# User:Carl Boettiger/Notebook/Comparative Phylogenetics/2010/03/02

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## Power in trees

### Exploration

• data created by brown simulation on the star tree in ouch matches the expected variance, $\displaystyle \sigma^2t$ . This variance is slightly reduced when simulated on the Felsenstein tree, as might be intuitively expected due to the increased correlations. The degree of decrease should be analytically attainable, but isn't obvious to me at the moment. Example using functions in felsenstein_tree.R, revision 210:

<syntaxhighlight lang="rsplus"> > bm <- init_brown_model(star_tree, 5) > E_trait_var(bm) [1] 24.52429 > bm <- init_brown_model(fels_tree, 5) > E_trait_var(bm) [1] 17.47615 > E_trait_var(bm, reps=1000) [1] 18.65548 </syntaxhighlight>

E_trait_var simulates reps replicates under the given model and computes the variance across the tips of each dataset generated, and averages this variance across the replicates.

• Meanwhile, data created by the hansen simulation on the star tree falls significantly short of $\displaystyle \sigma^2/2\alpha$ even when rates are certainly high enough to have reached the stationary distribution.

<syntaxhighlight lang="rsplus"> > ou <- init_hansen_model(star_tree, alpha=50, sigma=20) > E_trait_var(ou) [1] 0.0779675277 > E_trait_var(ou, 1000) [1] 0.0800359241 > # whereas theory would predict > 20^2/(2*50) [1] 4 </syntaxhighlight> not sure why this is. The simulations seems to think the steadystate variances is $\displaystyle \sigma^2/2\alpha^2$ instead. Using the deprecated function hansen.dev, I seem to recover the expected variance (averaging variance over 100 replicates):

<syntaxhighlight lang="rsplus"> > deviates <- as.data.frame(hansen.dev(100, star_tree@nodes, star_tree@ancestors, star_tree@times, regimes= NULL, alpha=5,sigma=3, theta=0)) Warning messages: 1: 'hansen.dev' is deprecated. Use 'simulate' instead. See help("Deprecated") and help("ouch-deprecated"). 2: 'is.valid.ouch.tree' is deprecated. Use 'ouchtree' instead. See help("Deprecated") and help("ouch-deprecated"). > out = sapply(1:100, function(i){var(deviates[i], na.rm=T)} ) > mean(out) [1] 0.905187 > #which agrees with theory; 3^2/(2*5) </syntaxhighlight>

• ouch has a convenient function to bootstrap a model
• ouch hansen trees give false impressions of accuracy around star (and hence nearly star) trees, where result is actually very sensitive to the original initialized search parameters for alpha and sigma.

### Brainstorming

• Current methods don't permit divergent selection models, for instance of parts of the tree. Again probably an information-limited arena, but interesting to think what one would do or do differently if you had geographic information or range overlaps along the tree (or at least for the tips).