Geometric Morphometrics A brief history

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

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

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

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

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

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

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

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)

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

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?

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

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

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

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