Effect of variations of the complexity of the target variable on the assessment of Wiener–Granger causality in cardiovascular control studies

Abstract

We hypothesized that Wiener-Granger causality (WGC) indexes might have different abilities in coping with modifications of the complexity of the target variable in the context of the assessment of the cardiovascular control from spontaneous fluctuations of heart period (HP), systolic arterial pressure (SAP) and respiratory activity (R). After having defined the universe of knowledge as the set Ω = {HP, SAP, R} and the unpredictability decrement (UPD) as the difference between the prediction error variances of the target signal computed in Ω after excluding the presumed cause (i.e. the restricted Ω) and in Ω, we computed the following frequently utilized WGC indexes: (i) the plain UPD; (ii) the fractional UPD (FUPD) by dividing UPD by the prediction error variance in the restricted Ω; (iii) the normalized UPD (NUPD) by dividing UPD by the prediction error variance in Ω; (iv) the log-unpredictability decrement (LUPD) by applying the logarithm transformation to the prediction error variances before computing the UPD. The hypothesis was tested over two experimental protocols known to produce modifications of the complexity of HP variability: graded head-up tilt (HUT) inducing a gradual decrease of the HP complexity with tilt table inclination and head-down tilt (HDT) inducing the opposite trend. We demonstrated that: (1) when the strength of the causal relations from SAP to HP during HUT and from R to HP during HDT is assessed in Ω, WGC indexes reach different conclusions; (2) UPD is biased by modifications of the complexity of HP dynamics; (3) FUPD, NUPD and LUPD are less sensitive to changes of the complexity of the target dynamic, even though they have slightly different statistical power, being the NUPD the weakest one and FUPD and LUPD the strongest ones. We conclude that UPD should be avoided when assessing WGC and FUPD and LUPD should be privileged over NUPD.

Information Decomposition in Multivariate Systems: Definitions, Implementation and Application to Cardiovascular Networks

“…In [30], different formulations of information transfer were compared, observing discrepancies between measures based on the difference in conditional variance and the ratio between the same conditional variances which are consistent with the differences observed here between variance-based and entropy-based measures. The agreement between our findings and those of [30] is confirmed by the fact that measuring the difference of conditional entropies equals to measuring the ratio of conditional variances.…”

“…Thus, the discrepancy between the two formulations arises from the shift towards markedly lower values of the conditional variances of H, and from the concave property of the logarithmic transformation that expands the differences between conditional variances producing higher differences in conditional entropy. Note that very similar trends of the information measures were found in a similar protocol in [30] in a different group of healthy subjects, indicating that these behaviours are a typical response of cardiovascular dynamics to head-up tilt. In [30], different formulations of information transfer were compared, observing discrepancies between measures based on the difference in conditional variance and the ratio between the same conditional variances which are consistent with the differences observed here between variance-based and entropy-based measures.…”

“…Similarly to what shown in simulations of Section 3.3, the mathematical explanation of these behaviors lies in the nonlinear transformation of a distribution of values performed by the logarithmic expression that relates conditional variance and conditional entropy. While in Section 3.3 this behavior is explained in terms of its consequences on the sign of interaction information measures for simulations, here we discuss its consequences on the variation of measures of information transfer for physiological time series, also drawing analogies with the findings of [30].…”

“…An important but not fully explored aspect is that the measures of information dynamics are often used in isolation, thus limiting their interpretational capability. Indeed, recent studies have pointed out the intertwined nature of the measures of information dynamics, and the need to combine their evaluation to avoid misinterpretations about the underlying network properties [4,12,30]. Moreover, the specificity of measures of information storage and transfer is often limited by the fact that their definition incorporates multiple aspects of the dynamical structure of network processes; the high flexibility of information-theoretic measures allows to overcome this limitation by expanding these measures into meaningful quantities [13,29].…”

“…Finally, from the point of view of their implementation, the outcome of analyses based on information dynamics can be strongly affected by the functional used to define and estimate information measures. Model-free approaches for the computation of these measures are more general but more difficult to implement, and often provide comparable results than simpler and less demanding model-based techniques [31,32]; even within the class of model-based approaches, prediction methods and entropy methods-though often used interchangeably to assess network dynamics-may lead to strongly different interpretations [16,30].…”