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