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

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.