The Statistical Distribution of Wrist Movements during Sleep 

Authors:

Víctor Gimeno, Teresa Sagales, Luchi Miguel, Mercedes Ballarin

Servei de Neurofisiologia,

I-Iospital Universitari Vall d'Hebron, Barcelona, Spain

Pharrnacoelectroencephalography

Original Paper

............................................... Key Words 

Sleep, Actigraphy, Series of events, Statistical distribution functions

.................................................................................................. Abstract

The purpose of this work was to describe the basic statistical properties of the process of production of movements measured with a wrist actimeter, along a complete sleep period in normal human subjects. Two distinct types of random magnitudes were considered to analyze the data, the times between successive groups of movements and the number of movements at each fixed time (1 min) measurement epoch. Suitable probabilistic models for the two variates were chosen, fitting theoretical distribution functions to the observed data. It is concluded that interval data fit a one-parameter exponential distribution, while the number of movements fit a two-parameter negative binomial distribution. The estimated values of these parameters, besides being necessary to perform further statistical analysis, are a measure of the intensity and frequency of the movements. Finally the relationship between polysomnography measures and the elicited parameters was studied.

......................

Introduction

Measurement of motility by wrist actigraphy has become a popular method in the study of human sleep ( 1 - 1 0), for it employs an inexpensive, easily wearable device, which is simple to use and allows ambulatory recordings to be taken. Actigraphic records consist of raw data usually presented as a comprehensible plot of number of movements versus time. However there is a lack of uniformity in the approach to the analysis of this data in the literature, which is mainly focused on the production of automatic sleep-wake scoring algorithms [l 1 ]. In fact, a precise characterization of the basic statistical properties of these series of events has yet to be done.

When measuring random data its basic properties are described in terms of statistical parameters, as the well known mean value and standard deviation pair in the case of random variables following the classical Gaussian distribution.

Movements can be considered as a series of random point events occurring on a time axis, but besides their position in time they are distinguished by an additional random quantity which is the number of movements appearing at each fìxed time measurement epoch. There are therefore two different types of magnitudes whose distributions must be considered, the times between successive appearances of movements and the number of movements. Properties of both descriptors, time intervals and counts, serve to characterize the process of production of movements. A joint analysis of these two different types of random variables is also necessary to elicit the possible association between both variates.

In this work, plausible theoretical distribution forms were fitted to the actimetric experimental data, estimating their parameters and testing their adequacy. Any further analysis of the actimetric data should take these results into account.

Subjects and Methods

Twenty-three healthy volunteers, 11 female, 12 male, between the ages of 24 and 47, were submitted to a polysomnography (PSG) during 2 consecutive nights. Subjects also wore an actimeter (Mini-Logger Series 2000), placed on their nondominant wrist, set to collect the number of movements in 1-min consecutive intervals. Actimeter and polygraphic recorder clocks were carefully synchronized to achieve a precise timing in both recordings. Stored actimeter data were read by a computer through a serial interface and stored in a disk file for further processing. After a first accommodation night, actigraphic records were analyzed during the second night, using data from only deep period time, i.e. discarding data prior to sleep onset and the epochs awake after the end of the sleep, as detected by the PSG. Extreme care was taken to minimize the presence of artifacts in the records, continuously checking the position of the device through a video camera to make sure that the influence from either breathing movements or the blocking of the arm for any postural reason produced no important artifacts [7, 12]. Hypnogram and actigraphic data were also jointly inspected to discard gross discrepancies between both records. Subjects presented no periodic limb movements and no sleep apneas. Three evident cases of irregular awakening and activity, two as a result of getting up to go to the toilet and another with breathing problems due to a cold were withdrawn from the study. PSG and actigraphic recordings of the remaining subjects were considered of good quality. Therefore the definite size of the sample was n = 20. Observing actigraphic recordings, an evident clustering of the movements can be clearly seen in most of them. Clustered distributions of this type (also called 'contagious' in statistics) present a variance substantially larger than the mean which precludes the possibility of a single Poisson process as the generating mechanism of the series [ 1 3], but clusters themselves can arise from a completely random (i.e. Poisson) process. In other words, times between the appearance of clusters are independent. This was the first hypothesis tested, regarding these intervals, or immobility periods, as the variate to analyze. If the distribution of clusters is consistent with a Poisson process, these periods will follow an exponential (continuous) distribution [ 1 4]. The parameter 'b' of this distribution (its estimated value will be denoted as'b') is easily estimated by the arithmetic mean of the intervals, i.e. the mean duration of immobility periods. Departures from this distribution for the interval data can then be used to reject this first hypothesis. A c2 test for the goodness of fit was used in this work [see 14 for a thorough discussion about tests for Poisson and for more general renewal processes]. With respect to the number of movements produced at each measurement epoch, i.e. every minute, a natural choice to model its distribution is to use a negative binomial (NB) distribution function [15^], which is a two-parameter (m, k) distribution with its mode usually near the origin and with a long tail (the definitions of this distribution vary slightly from one author to another). Clustered distributions frequently arise in various fields and in most instances an NB(m,k) form appears as an adeguate model. This was therefore the second hypothesis tested. Estimation of its two parameters can be made using several methods [ I 6]. The parameter 'm' is estimated by the mean, i.e. it is the average rate of occurrence of movements. Tbc other parameter, 'k', varies according to the proportion of time without movements, and its values were estimated in this work by the method of maximum likelihood. cgoodness-of-fit methods were also used to test this second hypothesis.

To gain a better insight into the meaning of these two parameters, Spearman's correlation coefficients between the obtained values of 'm' and the proportion of time with presence of movements as well as between the values of 'k' and the average length of the intervals were calculated.

Finally, the relation between position and magnitude (i.e. number of activity counts in a cluster) of the movements was investigated by the estimation of the proportion of movements that were directly associated with the time of occurrence. Standard regression techniques were used for the analysis, considering the time of origin of each cluster as the independent variable and the number of movements in the cluster as the dependent variable.

A study of the relationship between the structure of sleep as assessed from PSG and the elicited parameters 'b', 'm', 'k', was also carried out. Sleep efficiency (Eff%) and accumulated time awake after sleep onset (WASO) are the parameters that can be most closely estimated by present actigraphic methods [5, 17-19^]. In this work these two parameters and the number of awakenings (AW) during the sleep period time were taken as sleep descriptors. Entering 'b', 'm', 'k', as independent variables, multiple linear regression analyses were performed for these PSG measures to test the hypothesis that there is a significant relationsbip between PSG descriptors and actigraphic parameters.

Results

The pattern of a typical actigraphic recording presents random production times combined with random number of movements. We used a graphic presentation that includes both cumulative (fig. 1a) and individual (fig. 1 b) number of movements versus time. It is worth noting that the departure of the cumulative plot from a straight line measures the lack of uniformity in the appearance of movements, as the slope of the line between any two points represents the mean number of movements per unit time for that interval.

Goodness-of-fit tests for the exponential distribution, fitted to the interval data, lead to the conclusion that this model can be considered adeguate in all but 2 cases. However, in those cases, coefficients of variation had sample values near unity and serial correlation coefficients were calculated giving values which were also compatible with the hypothesis of independence of the intervals [14^]. Regarding the number of movements produced at each point, it was also possible to fit the sample data to a negative binomial distribution by an appropriate choice of its two parameters, 'm', 'k'. This was checked by c2 analysis.

 Only in 1 subject presenting a very low sleep efficiency was the fit rejected.

The elicited Spearman's correlation coefficients yielded high values, significantly different from zero: r = 0.8303 between 'm' and the proportion of time with movements and a negative correlation of r = -0.7829 between 'k' and the average length of the intervals. This correlation was improved considering reciprocal 'k' values (r = 0.9482).

Regression analysis for the different series showed that the number of movements appearing at each cluster is not related to its time value, i.e. the number of movements does not depend on the time of observation (coefficients of determination were as low as 0.1578). Having examined scatter diagrams (fig. 2) of the number of movements in each cluster against the time interval between the beginning of the cluster and the end of the previous one, it was also found that there is no apparent relationship.

  Variability in our numerical findings is summarized in table 1.

The results of the regression analyses are presented in table 2. They show that there is a significant dependence (p < 0.05) of each PSG deseriptor on the three statistical parameters.

Tbc proportions of variance in PSG data attributable to the dependence of PSG on 'b', 'm', 'k', are 64, 44 and 66% for Eff%, AW and WASO, respectively. Standard partial regression coefficients were used as indications of relative importance of the parameters 'b', Gm', 'k', in determining the value of PSG deseriptors. It was found that 'm' is the most irnportant parameter in determining the value of Eff% and WASO, and 'b' is the most impor- tant parameter in determining the value of AW.

Although the parameter 'k' has a low relative importance, separate simple correlation analyses between PSG descriptors and 'k' indicate that there is a signifícant relationship between them. Results are shown in table 3.

Discussion

There are currently a number of automatic scoring algorithms of the actimetric data to estimate several sleep parameters [2, 10, 20-22]. However, there has been little work on the investigation of actigraphic data throug, mathematical models carried out to date, although it is of great statistical importance to have reliable models that serve to describe the experimental measurements [23] Wallner [24] was able to estimate sleep efficiency and total sleep time analysing the body movement density function, and perfomed spectral analysis of data from sequence of several days to study circadian and ultradian rhythms. Home et al [25] used a filtered version of the actigram to convert it into a binary signal to study the effect of noise on sleep.

In our work, although other distribution functions could have been chosen to fit actimetric data, e.g. a more general gamma form for the intervals, the chosen models were simple and it was shown that they were also adequate. The estimated parameters are easily related to natural descriptors of actigraphic records. The parameter 'b' of the exponential distribution represents the mean duration of the immobility periods, its reciprocal value therefore giving information about the frequency of the process. As far as the NB distribution is concerned, its parameter 'm' (the average rate of occurrence of movements) was found to be highly correlated with the proportion of time with presence of movements, thus being a measure of the average intensity of the process. The other parameter,'k', was found to be closely related to the inverse of the average length of the intervals between movements, therefore it is also a measure of the frequency of the process. In fact, similar parameters can be found among the large number of variables that are used in the literature [26]. However, our different approach, based on the statistical analysis of the data, leads us to conclude that the triplet 'b', 'm', 'k', must be considered as a natural necessary minimum set of activity descriptors to be elicited as the first step in the analysis of nocturnal actigraphic data.

Moreover, the finding that the clusters of movements appeared randomly in time, added to the fact that the number of movements did not depend on the time of observation, makes the hypothesis of two different underlying processes as the generating mechanisms of the series of movements feasible. One would drive the timing of the series or, in other words, its frequency, and the other its intensity. However, it is not possible to draw reliable conclusions on the role of these processes in the regulation of movements during sleep and this question has yet to be studied.