Bias and error in estimates of mean shape in geometric morphometrics F. James Rohlf Department of Ecology and Evolution, State University of New York, Stony Brook, New York 11794-5245, USA Received 5 November 2002; accepted 8 March 2003 Abstract Sampling experiments were performed to investigate mean square error and bias in estimates of mean shape produced by different geometric morphometric methods. The experiments use the isotropic error model, which assumes equal and independent variation at each landmark. The case of three landmarks in the plane (i.e., triangles) was emphasized because it could be investigated systematically and the results displayed on the printed page. The amount of error in the estimates was displayed as RMSE surfaces over the space of all possible configurations of three landmarks. Patterns of bias were shown as vector fields over this same space. Experiments were also performed using particular combinations of four or more landmarks in both two and three dimensions. It was found that the generalized Procrustes analysis method produced estimates with the least error and no pattern of bias. Averages of Bookstein shape coordinates performed well if the longest edge was used as the baseline. The method of moments (Stoyan, 1990, Model. Biomet. J. 32, 843) used in EDMA (Lele, 1993, Math. Geol. 25, 573) exhibits larger errors. When variation is not small, it also shows a pattern of bias for isosceles triangles with one side much shorter than the other two and for triangles whose vertices are approximately collinear causing them to resemble their own reflections. Similar problems were found for the log-distance method of Rao and Suryawanshi (1996, Proc. Nat. Acad. Sci. 95, 4121). These results and their implications for the application of different geometric morphometric methods are discussed.  2003 Elsevier Science Ltd. All rights reserved. Keywords: Shape coordinates; EDMA; Tangent space; Procrustes; Moment estimates; Sampling experiments; Reflection invariance Introduction In recent years there has been an increasing interest in the use of geometric morphometric methods rather than the traditional multivariate analysis of selected distance measurements, angles, and ratios to study variation in shape. Geometric morphometric methods usually begin with digi- tized coordinates of a number of landmark lo- cations. The effects of variation in the location, orientation, and scale of the specimens are elimi- nated and the differences that remain represent shape variation and are expressed with respect to a suite of shape variables. Statistically, the E-mail address: rohlf@life.bio.sunysb.edu (F.J. Rohlf). Journal of Human Evolution 44 (2003) 665–683 0047-2484/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0047-2484(03)00047-2