1. Introduction to Geometric Morphometrics
François Gould, Ph.D.

2. What is geometric morphometrics?
A increasingly common buzzword

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

4. 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

5. 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!

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

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

8. Visual: The deformation grid

9. Quantitative: multivariate biometrics

10. Quantitative representation of shape I
Role of coordinate points: the landmark concept

11. 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

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

13. 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)

14. 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

15. 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.

16. On landmarks
Pivotal in geometric morphometrics

17. 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

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

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

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

21. 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

22. Doing a Geometric Morphometric analyis

23. 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

24. 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?

25. 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

26. 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

27. 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
REFLECT BIASES OF AUTHORS!

28. 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.

29. 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

30. 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

31. Example: Ecomorphological pattern in distal femoral variation