambda (Symonds & Blomberg 2014), which can be selected as 0 (equivalent to OLS regression if the tree is ultrametric; a weighted OLS otherwise), or 1 (full PGLS) or the maximum-likelihood estimate based on regression residuals (Revell 2010).
The maximum-likelihood estimates for lambda (and their optimization profiles) are calculated for the first data column only (x), the second data column only (y), a combination of x and y (Freckleton et al. 2002) or the regression residuals (Revell 2010).
References
Freckleton, R.P., Harvey, P.H., Pagel, M. 2002. Phylogenetic analysis and comparative data: A test and review of evidence. American Naturalist 160:712-726.
Grafen, A. 1989. The phylogenetic regression. Philosophical Transactions of the Royal Society B 326:119–157.
Pagel, M. 1999. Inferring the historical patterns of biological evolution. Nature 401:877–884.
Revell, L.J. 2010. Phylogenetic signal and linear regression on species data. Methods in Ecology & Evolution 1:319–329.
Smaers, J.B., Rohlf, F.J. 2016. Testing species' deviation from allometric predictions using the phylogenetic regression. Evolution 70:1145–1149.
Symonds, M.R.E., Blomberg, S.P. 2014. A primer on phylogenetic least squares. Pp. 105-130 in: L. Z. Garamszegi (ed.), Modern Phylogenetic Comparative Methods and their Application in Evolutionary Biology. Springer-Verlag.