Probabilistic Joint and Individual Variation Explained (ProJIVE) for Data Integration

Raphiel J. Murden; Ganzhong Tian; Deqiang Qiu; Benajmin B. Risk

stat.MLMar 12, 2026

Probabilistic Joint and Individual Variation Explained (ProJIVE) for Data Integration

Authors: Raphiel J. Murden, Ganzhong Tian, Deqiang Qiu, Benajmin B. Risk

Organizations: Department of Biostatistics and Bioinformatics, Rollins School of Public Health, Emory University, 1518 Clifton Rd NE, Atlanta, GA 30322, USA · Department of Radiology and Imaging Sciences, Emory University School of Medicine, 1364 Clifton Rd NE, Atlanta, GA 30322, USA

Abstract

Collecting multiple types of data on the same set of subjects is common in modern scientific applications including, genomics, metabolomics, and neuroimaging. Joint and Individual Variance Explained (JIVE) seeks a low-rank approximation of the joint variation between two or more sets of features captured on common subjects and isolates this variation from that unique to eachset of features. We develop an expectation-maximization (EM) algorithm to estimate a probabilistic model for the JIVE framework. The model extends probabilistic principal components analysis to multiple data sets. Our maximum likelihood approach simultaneously estimates joint and individual components, which can lead to greater accuracy compared to other methods. We apply ProJIVE to measures of brain morphometry and cognition in Alzheimer's disease. ProJIVE learns biologically meaningful courses of variation, and the joint morphometry and cognition subject scores are strongly related to more expensive existing biomarkers. Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. Code to reproduce the analysis is available on our GitHub page.

Figures & tables

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

	$r_{J}$	$P_{2}$	$R_{J1}^{2}$	$R_{J2}^{2}$	ProJIVE	AJIVE	R.JIVE	GIPCA	D-CCA
					Mean (SD)
Joint Subject Scores	1	20	0.1	0.1	0.39 (0.01)	0.39 (0.01)	0.40 (0.02)	0.68 (0.23)	0.49 (0.17)
			0.1	0.5	0.16 (0.01)	0.28 (0.01)	0.30 (0.14)	0.26 (0.08)	0.38 (0.20)
			0.5	0.1	0.16 (0.01)	0.28 (0.01)	0.20 (0.06)	0.26 (0.08)	0.39 (0.17)
			0.5	0.5	0.12 (0.00)	0.12 (0.01)	0.12 (0.01)	0.16 (0.05)	0.26 (0.23)
		200	0.1	0.1	0.17 (0.00)	0.28 (0.01)	0.31 (0.01)	0.95 (0.10)	0.35 (0.13)

Appendix

Table 9

	$r_{J}$	$P_{2}$	$R_{J1}^{2}$	$R_{J2}^{2}$	ProJIVE	AJIVE	R.JIVE	GIPCA	D-CCA
					Mean (SD)
Joint Loadings X1	1	20	0.1	0.1	0.10 (0.02)	0.10 (0.02)	0.10 (0.02)	0.43 (0.30)	0.10 (0.02)
			0.1	0.5	0.09 (0.02)	0.09 (0.02)	0.14 (0.09)	0.10 (0.03)	0.10 (0.02)
			0.5	0.1	0.03 (0.01)	0.03 (0.01)	0.04 (0.02)	0.10 (0.05)	0.03 (0.01)
			0.5	0.5	0.03 (0.01)	0.03 (0.01)	0.03 (0.01)	0.05 (0.02)	0.03 (0.01)
		200	0.1	0.1	0.09 (0.02)	0.09 (0.02)	0.09 (0.02)	0.50 (0.28)	0.10 (0.02)

Appendix

Table 10

	$r_{J}$	$P_{2}$	$R_{J1}^{2}$	$R_{J2}^{2}$	ProJIVE	AJIVE	R.JIVE	GIPCA	D-CCA
					Mean (SD)
Individual $X_{1}$ Scores	1	20	0.1	0.1	0.54 (0.04)	0.54 (0.04)	0.54 (0.04)	0.64 (0.13)	0.54 (0.04)
			0.1	0.5	0.53 (0.04)	0.54 (0.04)	0.58 (0.05)	0.54 (0.04)	0.54 (0.04)
			0.5	0.1	0.38 (0.04)	0.44 (0.10)	0.40 (0.06)	0.41 (0.05)	0.65 (0.10)
			0.5	0.5	0.38 (0.04)	0.38 (0.04)	0.39 (0.04)	0.39 (0.06)	0.38 (0.04)
		200	0.1	0.1	0.53 (0.04)	0.54 (0.04)	0.54 (0.04)	0.71 (0.07)	0.54 (0.04)

Appendix

Table 11

	$r_{J}$	$P_{2}$	$R_{J1}^{2}$	$R_{J2}^{2}$	ProJIVE	AJIVE	R.JIVE	GIPCA	D-CCA
					Mean (SD)
Individual $X_{2}$ Scores	1	20	0.1	0.1	0.53 (0.04)	0.54 (0.04)	0.54 (0.04)	0.64 (0.12)	0.54 (0.04)
			0.1	0.5	0.38 (0.04)	0.44 (0.09)	0.47 (0.10)	0.41 (0.05)	0.63 (0.10)
			0.5	0.1	0.53 (0.04)	0.54 (0.04)	0.53 (0.04)	0.54 (0.05)	0.53 (0.04)
			0.5	0.5	0.38 (0.04)	0.38 (0.04)	0.38 (0.04)	0.39 (0.05)	0.38 (0.04)
		200	0.1	0.1	0.20 (0.01)	0.21 (0.01)	0.21 (0.01)	0.69 (0.08)	0.21 (0.01)

Appendix

Table 12

	$r_{J}$	$P_{2}$	$R_{J1}^{2}$	$R_{J2}^{2}$	ProJIVE	AJIVE	R.JIVE	GIPCA	D-CCA
					Mean (SD)
Joint Subject Scores	1	20	0.1	0.1	0.39 (0.02)	0.39 (0.02)	0.42 (0.02)	0.75 (0.22)	0.51 (0.18)
			0.1	0.5	0.16 (0.01)	0.28 (0.01)	0.33 (0.12)	0.31 (0.08)	0.43 (0.22)
			0.5	0.1	0.16 (0.01)	0.28 (0.01)	0.23 (0.02)	0.31 (0.09)	0.42 (0.20)
			0.5	0.5	0.12 (0.01)	0.12 (0.01)	0.18 (0.01)	0.22 (0.06)	0.28 (0.25)
		200	0.1	0.1	0.17 (0.01)	0.28 (0.01)	0.34 (0.01)	0.97 (0.06)	0.40 (0.19)

Appendix

Table 13

	$r_{J}$	$P_{2}$	$R_{J1}^{2}$	$R_{J2}^{2}$	ProJIVE	AJIVE	R.JIVE	GIPCA	D-CCA
					Mean (SD)
Joint Loadings $X_{1}$	1	20	0.1	0.1	0.10 (0.02)	0.10 (0.02)	0.11 (0.03)	0.48 (0.30)	0.10 (0.02)
			0.1	0.5	0.09 (0.02)	0.09 (0.02)	0.13 (0.07)	0.11 (0.10)	0.10 (0.02)
			0.5	0.1	0.03 (0.01)	0.03 (0.01)	0.04 (0.01)	0.12 (0.08)	0.03 (0.01)
			0.5	0.5	0.04 (0.05)	0.03 (0.01)	0.03 (0.01)	0.05 (0.04)	0.03 (0.01)
		200	0.1	0.1	0.09 (0.02)	0.09 (0.02)	0.09 (0.02)	0.50 (0.26)	0.10 (0.02)

Appendix

Table 14

	$r_{J}$	$P_{2}$	$R_{J1}^{2}$	$R_{J2}^{2}$	ProJIVE	AJIVE	R.JIVE	GIPCA	D-CCA
					Mean (SD)
Individual $X_{1}$ Scores	1	20	0.1	0.1	0.53 (0.02)	0.53 (0.02)	0.54 (0.02)	0.66 (0.11)	0.53 (0.02)
			0.1	0.5	0.53 (0.02)	0.53 (0.02)	0.58 (0.05)	0.54 (0.03)	0.53 (0.02)
			0.5	0.1	0.38 (0.02)	0.45 (0.06)	0.39 (0.02)	0.42 (0.06)	0.38 (0.02)
			0.5	0.5	0.38 (0.03)	0.38 (0.02)	0.38 (0.02)	0.39 (0.06)	0.38 (0.02)
		200	0.1	0.1	0.53 (0.02)	0.53 (0.02)	0.53 (0.02)	0.72 (0.05)	0.53 (0.02)

Appendix

Table 15

	$r_{J}$	$P_{2}$	$R_{J1}^{2}$	$R_{J2}^{2}$	ProJIVE	AJIVE	R.JIVE	GIPCA	D-CCA
					Mean (SD)
Individual $X_{2}$ Scores	1	20	0.1	0.1	0.53 (0.02)	0.54 (0.02)	0.54 (0.02)	0.67 (0.11)	0.54 (0.02)
			0.1	0.5	0.38 (0.02)	0.45 (0.07)	0.48 (0.09)	0.42 (0.06)	0.66 (0.06)
			0.5	0.1	0.53 (0.02)	0.53 (0.02)	0.53 (0.02)	0.54 (0.04)	0.53 (0.02)
			0.5	0.5	0.38 (0.03)	0.38 (0.03)	0.38 (0.02)	0.39 (0.05)	0.38 (0.02)
		200	0.1	0.1	0.20 (0.01)	0.20 (0.01)	0.21 (0.01)	0.70 (0.05)	0.20 (0.01)

Appendix

Table 16

	AD (N=88)	MCI (N=340)	CN (N=159)	Total (N=587)	p value
	Mean (S.D.) or N (%)
Age					0.006
	74.0 (7.92)	71.5 (7.57)	72.8 (5.85)	72.2 (7.25)
Gender					0.006
Female	28 (31.8%)	150 (44.1%)	84 (52.8%)	262 (44.6%)
ApoE4					<0.001

Appendix

Table S.1: Summary statistics for selected covariates of participants in ADNI-GO and ADNI2.

Cognition Score	Mean (SD)				p value
	AD (N=88)	MCI (N=340)	CN (N=159)	Total (N=587)
CDRSB	4.8 (1.84)	1.4 (0.97)	0.1 (0.33)	1.6 (1.79)	<0.001
ADAS11	20.4 (6.52)	9.1 (4.69)	5.5 (2.76)	9.8 (6.60)
ADAS13	30.8 (8.06)	14.3 (7.03)	8.5 (4.35)	15.2 (9.63)	<0.001
MMSE					<0.001
MMSE	23.1 (3.05)	27.8 (2.00)	29.0 (1.14)	27.4 (2.74)	<0.001

Appendix

Table S.2: Summary statistics for Cognition Measures.

Figure S.1: Scree plots shows which values were chosen as total signal ranks. Choices depended the ‘elbow’ of the scree plot or by accounting for at least 80%, 90% ,or 95% of total variance.

Figure S.2:

90^{th}

percentile of the absolute value of loadings onto each of the four components that compose the individual cognition subspace estimated via ProJIVE in the setting

K=2

(combining all brain morphometry measures as one dataset).

Figure S.3:

90^{th}

percentile of the absolute value of loadings onto each of the nine components that compose the individual morphometry subspace estimated via ProJIVE in the setting

K=2

Figure S.4: Subject scores from each component of the individual cognition subspace estimated via ProJIVE in the setting

K=2

Figure S.5: Subject scores from each component of the individual brain morphometry subspace estimated via ProJIVE in the setting

K=2

Figure S.6: Ten most extreme joint cognition loadings, six most extreme joint SVol loadings, and upper

85^{th}

percentile, respectively, of the absolute joint CT, CVol, and SA loadings estimated via ProJIVE in the setting

K=5

(treating each brain morphometry measure as a separate dataset).

Panel of Figure 25 — Figure S.7: Joint subject scores estimated via ProJIVE (for $K=5$ in which brain morphometry data are separated into four different datasets) show separation by (a) the count of ApoE4 allele counts and (b) diagnosis via raincloud plots. (c) Scatterplot of joint subject scores and AV45 at Baseline with OLS regression line and confidence interval. (d) Scatterplot of joint subject scores and FDG at Baseline with OLS regression line and confidence interval. Results resemble findings from $K=2$ in which all brain morphometry variables were combined into one dataset.

Figure S.8: subject scores individual cognition subspace estimated via ProJIVE in the setting

K=5

Panel of Figure 27 — Figure S.9: Variable loadings for the individual brain morphometry subspaces estimated via ProJIVE in the setting $K=5$ . (a) CT, (b) CVol, (c) SA, (d) SVol.

Latent Variable Models

Latent variable models infer hidden factors, classes, or states from observed data to explain structure, uncertainty, and variation.

See all 151 papers

A Quantitative Analysis of Multimodal Biomarkers in Alzheimer's Disease
Jun 16, 2026Antonio Scardace, Daniele RavìAlzheimer's Disease Disease Progression Modeling
Department of Mathematics and Computer Science University of Catania Catania, Italy · Department MIFT University of Messina Messina, Italy
CALM: Interpretable Cross-Modal Alignment for Biomarker Discovery from Unpaired Data
Jul 2, 2026Jueqi Wang, Zachary Jacokes, John Darrell Van Horn +3Cross-Modal Alignment Scientific Discovery
Department of Electrical and Computer Engineering, Boston University · School of Data Science, University of Virginia · Department of Psychology, University of Virginia +2
M $^2$ PFN: End-to-End Disentangled Alignment for Generalizable Multimodal In-Context Learning in Alzheimer's Disease
Sep 23, 2026Lujia Zhong, Shuo Huang, Jianwei Zhang +2Multimodal Disentangled Representation Learning Multimodal ICL
Stevens Neuroimaging and Informatics Institute, Keck School of Medicine, University of Southern California, Los Angeles, CA 90033, USA · Ming Hsieh Department of Electrical and Computer Engineering, Viterbi School of Engineering, University of Southern California, Los Angeles, CA 90089, USA · Alfred E. Mann Department of Biomedical Engineering, Viterbi School of Engineering, University of Southern California, Los Angeles, CA 90089, USA

Jun 16, 2026J/K move · Enter open · S save

A Quantitative Analysis of Multimodal Biomarkers in Alzheimer's Disease

Antonio Scardace, Daniele Ravì

Department of Mathematics and Computer Science University of Catania Catania, Italy · Department MIFT University of Messina Messina, Italy

Alzheimer's Disease Disease Progression Modeling Neuroimaging

Open paper AskPDF arXiv

Despite increasing adoption of multimodal approaches in Alzheimer's Disease (AD) research -- aimed at integrating molecular, structural, clinical, and genetic biomarkers to enhance disease characterization -- the relationships among these modalities remain poorly understood. A systematic analysis of their dynamic interaction is essential for improving disease modeling, identifying redundant assessments, and reducing patient burden and acquisition costs. In this paper, we present a quantitative analysis of multimodal AD biomarkers by integrating tau-PET, structural MRI, cognitive scores (MMSE and CDR), and APOE4 data from 789 subjects drawn from the ADNI dataset. In our analyses, we (A) quantify cross-modal mutual information and explained variance to assess redundancy and predictive dependencies; (B) examine associations between tau topologies and structural atrophy across brain regions to select informative ROIs; (C) perform a statistical decomposition of the tau-cognition association into atrophy-related and atrophy-independent components; (D) and identify a dominant neurodegenerative trajectory that aligns with cognitive decline. This study provides a systematic characterization of cross-modal relationships, improving the interpretability and selection of biomarkers in AD. Code is publicly available at: https://github.com/antonioscardace/Multimodal-AD.

Abstract

Latent Variable Models

Explore similar work

A Quantitative Analysis of Multimodal Biomarkers in Alzheimer's Disease

CALM: Interpretable Cross-Modal Alignment for Biomarker Discovery from Unpaired Data

M2^22PFN: End-to-End Disentangled Alignment for Generalizable Multimodal In-Context Learning in Alzheimer's Disease

M $^2$ PFN: End-to-End Disentangled Alignment for Generalizable Multimodal In-Context Learning in Alzheimer's Disease