2 ShapeThe geometric properties of a configuration of points that are invariant to changes in translation, rotation, and scale. Slice, et al., 1996
3 MorphometricsThe study of shape variation and its covariation with other variablesUseful tool for quantifying anatomical objects, and showing how morphology correlates with other biological factorsDescribes 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 considerationsData should capture and archive shape in repeatable manner for statistical analysesData should be sufficient to reconstruct graphical representation of structureData should be appropriate for hypotheses being addressed
6 Morphometric features: Primary data Primary (raw) data contains more than shapeMust account for non-shape variationRaw Data = morphology + non-morphologyMorphology = size + shape + color textural patternShape* = morphology – (non-shape, size, color, texture)* Modern morphometrics is concerned with shape
7 General morphometric protocol Quantify primary (raw) dataRemove non-shape variation and generate shape variables*Perform statistical analyses to address biological hypothesesGraphical 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)DisadvantagesSize is confounding factor, and size adjustment methods not all equivalentSame values can represent different shapesHomology difficult to assessCannot usually reconstruct graphical image of shape (i.e. geometry of structure is lost)
10 TrussSets of linear distances do not record their relative position on the organism, so aspects of shape are lostThe truss includes this informationSet of distance measurements forming an interconnected networkEncodes the relative position of the distancesCan generate graphical image of specimensAverages and other statistical estimates often cause difficulties
11 From distances to landmarks Radical shift in morphometric methodology in the 1980sLinear distances alone do not capture all of shapeThe truss was an attempt to record the additional information of the relative position of linear distancesThis information is inherent in the common endpoints (landmarks) of the linear distancesIf the endpoints represent the geometric information of shape, why not just use the endpoints themselves as data?
12 Landmark coordinatesQuantify anatomical features using x,y (or x, y, z) coordinatesRemove non-shape variation (size and others)Multivariate analyses of data
13 Advantages of landmarks Homology assessments are strong, so biological interpretations more completeGeometry 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 quantificationInstead, use their boundary information as dataQuantify contour of structure using x,y (or x,y,z) coordinatesRemove non-shape variation (size and others)Multivariate analyses of data
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