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Figure 2 | BMC Neuroscience

Figure 2

From: Fast construction of voxel-level functional connectivity graphs

Figure 2

Dichotomization and tetrachoric correlation estimation. Consider a sample ((x1,y1), (x2,y2),…,(x T ,y T )) of size T where (x,y) is distributed according to bivariate normality. Further, let x=x1,x2,…,x T and y=y1,y2,…,y T denote the samples from x and y, respectively. Using x ~ and y ~ , as thresholds, x and y can be dichotomized resulting in the binary samples x d and y d . A: As an example, the density of a bivariate normal distribution (ρ=0.7) is shown (top, 3D curve) along with a sample (bottom, points in the x-y-plane) drawn from that distribution. By virtue of the two lines x= x ~ and y= y ~ , the x-y-plane is divided into four quadrants, such that the counts of sample points per quadrant form a 2×2 contingency table. The (relative) frequencies in the contingency table can also be expressed in terms of x d and y d (e.g., n11 is the number of indices where both x d and y d are equal to 1, and p11=n11T−1). The probability masses corresponding to the table’s relative frequencies are equal to the respective partial volumes belonging to the four quadrants in the x-y-plane under the bivariate normal’s curve. The tetrachoric correlation coefficient r t , for which these partial volumes resemble the relative frequencies in a given table, is an estimate of the population correlation parameter ρ belonging to the underlying distribution. B: Relationship between p11 and r t . Given x d and y d , r t can be found using r t =− cos(2π p11). For details see text.

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