Assessing erectile neurogenic dysfunction from heart rate variability through a Generalized Linear Mixed Model framework

Abstract

Background

The low (LF) vs. high (HF) frequency energy ratio, computed from the spectral decomposition of heart beat intervals, has become a major tool in cardiac autonomic system control and sympatho–vagal balance studies. The (statistical) distributions of response variables designed from ratios of two quantities, such as the LF/HF ratio, are likely to non-normal, hence preventing e.g., from a relevant use of the t-test. Even using a non-parametric formulation, the solution may be not appropriate as the test statistics do not account for correlation and heteroskedasticity, such as those that can be observed when several measures are taken from the same patient.

Objectives

The analyses for such type of data require the application of statistical models which do not assume a priori independence. In this spirit, the present contribution proposes the use of the Generalized Linear Mixed Models (GLMMs) framework to assess differences between groups of measures performed over classes of patients.

Methods

Statistical linear mixed models allow the inclusion of at least one random effect, besides the error term, which induces correlation between observations from the same subject. Moreover, by using GLMM, practitioners could assume any probability distribution, within the exponential family, for the data, and naturally model heteroskedasticity. Here, the sympatho–vagal balance expressed as LF/HF ratio of patients suffering neurogenic erectile dysfunction under three different body positions was analyzed in a case–control protocol by means of a GLMM under gamma and Gaussian distributed responses assumptions.

Results

The gamma GLMM model was compared with the normal linear mixed model (LMM) approach conducted using raw and log transformed data. Both raw GLMM gamma and log transformed LMM allow better inference for factor effects, including correlations between observations from the same patient under different body position compared to the raw LMM. The gamma GLMM provides a more natural distribution assumption of a response expressed as a ratio.

Conclusions

A gamma distribution assumption intrinsically models quadratic relationships between the expected value and the variance of the data avoiding prior data transformation. SAS and R source code are available on request.

Introduction

Heart rate variability (HRV) consists of the fluctuation between the intervals of consecutive normal heartbeats (RR intervals). It has become increasingly important in physiological studies. A signal derived from the RR intervals could provide meaningful information regarding the neural regulation of the cardiovascular system [1]. The RR signals can be studied either in the time or frequency domains [2]. For the latter case, the RR spectra are usually split in (at least) three frequency bands: the Very Low Frequency (VLF), low frequency (LF) and high frequency (HF) bands. A number of studies (cf. e.g. [2], [3], [4]) suggest that the LF band reflects sympathetic and vagal modulations while the HF band consists of a marker for vagal modulation. Consequently, the LF/HF ratio is considered a mirror of the sympatho/vagal balance, hence characterizing their relationships and commonly used as a non-invasive way of studying the health state of the cardiovascular system [2], [3], [4], [5].

Erectile dysfunction is defined as the inability to achieve and maintain an erection sufficient to permit satisfactory sexual intercourse [1], [6], [7]. Neurogenic erectile dysfunctions are an important group of organic etiologies probably because a deficiency of neurotransmitters is the final common pathway in many diseases and conditions [1], [6], [8]. A research area of increasing interest consists of studying the benefits of using spectral analysis to screen neurogenic erectile dysfunction by means of the heart rate variability [9], [10], [11].

Dynamical state modifications such as those provoked by body position changes (i.e., supine, seated and standing) are usually used as tools to analyze the sympatho–vagal balance by means of the LF/HF index under different treatments or health conditions [12]. In these experiments the LF/HF ratio and the Normalized LF and HF values are calculated at each of the body position states for the same patient. Usually, the data are analyzed by means of a t-test (hence assuming independence amongst data) or of its non-parametric counterpart, the Mann–Whitney test [13]. In any case, prior to analyze the statistical significance of treatment group differences, some distributional properties must be assessed, such as independently distributed normal data (i.i.d.) for the classical t-test. If the normal distribution assumption cannot be considered valid, non-parametric methods are usually employed. Yet, i.i.d. and variance homogeneity assumptions are still required [14]. Several experimental situations can lead to unfulfilling these required assumptions. For example, when several measures are taken on the same patient under different conditions and/or at different time intervals, correlated data structures are expected over the response data. Also, different variances between subjects belonging to different groups or sampling times could be expected. The use of classical statistical tests is not longer appropriate in these cases [15], [16].

Nowadays, the statistical theory as well as commercial statistical software has been significantly enhanced, allowing researchers to better fit experimental data even under more complex situations. One of these new approaches is known as Generalized Linear Mixed Models [14]. This kind of models allows the exploration of different effects that could impact the data as well as the consideration of more appropriate distributional assumptions for the observed data and, at the same time, considering different types of variation and correlations by using random effects.

In the present work we show that for the assessment of the neurogenic erectile dysfunction using the LF/HF ratio, a model assuming the classical independent and normal distributed observations does not properly explain the underlying data structure, leading to a poor inference and thus yielding inadequate conclusions. To overcome this limitation two new approaches using random effects are suggested. The first one is based on a log transformation of the responses, which deals with the non-normal distribution and the observed heteroskedasticity. The second approach is based on directly assuming a gamma (non-normal) distribution for the LF/HF ratio data.