The purpose of this study was to establish a new statistical method for the analysis of noisy mandibular helical axis parameters, especially the position vector of the finite helical axis (FHA). The subjects were children with anterior cross-bite who had received orthodontic treatment. Maximum mouth-opening was measured by means of an opto-electronic motion analysis system. These movements were compared with similar movement in the same group after treatment of their anterior cross-bite. Each curve of FHA position vectors was modelled as a spline function with random coefficients. To determine the optimal number of knots, two criteria were used: deviance information criteria (DIC) and mean squared prediction error (MSE). We were interested in estimating a typical curve for a population. Self-modelling regression (SEMOR) was extended to three dimensions to model groups of three-dimensional curves. Each curve was modelled as a spline function using nine knots. Population average curves were created using SEMOR. This study provided detailed information about jaw movement for comparing cross-bite to normal occlusion by calculating the population mean curves of the position vector of the FHA. Our results suggested that the two population mean curves for the position vector of the FHA were significantly different in the closing phase. The combination of a spline function with random coefficients and SEMOR extended to three dimensions can be used not only for FHA analysis but also for the analysis of other jaw movements.
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