EntoGene 606 Comparative Methods 2 Approaches to analysis of continuous traits

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Ento/Gene 606Comparative Methods 2Approaches
to analysis of continuous traits
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Comparison of two continuous variables

• But what if one (or both) characters have a
  phylogenetic component?

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Important issues

• Model of evolution determines appropriate
  analytical approach
• Choice of method may depend upon understanding of
  mode of evolution
• For example, if phenotypic evolution is
  punctuational, branch lengths are irrelevant
• In any case, we wish to produce independent
  comparisons, with expectation of 0 difference and
  the same variance

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Brownian Motion model of character evolution

• Characters evolve randomly with some constant
  variance
• Amount of change is proportional to time
  available for change

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Maximum Likelihood Approaches to trait evolution

• Higher likelihood of trait change on long
  branches
• See Hardy (2006)

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Mark Pagels program Continuous

• Given bifurcating tree with branch lengths and a
  vector of continuous traits in terminal taxa
• Maximum likelihood estimation of paramters to
  distinguish between various models of evolution
• Brownian motion - variance of trait proportional
  to branch length
• Punctuational - branch lengths have no effect
• Accelerated - variance increases
  disproportionally as branch length increases

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Independent Contrasts (Independent Comparisons)
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Independent Contrasts

• We wish to compare two (or more) phenotypic
  traits, measured on continuous scale
• We need an hypothesis for cladistic relationships
  (need not be fully resolved, but we lose power
  with polytomies)
• Phylogenetic branch lengths in units of expected
  variance of character change
• (we rarely have this, so we use a surrogate)

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Branch Lengths

• Genetic or molecular distances
• Divergence times
• steps or substitutions from cladogram
  (morphology or sequences)
• Assume all branch lengths equal
• Various other models around for arbitrarily
  scaling branch lengths
• Most analytical issues seem to arise from
  treatment of branch lengths

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Felsenstein (1985)

• Independent comparisons
• Account for both common ancestry and branch length

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Computing Independent Contrasts

• Find two tips, i and j with common ancestor k
• Compute xi - xj
• Expected value is 0
• Variance proportional to branch lengths
  subtending xi and xj vi and vj
• Assign new value to K

Xk (1/vi) xi (1/vj) xj
1/vi 1/vj
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Independent Contrasts

• Xk is weighted average of xi and xj
• Weighted by inverse of branch lengths

Xk (1/vi) xi (1/vj) xj
1/vi 1/vj
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Independent Contrasts

• Xk is weighted average of xi and xj
• Weighted by inverse of branch lengths

Xk (1/vi) xi (1/vj) xj
1/vi 1/vj
Xk (vj) xi (vi) xj
vi vj
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Independent Contrasts

• Next scale branch length subtending node k
• vk vk vivj(vi vj)

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Felsenstein (1985)
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PDAP manual
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Independent Contrasts

• Should have expectations of 0 and same variance
• Raw Independent Contrasts are usually
  standardized by dividing by standard deviation
  before correlation or regression analysis
• Standardized contrasts will receive equal weight
  in regressions or correlations
• Sign is arbitrary, but always regress through
  origin

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Garland et al. (1992)

• Need to check to see that standardization is
  effective
• Plot standardized independent contrasts against
  their standard deviations
• (standard deviation is square root of sum of a
  contrasts branch lengths)
• Plot should be flat, with no significant positive
  or negative correlations
• If not, need to transform branch lengths or
  phenotypic data themselves
• for example, log transformation will equalize
  branch lengths

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Garland et al. (1992)
Significant negative correlation indicates that
contrasts with large standard deviations (long
branch lengths) are overstandardized
Absolute value of standardized contrast
s.d. of contrast
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Garland et al. (1992)

• Plot should be flat, with no significant positive
  or negative correlations
• If not, need to transform branch lengths or
  phenotypic data themselves
• for example, log transformation will tend to
  equalize contributions of long and short branch
  lengths to s.d.

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PDAP PDTREE

• Module for Mesquite
• Peter Midford, Ted Garland, Wayne Maddison
• Produces phylogenetically independent contrasts
  (PIC)
• Diagnostics to check assumptions of PIC
• Output text files (.fic) with raw independent
  contrasts and associated statistics

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Studying Selection

• Brownian motion model not appropriate for traits
  under selection (Felsenstein 1985)
• Selection persists through time and evolutionary
  changes on successive branches are correlated
• Different lineages subject to same selective
  regime

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Butler and King (2004)

• Hansen (1997) suggested using Ornstein-Uhlenbeck
  process to model evolution
• OU model has selective optimum and allows for
  directional selection
• ? selection strength parameter
• When ? 0, equivalent to Brownian motion
• ? drift parameter, intensity of random
  fluctuations

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(Butler and King 2004)
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(Butler and King 2004)
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(Butler and King 2004)
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References

• Butler, M.A. and A.A. King (2004). Phylogenetic
  comparative analysis a modeling approach for
  adaptive evolution. American Naturalist 164(6)
  683-695.
• Felsenstein, J. (1985). Phylogenies and the
  comparative method. American Naturalist 125
  1-15.
• Felsenstein, J. (1988). Phylogenies and
  quantitative methods. Annual Review of Ecology
  and Systematics 19 445-471.
• Garland Jr., T., P.H. Harvey and A.R. Ives.
  (1992). Procedures for the analysis of
  comparative data using phylogenetically
  independent contrasts. Systematic Biology 41(1)
  18-32.
• Hardy, C.R. (2006). Reconstructing ancestral
  ecologies challenges and possible solutions.
  Diversity and Distributions 12 7-19.
• Martins, E.P. and T.F. Hansen (1997).
  Phylogenies and the comparative method a general
  approach to incorporating phylogenetic
  information into the analysis of interspecific
  data. American Naturalist 149(4) 646-667.
• Rohlf, F.J. (2006). A comment on phylogenetic
  correction. Evolution 60(7) 1509-1515.

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Phylogeny and adaptive radiation of Neotropical
cichlids
Hernán López-Fernández
Section of Ecology, Evolutionary Biology and
Systematics Department of Wildlife and Fisheries
Sciences Texas AM University
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Searching for phenotype-environment correlations
Geophagine cichlids are benthic invertebrate
feeders par excellence
Many also eat other types of prey, but most
geophagines sift invertebrates from sandy and
muddy substrates, apparently using some form of
winnowing (Drucker Jensen, 1991), as many
other fishes do.
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Searching for phenotype-environment correlations
Pruned phylogeny to 23 taxa including outgroups,
Cichlasomatinae and all clades within
Geophaginae Diet and morphological data for each
taxon
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Searching for phenotype-environment correlations
Phylogeny-based comparative methods to estimate
the correlations between morphology and benthic
diet, and to reconstruct the history of their
associations.
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Searching for phenotype-environment correlations
Phylogenetically Independent Contrasts
(Felsenstein, 1985) calculated in Mesquite using
PDAP module. Branch lengths used untransformed,
squared, and log-transformed ICs chosen with
least significant correlation between absolute
contrast value and its standard deviation
(Garland et al., 1992).
Multiple regression through the origin (Garland
et al., 1992) with stepwise addition
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Multiple regression morphology vs. benthivory
Independent Contrasts
R2 0.81 (through the origin) P lt 0.0001 1st
Ceratobranchial gill raker length Mouth
position LPJ width Body depth
López-Fernández et al. In prep.
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Another Approach - Nested Anova

• Partition variance into components representing
  nested, hierarchical levels
• Can use existing taxonomy
• Partition variance into taxonomic categories
• ?2tot ?2s(g) ?2g(f) ?2f(o)

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From Harvey and Pagel (1991)
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From Harvey and Pagel (1991)
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Nested Anova

• Assumes simultaneous evolution of higher
  categories (no shared, branch lengths)
• Or punctuational evolutionary model (all change
  occurs at speciation)
• Higher taxonomic levels may not be independent
• Assumes equivalence of taxonomic rank