Ability of Geometric Morphometric Methods to Estimate a Known Covariance Matrix

Landmark-based morphometric methods must estimate the amounts of translation, rotation, and scaling (or, nuisance) parameters to remove nonshape variation from a set of digitized figures. Errors in estimates of these nuisance variables will be reflected in the covariance structure of the coordinates, such as the residuals from a superimposition, or any linear combination of the coordinates, such as the partial warp and standard uniform scores. A simulation experiment was used to compare the ability of the generalized resistant fit (GRF) and a relative warp analysis (RWA) to estimate known covariance matrices with various correlations and variance structures. Random covariance matrices were perturbed so as to vary the magnitude of the average correlation among coordinates, the number of landmarks with excessive variance, and the magnitude of the excessive variance. The covariance structure was applied to random figures with between 6 and 20 landmarks. The results show the expected performance of GRF and RWA across a broad spectrum of conditions. The performance of both GRF and RWA depended most strongly on the number of landmarks. RWA performance decreased slightly when one or a few landmarks had excessive variance. GRF performance peaked when approximately 25% of the landmarks had excessive variance. In general, both RWA and GRF performed better at estimating the direction of the first principal axis of the covariance matrix than the structure of the entire covariance matrix. RWA tended to outperform GRF when > approximately 75% of the coordinates had excessive variance. When < 75% of the coordinates had excessive variance, the relative performance of RWA and GRF depended on the magnitude of the excessive variance; when the landmarks with excessive variance had standard deviations (sigma) > or = 4 sigma minimum, GRF regularly outperformed RWA.