Beyond linearity in neuroimaging: Capturing nonlinear relationships with application to longitudinal studies

Abstract

The ubiquitous adoption of linearity for quantitative predictors in statistical modeling is likely attributable to its advantages of straightforward interpretation and computational feasibility. The linearity assumption may be a reasonable approximation especially when the variable is confined within a narrow range, but it can be problematic when the variable's effect is non-monotonic or complex. Furthermore, visualization and model assessment of a linear fit are usually omitted because of challenges at the whole brain level in neuroimaging. By adopting a principle of learning from the data in the presence of uncertainty to resolve the problematic aspects of conventional polynomial fitting, we introduce a flexible and adaptive approach of multilevel smoothing splines (MSS) to capture any nonlinearity of a quantitative predictor for population-level neuroimaging data analysis. With no prior knowledge regarding the underlying relationship other than a parsimonious assumption about the extent of smoothness (e.g., no sharp corners), we express the unknown relationship with a sufficient number of smoothing splines and use the data to adaptively determine the specifics of the nonlinearity. In addition to introducing the theoretical framework of MSS as an efficient approach with a counterbalance between flexibility and stability, we strive to (a) lay out the specific schemes for population-level nonlinear analyses that may involve task (e.g., contrasting conditions) and subject-grouping (e.g., patients vs controls) factors; (b) provide modeling accommodations to adaptively reveal, estimate and compare any nonlinear effects of a predictor across the brain, or to more accurately account for the effects (including nonlinear effects) of a quantitative confound; (c) offer the associated program 3dMSS to the neuroimaging community for whole-brain voxel-wise analysis as part of the AFNI suite; and (d) demonstrate the modeling approach and visualization processes with a longitudinal dataset of structural MRI scans.

Trial registration: ClinicalTrials.gov NCT01132885 NCT01434368.

Published by Elsevier Inc.

Figures

Fig. 1.

Comparisons of data fitting among three polynomial models and a smoothing spline approach. Linear fitting (black line) between BOLD response and age in a region, as typically adopted in neuroimaging analysis, sometimes may render a roughly acceptable performance with a general sign (positive or negative) for the association between x and y such as (a) here, but other times it could fail badly as shown in (b). Quadratic fitting (green) performs better, and the cubic model (orange) is a further improvement. Furthermore, the curve estimated through smoothing splines (blue) provides the best fit without any prior knowledge of parameters such as the order of polynomials. More crucially, smoothing spline modeling is largely self-adaptive without the daunting undertaking of order selection in polynomial fitting.

Fig. 2.

Basis functions b0(x), b1(x), …, bK−1(x) with K = 6. Two spline types are discussed here: cardinal cubic splines (a, c) and thin plate splines (b, d). Cubic splines are knot-based; with 6 knots at 0, 0.2, 0.4, 0.6, 0.8, and 1.0 within [0, 1], there are 6 cardinal basis functions (a). Each cardinal basis function (e.g., b4(x) in blue) peaks at the associated knot (x = 0.6) with a value of 1, and takes the value 0 at all other knots. Therefore, the fitted value at a knot corresponds to the weight for the associated basis function. In contrast, as shown in (b, d), each of the thin plate basis functions within [0, 1] is not knot-specific. As the baseline (intercept) is usually modeled separately in real practice, we only utilize K − 1 basis functions to maintain model identifiability (c, d).

Fig. 3.

Statistical evidence of asymmetry in age trends of cortical gray matter volume (GMV) shown at voxelwise p < .05 (top) and three selected regions (highlighted in cyan) filtered at voxelwise p < 10−5 (below), based on smoothing spline modeling of trends with age. Trajectories are averaged across voxels within the three selected regions, and their uncertainty bands extend one standard error above and below the trajectories. Adjusted R2 values (Radj2) indicating goodness-of-fit were higher for the smoothing spline model than for a linear model (ranges for voxels within regions are shown below). (a) Heschl’s gyrus, demonstrating left-lateralized asymmetry with trajectories for left and right hemispheres that have roughly similar shape across the age range 7–20 (Radj2 in (0.786, 0.997) for MSS vs. (−0.020, 0.353) for linearity); (b) Inferior temporal gyrus, also showing left-lateralized asymmetry, but with trajectories that reveal decreasing asymmetry over the same age range (Radj2 in (0.978, 0.998) for MSS vs. (0.065, 0.116) for linearity); (c) Anterior insula, where trajectories of right and left hemispheres can be seen to “cross over” (Radj2 in (0.951, 0.983) for MSS vs. (−0.049, −0.022) for linearity).

Fig. 4.

Statistical evidence of regions for which asymmetries were captured by a nonlinear trajectory (top, voxelwise threshold of p < 0.05) of cortical gray matter volume (GMV) and three selected regions (highlighted in cyan) (below, voxelwise p < 0.005). These regions were a subset of the general asymmetries shown in Fig. 3. The nonlinear trends are averaged within the three selected regions, and their uncertainty bands extend one standard error above and below the trajectories. (a) shows a right-lateralized region within the intraparietal sulcus where both hemispheres exhibit an increased rate of gray matter reduction in the 10–15 year range (Radj2 in (0.993, 0.996) for MSS vs. (0.001, 0.044) for linearity). (b) and (c) illustrate similarly right-lateralized regions within the temporo-parietal junction (Radj2 in (0.993, 0.998) for MSS vs. (−0.086, 0.018) for linearity) and frontal pole (Radj2 in (0.984, 0.991) for MSS vs. (0.051, 0.102) for linearity), respectively, with an increased rate of reduction over a similar age range.