1. Geometric Morphometrics
A brief history

2. Shape
The geometric properties of a configuration of points that are invariant to changes in translation, rotation, and scale. Slice, et al., 1996

3. Morphometrics
The study of shape variation and its covariation with other variables
Useful tool for quantifying anatomical objects, and showing how morphology correlates with other biological factors
Describes a PATTERN not a PROCESS

4. How to measure shape Problem: no natural units of shape
Represent shape as set of proxy variables (morphometric features)
Various methods exist, depending on data available, and biological hypotheses

5. Data considerations
Data should capture and archive shape in repeatable manner for statistical analyses
Data should be sufficient to reconstruct graphical representation of structure
Data should be appropriate for hypotheses being addressed

6. Morphometric features: Primary data
Primary (raw) data contains more than shape
Must account for non-shape variation
Raw Data = morphology + non-morphology
Morphology = size + shape + color textural pattern
Shape* = morphology – (non-shape, size, color, texture)
* Modern morphometrics is concerned with shape

7. General morphometric protocol
Quantify primary (raw) data
Remove non-shape variation and generate shape variables*
Perform statistical analyses to address biological hypotheses
Graphical depiction of results
* Methods for removing non-shape variation depend on type of primary data

8. Conventional measurements
Quantify linear distances, angles, etc. between anatomical points, or measurements of structures (e.g., length of femur)
Remove non-shape variation - ????
Multivariate analyses of data (specimens are points in multivariate data space)
This is now called Multivariate Morphometrics

9. Linear measurements Advantages
Useful for comparison to previous results (which were largely based upon linear measurements)
Intuitive variables (length of femur directly interpretable)
Disadvantages
Size is confounding factor, and size adjustment methods not all equivalent
Same values can represent different shapes
Homology difficult to assess
Cannot usually reconstruct graphical image of shape (i.e. geometry of structure is lost)

10. Truss
Sets of linear distances do not record their relative position on the organism, so aspects of shape are lost
The truss includes this information
Set of distance measurements forming an interconnected network
Encodes the relative position of the distances
Can generate graphical image of specimens
Averages and other statistical estimates often cause difficulties

11. From distances to landmarks
Radical shift in morphometric methodology in the 1980s
Linear distances alone do not capture all of shape
The truss was an attempt to record the additional information of the relative position of linear distances
This information is inherent in the common endpoints (landmarks) of the linear distances
If the endpoints represent the geometric information of shape, why not just use the endpoints themselves as data?

12. Landmark coordinates
Quantify anatomical features using x,y (or x, y, z) coordinates
Remove non-shape variation (size and others)
Multivariate analyses of data

13. Advantages of landmarks
Homology assessments are strong, so biological interpretations more complete
Geometry of structure preserved in relative landmark locations (connects image to data)
Can generate graphical representation of structure

14. Possible disadvantages of landmarks
Non-shape variation present in original coordinates (size, position, orientation)
Potential difficulties representing 3-D specimens with 2-D landmarks (distortion)
Some structures have no landmarks
*NOTE: These disadvantages are all alleviated through Geometric Morphometric methods and their extensions to other data types

15. Outlines and semilandmarks
Some structures do not have sufficient landmarks for quantification
Instead, use their boundary information as data
Quantify contour of structure using x,y (or x,y,z) coordinates
Remove non-shape variation (size and others)
Multivariate analyses of data