The problem of assessing landmark error in geometric morphometrics: theory, methods, and modifications

Geometric morphometric methods rely on the accurate identification and quantification of landmarks on biological specimens. As in any empirical analysis, the assessment of inter- and intra-observer error is desirable. A review of methods currently being employed to assess measurement error in geometric morphometrics was conducted and three general approaches to the problem were identified. One such approach employs Generalized Procrustes Analysis to superimpose repeatedly digitized landmark configurations, thereby establishing whether repeat measures fall within an acceptable range of variation. The potential problem of this error assessment method (the "Pinocchio effect") is demonstrated and its effect on error studies discussed. An alternative approach involves employing Euclidean distances between the configuration centroid and repeat measures of a landmark to assess the relative repeatability of individual landmarks. This method is also potentially problematic as the inherent geometric properties of the specimen can result in misleading estimates of measurement error. A third approach involved the repeated digitization of landmarks with the specimen held in a constant orientation to assess individual landmark precision. This latter approach is an ideal method for assessing individual landmark precision, but is restrictive in that it does not allow for the incorporation of instrumentally defined or Type III landmarks. Hence, a revised method for assessing landmark error is proposed and described with the aid of worked empirical examples.     

Bias and error in estimates of mean shape in geometric morphometrics

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.     

Analysis of covariance with pre-treatment measurements in randomized trials: comparison of equal and unequal slopes

In randomized trials, an analysis of covariance (ANCOVA) is often used to analyze post-treatment measurements with pre-treatment measurements as a covariate to compare two treatment groups. Random allocation guarantees only equal variances of pre-treatment measurements. We hence consider data with unequal covariances and variances of post-treatment measurements without assuming normality. Recently, we showed that the actual type I error rate of the usual ANCOVA assuming equal slopes and equal residual variances is asymptotically at a nominal level under equal sample sizes, and that of the ANCOVA with unequal variances is asymptotically at a nominal level, even under unequal sample sizes. In this paper, we investigated the asymptotic properties of the ANCOVA with unequal slopes for such data. The estimators of the treatment effect at the observed mean are identical between equal and unequal variance assumptions, and these are asymptotically normal estimators for the treatment effect at the true mean. However, the variances of these estimators based on standard formulas are biased, and the actual type I error rates are not at a nominal level, irrespective of variance assumptions. In equal sample sizes, the efficiency of the usual ANCOVA assuming equal slopes and equal variances is asymptotically the same as those of the ANCOVA with unequal slopes and higher than that of the ANCOVA with equal slopes and unequal variances. Therefore, the use of the usual ANCOVA is appropriate in equal sample sizes.     

Principles and methods of geometric morphometrics

The basic concepts, notions and methods of geometric morphometrics (GM) are considered. This approach implies multivariate analysis of landmark coordinates located following certain rules on the surface of a morphological object. The aim of GM is to reveal differences between morphological objects by their shapes as such, the "size factor" being excluded. The GM is based on the concept of Kendall's space (KS) defined as a hypersphere with points distributed on its surface. These points are the shapes defined as aligned landmark configurations. KS is a non-Euclidian space, its metrics called Procrustes is defined by landmark configuration of a reference shape relative to which other shapes are aligned and compared. The differences among shapes are measured as Procrustes distances between respective points. For the linear methods of multivariate statistics to be applied to comparison of shapes, the respective points are projected onto the tangent plane (tangent space), the tangent point being defined by the reference. There are two principal methods of shape comparisons in GM: the Procrustes superimposition (a version of the least squares analysis) and thin-plate spline analysis. In the first case, Procrustes residuals are the outcome shape variables which remain after isometric alignment of the shapes being compared. Their summation over all landmarks yields Procrustes distances among these shapes. The Procrustes distances can be used in multivariate analyses just as the Euclidian distances. In the second case, the shapes are fitted to the references by stretching/compressing and shearing until complete identity of their landmark configurations. Eigenvectors of resulting bending energy matrix are defined as new shape variables, principal warps which yield another shape space with the origin defined by the reference. Projections of the shapes being compared onto principal warps yield partial warps, and their covariance matrix decomposition into eigenvectors yields relative warps which are similar to principal components (in particular, they are mutually orthogonal). Both partial and relative warps can be used in many multivariate statistic analyses as quantitative shape variables. Results of thin-plate spline analysis can be represented graphically by transformation grid which displays type, amount and localization of the shape differences. Basis rules of sample composition and landmark positioning to be used in GM are considered. At present, rigid (with minimal degrees of freedom) 2D morphological objects are most suitable for GM applications. It is important to recognize three type of real landmarks, and additionally semi-landmarks and "virtual" landmarks. Some procedures of thin-plate spline analysis are considered exemplified by some study cases, as well as applications of some standard multivariate methods to GM results. They make it possible to evaluate correlation between different shapes, as well as between a shape and some non-shape variables (linear measurements etc); to evaluate the differences among organisms by shape of a morphological structure; to identify landmarks which most accounted for both correlation and differences between the shapes. An annotated list of most popular softwares for GM is provided.     

An Integrated Approach for Landmark-Based Resistant Shape Analysis in 3D

The study of shape changes in morphology has seen a significant renovation in the last 20 years, particularly as a consequence of the development of geometric morphometric methods based on Cartesian coordinates of points. In order to extract information about shape differences when Cartesian coordinates are used, it is necessary to establish a common reference frame or system for all specimens to be compared. Therefore, a central issue in coordinate-based methods is which criterion should be used to align these configurations of points, since shape differences highly depend on those alignments. This is usually accomplished by aligning the configurations in a way that the sum of squared distances between coordinates of homologous points (landmarks) is minimized: the least-squares superimposition method. However, it is widely recognized that this method has some limitations when shape differences are not homogeneous across landmarks. Here we present an integrated approach for the resistant shape comparison of 3D landmark sets. It includes a new ordinary resistant Procrustes superimposition and its corresponding generalized resistant Procrustes version. In addition, they are combined with existing resistant multivariate statistical techniques for depicting the results. We demonstrate, by using both simulated and real datasets, that resistant Procrustes better detects and measures localized shape variation whenever present in up to half but one of the landmarks. The resistant Procrustes results are highly concordant with a priori biological information, and might dramatically improve the quality of inferences on patterns of shape variation.     

A simple method for visualization of influential landmarks when using euclidean distance matrix analysis

Euclidean distance matrix analysis (EDMA) differs from most other morphometric methods for the analysis of landmark coordinate data in that it is coordinate-system invariant. However, strict adherence to coordinate-system invariance (for both biological and statistical reasons) introduces some difficulty in using graphic aids for the analysis and interpretation of EDMA results. We present a simple and effective graphic method to help localize important differences in form, growth, or shape by identifying “influential” landmarks. Examples are presented using simulated data and real data involving both children with craniofacial dysmorphologies and sexual dimorphism in adult Macaca fascicularis. Am J Phys Anthropol 107:273–283, 1998. © 1998 Wiley-Liss, Inc.     

Capturing data from three-dimensional surfaces using fuzzy landmarks

Anatomical landmarks are defined as biologically meaningful loci that can be unambiguously defined and repeatedly located with a high degree of accuracy and precision. The neurocranial surface is characteristically void of such loci. We define a new class of landmarks, termed fuzzy landmarks, that will allow us to represent the form of the neurocranium. A fuzzy landmark represents the position of a biological structure that is precisely delineated, but occupies an area that is larger than a single point in the observer's reference system. In this study, we present a test case in which the cranial bosses are evaluated as fuzzy landmarks. Five fuzzy landmarks (the cranial bosses) and three traditional landmarks were placed repeatedly by a single observer on three-dimensional (3D) computed tomography (CT) surface reconstructions of pediatric dry skulls and skulls of pediatric patients, and directly on four of the same dry skulls using a 3Space digitizer. Thirty landmark digitizing trials from CT scans show an average error of 1.15 mm local to each fuzzy landmark, while the average error for the last ten trials was 0.75 mm, suggesting a learning curve. Data collected with the 3Space digitizer was comparable. Measurement error of fuzzy landmarks is larger than that of traditional landmarks, but is acceptable, especially since fuzzy landmarks allow inclusion of areas that would otherwise go unsampled. The information obtained is valuable in growth studies, clinical evaluation, and volume measurements. Our method of fuzzy landmarking is not limited to cranial bosses, and can be applied to any other anatomical features with fuzzy boundaries. Am J Phys Anthropol 107:113–124, 1998. © 1998 Wiley-Liss, Inc.     

Factors Influencing Superimposition Error of 3D Cephalometric Landmarks by Plane Orientation Method Using 4 Reference Points: 4 Point Superimposition Error Regression Model

Superimposition has been used as a method to evaluate the changes of orthodontic or orthopedic treatment in the dental field. With the introduction of cone beam CT (CBCT), evaluating 3 dimensional changes after treatment became possible by superimposition. 4 point plane orientation is one of the simplest ways to achieve superimposition of 3 dimensional images. To find factors influencing superimposition error of cephalometric landmarks by 4 point plane orientation method and to evaluate the reproducibility of cephalometric landmarks for analyzing superimposition error, 20 patients were analyzed who had normal skeletal and occlusal relationship and took CBCT for diagnosis of temporomandibular disorder. The nasion, sella turcica, basion and midpoint between the left and the right most posterior point of the lesser wing of sphenoidal bone were used to define a three-dimensional (3D) anatomical reference co-ordinate system. Another 15 reference cephalometric points were also determined three times in the same image. Reorientation error of each landmark could be explained substantially (23%) by linear regression model, which consists of 3 factors describing position of each landmark towards reference axes and locating error. 4 point plane orientation system may produce an amount of reorientation error that may vary according to the perpendicular distance between the landmark and the x-axis; the reorientation error also increases as the locating error and shift of reference axes viewed from each landmark increases. Therefore, in order to reduce the reorientation error, accuracy of all landmarks including the reference points is important. Construction of the regression model using reference points of greater precision is required for the clinical application of this model.     

Testing different 3D techniques using geometric morphometrics: Implications for cranial fluctuating asymmetry in humans

This study aimed to test the performance of 3D digitizer, CT scanner, and surface scanner in detecting cranial fluctuating asymmetry. Sets of 32 landmarks (6 in the midline and 13 bilateral) were acquired from 14 archeological crania using a 3D digitizer, and from 3D models generated from a CT scanner and surface scanner using Viewbox 4. Levels of shape variation were analyzed in MorphoJ using Procrustes analysis of variance and Principal component analysis. Intra-observer error accounted for 1.7%, 1.8%, and 4.5% of total shape variation for 3D digitizer, CT scanner, and surface scanner respectively. Fluctuating asymmetry accounted for 15%–16% of total shape variation. Variation between techniques accounted for 18% of total shape variation. We found a higher level of missing landmarks in our surface scan data than for both 3D digitizer and CT scanner data, and both 3D model-based techniques sometimes obscured taphonomic damage. All three 3D techniques are appropriate for measuring cranial fluctuating asymmetry. We advise against combining data collected with different techniques.     

Transformations of count data for tests of interaction in factorial and split-plot experiments

In applied entomological experiments, when the response is a count-type variable, certain transformation remedies such as the square root, logarithm (log), or rank transformation are often used to normalize data before analysis of variance. In this study, we examine the usefulness of these transformations by reanalyzing field-collected data from a split-plot experiment and by performing a more comprehensive simulation study of factorial and split-plot experiments. For field-collected data, significant interactions were dependent upon the type of transformation. For the simulation study, Poisson distributed errors were used for a 2 by 2 factorial arrangement, in both randomized complete block and split-plot settings. Various sizes of main effects were induced, and type I error rates and powers of the tests for interaction were examined for the raw response values, log-, square root-, and rank-transformed responses. The aligned rank transformation also was investigated because it has been shown to perform well in testing interactions in factorial arrangements. We found that for testing interactions, the untransformed response and the aligned rank response performed best (preserved nominal type I error rates), whereas the other transformations had inflated error rates when main effects were present. No evaluations of the tests for main effects or simple effects have been conducted. Potentially these transformations will still be necessary when performing these tests.     

CAT & MAUS: A novel system for true dynamic motion measurement of underlying bony structures with compensation for soft tissue movement

Optoelectronic motion capture systems are widely employed to measure the movement of human joints. However, there can be a significant discrepancy between the data obtained by a motion capture system (MCS) and the actual movement of underlying bony structures, which is attributed to soft tissue artefact. In this paper, a computer-aided tracking and motion analysis with ultrasound (CAT & MAUS) system with an augmented globally optimal registration algorithm is presented to dynamically track the underlying bony structure during movement. The augmented registration part of CAT & MAUS was validated with a high system accuracy of 80%. The Euclidean distance between the marker-based bony landmark and the bony landmark tracked by CAT & MAUS was calculated to quantify the measurement error of an MCS caused by soft tissue artefact during movement. The average Euclidean distance between the target bony landmark measured by each of the CAT & MAUS system and the MCS alone varied from 8.32 mm to 16.87 mm in gait. This indicates the discrepancy between the MCS measured bony landmark and the actual underlying bony landmark. Moreover, Procrustes analysis was applied to demonstrate that CAT & MAUS reduces the deformation of the body segment shape modeled by markers during motion. The augmented CAT & MAUS system shows its potential to dynamically detect and locate actual underlying bony landmarks, which reduces the MCS measurement error caused by soft tissue artefact during movement.     

Advances in Geometric Morphometrics

Geometric morphometrics is the statistical analysis of form based on Cartesian landmark coordinates. After separating shape from overall size, position, and orientation of the landmark configurations, the resulting Procrustes shape coordinates can be used for statistical analysis. Kendall shape space, the mathematical space induced by the shape coordinates, is a metric space that can be approximated locally by a Euclidean tangent space. Thus, notions of distance (similarity) between shapes or of the length and direction of developmental and evolutionary trajectories can be meaningfully assessed in this space. Results of statistical techniques that preserve these convenient properties—such as principal component analysis, multivariate regression, or partial least squares analysis—can be visualized as actual shapes or shape deformations. The Procrustes distance between a shape and its relabeled reflection is a measure of bilateral asymmetry. Shape space can be extended to form space by augmenting the shape coordinates with the natural logarithm of Centroid Size, a measure of size in geometric morphometrics that is uncorrelated with shape for small isotropic landmark variation. The thin-plate spline interpolation function is the standard tool to compute deformation grids and 3D visualizations. It is also central to the estimation of missing landmarks and to the semilandmark algorithm, which permits to include outlines and surfaces in geometric morphometric analysis. The powerful visualization tools of geometric morphometrics and the typically large amount of shape variables give rise to a specific exploratory style of analysis, allowing the identification and quantification of previously unknown shape features.     

Sharing is caring? Measurement error and the issues arising from combining 3D morphometric datasets

Geometric morphometrics is routinely used in ecology and evolution and morphometric datasets are increasingly shared among researchers, allowing for more comprehensive studies and higher statistical power (as a consequence of increased sample size). However, sharing of morphometric data opens up the question of how much nonbiologically relevant variation (i.e., measurement error) is introduced in the resulting datasets and how this variation affects analyses. We perform a set of analyses based on an empirical 3D geometric morphometric dataset. In particular, we quantify the amount of error associated with combining data from multiple devices and digitized by multiple operators and test for the presence of bias. We also extend these analyses to a dataset obtained with a recently developed automated method, which does not require human‐digitized landmarks. Further, we analyze how measurement error affects estimates of phylogenetic signal and how its effect compares with the effect of phylogenetic uncertainty. We show that measurement error can be substantial when combining surface models produced by different devices and even more among landmarks digitized by different operators. We also document the presence of small, but significant, amounts of nonrandom error (i.e., bias). Measurement error is heavily reduced by excluding landmarks that are difficult to digitize. The automated method we tested had low levels of error, if used in combination with a procedure for dimensionality reduction. Estimates of phylogenetic signal can be more affected by measurement error than by phylogenetic uncertainty. Our results generally highlight the importance of landmark choice and the usefulness of estimating measurement error. Further, measurement error may limit comparisons of estimates of phylogenetic signal across studies if these have been performed using different devices or by different operators. Finally, we also show how widely held assumptions do not always hold true, particularly that measurement error affects inference more at a shallower phylogenetic scale and that automated methods perform worse than human digitization.     

Landmarking the Brain for Geometric Morphometric Analysis: An Error Study

Neuroanatomic phenotypes are often assessed using volumetric analysis. Although powerful and versatile, this approach is limited in that it is unable to quantify changes in shape, to describe how regions are interrelated, or to determine whether changes in size are global or local. Statistical shape analysis using coordinate data from biologically relevant landmarks is the preferred method for testing these aspects of phenotype. To date, approximately fifty landmarks have been used to study brain shape. Of the studies that have used landmark-based statistical shape analysis of the brain, most have not published protocols for landmark identification or the results of reliability studies on these landmarks. The primary aims of this study were two-fold: (1) to collaboratively develop detailed data collection protocols for a set of brain landmarks, and (2) to complete an intra- and inter-observer validation study of the set of landmarks. Detailed protocols were developed for 29 cortical and subcortical landmarks using a sample of 10 boys aged 12 years old. Average intra-observer error for the final set of landmarks was 1.9 mm with a range of 0.72 mm–5.6 mm. Average inter-observer error was 1.1 mm with a range of 0.40 mm–3.4 mm. This study successfully establishes landmark protocols with a minimal level of error that can be used by other researchers in the assessment of neuroanatomic phenotypes.     

Anatomy of emotion: a 3D study of facial mimicry

Alterations in facial motion severely impair the quality of life and social interaction of patients, and an objective grading of facial function is necessary. A method for the non-invasive detection of 3D facial movements was developed. Sequences of six standardized facial movements (maximum smile; free smile; surprise with closed mouth; surprise with open mouth; right side eye closure; left side eye closure) were recorded in 20 healthy young adults (10 men, 10 women) using an optoelectronic motion analyzer. For each subject, 21 cutaneous landmarks were identified by 2-mm reflective markers, and their 3D movements during each facial animation were computed. Three repetitions of each expression were recorded (within-session error), and four separate sessions were used (between-session error). To assess the within-session error, the technical error of the measurement (random error, TEM) was computed separately for each sex, movement and landmark. To assess the between-session repeatability, the standard deviation among the mean displacements of each landmark (four independent sessions) was computed for each movement. TEM for the single landmarks ranged between 0.3 and 9.42 mm (intrasession error). The sex- and movement-related differences were statistically significant (two-way analysis of variance, p=0.003 for sex comparison, p=0.009 for the six movements, p<0.001 for the sex x movement interaction). Among four different (independent) sessions, the left eye closure had the worst repeatability, the right eye closure had the best one; the differences among various movements were statistically significant (one-way analysis of variance, p=0.041). In conclusion, the current protocol demonstrated a sufficient repeatability for a future clinical application. Great care should be taken to assure a consistent marker positioning in all the subjects.     

Comparison of asymptotic population growth rates with heterogeneous variances

This report explores how the heterogeneity of variances affects randomization tests used to evaluate differences in the asymptotic population growth rate, λ. The probability of Type I error was calculated in four scenarios for populations with identical λ but different variance of λ: (1) Populations have different projection matrices: the same λ may be obtained from different sets of vital rates, which gives room for different variances of λ. (2) Populations have identical projection matrices but reproductive schemes differ and fecundity in one of the populations has a larger associated variance. The two other scenarios evaluate a sampling artifact as responsible for heterogeneity of variances. The same population is sampled twice, (3) with the same sampling design, or (4) with different sampling effort for different stages. Randomization tests were done with increasing differences in sample size between the two populations. This implies additional differences in the variance of λ. The probability of Type I error keeps at the nominal significance level (α = .05) in Scenario 3 and with identical sample sizes in the others. Tests were too liberal, or conservative, under a combination of variance heterogeneity and different sample sizes. Increased differences in sample size exacerbated the difference between observed Type I error and the nominal significance level. Type I error increases or decreases depending on which population has a larger sample size, the population with the smallest or the largest variance. However, by their own, sample size is not responsible for changes in Type I errors.     

Phylogeny,landmark analysis and the use of wing venation to study the evolution of social wasps (Hymenoptera: Vespidae: Vespinae)

Wing venation provides useful characters with which to classify extant and fossil insects. Recently, quantification of its shape using landmarks has increased the potential of wing venation to distinguish taxa. However, the use of wing landmarks in phylogenetic analyses remains largely unexplored. Here, we tested landmark analysis under parsimony (LAUP) to include wing shape data in a phylogenetic analysis of hornets and yellow jackets. Using 68 morphological characters, nine genes and wing landmarks, we produced the first total‐evidence phylogeny of Vespinae. We also tested the influence of LAUP parameters using simulated landmarks. Our data confirmed that optimization parameters, alignment method, landmark number and, under low optimization parameters, the initial orientation of aligned shapes can influence LAUP results. Furthermore, single landmark configurations never accurately reflected the topology used for data simulation, but results were significantly close when compared to random topologies. Thus, wing landmark configurations were unreliable phylogenetic characters when treated independently, but provided some useful insights when combined with other data. Our phylogeny corroborated the monophyly of most groups proposed on the basis of morphology and showed the fossil Palaeovespa is distantly related to extant genera. Unstable relationships among genera suggest that rapid radiations occurred in the early history of the Vespinae.     

Technical note: A new method for measuring long bone curvature using 3D landmarks and semi‐landmarks

Here we describe and evaluate a new method for quantifying long bone curvature using geometric morphometric and semi‐landmark analysis of the human femur. The technique is compared with traditional ways of measuring subtense and point of maximum curvature using either coordinate calipers or projection onto graph paper. Of the traditional methods the graph paper method is more reliable than using coordinate calipers. Measurement error is consistently lower for measuring point of maximum curvature than for measuring subtense. The results warrant caution when comparing data collected by the different traditional methods. Landmark data collection proves reliable and has a low measurement error. However, measurement error increases with the number of semi‐landmarks included in the analysis of curvature. Measurements of subtense can be estimated more reliably using 3D landmarks along the curve than using traditional techniques. We use equidistant semi‐landmarks to quantify the curve because sliding the semi‐landmarks masks the curvature signal. Principal components analysis of these equidistant semi‐landmarks provides the added benefit of describing the shape of the curve. These results are promising for functional and forensic analysis of long bone curvature in modern human populations and in the fossil record. Am J Phys Anthropol, 2010. © 2010 Wiley‐Liss, Inc.