Introduction to Geometric Morphometrics François Gould, Ph.D.

What is geometric morphometrics? A increasingly common buzzword

What is geometric morphometrics? A toolkit of methods for the numerical analysis of 2D and 3D shape variation. Several different approaches!

What does geometric morphometrics examine? Form: aspects of geometry invariant to rotation, translation, reflection Most geometric morphometric approaches also scale: leave “pure shape”. Size can be examined separately with a metric

About Size and Shape Key concepts in understanding the morphology of organisms. Size: absolute difference in magnitude between objects Shape: relative differences in geometry between organisms These concepts are tricky!

Allometry and Scaling The allometric relation is a power relation: y=m*xb or ln(y)=b*ln(x)+ln(m)

Where did geometric morphometrics come from? Result of a synthesis of two trends (Bookstein, 1991)

Visual: The deformation grid

Quantitative: multivariate biometrics

Quantitative representation of shape I Role of coordinate points: the landmark concept 3 0 5 -5 -5 7

Quantitative representation of shape II Mathematical theory of shape space A space where each point defines a single configuration of landmarks Classical shape space non-euclidean: projection

Getting into shape space: the Procrustes transform Translate, rotate, scale. Least squares fit Creates Procrustes coordinates

Analysis of Procrustes coordinates Project the shapes into a tangent space passing through the mean shape Calculate the variance-covariance matrix of the projected procrustes coordinates These can either be analysed directly (Principal components) or using the Thin Plate Spline (Partial and Relative warps)

The Procrustes transform: problems Assumptions about variance: equal distribution Iterative algorithm without true solution: data dependent May be statistically problematic: requires estimation of nuisance parameters

Other approaches Bookstein coordinates, Resistant fit: different variance assumptions EDMA: Euclidean distance matrix analysis Calculates all pairwise distances and compares them as ratios Does not require estimation of nuissance parameters Eigenshape approaches: Phi function (angle change). Ideal for outlines.

On landmarks Pivotal in geometric morphometrics

Criteria for landmark selection Landmark homology Classical three-tier formulation (Bookstein 1991) Type I: meeting of tissue types (“true” landmarks) Type II: maxima of curvature (orientation independent) Type III: extremal points

Limitations of the Bookstein paradigm Many structures cannot be reduced to type I landmarks

Methods for the analysis of curves and surfaces Semilandmarks approaches (Bookstein, 1997) Fourier transform Eigenshape approaches (Macleod and Rose, 1993, Macleod 1999)

Limitations of the Bookstein paradigm PERISSODACTYL ARTIODACTYL Cannot deal with novel structures.

What is landmark homology? Individual landmarks are not biologically homologous. Moving towards a recognition of importance of homology of the underlying biological structure. Even Bookstein now agrees! (Gunz et al., 2005) Think about the BIOLOGY, not the theory

Doing a Geometric Morphometric analyis

Uses of Geometric Morphometrics Data exploration Exploration of distribution of data (ordination) Exploration of coordinated shape change (visualisation) Source of hypothesis Hypothesis testing: Development studies (fluctuating asymmetry, integration) Evolutionary (modularity, morphological evolution) Ecomorphology

Choose the best tool What is your biological question? Type of data: Data exploration Hypothesis testing Type of data: 2D or 3D? Landmark? Outline? Surface? Sample size?

Collecting your data From specimens? From photographs From 3D models Microscribe From photographs ImageJ Be VERY careful about parallax From 3D models Laser scans CT scans

Measurement Error Morphometric data can be assessed for error Global measurement error Error associated with landmarks Need to assess each stage of data collection protocol for error Error less of a problem in cross-taxonomic studies

Transforming your data into shape coordinates: WISYWIG software Written by researchers, increasingly powerful and easy to use TPS suite MorphoJ WinEDMA Can be found at SUNY morphometrics website http://life.bio.sunysb.edu/morph/ REFLECT BIASES OF AUTHORS!

Transforming your data into shape coordinates: the hard way Can code analysis in Matlab, Mathematica and R. Full geometric morphometrics R package: Geomorph(Adams, 2012) Often necessary if working with analyses outside what other researchers do. Get on the morphmet listserv: active community.

Analysis Exploratory analysis Discrimination Hypothesis testing Ordination (PCA or Relative warps) Shape change visualisation Discrimination CVA Discriminant function Hypothesis testing MANOVA Regression 2 Block Partial least squares

Exploring your shape space All methods allow visualisations of changes in shape. HOWEVER, need to know if you are in a shape space or not: different approaches to modelling in shape space (e.g. PCA) versus non-shape space (e.g. CVA). Do not overinterpret your shapes: do not extrapolate beyond data

Example: Ecomorphological pattern in distal femoral variation