Quantitative shape analysis with weighted covariance estimates for increased statistical efficiency

Background: The introduction and statistical formalisation of landmark-based methods for analysing biological shape has made a major impact on comparative morphometric analyses. However, a satisfactory solution for including information from 2D/3D shapes represented by ‘semi-landmarks’ alongside well-defined landmarks into the analyses is still missing. Also, there has not been an integration of a statistical treatment of measurement error in the current approaches. Results: We propose a procedure based upon the description of landmarks with measurement covariance, which extends statistical linear modelling processes to semi-landmarks for further analysis. Our formulation is based upon a self consistent approach to the construction of likelihood-based parameter estimation and includes corrections for parameter bias, induced by the degrees of freedom within the linear model. The method has been implemented and tested on measurements from 2D fly wing, 2D mouse mandible and 3D mouse skull data. We use these data to explore possible advantages and disadvantages over the use of standard Procrustes/PCA analysis via a combination of Monte-Carlo studies and quantitative statistical tests. In the process we show how appropriate weighting provides not only greater stability but also more efficient use of the available landmark data. The set of new landmarks generated in our procedure (‘ghost points’) can then be used in any further downstream statistical analysis. Conclusions: Our approach provides a consistent way of including different forms of landmarks into an analysis and reduces instabilities due to poorly defined points. Our results suggest that the method has the potential to be utilised for the analysis of 2D/3D data, and in particular, for the inclusion of information from surfaces represented by multiple landmark points