# 5.5: Multivariate Brownian Motion (Summary)

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There are at least four methods for testing for an evolutionary correlation between continuous characters: likelihood ratio test, AIC model selection, PICs, and PGLS. These four methods as presented all make the same assumptions about the data and, therefore, have quite similar statistical properties. For example, if we simulate data under a multivariate Brownian motion model, both PICs and PGLS have appropriate Type I error rates and very similar power. Any of these are good choices for testing for the presence of an evolutionary correlation in your data.

## Section 5.7: Footnotes

1: We might also want to carry out linear regression, which is related to correlation analysis but distinct. We will show examples of phylogenetic regression at the end of this chapter.

2: Although the joint distribution of all species for a single trait is multivariate normal (see previous chapters), individual changes along a particular branch of a tree are univariate.

3: Another way to think about regression through the origin is to think of pairs of contrasts across any node in the tree as two-dimensional vectors. Calculating a vector correlation is equivalent to calculating a regression forced through the origin.

## References

Felsenstein, J. 1985. Phylogenies and the comparative method. Am. Nat. 125:1–15.

Garland, T., Jr. 1992. Rate tests for phenotypic evolution using phylogenetically independent contrasts. Am. Nat. 140:509–519.

Hansen, T. F. 1997. Stabilizing selection and the comparative analysis of adaptation. Evolution 51:1341–1351.

Hohenlohe, P. A., and S. J. Arnold. 2008. MIPoD: A hypothesis-testing framework for microevolutionary inference from patterns of divergence. Am. Nat. 171:366–385.

Revell, L. J., and L. J. Harmon. 2008. Testing quantitative genetic hypotheses about the evolutionary rate matrix for continuous characters. Evol. Ecol. Res. 10:311–331.