Center Persistency (CP)

Above, we described how DBS functions across initial cluster‐forming thresholds. Although this approach was intended to avoid selecting one arbitrary threshold, the result still depends on a specific range of thresholds. This prompted us to define a new measure, center persistency (CP), by borrowing the concept of persistence from algebraic topology [Horak et al., 2009; Zomorodian and Carlsson, 2005]. This CP score represents the persistent and significant association of a cluster center with an effect of interest in a wide possible range of initial cluster‐forming thresholds. This persistency score is estimated for each cluster. The CP is calculated by obtaining the sum of the weighted degrees for the entire possible range of thresholds.

The threshold range is determined in a data‐driven way. At first, the lower boundary of the threshold is set to the statistical value at a P‐value of 0.05 (P = 0.05 for the tested hypothesis is the conventional threshold value in univariate hypothesis testing). Then, the initial threshold is gradually raised to the value before the binary degree threshold becomes 2, which was acquired in the previous permutation step for the cluster‐wise inference in section “Degree‐based statistic (DBS): a correction procedure.” The reason for this range is that (1) the P‐value of 0.05 for the lower boundary is a commonly selected statistical threshold in one univariate hypothesis testing and (2) the higher boundary guarantees the minimum possibility of a cluster formation having three edges. Then, the CP score of the v _i node‐centered cluster, CP_vi, is measured for this range.

The significance of CP_vi is estimated by comparison with a null distribution of the maximum persistency score, CP_max. The null distribution is acquired empirically from permutation similar to the degree described in section “Degree‐based statistic (DBS): a correction procedure.” We measure CP_max from the same permutation set (e.g., CP_max,j is for the jth permutation). Then, the observed CP_vi is compared with the 95th percentile of the empirical null distribution of CP_max. Clusters with CP scores higher the 95th percentile are persistent clusters with a corrected significance of P < 0.05. In other words, the CP score represents the extent to which the persistent association of a cluster center with an effect of particular interest is significant, independent of the initial threshold selection. In addition, the normalized CP can be used for further comparison among clusters.

We suggest the use of a normalized CP to compare clusters among studies based on different imaging modalities (such as DTI and fMRI) or connectivity measures (such as mean FA and fiber number for DTI, and different correlations and coherences for fMRI). Because the weighted degree value, even its order of magnitude, could be different across studies, the original CP scores of clusters were not easily compatible. However, as shown in the above equation, this normalized CP is estimated by the ratio of the CP score of a cluster to the threshold selected for the significance. Thus, the above equation would enable us to determine which of the clusters (estimated from different imaging modalities and connectivity measures) is more persistent for the effect of interest with a fixed threshold.