TABLE VII.

Example studies using multivariate regression, which aim to reveal complex imaging genomics associations between multivariate SNP data and imaging QT data.

Ref	Notes
[106]	P-GLAW (pathways group lasso with adaptive weights), multi-SNP-single-QT, group lasso, SNPs grouped by pathway
[146]	TGSL (tree-guided sparse learning), multi-SNP-single-QT, group lasso, tree-like group structure (SNPs grouped by LD block, LD blocks grouped by gene)
[147]	DAMM (diagnosis-aligned multimodal regression), single-SNP-multi-QT, select ROIs with genetic effects at most modalities, learning diagnosis-related components in the projected space
[148]	G-SMuRFS (Group-Sparse Multi-task Regression and Feature Selection), multi-SNP-multi-QT, group l2,1 for feature selection at LD block level, l2,1 for feature selection at SNP level.
[151]	TCLSR (task-correlated longitudinal sparse regression), longitudinal imaging QTs to predict SNPs, each time point treated as a task, trace norm for weight matrix rank minimization, l2,1 norm for selecting imaging QTs with effects at most of the time points
[149]	TSAL (temporal structure auto-learning), longitudinal imaging QTs to predict SNPs, Schatten p-norm for weight matrix rank minimization, l2,0+ norm for selecting imaging QTs with effects at most of the time points
[153]	JPLSR (joint projection learning and sparse regression), multi-SNP-multi-QT, projecting SNP and QT data into a joint latent space, SNP and QT components aligned with diagnosis, l2,1 norm for selection of SNP and QT features
[154]	SRRR (sparse reduced rank regression), multi-SNP-multi-QT, reduced rank loss function, l1 norm for selecting SNP and QT features, evaluation on ROI-based simulation data
[155]	SRRR (sparse reduced rank regression), multi-SNP-multi-QT, reduced rank loss function, penalized LDA to select diagnosis-related QT, l1 norm and re-sampling for SNP identification, evaluation on voxelwise ADNI data
[107]	P-SRRR (pathways SRRR), integration of P-GLAW and SRRR, group lasso on SNP side, SNPs grouped by pathway, identifying QT-related pathways
[156]	S-SRRR (structured SRRR), reduced rank loss function, l2,1 norm for selecting SNP and QT features
[157]	GRS-SRRR (graph-regularized S-SRRR), incorporation of graph self-representation on the SNP side into S-SRRR
[159]	RGRS-SRRR (robust GRS-SRRR), robust version of reduced rank loss function and graph self-representation loss function
[161]	BGSMTR (Bayesian group sparse multi-task regression), variable selection at both SNP and gene level, full posterior inference
[164]	GLRR (Bayesian generalized low rank regression), low rank approximation of weight matrix, dynamic factor model for imaging covariance, efficient MCMC algorithm for posterior computation
[165]	L2R2 (Bayesian longitudinal low rank regression), SNP effects on longitudinal imaging QTs, low rank approximation of weight matrix and gene-age interaction, penalized splines for overall time effect, efficient MCMC algorithm for posterior computation
[166]	FNAM (Additive Model via Feedforward Neural networks with random weight), modeling non-linear associations, computational efficiency, flexibility and interpretability of additive models