I am applying phylogenetic methods for regression to study linear and polynomial correlations between morphological and ecological data. Both my dependent (DEP) and independent (IND) variables are depending on a third trait (3RD), so I should correct DEP over 3RD, before I perform the regression between DEP and IND. Of course, I need to interpolate phylogeny over this correction.
I compared two methods:
1) The one used by Blomberg et al. (2003) http://onlinelibrary.wiley.com/doi/10.1111/j.0014-3820.2003.tb00285.x/abstract (pages 721-722). This is basically r = log(DEP/(3RD^b)), where b is the slope of the Phylogenetic Generalised Least Squares (PGLS) between DEP and 3RD.
2) The one used by Revell (2009) http://www.stat.wisc.edu/~larget/botany940/Revell2009.pdf (page 3260). This is r = DEP−(3RD*b), where b is a factor representing the least squares estimates of intercept and slope of the PGLS between DEP and 3RD (see previous link for details).
I would trust the second method, since it is not based on a ratio between DEP and 3RD. Indeed, from a mathematical point of view, correcting by the ratio DEP/3RD (let's call this corrected value R) would enhance the correlation between R and IND, since IND is in relation with 3RD as well. I would avoid to correct IND as well, since the IND data is characterised by high levels of error.
Nonetheless, from an a posteriori perspective, I see results with the first methods are in accordance with previous studies in the same topic.
Any suggestion?
Aren't them basically the same? Either (3RD*b) or (3RD^b) as you described are only expected value for DEP. Revel is just obtaining the standard residual, but if you transform Bloomberg's residual into its expected log format you get
r = log(DEP/(3RD^b)) => log(r)= log(DEP)-log(3RD)*b
It seems a bit messy because you are calling r both the one estimated in log and not on log scale. The same goes for DEP. In any way, they have exactly the same format. I guess both are equaly valid, and your choice should depend on the kind of data. For morphometric data, i would say that the log-form is better, but that can vary.
But I would investigate the relationship between both variables directly through an ANCOVA. Martins just published a new method that might help
Fuentes-G, J. A., Housworth, E. A., Weber, A., & Martins, E. P. (2016). Phylogenetic ANCOVA: Estimating Changes in Evolutionary Rates as Well as Relationships between Traits, 1–13. http://doi.org/10.5061/dryad.2r504.
best
First of all, I apologise for the use of "r" and "b" for both the type of residulas.
I don't think they are equal, since the "b" used by Revell is not simply a slope, but holds information about the intercept and, in general, the error structure of the phylogenetic regression.
You are right, though. An ANCOVA may clarify and many thanks for the link.
Best
You may want to investigate Phylogenetic Principal Components Analysis (pPCA) as a method for ordinating multivariate data that accounts for the phylogenetic non-independence among species means. Have a look at the Polly et al (2013) article Phylogenetic Principal Components Analysis and Geometric Morphometrics. The code for carrying out pPCA ordination and projecting the phylogenetic tree into it is available in the Geometric Morphometrics for Mathematica package. Hope that this helps.
I would also (in the sense of Marc Meyer) suggest that a multivariate approach (regression of 1 dependent variable on 2 independent variables) is better. If the 2 independent variables are themselves dependent, then, rather than using pPCA, SVD of the entries would eliminate any correlations between the 2 independent variables. The only hitch: the data matrix must be complete. However, there exist many missing-data reconstruction algorithms that are at a researcher's disposal.
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