The Annals of Applied Statistics

A nonlinear mixed effects directional model for the estimation of the rotation axes of the human ankle

Mohammed Haddou, Louis-Paul Rivest, and Michael Pierrynowski

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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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Ann. Appl. Stat., Volume 4, Number 4 (2010), 1892-1912.

First available in Project Euclid: 4 January 2011

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


Haddou, Mohammed; Rivest, Louis-Paul; Pierrynowski, Michael. A nonlinear mixed effects directional model for the estimation of the rotation axes of the human ankle. Ann. Appl. Stat. 4 (2010), no. 4, 1892--1912. doi:10.1214/10-AOAS342.

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  • Bingham, C., Chang, T. and Richards, D. (1992). Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and procrustes analysis. J. Multivariate Anal. 41 314–337.
  • Chirikjian, G. S. and Kyatkin, A. B. (2001). Engineering Applications of Noncommutative Harmonic Analysis. CRC Press, Boca Raton, FL.
  • Grood, E. S. and Suntay, W. J. (1983). A joint coordinate system for the clinical description of three dimensional motion: Application to the knee. J. Biomech. Eng. 105 136–144.
  • Inman, V. T. (1976). The Joints of the Ankle. Williams and Wilkins, Baltimore, MD.
  • Kim, P. T. (1991). Decision theoretic analysis of spherical regression. J. Multivariate Anal. 38 233–240.
  • León, C., Massé, J.-C. and Rivest, L.-P. (2006). A statistical model for random rotations. J. Multivariate Anal. 97 412–430.
  • Lewis, G. S., Cohen, T. L., Seisler, A. R., Kirby, K. A., Sheehan, F. T. and Piazza, S. J. (2009). In vivo test of an improved method for functional location of the subtalar joint axis. J. Biomechanics 42 146–151.
  • Lewis, G. S., Sommer, H. J. and Piazza, S. J. (2006). In vitro assessment of a motion-based optimization method for locating the talocrural and subtalar joint axes. J. Biomech. Eng. 128 596–603.
  • Lindstrom, M. J. and Bates, D. M. (1990). Nonlinear mixed-effects models for repeated measures data. Biometrics 46 673–687.
  • Mansour, E., Begon, M., Farahpour, N. and Allard, P. (2007). Forefoot–rearfoot coupling patterns and tibial internal rotation during stance phase of barefoot versus shod running. Clin. Biomech. 22 74–80.
  • Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics. Wiley, New York.
  • McCarthy, J. M. (1990). Introduction to Theoretical Kinematics. MIT Press, Cambridge, MA.
  • Pierrynowski, M. R., Finstad, E., Kemecsey, M. and Simpson, J. (2003). Relationship between the subtalar joint inclination angle and the location of lower-extremity injuries. J. Amer. Pediatr. Med. Assoc. 93 481–484.
  • Pinheiro, J. C. and Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. Springer, New York.
  • Rancourt, D., Rivest, L. P. and Asselin, J. (2000). Using orientation statistics to investigate variations in human kinematics. J. Roy. Statist. Soc. Ser. C 49 81–94.
  • Rivest, L.-P., Baillargeon, S. and Pierrynowski, M. (2008). A directional model for the estimation of the rotation axes of the ankle joint. J. Amer. Statist. Assoc. 103 1060–1069.
  • Rivest, L.-P. and Chang, T. (2006). Regression and correlation for 3×3 rotation matrices. Canad. J. Statist. 34 184–202.
  • van den Bogert, A. J., Smith, G. D. and Nigg, B. M. (1994). In vivo determination of the anatomical axes of the ankle joint complex: An optimization approach. J. Biomechanics 27 1477–1488.