ApEn
Generally speaking, the ApEn algorithm quantifies randomness by determining the extent to which short sequences of data points are repeated in a time series. More precisely, the ApEn algorithm calculates the logarithmic probability that runs of patterns that are close (i.e., within error tolerance
r) for
m observations remain close on subsequent incremental comparisons. To calculate ApEn for a time series containing
N data points,
u(1), u(2), ..., u(N), an operator inputs (1)
m, a pattern length, and (2)
r, an error tolerance. The first step is to form vector sequences
x(1) through
x(N - m - 1) from the
{u(i)}, defined by
x(i) = [u(i), ..., u(i + m - 1)]. These vectors are basically
m consecutive
u values, beginning with the
i-th point. The second step is to define the distance
d [x(i),x(j)] between vectors
x(i) and
x(j) as the largest difference in their respective scalar components. The third step is to use the vector sequences
x(1) through
x(N - m - 1) to create (for each
i ≤ N - m + 1)
(1)
The
values measure (within the tolerance
r) the regularity of patterns similar to a given pattern of window length
m. The fourth step is to define Φ
m
(
r) as the average value of ln
, where ln is the natural logarithm. Lastly, we define Approximate Entropy as
ApEn(m,r,N) = Φ
m
(r) - Φm+1(r)
ApEn generates a unit-less real number from 0 to 2 [
30]. Smaller ApEn values indicate a higher probability of regularly repeating sequences of
m observations. An ApEn value of zero, for example, corresponds to a time series that is perfectly repeatable (i.e., sine wave). An ApEn value of 2 is produced by random time series, for which any repeating sequences of points occur by chance alone (i.e., Gaussian noise).
Using Matlab software (Mathworks, Natick, MA), we calculated separate ApEn values for the AP and ML components of the COP coordinate time series (N = 2000) from each test trial. Input parameters for the ApEn calculation were (1) a pattern length (
m) of 2 data points, (2) a tolerance window (
r) normalized to 0.2 times the standard deviation of individual time series, and (3) a lag value of 10. The pattern length (m) and tolerance value (r) were selected based on previous work [
21,
31‐
33]. The lag value of 10 dictated that the ApEn calculation include every 10
th point the raw time series. We chose this lag value to lower the effective sampling frequency of the algorithm from 100 Hz to 10 Hz, thereby reducing the influence of extraneous noise in the data.
As a necessary component of nonlinear dynamics methodology, we also applied a surrogation (phase randomization) procedure to verify that COP data were derived from a deterministic source [
34]. Surrogate AP and ML time series were created having identical means, standard deviations, and power spectra to the original data but with randomly generated order. This procedure also was performed in Matlab using the algorithms developed by Theiler et al [
34‐
36]. ApEn values from the original data and their surrogated counterparts were compared using the Student t-test (α = .05). The procedure revealed that ApEn values for the original time series were significantly less than for their respective surrogated counterparts, indicating that the original data were not randomly derived, and therefore, were deterministic in nature.
Equilibrium score
An ES was generated for each trial based on an algorithm developed for the SOT [
37]. The algorithm uses the peak-to-peak amplitude (range) of COP AP displacement to estimate the amount of postural sway in the AP plane. Scores are calculated as the angular difference, expressed as a percentage, between the amount of estimated AP postural sway and the theoretical limit of stability, approximately 12.5° in the AP plane [
37]. Lower amplitudes of postural sway require less COP displacement to control and produce higher percentage differences from the theoretical limit. Thus, a higher ES indicates greater postural stability in the AP plane. No analogous ES exists for the COP ML component. Although similar in construct to RMS values, we chose to analyze ES because of its common clinical use in conjunction with the SOT. Like RMS, ES values also have been previously used in dual task research [
10].
All statistical analyses were conducted using SPSS 11.0 software (SPSS, Inc., Chicago, IL). We applied separate 2 × 6 (cognitive task × sensory condition) repeated measures analyses of variance (ANOVA) for ApEn values, RMS displacement and ES (α = 0.05) generated from single and dual task trials. Due to violations of Mauchly's sphericity assumption, we adjusted the ANOVA results using the more conservative Geisser-Greenhouse F-test.