GLOW: General Linear Models Optimized with Ward's Method

Matt Higger

cs.LGOct 16, 2026

GLOW: General Linear Models Optimized with Ward's Method

Authors: Matt Higger

Organizations: Khoury College of Computer Sciences, Northeastern University, Boston, MA 02115, USA

Abstract

Neuroimaging seeks brain regions whose imaging features correlate with variables of interest (e.g., diagnosis, age), usually via the General Linear Model (GLM). Region-Based Analysis dilutes or splits any effect that does not fill exactly one atlas region. Voxel-Based Analysis, with Cluster Extent Thresholding and Threshold-Free Cluster Enhancement built on it, fits a GLM per voxel and groups adjacent significant voxels. But neighboring voxels can share a statistic without sharing a GLM, which a per-voxel statistic cannot detect. GLOW (General Linear Models Optimized with Ward's method) instead builds a hierarchy of nested regions by greedily minimizing GLM error (provably a special case of Ward's clustering). Maximum-statistic (Westfall-Young) permutation controls family-wise error over all regions. Significance spreads to an effect's nested fragments and the regions containing it; a final pruning step seeks whole, pure effect extents. We plant synthetic effects of known extent and varying strength in Human Connectome Project (HCP) diffusion images and in white noise. Pruning delivers wholeness: GLOW returns a spatially consistent effect as one region on 57% of HCP trials, the best baseline on 35%; baselines recover comparable volume, but in many pure pieces. The cost is positive predictive value (PPV): GLOW is more sensitive (by up to 0.29), but its regions overrun the effect, mostly because no hierarchy region isolates it, not because of pruning. On HCP, GLOW's mean Dice gains 0.059 over the best baseline at weaker effects and gives back 0.005 at stronger ones. On white noise, GLOW trails by up to 0.61 Dice: its hierarchy holds the effect, but its permutation test does not certify it. Runtime grows roughly linearly with voxel count, about 4.5 times the voxel-wise methods' on the full HCP volume. The PPV loss may trace to the assumption of spatially independent residuals.

Figures & tables

Figure 1: Toy example: three voxels whose effects (A, B, C) are distinct yet share the same

F

-statistic. Pooling across voxels reveals their heterogeneity.

Figure 2: GLOW operating on a

1\times 4

array of voxels. Regions 0 to 3 are single voxels, regions 4 to 6 their merges. GLOW estimates that region 4 (voxels 0 and 1) contains an effect. Regions 0, 1, 4 and 6 were all found significant, but only region 4 is output to the user; the rest are pruned away, their significance explained by their intersection with region 4. The viewer at glow-viewer.web.app is another helpful entry point.

Method	Objective	Removes components of …
Naive	$\displaystyle\sum_{r}\|r\|\,\\|\bar{Y}_{r}\\|_{F}^{2}$	none
GLM Error	$\displaystyle\sum_{r}\|r\|\,\\|Q^{T}\bar{Y}_{r}\\|_{F}^{2}$	GLM residuals
Focus	$\displaystyle\sum_{r}\|r\|\,\\|Q_{1}^{T}\bar{Y}_{r}\\|_{F}^{2}$	GLM residuals & Covariates

Table 1: Segmentation objectives: three varieties of Ward’s method.

Figure 3: GLOW’s nested permutation. Each outer Westfall-Young draw (

j=1\ldots n_{\mathrm{fwer}}

) re-segments the permuted images and records

z_{j}=\max_{r}\hat{z}_{r}

. Calibrating a region (shaded) nests an inner loop (

k=1\ldots n_{\mathrm{inner}}

) that recomputes the region’s statistic as

\tau_{r,k}

to form its null

\{\tau_{r,k}\}

, then z-scores the observed

\tau_{r}

against it, making

\hat{z}_{r}

comparable across region sizes.

Figure 4: The pruning result for the running example from Fig 2 . Region 4 is bloomed (output to the user). The smaller regions nested inside it (0 and 1) and the larger region containing it (6) are pruned , their significance explained away by their overlap with region 4.

Figure 5: The two extent shapes on each source, each of 2,500 voxels, planted on the same images: one seed’s extents (red) on a shared axial slice of one of those images (HCP: FA; WGN: noise).

Figure 6: Imposing a synthetic effect. Green and red mark moderate and severe effects imposed onto the original image set, in blue. The left column gives one image of the original set alongside its modified versions at each effect strength, each labeled with the FA offset the effect induces, $\beta x$ . The right column repeats this for another image, the one with maximum $x$ .

Figure 7: Segmentation quality. Oracle Dice of the best-matching region in the hierarchy for each of the three Ward’s objectives (Table 1 ). Markers show the mean across random seeds; error bars give its

95\%

confidence interval (

\pm 1.96

standard errors of the mean).

Figure 8: Pruning rule comparison, under the Focus segmentation objective. Markers show the mean across random seeds; error bars give its

95\%

confidence interval.

Figure 9: FWER control. Shading is the pointwise

95\%

interval for the true rejection rate given the 1,000 null seeds, so a calibrated method’s band brackets the dashed ideal.

	HCP				WGN
	MinVar		Sphere		MinVar		Sphere
LLR $/\|r\|$	$b{=}1$	$b{=}2$	$b{=}1$	$b{=}2$	$b{=}1$	$b{=}2$	$b{=}1$	$b{=}2$
$0.0189$	1	0	1	0	0	0	3	0
$0.0300$	1	0	1	1	0	0	2	0
$0.0475$	1	1	1	0	0	0	5	1
$0.0754$	1	0	3	1	7	2	4	1

Table 2: GLOW trials (of 50) returning a region which holds no planted voxel, a false rejection of a true null. Bold where the count exceeds

\alpha=0.05

(one-sided binomial

p<0.05

, i.e. 6 or more). The four weakest strengths of the grid (

0.0030

0.0119

) are omitted: every count there is 0 or 1.

Figure 10: GLOW’s Dice minus the best baseline’s, by source, extent shape, $b$ and effect strength. The best baseline is the one with the highest mean Dice in that cell (letter: V for VBA, T for VBA-TFCE, C for CET), and each entry is the mean of the 50 seed-paired differences, bold where its $95\%$ confidence interval excludes zero.

Figure 11: Detection performance versus effect strength (

b=1

), one metric per row and one source and extent shape per column. Each curve is the mean across random seeds, shaded with the

95\%

confidence interval of that mean. PPV, completeness and homogeneity average over the trials which detect some effect voxel.

Figure 12: Fig 11 at

b=2

Figure 13: The PPV available to a pruning rule, and to the tree it prunes, for the MinVar extent at

b=1

, over the same seeds as Fig 11 . Each line gives the PPV of the union of at most

k

regions, chosen with knowledge of the ground truth to maximize Dice with the planted extent. Solid lines choose from the FWER-significant regions alone, the pool GLOW’s greedy rule ranks; dashed lines choose from every node of the Ward tree. Circles mark GLOW’s own operating point, its mean region count against its mean PPV. All three are averaged over the trials in which GLOW discovered something. Both Oracles maximize Dice rather than PPV, so the wider pool is not guaranteed to be the more precise one.

Figure 14: Wall-clock fit time versus analyzed voxel count, on log-log axes. Each curve is the median over three HCP crops (shaded band: min to max). Each legend entry gives that method’s fitted log-log slope over the largest voxel counts (22,462 voxels and up). Note these times use

n_{\mathrm{fwer}}=500

outer permutations.

Figure 15: GLOW’s per-permutation cost, timed on one core with no GPU and BLAS pinned to one thread. Panels sweep voxel count

|V|

, inner draws per tree

n_{\mathrm{inner}}

, imaging features

b

and subjects

N

, each holding the rest at the shared center (

25{,}000

voxels, one outer permutation,

b=1

, the full cohort). Median over three HCP crops (band: min to max) on a shared wall-time axis; titles give the fitted log-log slope, for

|V|

over its largest decade as in Fig 14 .

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

		FWHM (mm)
Method	LLR $/\|r\|$	0	2	4	8	12
VBA	0.003	0.002	0.004	0.011	0.023	0.031
	0.0075	0.031	0.041	0.087	0.116	0.118
	0.03	0.242	0.277	0.393	0.430	0.397
	0.12	0.848	0.866	0.787	0.674	0.597
	0.3	0.995	0.992	0.810	0.651	0.559

Appendix

Table 3: Mean Dice against kernel width over the 50 tuning experiments (HCP, MinVar extent,

b=1

), each baseline at its chosen statistic, per effect strength (10 seeds per entry) and pooled over strengths (mean rows). Bolded entries are each row’s maximum. Each strength row is shaded in its baseline’s color, linearly in Dice, from white at the row’s minimum to darkest at its maximum.

Method	Scaling	LLR	Wilks	Pillai	Hotelling	Roy
VBA	raw	0.411	0.411	0.411	0.411	0.411
	z-scored	0.409	0.409	0.409	0.410	0.410
VBA-TFCE	raw	0.464	0.493	0.493	0.411	0.411
	z-scored	0.503	0.509	0.509	0.498	0.498
CET	raw	0.492	0.492	0.492	0.492	0.492
	z-scored	0.490	0.490	0.490	0.490	0.490

Appendix

Table 4: Mean Dice over 250 unsmoothed experiments (HCP, MinVar extent,

b=2

, five effect strengths

\times

50 seeds).

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