Correlated trait evolution
Maximum likelihood approach(Pagel and Milligan)
0,0 0,1 1,0 1,1
0,0 q12 q13 0
0,1 q21 0 q24
1,0 q31 0 q34
1,1 0 q42 q43
Procedure
• Estimate the set of rates in the q-matrix that maximize the likelihood of the data and calculate that likelihood
• Constrain the matrix so that it represents independence (q12 = q34; q13 = q24; q21 = q43; q31 = q42) and repeat the calculation
• Use a likelihood ratio test to evaluate significance
Fruit color/size in
figsLomascolo et
al. (2008) OECOLOGIA
likelihood ratio = 4.889; P value
= 0.027
Wind-pollination correlated with….?
Friedman and Barrett (2008) IJPS
Wind-pollination correlated with….?
Friedman and Barrett (2008) IJPS
Issues to consider
• Rejection of independence does not tell you what kind of non-independence you have
• You need reasonable branch lengths
• Sampling matters (if perhaps less than parsimony)
Continuous traits
• All morphological traits can be treated as continuous variables
• Often people have wanted to look at the correlation of such variables across species
Sperm competition in primates
But, species are not independent!
Why phylogeny should be considered
.
Trait 1 Trait 1
Major Available Methods
• Linear/square-change Parsimony
• Independent contrasts
• Phylogenetic Generalized Least Squares
Linear/Square-change parsimony
Tips = nBranches = 2n-2
a b c d
e f
g
Linear Parsimony
• Find the set of ancestral states such that the absolute amount of change summed across branches is minimized
• Each internal node is the average of the three surrounding nodes
10
10
0
Change in x
Ch
ange
in y
Graph the changes
-10
-10
0
a b c d
e f
g1
2 3 5
4
6
1
6
2
5 3
4
Independent contrasts
Tips = nContrasts = n-1
a b c d
e f
g
24 30 40209 14 207
8 1722 35
value of x value of y
Independent Contrasts
Independent Contrasts• Calculations:
– by convention, contrasts for independent variable are positive
– contrasts for dependent variable may be positive or negative
• correlate contrasts– no correlation = no causal relationship– significant correlation = causal relationship
(negative or positive)
24 30 40209 14 207
8 1722 35
value of x value of y
Independent Contrasts
x y
d1 2 4
d2 6 10
d3 9 13
Independent Contrasts
x y
d1 2 4
d2 6 10
d3 9 13
15
10
0
x contrast
y co
ntra
st
Assumptions of independent contrasts
• We know branch lengths
• We know tip values with certainty
• Traits values evolve by Brownian motion– Needed to calculate ancestral states– Needed to accommodate error in the estimation
of ancestral states
.
Trait (x)
Expected change in time, t, is 0 with var. t
Assumptions of independent contrasts
• We know branch lengths
• Traits values evolve by Brownian motion
• Strength of correlation is the same across the tree
.
Traits (x, y)
ρ = 0.0ρ = 0.9
Broader objections
• The tip correlation may be what we care about
• No characters evolve by Brownian motion
• The assumption of a constant “correlation” is biologically unrealistic