Phylogenetic signal¶

Several metrics are available to measure "phylogenetic signal" in a trait value among the tips of a phylogeny. These metrics test the extent to which traits exhibit phylogenetic inertia, such that their values can be explained by a random walk (Brownian motion model of evolution) along the edges of a tree. A trait exhibiting low phylogenetic signal is poorly explained by the this model and the specified tree, whereas a trait exhibiting high phylogenetic signal fits well to this model of evolution on the tree.

In [1]:

import toytree
import numpy as np

import toytree import numpy as np

Example dataset¶

Let's simulate some test data. We generate a random 20 tip tree with uniform internal edge lengths and then simulate two traits ("t0" and "t1") on the tree under the Brownian motion (BM) model, and another trait ("t2") is generated by assigning random uniform values to tips of the tree (i.e., it is not simulated under a BM model on the tree). We should expect the first two traits to exhibit greater phylogenetic signal that the latter trait.

In [2]:

# generate a random tree
tree = toytree.rtree.unittree(ntips=20, treeheight=10.0, seed=123)

# generate two traits under BM model and store to tree
tree.pcm.simulate_continuous_bm(rates=[5.5, 0.05], seed=123, tips_only=True, inplace=True)

# generate a random uniform trait (i.e., not evolved under BM)
uniform = np.random.default_rng(seed=123).uniform(-50, 50, size=tree.ntips)
tree.set_node_data("t2", {i: j for (i, j) in enumerate(uniform)}, inplace=True)
                   
# show all feature data for tip nodes
tree.get_node_data()[:tree.ntips]

# generate a random tree tree = toytree.rtree.unittree(ntips=20, treeheight=10.0, seed=123) # generate two traits under BM model and store to tree tree.pcm.simulate_continuous_bm(rates=[5.5, 0.05], seed=123, tips_only=True, inplace=True) # generate a random uniform trait (i.e., not evolved under BM) uniform = np.random.default_rng(seed=123).uniform(-50, 50, size=tree.ntips) tree.set_node_data("t2", {i: j for (i, j) in enumerate(uniform)}, inplace=True) # show all feature data for tip nodes tree.get_node_data()[:tree.ntips]

Out[2]:

	idx	name	height	dist	support	t0	t1	t2
0	0	r0	0.0	7.0	NaN	77.160413	-0.245377	18.235186
1	1	r1	0.0	6.0	NaN	40.971918	0.305800	-44.617898
2	2	r2	0.0	6.0	NaN	2.540054	-0.040868	-27.964013
3	3	r3	0.0	6.0	NaN	28.549970	-0.207394	-31.562819
4	4	r4	0.0	3.0	NaN	19.181300	-0.070625	-32.409410
5	5	r5	0.0	2.0	NaN	-1.701751	0.172896	31.209451
6	6	r6	0.0	1.0	NaN	10.366843	-0.048158	42.334500
7	7	r7	0.0	1.0	NaN	9.120536	-0.053057	-22.342560
8	8	r8	0.0	3.0	NaN	16.935370	0.186425	31.975456
9	9	r9	0.0	2.0	NaN	8.033774	-0.117348	38.989269
10	10	r10	0.0	2.0	NaN	12.220348	-0.026871	1.297046
11	11	r11	0.0	5.0	NaN	17.805914	-0.045131	-25.503540
12	12	r12	0.0	7.0	NaN	11.572519	-0.033040	32.424160
13	13	r13	0.0	8.0	NaN	99.549183	-0.379688	-28.623704
14	14	r14	0.0	8.0	NaN	-3.906753	-0.249530	24.146705
15	15	r15	0.0	7.0	NaN	-19.235036	-0.203506	12.994020
16	16	r16	0.0	6.0	NaN	50.392113	0.000320	42.740726
17	17	r17	0.0	6.0	NaN	27.232351	-0.331218	-26.809181
18	18	r18	0.0	8.0	NaN	45.349668	-0.125299	29.912513
19	19	r19	0.0	9.0	NaN	37.123951	0.774358	1.816504

In [3]:

# draw tree with extra space reserved to the right 
canvas, axes, mark = tree.draw(shrink=50, label="example tree w/ 3 traits");

# draw traits color mapped on the same scale (-50, 50)
tree.annotate.add_tip_markers(axes, marker="s", xshift=45, color=("t0", "Greys"))
tree.annotate.add_tip_markers(axes, marker="s", xshift=60, color=("t1", "Greys"));
tree.annotate.add_tip_markers(axes, marker="s", xshift=75, color=("t2", "Greys"));

# draw tree with extra space reserved to the right canvas, axes, mark = tree.draw(shrink=50, label="example tree w/ 3 traits"); # draw traits color mapped on the same scale (-50, 50) tree.annotate.add_tip_markers(axes, marker="s", xshift=45, color=("t0", "Greys")) tree.annotate.add_tip_markers(axes, marker="s", xshift=60, color=("t1", "Greys")); tree.annotate.add_tip_markers(axes, marker="s", xshift=75, color=("t2", "Greys"));

Blomberg's K¶

Blomberg's K (Blomberg et al. 2003) is used to quantify phylogenetic signal relative in trait evolution relative to a Brownian motion model. Values of K>1 indicate samples are less similar than expected, whereas K<1 indicates that they are more similar than expected. Permutations can be used to perform a significance test.

Examples¶

As an example, when K is calculated for the trait "t0" that was simulated under a model of Brownian motion we recover a K statistic very close to 1.0. By contrast, when K is calculated for trait "t2" which has uniform random values assigned to the tips, we get a K of 0.4. From this we can say that the phylogenetic signal in "t0" is greater than that of "t2". However, we don't yet know how much we expect either trait to deviate from the null expectation (K=1) by chance. This deviation depends on the variance in the trait value and on the shape and size of the tree, thus we use a permutation approach below to perform a significance test.

In [4]:

# measure K for BM trait 't0'
toytree.pcm.phylogenetic_signal_k(tree, "t0")

# measure K for BM trait 't0' toytree.pcm.phylogenetic_signal_k(tree, "t0")

Out[4]:

{'K': 1.0177977520950325, 'P-value': nan, 'permutations': nan}

In [5]:

# measure K for non-BM trait 't2'
toytree.pcm.phylogenetic_signal_k(tree, "t2")

# measure K for non-BM trait 't2' toytree.pcm.phylogenetic_signal_k(tree, "t2")

Out[5]:

{'K': 0.4059842428938925, 'P-value': nan, 'permutations': nan}

Significance test¶

We can perform a permutation test to calculate the probability that the phylogenetic signal in a trait value is greater than expected by chance given the tree and variance in the trait data. This shuffles the trait values among the tips and recalculates K many times. The P-value represents the number of permutations that generate a K value with as much phylogenetic signal as the original trait data. A P-value < 0.05 is typically considered significance evidence of phylogenetic signal.

In [6]:

# measure K and perform significance test
toytree.pcm.phylogenetic_signal_k(tree, "t0", test=True)

# measure K and perform significance test toytree.pcm.phylogenetic_signal_k(tree, "t0", test=True)

Out[6]:

{'K': 1.0177977520950325, 'P-value': 0.007, 'permutations': 1000}

In [7]:

# measure K and perform significance test
toytree.pcm.phylogenetic_signal_k(tree, "t2", test=True)

# measure K and perform significance test toytree.pcm.phylogenetic_signal_k(tree, "t2", test=True)

Out[7]:

{'K': 0.4059842428938925, 'P-value': 0.868, 'permutations': 1000}

Variation (Std. Err.)¶

If a trait is measured from many individuals then you can represent its value as both a mean and standard error, and the variation can be taken into account when calculating K (Ives et al. 2007). Here a model is fit to estimate the Brownian rate parameter ($\sigma^2$), which is also returned along with the log-likelihood.

In [8]:

# measure K for trait w/ standard error and perform significance test
toytree.pcm.phylogenetic_signal_k(tree, data="t0", error="t1", test=True)

# measure K for trait w/ standard error and perform significance test toytree.pcm.phylogenetic_signal_k(tree, data="t0", error="t1", test=True)

Out[8]:

{'K': 1.0178023908892584,
 'P-value': 0.011,
 'permutations': 1000,
 'log-likelihood': -92.34654755664657,
 'sig2': 94.18633873360999,
 'convergence': True}

Multivariate K¶

Adams...

In [9]:

# TODO...

# TODO...

Lambda¶

In [10]:

# TODO...

# TODO...