DISCUSSION
stroke, connectome‐based
REGRESSION, STROKE
In this manuscript, a new framework for analyzing functional connectivity was explored. Functional data analysis of log quantile connectivity densities investigates possible nonlocalized effects associated with subject‐level variables. It is clear that our method can be directly applied to other kinds of numerical measurements. For example, partial correlations or entropy‐based measures. However, it continues to be only useful suitable if connection exchangeability represents a useful model. A sizable by‐product of this style of analysis is the drastic reduction of multiplicity considerations. This is of great importance in connectivity analysis, where the number of comparisons grows at a rate of the square of the number of foci being considered. In the data application, we find associations between stimulation and connectivity density. In contrast, edgewise methods fail to find any results, because of multiplicity issues. This is partially due to a wide search of all possible region pairs from the parcellation. Of course, one could also reduce multiplicity concerns by restricting attention to regions associated with a priori hypotheses of interest, as was done in Ficek et al. (Density regression, as a prediction model, can be view as a generalization of connectome‐based predictive modeling (Shen et al., An interesting direction to pursue with connectivity density methods is to consider potential robustness to spatial registration (Oliveira & Tavares, However, to reiterate, ignoring potentially useful localization information can reduce power and sensitivity. Surely, the optimal strategy removes subject‐specific artifacts and reduces the search space with—correct—strong a priori hypotheses and then tests only those edges. However, in the absence of this ideal case, one is often confronted with a massive unstructured search problem with localization analyses. In contrast, density regression is more akin to an omnibus We used functional data analysis to relate connection densities to outcomes. Functional data analysis tools (Ramsay & Silverman, Utilizing functional regression also has the benefit of producing more interpretable models as compared with machine learning approaches. However, this is achieved at a likely cost of prediction performance. It is possible that ML approaches could navigate the trade‐offs between localization and exchangeability non‐parametrically and possibly achieve better prediction performance. Thus, we view density regression as a parsimonious modeling choice rather than a method to optimize prediction performance.Statistically, we assumed independence between subjects and relied on the randomization to invert the predictor/response relationship using logit models. This borrows techniques from case‐referent sampling from epidemiology dating back to the seminal work of Cornfield (see Breslow, Nonetheless, we reiterate that the use of connectivity density as a regressor remains useful, even if one prefers not to flip the predictor/response relationship. For example, in our tDCS example, connecting the connectivity density to behavioral outcomes would be relevant, where the natural predictor would be functional connectivity.Independence between subjects was used for inference. We also used density estimates for connection densities, techniques that implicitly require sampling assumptions for theoretical convergence. However, we contend that connectivity densities are intrinsically of interest, and therefore no appeals to super‐population inference and sampling assumptions are needed for estimation. This is analogous to spatial group ICA, where productive estimates are obtained via independence assumptions on voxels over space, without a true sampling or super‐population model for inference (Calhoun et al., Our recommended approach uses log quantile densities as the functional predictor, rather than the density, distribution function, or quantile function directly (Petersen & Muüller, Our simulations and data results focus on settings that highlight the benefits of an omnibus density regression approach. In the simulations, we investigated a non‐localized caricature of typical effects. Similarly, in our data analysis, we performed no filtering of regions prior to analysis (thus magnifying multiple comparison concerns). It was shown in the simulation, that functional density regression approaches can find real non‐localized effects, whereas, as expected, edgewise methods do not find any. It should be emphasized that the performance of the density regression approach is invariant to the distribution of effects across subjects, whereas edgewise approaches become viable as the degree of localization increases.In addition, the flexibility of the approach finds effects in the real data, even though there are a great deal of irrelevant connections (i.e., unnecessarily included region pairs) being studied. Edgewise and other regression approaches are highly sensitive to unnecessary null connections being included in the analysis. A benefit of the data being considered is the likely existence of an effect related to the stimulation. However, we emphasize that a single omnibus approach does not represent a full analysis of the data. We recommend this approach as a global analysis to be performed prior to edgewise or other localization methods. This mirrors the classic ANOVA (analysis of variance) approach of performing an overall F test before investigating pairs of explanatory factor levels. It would be most useful in exploratory model building where foci selection is not restrictive. In cases of tightly coupled statistical hypotheses involving relatively few regions or foci, density regression would not be needed or particularly helpful.This methodology raises many avenues for future research. For example, one the idea of non‐localized effects in dynamic connectivity (Hutchison et al., Finally, there is the role that connectivity‐density methods could play in fMRI analysis of subjects with missing brain tissue, such as studies of stroke or surgical interventions. Connectivity density methods may be resilient to the missing data impact of differential brain structure in a way that localization methods are not. In fact, it is interesting to conjecture what localization methods even mean in these settings where a subset of subjects are missing areas of localization. In contrast, density methods may provide a more robust and well‐defined methodology. It is worthy of note that components of graph methodology (Bullmore & Sporns, 