The misuse of the confidence ellipse in evaluating statokinesigram Marco Bruno Luigi Rocchi1, Davide Sisti2, Massimiliano Ditroilo3, Annarita Calavalle3, Renato Panebianco1 Istituto di Biomatematica, Università degli Studi di Urbino “Carlo Bo” Facoltà di Scienze Matematiche, Fisiche e Naturali, Università degli Studi di Urbino “Carlo Bo” 3 Istituto di Ricerca sull’Attività Motoria, Università degli Studi di Urbino “Carlo Bo” 1 2
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ABSTRACT Rocchi MBL, Sisti D, Ditroilo M, Calavalle A, Panebianco R The misuse of the confidence ellipse in evaluating statokinesigram Ital J Sport Sci 2005: 12: 169-172 The aim of this brief technical note was to focus on a widely used method for evaluating the statokinesigram: the confidence ellipse. We pointed out that several authors misinterpret the meaning of CE, which is actually well defined in statistics. Anyway we illustrated some limitations when using confidence ellipse in evaluating statokinesigram. Finally we suggested substituting confidence ellipse with the standard ellipse.
The problem of quantifying the oscillation of an immobile standing subject in order to evaluate his balance ability is a major issue in posturography and stabilometry. The methods employed to assess balance behavior vary widely among authors. The analysis of the socalled statokinesigram, i.e. the projection of the center of pressure (CoP) on a bidimensional plane by means of a force platform, is often used. Circles and rectangles have been fitted to the statokinesigrams in the xy plane [1-3], but the most accepted method is the analysis of the 90% or 95% confidence ellipses; we will generically denote them as (1- α% confidence ellipses) of the postural sway [4-7]. This parameter is often used alone, or combined with the length of the sway to obtain the LFS (Length as Function of Surface) index [8-9]. In both cases, the area of the confidence ellipse is used as a measure of energetic expenditure of the subject to maintain his balance [8-10]. We think it is important to underline some basic statistical aspects of the confidence ellipse, to illustrate some of its limitations in evaluating statokinesigram. 1. First of all, the (1- α)% confidence ellipse is often misinterpreted as the ellipse that contains the (1- α)% of the points of the sway. For example, Prieto et al. VOL. 12 - NUMERO 2 2005
[11] wrote: “The 95% confidence ellipse should enclose approximately 95% of the COP point”. This definition is incorrect, not only because an ellipse that contains a certain percentage of the points cannot be univocally defined, but also because this interpretation is conceptually flawed. In fact, the (1- a)% confidence ellipse is correctly defined as the ellipse that, with the (1- α)% of probability, contains the center of the points of the sway. In more general terms, a confidence ellipse is a region that covers the center of a sample with a given probability. In formal terms, if we denote with: n: the number of points of the sway, (xi, yi) the points of the sway, : the mean position on lateral projection,
: the mean position on longitudinal projection, (x¯, ¯y) : the midpoint of the sway (i.e. the center of the points of the sway), : the variance of the points on lateral projection,
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KEY WORDS: posturography, statokinesigram, confidence ellipse, standard ellipse
: the variance of the points on longitudinal projection, : the correlation coefficient between the xi and the yi, then, the (1-α)% confidence ellipse is defined by the equation:
where denotes the critical value of the Fisher distribution with 2 and n – 2 degrees of freedom, for a significance level α [12,13]. It is then possible to compute the area of the (1-α)% confidence ellipse (CEA). In order to facilitate this computation, we provide the following notations: ;
val (of which the concept of confidence ellipse is a natural bidimensional extension), and the equations that define the ellipse, we derive another important observation: the area of the confidence ellipse strongly depends on the number of points of the sway (it decreases when the number of points increases). Batschelet [12] emphasizis that “the confidence ellipse is not only subject to random fluctuations from sample to sample, it clearly depends on the sample size, n. If, e.g., the sample size is increased by the factor 4, the semi-axes, a and b, are reduced to one-half. Thus, the confidence ellipse does not describe the variability of the individual sample points”. From a practical standpoint, in our field of application, this means that the area can vary not only according to the balance ability of the subject, but also according to the duration and the sampling frequency of the points. This make this parameter absolutely unsuitable to compare data from different trials and conditions, unless standard international procedures are adopted.
; ; . We can now calculate the semi-axes a, b (the major and minor semi-axis, respectively) and the angle θ by which the major axis is inclined versus the x-axis: ;
;
.
4. If the area of the sway is considered essential, then we suggest substituting the confidence ellipse with the standard ellipse. Unlike the confidence ellipse, which strongly depends on sample size because of its inferential purpose, the standard ellipse only has a descriptive purpose, so that it does not at all depend on the number of points of the sway . We again quote Batschelet [12]: “The standard ellipse is subject to random fluctuations from sample to sample, but it does not depend on the sample size, n”. Moreover, the Normal bivariate distribution of the points of the sway is not required. The equation of the standard ellipse is similar to the equation of the confidence ellipse (note that only the right member of the equation is modified); it is: .
Lastly, we can compute the CEA through the formula: CEA = πab For further details on these geometrical and statistical passages, we suggest consulting Batschelet [12].
2. The approach of the confidence ellipse is based on the assumption that the distribution of the points is a Normal bivariate distribution [12]. This assumption is not demonstrated for the points of the sway.
To calculate the standard ellipse area (SEA) we can use the same formulas already described to compute the CEA, simply substituting the following expression for D: . In conclusion we think that CEA should be abandoned and replaced with SEA, to allow for the comparison of data collected under different conditions in term of the duration and frequency of acquisition of the postural sway.
3. Directly from both the concept of confidence inter-
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Esperto nella riabilitazione sportiva
BAROPODOMETRIA E ATTIVITÀ SPORTIVA È ormai consuetudine che gli atleti, sia dilettanti che professionisti, siano controllati e monitorati con sempre maggiore attenzione. L’attività agonistica ed il gesto tecnico determinano traumi e sovraccarichi che i preparatori ed i tecnici devono prevenire. Una parte consistente di questo lavoro è rappresentata dal sostegno, dal recupero e soprattutto dallo studio della postura. L’analisi del gesto tecnico, della postura e dell’equilibrio permettono la prevenzione degli infortuni ed il miglioramento delle capacità tecniche dell’atleta. È determinante, che lo sportivo sia analizzato e monitorato durante tutte le fasi della preparazione e della prestazione sportiva con strumentazioni in grado di determinare l’equilibrio posturale, il confort e la riuscita del gesto sportivo senza sovraccarichi inutili e dannosi. È estremamente interessante una strumentazione computerizzata (sistema computerizzato di sensori) costruita da poche aziende al mondo in grado di valutare le fasi di spinta del piede mentre l’atleta effettua il gesto sportivo. Si tratta di una serie di sensori applicati all’interno della calzatura a contatto con la pianta del piede, con o senza il
plantare, in grado di registrare ed analizzare l’appoggio durante l’allenamento, nel momento stesso della prestazione; questo consente di verificare direttamente le anomalie delle zone di spinta ed esattamente dove intervenire sia con un allenamento specifico, che tramite il supporto di un plantare idoneo. Il giusto plantare assicura un alto grado di ammortizzazione ed una corretta distribuzione dell’appoggio al suolo, al fine di migliorare ed ottimizzare le prestazioni dell’atleta ed evitare traumi ai legamenti ed alla struttura muscolare. Si tratta di tecnologie indispensabili per chi vuole approfondire lo studio della postura degli atleti e anche della gente comune.
Sistema di sensori da posizionare sotto la pianta dei piedi
