
Research Article


Modelling the Duration of Hypopnea 

Nur Zakiah Mohd Saat
and
Abdul Aziz Jemain



ABSTRACT

In this study, we fit a parametric model to the duration of hypopnea. Hypopnea is a partial apnea. A total of two hundred and seven events of duration of hypopnea is considered in this study. The parametric model that has been used is gamma, Weibull, lognormal and loglogistic distribution. The most appropriate distribution to fit the duration of hypopnea is analyzed according to sleep stages. The results indicated that Weibull distribution is the best model for light, deep and Rapid Eye Movement (REM) sleep stages.







INTRODUCTION
Apnea describes the process of complete cessation of airflow during sleep which
last between 10 and 60 sec (Tsai et al., 1999).
Notably, this type of sleep disorder has been associated with factors such as
increasing age, obesity, gender and smoking habits (Guilleminault
and Dement, 1976; Worsnop et al., 1998). Prior
research on the modelling of apnea has focused on the occurrence of apnea during
sleep and wake duration (Lo et al., 2002). On
the other hand, a hypopnea is a partial apnea (Moser et
al., 1994; Guilleminault et al., 2001).
Hypopnea is defined as 50% cessation of airflow (Tsai et
al., 1999). Also, research on the modelling of sleep disorder has extended
to examine sleep wake transitions. For instance, study by
Yang and Hursch (1973) gathered that the suitability of Markov model is
depending on sleep stages [i.e., light sleep, deep sleep and Rapid Eye Movement
(REM)] and age group.
Patients that are suffering from apnea usually experienced symptoms such as
fatigue and sleepiness during the day (Black, 2003).
To ensure appropriate treatment of the illness, the medical practitioners would
refer these patients to a sleep clinic. At the clinic, the patients will be
observed during their sleep in order to determine the severity of their apnea.
Study on apnea duration is important as apnea is related to hypertension, cardiovascular
disease, automobile accident and excessive day time sleepiness (Hla
et al., 1994; Nieto et al., 2000;
Hossain and Shapiro, 2002). Consequently, it is essential
to study the distribution of apnea and hypopnea. Parametric model is one of
the methods to identify the distribution of apnea and hypopnea. More importantly,
it will allow us to understand the patterns of apnea.
Parametric models have been used in the study of rainfall in terms of rain
rate (Kedem et al., 1994). The probability distribution
that has been applied to model the rainfall data is gamma and lognormal distribution.
It was found that the lognormal distribution was sensible for the rain rate
in Darwin and Florida. Furthermore, Onoz and Bayazit (1995)
employed pareto, lognormal and Generalized Extreme Value (GEV) distribution
to fit the data of flood samples. It was found that GEV is the sensible model
compared the other model selected. A number of authors applied parametric model
in medical research. Atienza et al. (2008) used
a mixture model which is gamma, Weibull and lognormal to model the length of
stay in the hospital.
The distribution of apnea is truncated because an apnea is a cessation of airflow for at least 10 sec. It indicates that cessation of airflow that less than 10 sec is discarded from the recording of apnea event. Therefore, it is interesting to determine the effect of truncation towards the probability distribution of apnea. Using a probability distribution approach is essential in determining the distribution of apnea. A parametric model is one of the methods to identify the distribution of apnea. This will lead us to understand the process of apnea.
Although, there are many studies explaining regarding the duration of sleep
(Hasler et al., 2004; Rechtschaffen
et al., 1999; Wehr et al., 1998),
there are limited studies on the duration of apnea. In this study, we consider
the duration of hypopnea. The objective of this study is to determine the best
model to fit the data of hyponea. This is essential in order to understand the
pattern of apnea and to find the best model for the duration of apnea.
PARAMETRIC MODELS
The data on apnea is obtained from a polysomnography which was originated from
Rigney et al. (1994). This database involves collaboration
between Massachusetts Institute of Technology and Beth Israel Hospital (MITBIH)
in Boston. The database consists of 14 subjects records of polysomnography for
14 males of age between 32 and 56 years. The database was extracted from http://www.physionet.org/physiobank/
(Goldberger et al., 2000).
A hypopnea event is defined as reduction in breathing that remain for at least
10 sec (Guilleminault et al., 1976, 2001).
However, reductions in breathing that last for less than 10 sec are not recorded.
Let X denote the duration of a periods of reduced breathing. We assume that
the duration of successive events are independent random variables from a distribution
with probability density function (pdf), f(x; θ), where, θ is a vector
of parameters to be estimated. The observed data are independent observations
from this distribution lefttruncated at X = 10. Therefore the pdf of the duration
of an observed hypopnea is:
Where:
Let x_{1}, x_{2},..,x_{n} enote the durations of a
sequence of observed hypopneas. Because of the assumption of independence, the
loglikelihood function of θ is:
The following distributions of durations have been investigated
Gamma distribution
where, α>0 and β>0
In reference to Hogg and Tanis (2005), the mean and variance
of the distribution is αβ and αβ^{2}, respectively.
Lognormal distribution
The mean of the distribution is exp[μ+(σ^{2}/2)] and the
variance is exp[2μ+σ^{2}] [exp(σ^{2})1] (Hosmer
and Lemeshow, 1999).
Weibull distribution
where κ, λ>0 and x>0. T he mean of the distribution is λΓ
(1+1/κ) and the variance is (Hosmer and Lemeshow, 1999).
Loglogistic distribution
where, ∞<μ<∞ and σ>0
The mean of the distribution is exp (μ) [Γ (1+σ) Γ (1σ)
and the variance is exp (2μ) [Γ(1+σ) Γ(12σ) Γ^{2}(1+σ)
Γ^{2}(1σ)] (Law and Kelton, 1982).
RESULTS
Distribution of hypopnea event: There are 207 hypopnea events, with
108 during REM sleep. The mean duration of a hypopnea event is 15.88 sec and
maximum is 41.20 sec. A histogram of duration for all sleep stages categories
was constructed and specifically focusing on the hypopnea events as it has most
events compared to other events(arousal, false oxygen saturation, obstructive
sleep apnea and central sleep apnea). Figure 1 shows that
hypopnea events are skewed to the right with most of the hypopnea duration are
10 to 20 sec.

Fig. 1: 
Histogram of the duration of hypopnea 
A standard survival models which is gamma, lognormal, loglogistic
and Weibull was fitted to the data. The statistical analysis was performed using
SPLUS 6.0 for Windows, Insightful Corp (2000).
Goodness of fit test which is to discriminate between continuous distribution was done based on empirical distribution function KolmogorovSmirnov (KS). We compare between distributions of light sleep and deep sleep to determine the differences between both distributions. There is slightly difference for cumulative density function (cdf) between NREM and REM. Based on KS test NREM and REM are not significantly different (KS = 0.181, p = 0.052) and there was no significantly different for cdf of light sleep and deep sleep (KS = 0.191, p = 0.538). This indicated that there are small difference of distribution when we divide NREM to two groups because the pvalue is almost less than 0.05. In order, to investigate a better understanding about the differences between distribution of light sleep and deep sleep Maximum Likelihood Estimation (MLE) is used.
Maximum Likelihood (ML) estimates of parameters of two parameter lefttruncated probability distributions were obtained. The data is left truncated because it is truncated at 10 sec. The objective of the study is to fit the four distribution (gamma, lognormal, Weibull and loglogistic) to the duration of the hypopnea. The particular distributions were selected because the data set are positive distribution and skewed to the right. Another purpose of the study is to fit the distribution of duration of hypopnea according to sleep stages.
FITTING A PARAMETRIC DISTRIBUTION FOR DURATION OF HYPOPNEA
The loglikelihood is used to select the best model that described the duration
of hypopnea according to sleep stages.
Table 1: 
ML estimates of parameters of duration of hypopnea event 

Table 2: 
ML estimates of parameters of nonREM (NREM) sleep stages
(n = 99) 

Table 3: 
ML estimates of parameters for REM sleep stages (n = 108) 

Table 4: 
ML estimates of parameters of light sleep stages (n = 80) 

The loglikelihood of fitting the data of duration of hypopnea are shown in Table 1. The results indicated almost similar value of loglikelihood for the four distributions. The loglikelihood according to NREM sleep stages is shown in Table 2. Generally, the loglikelihood is decreased compared to the loglikelihood of overall duration of hypopnea. This is due to the reduction of sample size in NREM sleep stages compare to overall duration of hypopnea. The loglikelihood for Weibull distribution is the lowest while the loglogistic distribution shown the highest loglikelihood indicating that loglogistic is poorly fit for duration of hypopnea during NREM.
Table 3 shows the loglikelihood of fitting the data during REM sleep stages. Most of hypopnea events occurs during REM, with the number of hypopnea events is 108. Similarly, the loglikelihood of the four distribution is almost equal with the most difference is less than two. The parameter of gamma, lognormal and Weibull decreased in REM group compared to the NREM group. However the parameter of μ and σ from the logistic distribution is slightly increased.
In Table 4, it is showed the loglikelihood values for light sleep stage group. There was higher increased in the shape parameter which is parameter α compared to REM and NREM sleep stages.
Table 5: 
ML estimates of parameters for deep sleep stages (n = 19) 

Similarly, there was a slight increased in parameter scale for Weibull distribution. However, there was a decrease in parameter lognormal and loglogistic distribution. There was a reduction almost 100 in terms of loglikelihood compared to loglikelihood in REM sleep stages. Results about deep sleep stages is shown in Table 5. Hypopnea events were less in deep sleep stages compared to REM and light sleep stages. The value of loglikelihood is almost similar for all the distributions. Moreover, there was an increase in the value of scale parameter for Weibull distribution.
The KS test has been used to determine the goodness of fit test. The results
from Table 1 until Table 5 show that there
was no appropriate distribution from the four distributions to fit the data.
It is due to the limitation of the KS test. KS test is not preferable for
the skewed data. Another limitation, the KS test is not suitable for the distributions
that the parameter is estimated from the data as it can determine the goodness
of fit better if the parameter is estimated from the simulation data. It is
believe due to the limitation in KS test, the distribution that is appropriate
for the data may not be determined (D’Agostino and Stephens,
1986).
The likelihood ratio in given by LR = 2 ln(θ). θ is l_{o}/l_{1},
l_{o} is the loglikelihood for the first model and l_{1}
is the loglikelihood for the second model (Hosmer and Lemeshow,
1999). The value of LR is compared with the Chisquare with 5% significance
level and degree of freedom is two. The results obtained in Table
6 comparing between light and deep sleep showed a reduction in maximized
likelihood, for example for gamma distribution from 563.14 to 562.10. The LR
test showed that there were significantly different between light and deep sleep
stages for Weibull distributions (test statistic = 7.66, critical value = 5.99,
p<0.05). However, for gamma, lognormal and loglogistic the LR test is similar
between light and deep sleep stages. Interestingly, the decreased in maximized
likelihood is due to subdivide the NREM to light and deep sleep. Loglogistic
distribution showed a clearly higher maximized likelihood values compared to
other distributions. Numerical optimization was used to maximize the loglikelihood
which is nonlinear least squares functions (Venables and Ripley,
2002).
In Table 7, we reported a comparison between NREM and REM.
The LR test revealed that the loglikelihood in combining NREM and REM are significantly
higher compared to the full model (p< 0.05). It shows that gamma, lognormal,
Weibull and loglogistic distribution is significantly different between NREM
and REM sleep stages.
Table 6: 
Likelihood ratio for light and deep sleep stages 

Table 7: 
Likelihood ratio for NREM and REM 

The likelihood ratio test presents that Weibull is appear to provide the best fit for the NREM and REM because it has the highest LR test, compared to other three distributions. Therefore, Weibull distribution is selected as the best model to fit the REM and NREM sleep stages. This indicated that there was a significantly different between REM and NREM for the duration of hypopnea, especially when Weibull is fitted to the data.
DISCUSSION
Many studies have shown that apnea have a relationship with cardiovascular
disease, obesity and hypertension (Ichimaru et al.,
1990). Therefore study on apnea is important in order to understand the
underlying process of apnea. Researcher has been searching a statistical analysis
that is appropriate to determine patients that has severe apnea. Measurement
that has been used include percentage of apnea in REM, light sleep and deep
sleep stages. Study by Goh et al. (2000) has
found that most of apnea event is during REM. This is consistent with present
study as most of hypopnea event is also in REM sleep stages.
Earlier investigators has considered other variables such as ApneaHypopnea
Index (AHI) to compare between age group, gender and race. Most of the study
conclude that AHI is differ according to demographic factors (Redline
et al., 1997; Bixler et al., 2001;
Stradling and Davies, 2004). However, there was very
few study on the distribution of apnea according to subjects, age group, gender
and other demographic factors. Consequently, the appropriate model to describe
the distribution of apnea is essential to give more explanations about the comparison.
This will allow the researcher to distinguish the apnea between the sleep stages
rather than percentage results only. This study highlighted the distribution
of hypopnea, which is 50% reduction of airflow (Tsai et
al., 1999). In this study, a variety of parametric distributions, particularly
hypopnea is used in order to find a model that is suitable in presenting the
duration of hypopnea distribution. The histogram of the duration of hypopnea
is skewed to the right with a long tail. However, which distributions would
provide the best fit was not obvious from the histogram. The normal distribution
was not considered in this study because it may give a poor fit because the
data is skewed. Present results demonstrate that most of the hypopnea event
is during REM sleep stages and very seldom during deep sleep stage. This finding
consistent with the study by Goh et al. (2000),
who founds that most of the hypopnea events, is during REM compared to NREM
sleep stages and very few during deep sleep stage.
The most important result of this study showed that an almost similar loglikelihood value for gamma, lognormal, Weibull and loglogistic distribution. Secondly, the appropriate distribution for the overall data is lognormal distribution and Weibull distribution is the best fit for light, deep and REM sleep stages. Finally another important finding is that there was improvement in fit when the data is subdivided into light, deep and REM sleep stages from the LR test. Models that is fit badly such as loglogistic model gave an almost similarly loglikelihood for all the three categories of sleep stages. This result may be due to the truncation as hypopnea is an event that at least 10 sec. Therefore further studies on simulation of hypopnea data is suggested to considering the effect of truncation to the duration of hypopnea.
The results that are obtained may also be influenced by other factors such
as age group, overweight, smoking habits, race and alcohol consumption. Many
studies have shown that there were more frequent apnea per hour in patients
that male, older adults, overweight and smokers (Sakakibara
et al., 1999; Redline et al., 1997;
Stradling et al., 2004). Therefore it is also
suggested that further study to determine the best model to describe the duration
of apnea according to age group, gender, overweight, smoking habits and alcohol
consumption.
ACKNOWLEDGMENTS
The authors acknowledge the Ministry of Higher Education Malaysia for financial support. The corresponding author thanks M.S. Ridout for the comments on the results.

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