Contrasts can also be calculated for deeper nodes in the tree if we estimate the phenotypes of hypothetical ancestors. For node 4, the appropriate value (assuming Brownian motion evolution with no directional trend) would be the average of its daughter species A and B, i.e. a value of 2.5 for trait 1. Similarly, for node 3, the value is 7. The raw contrast (difference) between nodes 4 and 3 is thus a value of 2.5–7=–4.5. This value would then be divided by its standard deviation. However, this contrast is different from those calculated at the tips of the tree. The contrast of node 4 vs 3 uses two estimated phenotypes, not actual measured data. Therefore, we should devalue it relative to the tip contrasts. This is accomplished by lengthening the branches leading to internal nodes. As explained in Felsenstein (1985), Under Brownian motion, the amount of lengthening is computed as: (daughter branch length 1 x daughter branch length 2) /(daughter branch length 1 + daughter branch length 2). Thus, the branch leading to node 4 is lengthened by 1, giving a corrected length of 4. The branch to node 3 is lengthened by 2, for a corrected length of 3. These corrected values are then used to compute the standard deviation of contrast 4 vs 3.

To compute the contrast between internal node 2 and terminal node E, the value at node 2 must be estimated. It is taken as a weighted average, with weights inversely proportional to the lengths of the branches leading to its daughter nodes 3 and 4. The value computed for node 2 will, therefore, be more similar to the value at node 3 because the branch leading to it (corrected length=3) is shorter than the one leading to node 4 (corrected length=4). The branch leading to node 2 is also lengthened, with computations that use the already-lengthened branches above node 2.

The IC algorithm involves iterative calculations best accomplished by computer programs, such as the PDTREE module of the Phenotypic Diversity Analysis Programs (available from T.G.) or the corresponding module in Mesquite (http://mesquiteproject.org/mesquite/mesquite.html). Readers are cautioned that some computer programs do not properly lengthen internal branches or have other errors. One way to check whether a program is performing the correct calculations is to collapse the internal branches of the phylogeny to be length zero, and then make all of the terminal branches (those leading to tips) equal in length. This yields a `star' phylogeny (e.g. Figs 3A and 4A). This star phylogeny can then be used to compute statistics with independent contrasts, such as the Pearson correlation (through the origin) between two traits. This value should be identical to the value obtained with a conventional statistical program (not through the origin). If the values differ, the IC program is doing something incorrectly.

Most programs for independent contrasts will output the contrasts and some simple statistics, such as the correlation between two sets of contrasts. For more complicated analyses, such as multiple regression or principal components analysis, one will generally need to write out the contrasts and then read them back into a commercial statistics package, most of which will allow the user to specify that calculations of correlations, regressions, etc., be done through the origin.

If standardized contrasts are calculated for the second trait shown in the figure, then the Pearson correlation (through the origin) with trait 1 is 0.85289. This would be considered not statistically significant as compared with a critical value of 0.878 for a two-tailed test with three degrees of freedom. For comparison, the conventional Pearson correlation of the tip data (mathematically equivalent to performing the contrasts calculations on a star phylogeny) is 0.88024, which would be considered marginally significant. Obviously, one should not get too exorcised about such small differences in parameter estimates or P values, but this example serves to make the point that results will always be somewhat different when the phylogeny is specified as being hierarchical rather than a star. For the least-squares linear regression slope, values are 0.55003 for IC and 0.60057 for conventional; corresponding y-intercepts are 1.00724 (calculated as described in Garland et al., 1993; Garland and Ives, 2000) and 0.57759.

Note that the value at the very bottom of the tree, the basal or root node, can also be estimated, but for this data set there is nothing with which to contrast it. Therefore, Felsenstein (1985) did not show calculation of the root node value in his example. However, this value is of interest because it represents (1) the phylogenetically correct estimate of the mean value for all five species and also (2) an estimate of the value for the hypothetical ancestor. As discussed elsewhere (e.g. Garland et al., 1999), both interpretations hinge on the Brownian motion assumption, and the latter also requires that no directional trend in character evolution has occurred. One can also compute standard errors and confidence intervals for the root node (Garland et al., 1999). For trait 1, the value computed by our PDTREE program is 4.115 with a standard error (S.E.) of ±1.7055 and a 95% confidence interval of –0.6204 to 8.8499. Conventional values (which are identical to those obtained after collapsing the phylogeny to be a star) are 4.200 with S.E. ±1.3191 and 95% CI of 0.5376 to 7.8624. Thus, the phylogenetic point estimate is slightly different from the conventional one, but the S.E. and CI are substantially wider. As also noted in the text, this general pattern of wide S.E. values and CIs for estimates at basal nodes has been pointed out before (Garland et al., 1999), and emphasizes that it is often hard to infer ancestral values with much confidence (Schluter et al., 1997; Cunningham et al., 1998). Finally, note that the values given above, which are derived literally by the algebra of phylogenetically independent contrasts in the PDTREE program (Garland et al., 1999), are the same as those obtained by GLS (e.g. see box 3 in Cunningham et al., 1998), and can also be obtained from our REGRESSION.M Matlab program (Blomberg et al., 2003).

Importantly, for nodes not at the base of the phylogeny, the values estimated during the IC algorithm have no special meaning. Rather, they should be viewed merely as intermediate steps in the calculations. Only the value for the root node is equivalent to the maximum likelihood estimate, which can be obtained by GLS or by the `squared-change parsimony' algorithm (Schluter et al., 1997; Cunningham et al., 1998; Garland et al., 1999). However, an interesting property of the IC algorithm is that if you reroot the tree at an internal node (as can be done in our PDTREE program) then the value obtained for that node will be the same as the GLS value (Garland et al., 1999; empirical examples there and in Laurin, 2004). Although such rerooting when performing the IC algorithm affects the values estimated at nodes, it does not affect the estimated correlation or slope between two traits.

For this example data set, the diagnostic Pearson correlation (not through the origin) between the absolute values of the standardized contrasts and their standard deviations (Garland et al., 1992) is 0.663 for trait 1 and 0.144 for trait 2, neither of which is statistically significant (two-tailed critical value for two d.f.=0.950). This would suggest that the branch lengths are adequate for the Brownian motion assumption of independent contrasts, but we should be cautious with so few data points. (Note also that the hypothetical tip values shown in the figure were just made up, not actually produced by a Brownian motion process.)

Blomberg et al. (2003) derived a statistic, termed K, which indicates the amount of `phylogenetic signal' in the tip data relative to the expectation for a trait that evolved by Brownian motion along the specified topology and branch lengths. A value of 1.00 indicates exactly the amount expected. Values <1 indicate less resemblance of relatives than expected, and values >1 indicate stronger-than-expected phylogenetic `clumping' of trait values. For trait 1, K=1.084, indicating that the tendency for related species to resemble each other is, averaged over the entire tree, just about what one would expect if the traits had actually evolved by a process similar to Brownian motion. The PHYSIG.M Matlab program can be obtained from T.G. on request.

As discussed in the text, Blomberg et al. (2003) also present a randomization test for the presence of phylogenetic signal (also implemented in PHYSIG.M). For trait 1, 247 of 1000 randomized data sets (tip values shuffled randomly) yielded a mean squared error that was less than or equal to the value of 8.0264 for the data in their correct position. Thus, the presence of signal would not be considered statistically significant at the usual P<0.05. However, computer simulations demonstrate that approximately 20 species are required to achieve a statistical power of 80% for detecting signal when it is present (Blomberg et al., 2003), so this result should not be taken to suggest that phylogenetic signal is actually lacking for these example data. (Note that the K statistic does not depend on sample size and is a valid descriptive statistic for the amount of signal even for small data sets such as this.)

Finally, the matrix derived from this tree that would be used in GLS calculations (see text) is as follows (as produced by our PDDIST program as a DSC file):

Other examples of this sort of matrix can be found in Cunningham et al. (1998, box 3) and Freckleton et al. (2002, fig. 1).