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Dependence and Independence: Structures, Testing, and Measuring
Published in Albert Vexler, Alan D. Hutson, Xiwei Chen, Statistical Testing Strategies in the Health Sciences, 2017
Albert Vexler, Alan D. Hutson, Xiwei Chen
Székely and Rizzo (2009) proposed new approaches to measuring multivariate dependence and testing the joint independence of random vectors in arbitrary dimension, including distance correlation as well as distance and Brownian covariance with respect to a stochastic process. Distance covariance and distance correlation are analogous to product–moment covariance and correlation, but generalize and extend these classical bivariate measures of dependence. The distance covariance statistic is defined as the square root of , where Akl and Bkl are simple linear functions of the pairwise distances between sample elements. Distance correlation, a standardized version of distance covariance, is zero if and only if the random vectors are independent. It was shown that population distance covariance coincides with the covariance with respect to Brownian motion; thus, both can be called Brownian distance covariance. In the bivariate case, Brownian covariance is the natural extension of product–moment covariance, since Pearson product–moment covariance is obtained by replacing the Brownian motion in the definition with identity. Note that while uncorrelatedness can sometimes replace independence, uncorrelatedness is too weak to imply a central limit theorem even for strongly stationary summands. Therefore, Brownian covariance and correlation can be recommended compared to the classical Pearson covariance and correlation.
ECoG-Based BCIs
Published in Chang S. Nam, Anton Nijholt, Fabien Lotte, Brain–Computer Interfaces Handbook, 2018
Another limitation of ECoG studies performed with this patient population is the electrode size and interelectrode distance of the clinical arrays that are currently being used. As discussed in previous subsections, conventional clinical grids are often configured as an 8-by-8 array with 1 cm interelectrode spacing and an exposed contact diameter of 2–3 mm. These arrays are designed with such large contact area surfaces to yield low impedance values (around several hundred ohms), which prove to be advantageous in the noisy environment of a hospital room. A grid can spatially cover a large area of a lateral hemisphere but samples sparsely because of the large interelectrode distance, typically from only a few electrodes on a given gyrus (Chang 2015). The optimal electrode size and interelectrode distance to maximize the amount of information extracted through ECoG is one of the most important open questions in the field. While smaller electrodes, sometimes referred to as micro-ECoG, can bring about improved spatial selectivity, this comes at the expense of increased impedance. Moreover, while more channels may promise increased spatial sampling, an increased number of channels provides practical and clinical challenges. Some studies have utilized micro-ECoG arrays to show higher information extraction in the superior temporal gyrus to study speech generation (Bouchard et al. 2016; Chang et al. 2010; Mesgarani et al. 2014) and to show improved BCI performance (Kellis et al. 2010, 2016; Leuthardt et al. 2009). A computational finite element modeling study that predicted the biophysical correlation between electrodes at various distances suggested a minimum spacing of 1.7–1.8 mm for subdural recordings (Slutzky et al. 2010), which is still larger than the size of cortical ocular dominance columns in the human visual cortex that are about 1 mm wide (Adam and Horton 2008). It should be noted that correlation between electrode recordings, and thus optimal electrode spacing, is dependent on frequency. As discussed earlier, lower frequencies are more broadly distributed while high frequencies are spatially more focal. Thus, it is not surprising that when Chang (2015) computed the correlation of different spectral bands as a function of electrode distance, correlation was lowest for the broadband gamma band at the lowest electrode spacing and higher for lower frequencies (see Figure 16.18). Rouse et al. (Rouse et al. 2016) utilized micro-ECoG electrodes with 300 m diameter for one-dimensional and two-dimensional BCI tasks in primates. They tested the effect of interelectrode distance on BCI control between 3 and 15 mm. The primates achieved successful BCI control with two electrodes separated by 9 and 15 mm. Performance decreased and the signals became more correlated when the electrodes were only 3 mm apart. Overall, more systematic studies informed by computational models are imperative to determine the optimal electrode design for BCI utility.
Correlation Database of 60 Cross-Disciplinary Surveys and Cognitive Tasks Assessing Self-Regulation
Published in Journal of Personality Assessment, 2021
Gina L. Mazza, Heather L. Smyth, Patrick G. Bissett, Jessica R. Canning, Ian W. Eisenberg, A. Zeynep Enkavi, Oscar Gonzalez, Sunny Jung Kim, Stephen A. Metcalf, Felix Muniz, William E. Pelham, Emily A. Scherer, Matthew J. Valente, Haiyi Xie, Russell A. Poldrack, Lisa A. Marsch, David P. MacKinnon
Product-moment and distance correlations were computed among the 66 variables derived from the 23 surveys, 138 variables derived from the 37 cognitive tasks, and 11 variables pertaining to health and substance use (total = 215 variables). These 215 variables resulted in 215C2 = 23,005 (product-moment or distance) correlations computed using pairwise deletion. The product-moment correlations were estimated using the cor function in R, and the distance correlations were estimated using the dcor function available through the energy package in R (Rizzo & Székely, 2016). Whereas the product-moment correlation measures the strength and direction of the linear association between two variables any dependence (i.e., linear and nonlinear associations) between 2007). The distance correlation ranges from 0 to 1 and equals 0 only if
Kernel partial correlation: a novel approach to capturing conditional independence in graphical models for noisy data
Published in Journal of Applied Statistics, 2018
Jihwan Oh, Faye Zheng, R. W. Doerge, Hyonho Chun
Several other researchers have proposed GM approaches that use the distance information among observations, such as kernels. Kernel methods detect flexible dependence among variables by using linear combinations of kernel functions evaluated at the data points [5]. For example, Fukumizu et al. [18] suggested using the Hilbert–Schmidt norm of the normalized conditional cross-covariance operator for measuring nonlinear conditional dependence. Li et al. [33] introduced additive semigraphoid models using the notion of additive conditional independence and estimated it by using reproducing kernel Hilbert space (RKHS) estimators. Besides, Székely and Rizzo [52] introduced partial distance correlation (pdCor) using Euclidean distances among observations, and Wang et al. [57] proposed conditional distance correlation that also utilized the Euclidean distances. We remark that the distance-based approaches are viewed as kernel approaches due to the equivalence between kernels and the Euclidean distances [43]. All these kernel-based approaches are flexible enough to capture nonlinear associations.
M1-like TAMs are required for the efficacy of PD-L1/PD-1 blockades in gastric cancer
Published in OncoImmunology, 2021
Rui Zhao, Qianyi Wan, Yong Wang, Yutao Wu, Shuomeng Xiao, Qiqi Li, Xiaoding Shen, Wen Zhuang, Yong Zhou, Lin Xia, Yinghan Song, Yi Chen, Hanshuo Yang, Xiaoting Wu
Differences between continuous variables were analyzed through Student’s two-tailed test, Mann-Whitney U test or one-way analysis of variance, and categorical variables were analyzed using Pearson’s Chi-square test or Fisher’s exact test. Correlation coefficients were computed by Spearman and distance correlation analyses. For corresponding survival data attached to gene-expression profiles, survival curves were generated by the Kaplan–Meier method, and the log-rank (Mantel-Cox) test was used to determine the statistical significance of differences. All P values were two-tailed, and the P value of < 0.05 was considered statistically significant. The software R 3.4.0, SPSS 25.0, and GraphPad Prism 8.0 were used for data analysis and image presentation.