## The Annals of Applied Statistics

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

### 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

#### Abstract

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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 4 January 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.aoas/1294167803

**Digital Object Identifier**

doi:10.1214/10-AOAS342

**Mathematical Reviews number (MathSciNet)**

MR2829940

**Zentralblatt MATH identifier**

1220.62136

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aoas/1294167803