Probabilistic Joint and Individual Variation Explained (ProJIVE) for Data Integration
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
| 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) |
| 0.5 | 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 | 0.12 (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) | ||
| 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) |
| 0.5 | 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 | 0.03 (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) | ||
| ProJIVE | AJIVE | R.JIVE | GIPCA | D-CCA | |||||
| Mean (SD) | |||||||||
| Individual 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) |
| 0.5 | 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 | 0.38 (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) | ||
| ProJIVE | AJIVE | R.JIVE | GIPCA | D-CCA | |||||
| Mean (SD) | |||||||||
| Individual 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) |
| 0.5 | 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 | 0.38 (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) | ||
| 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) |
| 0.5 | 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 | 0.12 (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) | ||
| ProJIVE | AJIVE | R.JIVE | GIPCA | D-CCA | |||||
| Mean (SD) | |||||||||
| Joint Loadings | 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) |
| 0.5 | 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 | 0.04 (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) | ||
| ProJIVE | AJIVE | R.JIVE | GIPCA | D-CCA | |||||
| Mean (SD) | |||||||||
| Individual 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) |
| 0.5 | 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 | 0.38 (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) | ||
| ProJIVE | AJIVE | R.JIVE | GIPCA | D-CCA | |||||
| Mean (SD) | |||||||||
| Individual 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) |
| 0.5 | 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 | 0.38 (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) | ||
| 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 | ||||
| AD (N=88) | MCI (N=340) | CN (N=159) | Total (N=587) | ||
|---|---|---|---|---|---|
| Cognition Score | Mean (SD) | p value | |||
| 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 |
Latent Variable Models
Latent variable models infer hidden factors, classes, or states from observed data to explain structure, uncertainty, and variation.
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