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December 2010 A nonlinear mixed effects directional model for the estimation of the rotation axes of the human ankle
Mohammed Haddou, Louis-Paul Rivest, Michael Pierrynowski
Ann. Appl. Stat. 4(4): 1892-1912 (December 2010). DOI: 10.1214/10-AOAS342


This paper suggests a nonlinear mixed effects model for data points in SO(3), the set of 3×3 rotation matrices, collected according to a repeated measure design. Each sample individual contributes a sequence of rotation matrices giving the relative orientations of the right foot with respect to the right lower leg as its ankle moves. The random effects are the five angles characterizing the orientation of the two rotation axes of a subject’s right ankle. The fixed parameters are the average value of these angles and their variances within the population. The algorithms to fit nonlinear mixed effects models presented in Pinheiro and Bates (2000) are adapted to the new directional model. The estimation of the random effects are of interest since they give predictions of the rotation axes of an individual ankle. The performance of these algorithms is investigated in a Monte Carlo study. The analysis of two data sets is presented. In the biomechanical literature, there is no consensus on an in vivo method to estimate the two rotation axes of the ankle. The new model is promising. The estimates obtained from a sample of volunteers are shown to be in agreement with the clinically accepted results of Inman (1976), obtained by manipulating cadavers. The repeated measure directional model presented in this paper is developed for a particular application. The approach is, however, general and might be applied to other models provided that the random directional effects are clustered around their mean values.


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Mohammed Haddou. Louis-Paul Rivest. Michael Pierrynowski. "A nonlinear mixed effects directional model for the estimation of the rotation axes of the human ankle." Ann. Appl. Stat. 4 (4) 1892 - 1912, December 2010.


Published: December 2010
First available in Project Euclid: 4 January 2011

zbMATH: 1220.62136
MathSciNet: MR2829940
Digital Object Identifier: 10.1214/10-AOAS342

Keywords: Bayesian analysis , directional data , joint kinematics , Mixed effects model , penalized likelihood , rotation matrices

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.4 • No. 4 • December 2010
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