The misuse of the confidence ellipse in evaluating statokinesigram

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
[email protected]
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,
(x̄, ȳ) : 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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STRUMENTI E METODI
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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ITALIAN JOURNAL of SPORT SCIENCES
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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