Introduction
Whenever we analyse comparative data, we have to make assumptions. As with any other kind of analysis, it is important to check whenever possible that our assumptions are reasonable. This page describes the ways in which CAIC makes such 'reality checks' and provides suggestions for assumption tests in a wider context.
Evolutionary assumptions
Felsenstein's (1985) model assumes that the evolution of continuous characters can be modelled as a random walk process. This provides a strong reason for logarithmic transformation of many data in comparative analyses. An increase in size of one kilogram is much more likely in a whale lineage than in a lineage of shrews. Log transformation of size data make the more reasonable assumption that different lineages are equally likely to make the same proportional change in size
The model predicts that the absolute value of the standardised contrast should be independent of the estimated value of the character for the node at which the contrast was taken. Regressions (not through the origin) of the absolute values of contrasts on the estimated nodal values should not have slopes significantly different from zero. If the slope is significant you may need to apply a transformation to your data or branch lengths (see below).
Statistical assumptions
Regression models make the assumption that the residual variation or scatter around the regression line has the same mean and variance at all points along the line. The aim of estimating branch lengths and calculating standardised, rather than raw, contrasts is to produce contrasts for continuous characters which fit this criterion and are therefore suitable for regression analysis. CAIC applies Felsenstein's (1985) approach to the branch lengths used; i.e. it assumes equal rates of evolutionary change per unit branch length in all branches of the phylogeny. With this assumption, we can calculate the variance of a raw contrast. If these raw contrasts are then scaled by dividing them by the square root of their expected variances, the results should be suitable for use in regression analysis. However, if evolution has not proceeded in a way that conforms to Felsenstein's model, or if the approximate branch lengths are systematically biased, then these scalings will not be correct; there will be heterogeneity of variance in the residuals. See Diaz-Uriate & Garland (1996) for a discussion of branch length transformations.
Two sorts of test are relevant. First, we can test whether the absolute values of scaled (standardised) contrasts are independent of both the age of the node ("Height") and the square root of the expected variance ("SD"). Secondly we can look directly for heterogenity of variance in the residuals. Since the residuals were derived by regression through the origin, the predicted values of the dependent (Y) variable are directly proportional to the contrasts in the predictor variable. Thus we can regress the absolute values of the residuals against the standardised contrasts in the independent (X) variable.
What to do if the assumptions are violated?
One useful rule of thumb that has been found to be useful is to repeat statistical analysis following deletion of all contrasts with studentised residuals greater than 3 (see Jones & Purvis 1997). This is especially useful in multivariate analyses. However, for obvious reasons this should be done sparingly and you should check that the residual variance is reduced without much change in slope.
In the face of persistent heterogeneity of variance, you may want to do a weighted regression. Weighted regressions stretch or compress each of the data points as a way of equalizing the residual variances. Many statistical packages have a weighted regression option.
Multiple nodes are another possible cause of heterogeneity of variance: they might have systematically different variances from bifurcating nodes. To test for such an effect after performing a regression, calculate the residuals about the regression line and do an analysis of variance on the residuals using the 'Number of sub-taxa' column from the output as the grouping factor. The variance of residuals (not the mean) by number of sub-taxa is of interest. If they differ significantly (testable by variance ratio), then tests of significance may lose some sensitivity. If a test expected to be significant comes out non-significant in the presence of heterogeneity of variance, it may be useful to explore the data to see if one or a few taxonomic groups are responsible.
Assumption Testing in CAIC
The above tests are conducted automatically in CAIC 2.6.x when two columns of data are chosen for analysis. If any assumption is violated a message appears onscreen and the result is written to the stats file as follows:
| Result | Written to Stats file |
| p<0.05 | Slope of regression equation |
| 0.05<p<0.1 | Direction of slope |
| p>0.1 | "n.s" |
The slope (rather than the p-value) is written so as to be informative in correcting the violation. A positive slope when testing the evolutionary assumptions means that large changes are associated with a large values. The slope can be lowered by applying a more severe transformation, e.g. by moving up the scale from Raw (x) to Sqrt(x) to Log(x) to (-1/x). A negative slope can be corrected by moving down the scale.
Although these tests are fiddly to conduct by hand and only tested automatically in a limited set of circumstances, the software can be adapted to suit every need. For instance, in order to work out what the correct transformation might be, try setting up dummy datasets containing the raw value of each character and various candidate transformations. Run the columns in pairs and see which one doesn't produce a violation.
Note that the p-values of the assumption checks may differ slightly depending on which variable is the main predictor. This is due to the way CAIC handles polytomies.
Note that CAIC v2.6.x does NOT represent a comprehensive comparative modelling package and we recommend that users also validate their analyses with traditional model checks such as diagnostic plots.
Assumption Testing in MacroCAIC
All tests in CAIC are also performed in MacroCAIC, although only when a single column is chosen for analysis. However, additional assumptions are necessary to calculate contrasts in clade growth rate. These are tested by regressing the absolute value of the clade richness contrasts against the total clade size (species richness) at the node in question. The results are written in the same format as other assumption tests.
References
Diaz-Uriate, R. & Garland ,T. (1996). Testing hypotheses of correlated evolution using phylogenetically independent contrasts: sensitivity to deviations from Brownian motion. Systematic Biology 45, 27-47.
Felsenstein, J. (1985). Phylogenies and the comparative method. American Naturalist 125, 1-15.
Jones, K.E. & Purvis, A. (1997). An optimum body size for mammals? Comparative evidence from bats. Functional Ecology 11, 751-756.
Pagel, M. D. (1992). A method for the analysis of comparative data. Journal of Theoretical Biology 156, 431-442.