ࡱ > 3 5 2 | ~ B D F H J L N P
R T V
X Z y M bjbj 8R { { ) E p p 8 d z P 0 ( 6 6 6 9P ;P ;P ;P ;P ;P ;P $ R U _P 9 6 6 6 6 6 _P H P ," ," ," 6 K ," 6 9P ," ," 1 73 @B- T 72 K P 0 P K2 FV B FV ( 73 73 FV 3 6 6 ," 6 6 6 6 6 _P _P f 6 6 6 P 6 6 6 6 FV 6 6 6 6 6 6 6 6 6 p : MACROBUTTON MTEditEquationSection2 Equation Chapter 1 Section 1 SEQ MTEqn \r \h \* MERGEFORMAT SEQ MTSec \r 1 \h \* MERGEFORMAT SEQ MTChap \r 1 \h \* MERGEFORMAT
Supplementary Material from D.C. Adams, Comparing evolutionary rates for different phenotypic traits on a phylogeny using likelihood. Systematic Biology.
Type I error and Statistical Power
Here I use computer simulations to show the statistical properties of the proposed method for comparing evolutionary rates for different traits using likelihood. These simulations demonstrate that the proposed method has appropriate Type I error rates, and reasonable statistical power. Three sets of simulations were performed. The first set was conducted on four perfectly balanced phylogenetic trees that differed in their number of taxa: N = 16, 32, 64, 128. The second set was conducted on randomly generated trees that differed in their number of taxa: N = 16, 32, 64, 128. The final set of simulations was conducted on perfectly balanced phylogenetic trees containing 32 taxa.
For each simulation, two phenotypic traits were evolved along the phylogeny. Changes in the first trait followed a Brownian motion model of evolution with a rate parameter of EMBED Equation.DSMT4 . Changes in the second trait also followed a Brownian motion model of evolution, but with differing evolutionary rates that depended on the type of simulation. For Type I error rate simulations, the evolutionary rates for both traits were equal: EMBED Equation.DSMT4 . For the power simulations, the evolutionary rate for the second trait was higher than for that of the first trait ( EMBED Equation.DSMT4 ), in increments up to a four-fold difference in relative rates ( EMBED Equation.DSMT4 ). For each scenario, 1000 data sets were simulated on the phylogeny, using sim.char function in geiger ( ADDIN EN.CITE Harmon200827027027017Harmon, L.J.J. WeirC. BrockR.E. GlorW. ChallengerGEIGER: Investigating evolutionary radiationsBioinformaticsBioinformaticsBioinformaticsBioinformatics129-131242008Harmon et al., 2008). For the case of randomly generated trees, a new phylogeny was simulated for each data set.
Evolutionary rates and were then estimated using equation 1 in the text, and the proposed likelihood ratio test was used to determine whether the two traits differed significantly in their evolutionary rates. Additionally, two alternative approaches were used to test for differences in evolutionary rates. First, phylogenetically independent contrasts were obtained for each trait, and evolutionary rates for the two traits were compared using a t-test on the absolute value of the contrasts ( ADDIN EN.CITE Garland199224824824817Garland, T.J.Rate tests for phenotypic evolution using phylogenetically independent contrastsAmerican NaturalistAm. Nat.American NaturalistAm. Nat.American NaturalistAm. Nat.2104-21111401992Garland, 1992). Second, 95% confidence intervals were estimated for each evolutionary rate, and were compared to one another for potential overlap. Standard errors of each evolutionary rate were obtained from the Hessian matrix, which is a matrix of second-order partial derivatives of the likelihood function ( ADDIN EN.CITE Harville19972872872876Harville, D.A.Matrix algebra from a statistician's perspective1997New YorkSpringerHarville, 1997). The negative inverse of the Hessian matrix is obtained, and standard errors of the model parameters are the square-root of the diagonals of this inverse matrix. 95% CI can were then derived from these standard errors. Across all conditions, simulations where EMBED Equation.DSMT4 represented Type I error rate assessments (i.e., no difference in their evolutionary rates), while simulations where EMBED Equation.DSMT4 assessed statistical power. All three methods were used to evaluate data from the first set of simulations.
The second set of simulations was performed in a manner identical to the first set. However, here I compared the two implementations of the proposed likelihood approach to evaluate their performance. Specifically, I compared results based on the implementation that includes trait covariation to results from the implementation where traits were assumed to be independent. As before, across all conditions, simulations where EMBED Equation.DSMT4 represented Type I error rate assessments (i.e., no difference in their evolutionary rates), while simulations where EMBED Equation.DSMT4 assessed statistical power.
The third set of simulations evaluated the effect of incorporating within-species trait covariation when generating the data. Here I simulated data as described above, but included increasing amounts of input covariation between traits (r = 0.0, 0.1, 0.25, 0.5, 0.75). The Type I error rate and statistical power of the proposed likelihood approach was then evaluated as described above. In addition, I examined parameter estimates from a subset of simulations with and without trait covariation (i.e., r = 0.0 and 0.5).
Results: For all simulation conditions, the likelihood-based method for comparing evolutionary rates between traits displayed appropriate Type I error rates near a = 0 . 0 5 ( f i g . A 1 ) . A d d i t i o n a l l y , w h e n e v o l u t i o n a r y r a t e s d i f f e r e d b e t w e e n t r a i t s ( E M B E D E q u a t i o n . D S M T 4 ) , t h e l i k e l i h o o d - b a s e d m e t h o d d e m o n s t r a t e d a c c e p t a b l e s t a t i s t i c a l p o w e r , a n d w a s c a p a b l e o f i d e n t i f y i n g e v e n s m a l l d i f f e r e n c e s i n E M B E D E q u a t i o n .DSMT4 between traits. The power of the test rose rapidly as the difference between evolutionary rates increased. This pattern became more acute as the number of species in the phylogeny increased (fig. A1).
When compared to the independent contrasts method, the likelihood-based approach displayed higher statistical power for all conditions. This was the case for both balanced and random trees, across all levels of species richness, and for all differences in relative rates (fig. A1). Similarly, the likelihood-based approach outperformed the method based on 95% CI. Here, the latter displayed lower type I error rates than expected, and low power for detecting small differences in evolutionary rates (fig. A1).
In terms of the two implementations of the likelihood approach, results for the likelihood method incorporating trait covariation attained slightly higher statistical power as compared to the method where traits were assumed to evolve independently (fig. A2). Thus, the likelihood model incorporating trait covariation is generally preferred.
The statistical performance of the likelihood approach was similar when traits were simulated with increasing amounts of trait covariation (fig. A3). Additionally, parameter estimates representing evolutionary rates and trait covariances were similar to input values for all simulations (fig. A4).
Fig. A1. Statistical power curves for tests comparing rates of evolution based on computer simulations. For each plot the abscissa shows the relative difference in evolutionary rates between the two traits while the ordinate displays the relative power based on the percentage of 1000 simulations found to be significant. Results for the likelihood-based approach are shown as black dots, results for the independent contrasts approach are shown as grey squares, and results from a comparison of 95% confidence intervals found from parameter standard errors are shown as white diamonds.
Fig. A2. Statistical power curves for two implementations of the likelihood approach for comparing compare rates of evolution revealed by computer simulations. Results for the likelihood-based approach incorporating trait covariation are shown as black dots, results for the likelihood approach assuming trait independence are shown as grey squares.
Fig. A3. Statistical power curves for the likelihood approach for comparing compare rates of evolution revealed by computer simulations. Power curve for traits simulated with no covariation are shown as black dots. Power curves for data simulated with increasing amounts of trait covariation are as follows: r = 0.1 (black squares); r = 0.25 (black diamonds); r = 0.5 (black triangles); r = 0.75 (black down-triangles).
Fig. A4. Parameter estimates from simulations for two traits under independence (r = 0.0: left panel) and including trait covariation (r = 0.5: right panel). Parameters are designated as: rate for trait 1 (black circles), rate for trait 2 (black diamonds), covariation between traits (gray squares).
Literature Cited
% A B b d # 6 7 I K L n v ȸuh\P\D h W CJ OJ QJ aJ hv! CJ OJ QJ aJ hbL CJ OJ QJ aJ hbL 5CJ OJ QJ aJ hP, ;CJ OJ QJ aJ hU 56CJ OJ QJ aJ hR 5CJ OJ QJ aJ hU 5CJ OJ QJ aJ h$ 5CJ OJ QJ aJ h]u hD 5CJ OJ QJ aJ h?s h?K ;CJ OJ QJ aJ h[ h[ 0J h[ ;CJ OJ QJ aJ "j h[ ;CJ OJ QJ UaJ K L o r {" |" &