Abstract
Motivation
With the rapid development of modern technologies, massive data are available for the systematic study of Alzheimer’s disease (AD). Though many existing AD studies mainly focus on single-modality omics data, multi-omics datasets can provide a more comprehensive understanding of AD. To bridge this gap, we proposed a novel structural Bayesian factor analysis framework (SBFA) to extract the information shared by multi-omics data through the aggregation of genotyping data, gene expression data, neuroimaging phenotypes and prior biological network knowledge. Our approach can extract common information shared by different modalities and encourage biologically related features to be selected, guiding future AD research in a biologically meaningful way.
Method
Our SBFA model decomposes the mean parameters of the data into a sparse factor loading matrix and a factor matrix, where the factor matrix represents the common information extracted from multi-omics and imaging data. Our framework is designed to incorporate prior biological network information. Our simulation study demonstrated that our proposed SBFA framework could achieve the best performance compared with the other state-of-the-art factor-analysis-based integrative analysis methods.
Results
We apply our proposed SBFA model together with several state-of-the-art factor analysis models to extract the latent common information from genotyping, gene expression and brain imaging data simultaneously from the ADNI biobank database. The latent information is then used to predict the functional activities questionnaire score, an important measurement for diagnosis of AD quantifying subjects’ abilities in daily life. Our SBFA model shows the best prediction performance compared with the other factor analysis models.
Availability
Code are publicly available at https://github.com/JingxuanBao/SBFA.
Contact
qlong@upenn.edu
Keywords: structural Bayesian factor analysis, multi-omics, Alzheimer’s disease, biological network
Introduction
With the rapid advances in modern biotechnologies such as high-throughput genotyping and sequencing, and multimodal neuroimaging, researchers can investigate disease mechanisms from multiple perspectives simultaneously. Multi-omics data integration enables us to discover new insights into biological mechanism of complex diseases such as cancer and Alzheimer’s disease (AD) [1].
AD diagnosis prediction or AD-related cognitive score prediction can help uncover novel therapeutic targets for AD. Many studies seek to predict the AD diagnosis or cognitive scores using only imaging data [2–8] or only genetics data [9]. However, due to the pathological heterogeneity of AD [10, 11], overly relying on one type of data may undermine the power of analyses. To boost the power of the existing AD research, studying AD from both imaging and genetics/genomics perspectives may improve prediction accuracy and reduce false negative signals [12–17]. In our study, we proposed a novel structural Bayesian factor analysis (SBFA) framework to extract the common information shared by both imaging and genetics/genomics information, which can be used to guide future AD research in a biologically meaningful way.
Factor analysis is a popular statistical method for integrative analysis of multi-modal data. The factor analysis aims to describe variability among high-dimensional variables in terms of a substantially lower number of unobserved variables called latent factors. Many state-of-the-art factor-analysis-based methods have been proposed and applied to multi-omics data [18], for example to integratively define the subtypes of diseases such as breast cancer [19–23] and lung cancer [19, 20]. The iCluster [19] is a widely used factor analysis framework based on the joint latent variable model. It can flexibly model different types of genomics data in a single framework. It successfully models the tumor subtypes as latent variables integrating the information from multiple datasets. Later, Mo et al. [24] extended the iCluster to the iCluster+ framework by allowing the simultaneous integration of continuous and discrete datasets. Another highly cited integrative factor analysis method based on the non-negative matrix factorization is joint and individual variation explained (JIVE) factor analysis [25], which extends iCluster by adding a data-specific term [21]. A recently published integrative factor analysis method, structural learning and integrative decomposition of multi-view data (SLIDE) [26], represents each data view via shared and individual components. SLIDE has been successfully applied to multiple applications including exploratory dimension reduction, association analysis and consensus clustering.
There has been limited work on integrative analysis of multi-omics and imaging data in a principled modeling framework in AD studies. Some existing AD studies applied factor analysis models trying to define AD subtypes solely in terms of neuropathology [27, 28], molecular signatures [29, 30] or solely through imaging phenotype [31, 32]. Additionally, none of these studies takes prior biological network information into consideration. Network information encourages the selection of related features (e.g. genes belonging to the same pathway) and produces a more biologically interpretable result.
In AD studies, the interactions or associations within or across molecular pathways can be treated as network information. Many databases can provide pathway-level interactions and associations such as Integrative Multi-species Prediction [33] and Kyoto Encyclopedia of Genes and Genomes [34–36]. Some databases can provide tissue-specific interactions and associations such as the HumanBase [37–40]. Several existing studies achieved the incorporation of the biological network information using network-based penalties [41, 42], or proper priors in the Bayesian statistics framework [43–46]. One factor-analysis-based Bayesian integrative study of multi-omics data proposed a generalized Bayesian factor analysis (GBFA) framework by extending the iCluster+ framework [47]. They incorporate the prior biological network information by employing the Markov random field prior. However, one limitation of the GBFA framework is the phase transition problem [48, 49], which may require a massive hyperparameter tuning process.
In this paper, we propose an SBFA model that assigns the Laplace prior to the loading matrix for feature selection. It also assigns a log-normal prior to the shrinkage parameter in the Laplace prior and assigns a graph Laplacian prior [50, 51] to the precision matrix in the log-normal prior. Our model formulation enables smoothing shrinkage parameters for variables that are connected in the biological graph. Our major contributions are summarized as follows:
We propose a novel structural Bayesian factor analysis framework as a dimension reduction technique that yields latent factors by integrating the genotyping data, gene expression data, region-level brain imaging amyloid deposition data and biological network information. Our SBFA framework is biologically interpretable. As such, the common information shared among multi-omics data learned from our model can be used for various subsequent analyses.
Our model can handle both continuous (e.g. gene expression and imaging phenotypic traits) and discrete [e.g. genotyping values for single nucleotide polymorphisms (SNPs)] data simultaneously. We successfully incorporate prior biological information without suffering the phase transition problem of the existing GBFA framework [47].
Our work bridges the gap that there are few multi-omics studies applied to AD. We extracted more biologically meaningful information from multiple modalities, and our framework achieves the best prediction accuracy for the subsequent machine learning prediction problems compared with the other state-of-the-art factor-analysis-based methods.
The rest of the paper is organized as follows. We introduce our proposed method including model formulation and computation in Section 2. We present and discuss the analyses of real data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) in Section 3. We present our simulation experiment results in Section 4 and conclude the paper with some discussion remarks in Section 5.
Methodology
We propose a hierarchical Structural Bayesian Factor Analysis model named SBFA (Figure 1). Our model serves as a dimension reduction technique to extract common information from multi-modal data.
Figure 1.
Structural Bayesian factor analysis framework. 
represents the number of multi-modal data sets with each dataset denoted as 
, 
. The concatenation of all 
data sets is denoted by 
. We assume both continuous and discrete underlying distribution for each dataset dominated by mean parameter 
, 
where 
is number of samples. 
represents the 
th column of factor matrix 
where 
. 
and 
represent the 
th row and 
th column of loading matrix 
and shrinkage parameter for Laplacian prior where 
and 
is the total number of features. 
represents the log-normal distribution. 
and 
denote the mean and precision matrix for log-normal prior, respectively.
Notation
All vectors and matrices in this paper are denoted by bold lowercase letters or bold Greek letters. We use unbold lowercase letters or Greek letters to denote the corresponding elements of the vector/matrix. The columns and rows of the matrix are indicated by the subscription. For example, 
where 
denotes the 
th element of vector 
of length 
; 
where 
and 
denotes the element at the 
th row and 
th column of the matrix 
with dimension 
; 
represents the 
th column of matrix 
and the 
represents the 
th row of the matrix 
.
Structural Bayesian factor analysis model
Suppose we have multi-modal data with 
modalities generated from 
different technologies. We denote each data modality by 
for 
, where 
can be any omics data such as genotyping data, gene expression data, DNA methylation data and metabolomics data. Additionally, it can be imaging phenotypic traits such as brain gray matter density, brain volume, protein aggregates and metabolism level in a specific area of the brain. We denote the vertical concatenation of the multi-omics data to be
 |
where 
and 
is number of features in dataset 
, 
representing the 
th feature, and 
representing the 
th subject.
We assume that each feature of the dataset 
has the same underlying distribution family 
governed by the mean parameter 
for each sample 
. Under the assumption that every subject is independent of each other and every omic dataset is independent of each other conditioning on 
, we can write the following likelihood function:
 |
Our SBFA framework can integrate multi-modal datasets with both continuous and discrete data types simultaneously, including Gaussian data, binomial data and negative binomial data.
-
Gaussian distribution:
Continuous
is assumed to be normally distributed with mean 
and precision 
,
where a gamma prior
will be assigned to 
. -
Binomial distribution:
If the random variable
follows a binomial distribution with parameters 
trials and success probability
we will use the logit model with probability mass function (pmf) shown as follows:
The parameter
(1) 
is determined by data. Bernoulli distribution is a special case for the binomial distribution with 
to be 1. -
Negative binomial distribution:
To account for over-dispersion in counting data, we accommodate the negative binomial distribution
where parameter
(2) 
is provided by data. The geometric distribution is a special case for the negative binomial distribution with 
to be 1.
The mean parameter 
are modeled by the generalized factor analysis framework as
 |
which, in matrix form, can be written as follows:
 |
where 
is the 
latent factor shared by all modalities; 
is the 
factor loading matrix; 
is the 
th row of 
and 
is the 
th column of 
; 
, and 
. Our goal is to estimate a biologically meaningful low-rank representation (
) of 
to reduce the dimension of original multi-modal data while preserving the common information shared by multiple modalities. Such dimension reduction can reduce computational expenses and prevent the potential overfitting issue.
represents a low-dimensional latent driver for all 
omics and imaging modalities and can be used for subsequent analysis in the applications on AD. For example, it can be used to cluster the subjects to define different AD subtypes. On the other hand, it can also serve as a predictor to predict the diagnosis-related cognitive traits such as the functional assessment questionnaire (FAQ) in the ADNI database, which will be described in Section 3.
Priors
Priors for W and Z
To obtain sparse estimates for 
, we employ the Laplace prior for 
 |
where 
is the shrinkage parameter. The prior distributions for the entries in 
are the standard Gaussian distribution
 |
Prior for 
: incorporating graphical knowledge
One key contribution of our work is to incorporate biological knowledge 
via a structured prior for the shrinkage parameter 
. In our setting, the graph information for each omics dataset 
is represented by 
, where 
denotes the nodes and 
denotes the edges. The presence of an edge indicates that the corresponding pair of nodes are correlated. We denote the 
as the adjacency matrix representation of 
. We combine the 
to get the graph information for the multiomics data 
by diagonally concatenating the adjacency matrix for each omics data 
. Now, the graph information of multiomics data can be represented by 
where 
with 
and 
with 
referring to the index of the 
th feature of modality 
in the concatenated graph 
. Such a setup can help us to achieve the goal that given independent latent factors, the only way a pair of variables can be correlated conditionally on the rest is that they must load at least one common factor. Mathematically, if 
and 
are connected in 
and 
for some 
, then 
is encouraged to be nonzero. To this end, we employed a log-normal prior to 
with graph Laplacian on the precision matrix
 |
(3) |
where scalar 
, the mean of 
, is a tuning parameter that controls the overall shrinkage level of 
. Scalar 
, variance of 
, controls the extent of individual shrinkage parameters 
adapted to 
and 
. If 
, we have 
and the prior reduces to the Bayesian lasso. As 
increases, 
are more adapted to 
and 
. In our study, we fix the relative variance, known as the adaptivity, to one to allow sufficient variability for 
for all shrinkage levels [47], i.e.
 |
or equivalently,
 |
The intuition behind the choice of log-normal prior is as follows: (i) we employ the log-normal distribution for 
to incorporate the graph information via correlation and (ii) the shrinkage parameter 
needs to be positive.
Prior for 
: incorporating biological knowledge
We use the graph Laplacian prior [45] for 
to incorporate the graph information 
, which is a symmetric and diagonally dominant matrix
 |
where the elements in matrix 
, 
, are assigned by the following prior:
 |
where 
is the Dirac delta function concentrated at 
and 
is the indicator function. 
represents all connected edges in prior biological graph information. In real data application, the graph information can be summarized from gene–gene interaction networks for genetics data and from brain functional pathway information for neuroimaging modality.
There are several mathematical intuitions behind the choice of 
. First of all, the diagonal dominance of the precision matrix 
ensures its positive definiteness. Second, to ensure the positiveness of 
on the connected pairs, we incorporate a gamma-like distribution on all the connected pairs and 
on those pairs without an edge. The overall tendency of positiveness of the variance-covariance matrix ensures that increasing 
for some 
encourages an increase of the corresponding correlation.
Model for heterogeneous data types
Our model can handle heterogeneous data types simultaneously. In our framework, we use the exponential family for modeling heterogeneous data types in 
. For example, gene expression and neuroimaging data can be modeled as continuous data with normal distribution. SNP data can be modeled as two Bernoulli random variables where each SNP feature can be represented by the presence of homozygous major allele and the presence of homozygous minor allele, respectively.
With three different underlying data distributions (Gaussian, binomial, negative binomial), we can integrate the discrete exponential family random variables by unifying the likelihood function via the following identity [52]:
 |
where 
and 
( Pólya-Gamma distribution). By applying the identity to the binomial (Eq. (1)) and negative binomial (Eq. (2)) likelihood functions, we obtain the unified likelihood function:
 |
where the unknown parameters are shown in Table 1. Now, the full posterior density can be written as
Table 1.
Parameters for unified likelihood.
| Data Type |
|
|
|
|
|---|---|---|---|---|
| Gaussian |
|
0 | NA |
|
| Binomial | 0 |
|
|
|
| Negative Binomial | 0 |
|
|
|
 |
where 
.
Feature representation
In our model, new feature representation is learned from genotyping, gene expression and imaging datasets with biological graph information. The high dimensional information is summarized into the low dimensional latent variable 
for subsequent analyses.
For example, our SBFA framework estimated low dimensional latent features, which can serve as a predictor for machine learning algorithms. For unsupervised learning such as clustering, the low dimensional features derived from our algorithm are considered to be less noisy. We can apply the existing clustering algorithm to cluster the subjects based on the new feature representation. Supervised learning such as prediction often suffers from overfitting with high dimensional input data. With the low dimension feature representation derived from our algorithm, we can reduce the risk of overfitting. Compared with the other dimension reduction techniques, our SBFA is more biologically interpretable.
Parameter estimation and computation
We estimate the factor loading matrix 
and the location parameter 
by finding the mode of their marginal posterior density, i.e. the maximum a posteriori (MAP) estimator. Specifically, denote the parameters to be estimated by 
and the nuisance parameters by 
. Then, the MAP estimator for 
and 
is defined as
 |
where 
is the (joint) posterior density. To find the MAP, we could utilize the expectation maximization (EM) algorithm [53], which iteratively maximizes conditional expectations as follows:
 |
where 
denotes the expectation with respect to the conditional distribution 
.
Unfortunately, the conditional expectation of 
in our model does not have a closed-form solution. Therefore, we choose to use the variational EM algorithm [54, 55], where the conditional distribution 
is approximated by a more tractable distribution family 
parameterized by 
. The variational parameter 
can be determined by minimizing the Kullback–Leibler (KL) divergence
 |
where 
denotes the expectation under 
. Thus, the variational EM is equivalent to maximizing
 |
which is called the evidence lower bound (ELBO) [55] as it always satisfies 
, with respect to 
and 
.
In our model, we use a product measure on 
as the variational (approximating) measure, i.e. 
where
 |
where 
’s, 
’s, 
’s 
’s, 
’s, 
’s and 
’s are variational parameters. Note that 
is called tilted Pólya-Gamma distribution.
E-step for 
:
We update the variational parameters 
and 
by matching the form of the equations between the expectation for log variational measure and the expectation for log posterior measure
 |
where 
and 
; 
is the mean of the variational measure for the 
th element of 
; 
is the 
th row and 
th column of the covariance matrix of the variational measure for 
.
We then update the expectation of the 
using the mean formula for gamma distribution by
 |
for 
and 
. Now, we have a graph Laplacian matrix
 |
E-Step for 
:
Since there exists an 
term in the log posterior distribution but not in the log variational measure, we cannot update the variational parameters by matching the form of equations as we did when updating the 
(E-step for 
). Instead, we need to maximize the ELBO function and the results give us that
 |
where 
, 
, 
represents the Lambert W function [56] and
 |
and 
is an operator taking the diagonal elements of a square matrix as a column vector; 
represents the 
th row and 
th column element of the matrix 
; 
represents the mean matrix for 
; 
represents the covariance matrix for 
th column of 
; 
represents the 
th element of the vector 
.
The update rule for the covariance matrix of 
, i.e., 
, is separated into two parts. In the first part, we update the off-diagonal element of the 
by
 |
(4) |
for each 
, where
 |
The 
operator defines the trace of the matrix, and the minus sign in the subscript represents the matrix with the corresponding row or column removed, or vector with the corresponding element removed.
In the second part, we update the diagonal element of the 
as follows:
 |
(5) |
for each 
, where the r-Lambert function 
is defined as the inverse of the function 
[57] and
 |
where 
is the inverse operator for a square matrix; 
is the 
th row and 
th column of the loading matrix 
. We iteratively update 
for each 
and 
.
E-step for 
:
We update the precision matrix of the 
th column of factor matrix 
, i.e. 
, by matching the form of equations between log posterior distribution and the log variational measure. For each 
, we have
 |
where 
is identity matrix; 
is the mean for parameter 
and 
is the 
th row of loading matrix 
. For each 
, we update the mean parameter for 
by
 |
E-step for 
:
By matching the form of expectation of log posterior and expectation of log variational measure, the variational parameter 
for each 
and 
for 
can be updated by
 |
where 
is the 
th row of loading matrix 
and 
is the variance-covariance matrix for variational measure of 
.
If 
is a discrete random variable, we have
 |
whereas if 
follows Gaussian distribution, we have
 |
M-step for 
:
We update the loading matrix row by row via maximizing the ELBO function with respect to 
for each 
. The optimization problem is solved by the dynamic weighted lasso algorithm [58].
M-step for 
:
The location parameter 
is updated by maximizing the ELBO function with respect to 
for each 
 |
Initialization
For the initialization, we define the matrix 
as an approximation of the mean matrix 
as follows [47]:
 |
The location vector 
can be initialized by a fixed pre-specified constant vector. If not, for each 
, 
can be initialized by the trimmed mean, median or mean of 
.
In our setting, we handle the problems by choosing 
and 
matrices close to orthogonal matrices via singular value decomposition. Specifically, we have
 |
where 
and 
are orthonormal matrices and 
is diagonal matrix.
Hyperparameter tunning
We used the Bayesian information criterion (BIC) to tune the hyperparameters. We select the hyperparameters with the lowest BIC criteria
 |
where 
is the estimated mean matrix of 
; 
is the number of subjects and 
denotes the degrees of freedom, which is the number of non-zero elements in estimated factor loading matrix. We try different sets of hyperparameters (
) in both simulation and real data analysis.
Analysis of ADNI Data
To demonstrate the usefulness of our proposed method in real data analysis, we aimed to predict the FAQ cognitive score using the latent factors learned through integrative analysis of the genotyping, gene expression, brain regional level amyloid deposition, gene-gene interaction network and brain functional network data from the ADNI biobank database. We also performed the same analysis using existing factor analysis models (JIVE [25], SLIDE [26], iCluster+ [24], GBFA [47]) as the benchmark.
Data description and data preprocessing
The genotyping data, gene expression data, demographic data and imaging data used in the preparation of this article were obtained from the ADNI database (adni.loni.usc.edu) [59–61]. The ADNI was launched in 2003 as a public–private partnership, led by Principal Investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging, positron emission tomography (PET), other biological markers and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early AD. Up-to-date information about the ADNI is available at www.adni-info.org.
After data preprocessing steps and the common subjects matching in each omics data, we have 604 subjects left with main characteristics summarized in Table 2.
Table 2.
Participant characteristics. Total number of subjects, age and sex are shown in this table. The mean 
sd for the age of all subjects within each diagnosis group is reported. The number of male/female subjects within each diagnosis group is also shown.
| Diagnosis | CN | EMCI | LMCI | AD | Overall |
|---|---|---|---|---|---|
| Number | 199 | 198 | 165 | 42 | 604 |
Age ( ) |
|
|
|
|
|
| Sex (M/F) |
|
|
|
|
|
Demographic data preprocessing and cognitive score
Our demographic data and the cognitive score are extracted from Quantitative Templates for the Progression of Alzheimer’s Disease project in the Alzheimer’s Disease Modeling Challenge (www.pi4cs.org/qt-pad-challenge). We use baseline age, gender, education and the presence of APOE4 as our covariates. A test of the subjects’ daily living activities, the FAQ [62], is extracted as our dependent variable of real data analysis. All subjects with missing values are excluded from our study.
Genotyping data preprocessing
For the genotyping data, we performed quality control (QC) using the following criteria: genotyping call rate > 95%, minor allele frequency > 5% and Hardy–Weinberg Equilibrium > 
. After the QC, rs429358 (APOE) SNP was added to the genotyping data. We prioritized the SNP markers based on the existing findings from the major AD GWAS articles [63–67]. Our raw data were obtained by additive recoding the number of minor alleles (–recode A in PLINK1.9 [68]). Given that we want to model our genotyping data as Bernoulli distribution, we further recoded the genotyping data by the presence of homozygous major allele and the presence of homozygous minor allele, which doubled the number of input features for the genotyping dataset.
Gene expression data preprocessing
Gene expression data were produced using Affymetrix Human Genome U219 Array (Affymetrix, Santa Clara, CA) for expression profiling where the blood sample extraction, hybridization and QC process could be found in [59]. In the raw data, one gene can correspond to multiple probes. In our real data analysis, we averaged all the probes within each gene across the subjects to obtain the corresponding gene expression. We prioritize the AD-related genes according to Disease Gene Network (DisGeNet) [69]. We extract all the genes that have gene–disease association scores [69] greater than 0.15.
Imaging data preprocessing
For the neuroimaging data, participants with complete [
F]florbetapir (AV45) PET data (measuring amyloid burden) were included in our analysis. The AV45 PET scans were registered to the Montreal Neurological Institute space, and the standard uptake value ratio was computed by intensity normalization using the cerebellar curs reference region. Region-of-interest-level (ROI-level) AV45 measures were extracted based on the Glasser atlas [70], where 360 ROI-level quantitative traits (QTs) were obtained by averaging all the voxel-level measures within each ROI.
Biological network
We download the tissue-specific gene–gene interaction network from GIANT [40] in HumanBase database (hb.flatironinstitute.org). The imaging network is obtained from Glasser et al. [70], which utilizes a supervised machine learning classifier to automatically delineate and identify each cortical area from its neighbors across a large majority of individual subjects based on multi-modal information.
Real data analysis
In our real data analysis, we evaluate our SBFA framework by predicting the FAQ score using our extracted latent factors, which can be considered as a low-dimensional representation of the multi-omics data sets. Our real data analysis pipeline is shown in Figure 2.
Figure 2.
Workflow for real data analysis. 
and 
are genotyping data encoded by the presence of homozygous major allele and the presence of homozygous alternative allele, respectively. 
and 
are normalized QTs representing the gene expression and neuroimaging data. Starting from the multi-modal data, different factor analysis methods are used to extract the latent factors. The number of latent dimensions is tuned either through the implemented function from the package or using BIC. After obtaining the latent factor, we use the latent factor as the input predictor to predict the FAQ score. Linear regression with lasso, ridge and elastic net regularizations are used. The raw data (transpose of multi-omics data), covariates and raw data plus covariates are also used for comparison purpose.
Our framework starts from the dimention reduction of multi-omics data including indicator variables for homozygous dominant genotyping data 
, indicator variables for homozygous recessive genotyping data 
, AV45 imaging measurements 
and gene expression 
. Specifically, 
and 
are binary matrices with dimension 
, the entries of which are assumed to follow the Bernoulli distribution. 
and 
are normalized QTs with dimension 
and 
, which are assumed to be normally distributed.
We estimate the latent factors from the multi-omics and imaging dataset using our proposed SBFA method, in comparison with several existing methods including iCluster+ [19, 71], JIVE [25], SLIDE [26] and GBFA [47]. For the hyperparameters, we fix 
and tune 
. The best tuning results are obtained by minimizing the BIC criteria, which is presented in Figure 2. The best-tuned 
is indicated by 
, which is the number of columns of the corresponding latent factors. For our proposed SBFA model, the best-tuned hyperparameter set is 
. We then apply linear regression with lasso, ridge and elastic net regularizations models to predict the FAQ score using the latent factors learned from the aforementioned methods as predictors. For comparison purposes, we also include three other approaches that use (i) the demographic covariates, (ii) the multi-omics and imaging data and (iii) the multi-omics and imaging data together with the demographic covariates as the input of machine learning models, respectively; see Table 3.
Table 3.
Real data performance. The averaged MSE between the predicted FAQ score and the true FAQ score is reported across 100 random splittings. All standard errors are rounded to 0 after preserving 3 digits. For comparison purposes, we also include three other approaches that use (A) the demographic covariates, (B) the multi-omics and imaging data and (C) the multi-omics and imaging data together with the demographic covariates as the input of machine learning models.
Feature Method |
Lasso | Ridge | ElasticNet | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Train | Validation | Test | Train | Validation | Test | Train | Validation | Test | |
| Raw data and clinical variables | |||||||||
| Demographic covariates | 0.626 | 0.955 | 0.953 | 0.592 | 0.934 | 0.949 | 0.596 | 0.938 | 0.945 |
| Multi-omics and imaging | 0.626 | 0.958 | 0.959 | 0.334 | 0.883 | 0.921 | 0.516 | 0.864 | 0.940 |
| Multi-omics, imaging and covariates | 0.622 | 0.957 | 0.956 | 0.333 | 0.886 | 0.933 | 0.517 | 0.864 | 0.940 |
| Latent factors | |||||||||
| JIVE | 0.638 | 0.964 | 0.963 | 0.638 | 0.964 | 0.963 | 0.638 | 0.964 | 0.963 |
| SLIDE | 0.638 | 0.964 | 0.963 | 0.478 | 0.920 | 0.908 | 0.638 | 0.963 | 0.963 |
| iCluster+ | 0.638 | 0.964 | 0.963 | 0.637 | 0.974 | 0.961 | 0.638 | 0.964 | 0.963 |
| GBFA | 0.528 | 0.894 | 0.918 | 0.452 | 0.803 | 0.911 | 0.483 | 0.842 | 0.918 |
| SBFA (proposed) | 0.538 | 0.896 | 0.893 | 0.454 | 0.850 | 0.881 | 0.466 | 0.853 | 0.895 |
For our training process, we first evenly split the subjects into five folds where 4-folds are treated as the training-validation set and 1-fold is treated as the testing set. Next, we apply 4-fold cross-validation in our training-validation set to select the best tuning hyperparameters according to the mean squared error (MSE). The best tuning models are obtained from the validation set and applied to the testing set to get the testing performance. We repeat the splitting and tuning process 100 times by randomly splitting the data 100 times with 100 different random seeds. We summarized our real data analysis results in Table 3. The real data analysis performance is reported using the mean of the training, validation and testing MSE across 100 random splits. The standard errors for the training and testing MSE are rounded to zero after preserving three digits.
As shown in Table 3, compared with the results using original multi-omics and imaging data and clinical variables, our SBFA method does a better job in mitigating the overfitting problem. In particular, the three prediction models trained using estimated factors from our SBFA method outperform the models trained using the original multi-omics, imaging and demographic variables, in spite of the fact that our factor analysis used only multi-omics and imaging data, not demographic variables. Compared with other existing factor analysis methods, our SBFA method yields the best performance (smallest MSE) across all comparisons (i.e. linear regression model with lasso, ridge and elastic net regularizations).
Simulation Study
We compared the performance of our proposed SBFA framework with other factor analysis frameworks including JIVE [25], SLIDE [26] and GBFA model with spike and slab prior and Markov random field prior [47] in the simulation study.
Simulation design
We evaluated our proposed SBFA framework in two different simulation scenarios—low dimensional scenario and high dimensional scenario with the number of features 
and 
, respectively. We used the same number of samples in both simulation scenarios with 
. To simulate multi-omics data, we used the following model:
 |
(6) |
where 
, 
,
and 
. We choose the ground truth latent variable 
and the number of datasets 
.
We design our ground-truth loading matrix 
as shown in Figure 3, where each row represents a pathway (
). The shaded areas are generated by continuous uniform distribution with 
and every non-zero element of the loading matrix 
has a probability of 
to flip its sign.
Figure 3.
Groundtruth setting for factor loading matrix 
. Numbers labeled on each column represent the latent variable. Numbers labeled on rows represent the number of rows from the first row to the corresponding row. All shaded area represents elements generated by 
with probability 
to be multiplied by 
. All the other area contains only zero elements.
The ground truth factor matrix 
is simply generated by
 |
where 
represents the zero vector, and 
represents the identity matrix.
The ground truth mean parameter for data 
, i.e. 
, is obtained by
 |
where we assume location parameter 
. For the first setting in each simulation scenario, we assume only Gaussian distribution with mean 
and standard deviation 
for all 
. For the second setting in each simulation scenario, we assume the first two omic datasets with 
features following Bernoulli distribution with each element having probability 
to be 
. It mimics the data structure for the allele homozygosity encoded genotyping dataset. For the second two omics datasets with 
features, we assume that they are generated by a binomial distribution with trail parameter independently sampled from the set 
. For each feature and each subject, we assume the success probability parameter 
. For the third two omics datasets with 
features, we assume that they are generated to mimic the gene expression and imaging data, which follows the Gaussian distribution with mean 
and standard deviation 
.
For the graph information, we evaluate our algorithm with 
different settings:
Within-pathway edges only: For the first graph information setting, we randomly generate
edges within each pathway and do not add across pathway edges (i.e. in total, we have 
within pathway edges for 
features). This is a simulation of an ideal case.Within-pathway edges and cross pathway edges: For the second graph information setting, besides the
within pathway edges generated in the first setting, we also randomly generate 
cross-pathway edges within each modality (i.e. in total, we have 
within pathway edges and 
cross-pathway edges for 
features).
The graph with within-pathway edges only represents the ideal case where the graphical structure coincides with the sparsity structure in 
, while the cross-pathway edges mimic the real data.
Results
In our simulation study, we seek to reconstruct the mean matrix from the extracted low-rank representation. The performance is evaluated by the relative reconstruction error (RRE) between estimated 
and true 
using Frobenius norm
 |
(7) |
where 
denotes the ground truth mean parameter; 
represents the estimated mean parameter.
To reduce the randomness of the performance, we generate 
different datasets and apply our proposed algorithm to obtain the new feature representation for each dataset. We fixed 
and tuned the 
parameters using BIC. We compare our SBFA framework with four different baseline factor analysis algorithms including JIVE, SLIDE, iCluster+ and GBFA.
We first evaluate our results by calculating the mean and the standard error of the RRE across 
simulation datasets. It is worth noting that the best-tuned 
based on BIC is always equal to or larger than the ground truth. For those 
greater than the ground truth, compared with the other baseline methods, the estimated loadings by SBFA for the extra latent dimension(s) are exactly 0 except for a small number of nonzero entries (supplementary information Figure 1). The best-tuned results for each method and each simulation setting are summarized in Tables 4 and 5. Our proposed SBFA model always outperforms the other baseline methods in both the Gaussian distribution setting (Table 4) and the mixture distribution setting (Table 5).
Figure e1.
Estimated factor loadings for the simulated dataset. The heatmap shows the factor loadings from iCluster, JIVE, SLIDE and GBFA without graph information (GBFA(NE)), GBFA with graph information (GBFA(E)), SBFA without graph information (SBFA(NE)), SBFA with graph information (SBFA(E)) and the ground truth. The results are derived from one simulated dataset in the low-dimensional scenario (
). The optimal latent dimension (
) for each model is indicated by the number of circles in the corresponding method track. Numbers beyond the range of (
) have the same color as the boundaries (
and 1).
Table 4.
Simulation performance results for Gaussian only setting. ‘Dimension’ denotes our simulation scenario where ‘Low’ indicates the simulation scenario with 
and ‘High’ indicates the simulation scenario with 
. ‘Edge’ represents different graph information where ‘Within’ denotes the ideal case that we use the graph information which only connects the features within each pathway, and ‘Cross’ denotes the case that we use the graph information that connects the features not only within each pathway but also some randomly selected nodes between pathways within each modality. ‘RRE (mean)’ is the average of the reconstruction error between estimated 
and the ground truth 
ratio to the true 
across 100 simulations. ‘RRE (se)’ is the standard error of the reconstruction error between estimated 
and the ground truth 
across 100 simulations.
| Method | Dimension | Edge | RRE (mean) | RRE (se) |
|---|---|---|---|---|
| JIVE | low | 0.236552 | 0.006583 | |
| SLIDE | low | 0.167271 | 0.000722 | |
| iCluster | low | 0.221568 | 0.000935 | |
GBFA ( ) |
low | Within | 0.198160 | 0.000848 |
GBFA ( ) |
low | Within | 0.190166 | 0.001088 |
SBFA (No  ) |
low | Within | 0.182882 | 0.000743 |
SBFA (With  ) |
low | Within | 0.180844 | 0.000699 |
GBFA ( ) |
low | Cross | 0.198160 | 0.000848 |
GBFA ( ) |
low | Cross | 0.185953 | 0.001015 |
SBFA (No  ) |
low | Cross | 0.182882 | 0.000743 |
SBFA (With  ) |
low | Cross | 0.181908 | 0.000704 |
| JIVE | high | 0.180514 | 0.001741 | |
| SLIDE | high | 0.154596 | 0.000727 | |
| iCluster | high | 0.188864 | 0.000806 | |
GBFA ( ) |
high | Within | 0.166760 | 0.000663 |
GBFA ( ) |
high | Within | 0.166246 | 0.000772 |
SBFA (No  ) |
high | Within | 0.153240 | 0.000686 |
SBFA (With  ) |
high | Within | 0.151277 | 0.000665 |
GBFA ( ) |
high | Cross | 0.166760 | 0.000663 |
GBFA ( ) |
high | Cross | 0.158246 | 0.000882 |
SBFA (No  ) |
high | Cross | 0.153240 | 0.000686 |
SBFA (With  ) |
high | Cross | 0.152455 | 0.000672 |
Table 5.
Simulation performance results for mixture distribution setting. ‘Dimension’ denotes our simulation scenario where ‘Low’ indicates the simulation scenario with 
and ‘High’ indicates the simulation scenario with 
. ‘Edge’ represents different graph information where ‘Within’ denotes the ideal case that we use the graph information which only connects the features within each pathway, and ‘Cross’ denotes the case that we use the graph information that connects the features not only within each pathway but also some randomly selected nodes between pathways within each modality. ‘RRE (mean)’ is the average of the reconstruction error between estimated 
and the ground truth 
ratio to the true 
across 100 simulations. ‘RRE (se)’ is the standard error of the reconstruction error between estimated 
and the ground truth 
across 100 simulations.
| Method | Dataset | Edge | RRE (mean) | RRE (se) |
|---|---|---|---|---|
| JIVE | low | 1.539543 | 0.014274 | |
| SLIDE | low | 1.519234 | 0.010254 | |
| iCluster | low | 1.407917 | 0.011129 | |
GBFA ( ) |
low | Within | 0.220563 | 0.000961 |
GBFA ( ) |
low | Within | 0.220918 | 0.000993 |
SBFA (No  ) |
low | Within | 0.197850 | 0.000877 |
SBFA (With  ) |
low | Within | 0.196208 | 0.000864 |
GBFA ( ) |
low | Cross | 0.220563 | 0.000961 |
GBFA ( ) |
low | Cross | 0.217769 | 0.001006 |
SBFA (No  ) |
low | Cross | 0.197850 | 0.000877 |
SBFA (With  ) |
low | Cross | 0.199828 | 0.000874 |
| JIVE | high | 1.551844 | 0.007424 | |
| SLIDE | high | 1.508521 | 0.006791 | |
| iCluster | high | 1.397968 | 0.007480 | |
GBFA ( ) |
high | Within | 0.190666 | 0.000886 |
GBFA ( ) |
high | Within | 0.188583 | 0.000733 |
SBFA (No  ) |
high | Within | 0.175308 | 0.000773 |
SBFA (With  ) |
high | Within | 0.173823 | 0.000733 |
GBFA ( ) |
high | Cross | 0.190666 | 0.000886 |
GBFA ( ) |
high | Cross | 0.189722 | 0.000890 |
SBFA (No  ) |
high | Cross | 0.175308 | 0.000773 |
SBFA (With  ) |
high | Cross | 0.177781 | 0.000764 |
In the Gaussian distribution setting (Table 4), without incorporating the graph information, our proposed model (SBFA No 
) shows significant improvement compared with JIVE, SLIDE, iCluster+ and GBFA without graph information (GBFA 
). By taking the graph information into consideration, our proposed model (SBFA With 
) can further achieve a smaller RRE and significantly outperforms the GBFA model (GBFA 
) both with and without cross pathway noise.
In the mixture distribution setting (Table 5), JIVE, SLIDE and iCluster yield a large RRE compared with GBFA and SBFA models. This is expected because those methods handle the discrete data by treating them as continuous random variables. Although the GBFA model and our proposed SBFA model can both handle discrete and continuous data simultaneously, our SBFA model outperforms the GBFA model in both low-dimensional simulation and high-dimensional simulation scenarios with or without incorporating the graph information.
In summary, our proposed SBFA model can handle discrete and continuous data simultaneously. Our simulation study demonstrates that our proposed SBFA model can outperform the most state-of-the-art factor analysis models. Moreover, our SBFA model can incorporate the biological graph information as prior knowledge so that our model can produce biologically interpretable results.
Conclusion
In this article, we proposed a structural Bayesian factor analysis model which decomposes the underlying mean parameters of the data into a sparse factor loading matrix and factor matrix. Our model employed the Laplace prior to achieve the sparse estimation of the loading matrix and used a graph Laplacian prior to incorporate the biological network information. Our method can handle both discrete and continuous data simultaneously. We derived an efficient variational EM algorithm to obtain the MAP estimators of the parameters of interest.
Our SBFA framework is a useful dimension reduction technique that yields latent factors that are biologically interpretable. As such, the common information shared among multi-modal data learned from our model can be used for various downstream machine learning tasks including, but not limited to, prediction. For example, another potential application is to use learned factors for clustering which can be used to define disease subtypes based on genotyping data, imaging modalities and biological pathways.
Our simulation study demonstrates that our proposed SBFA framework achieves the best performance compared with the other state-of-the-art factor-analysis-based integrative methods. Our work bridges the gap that there are few multi-omics studies applied to AD. We used the multi-omics data from the ADNI dataset to predict the FAQ score, which is an important evaluation for the diagnosis of AD. The real data analysis results show the advantages of our framework by comparing the prediction performance between our SBFA framework and other state-of-the-art factor-analysis-based integrative methods. Our method exhibited the best prediction performance.
Key Points
We propose a novel structural Bayesian factor analysis framework as a dimension reduction technique that yields latent factors by integrating the genotyping data, gene expression data, region-level brain imaging amyloid deposition data and biological network information. Our SBFA framework is biologically interpretable. The common information shared among multi-omics data learned from our model can be used for various subsequent analyses such as supervised and unsupervised machine learning.
Our structural Bayesian factor analysis model can handle both continuous (e.g. gene expression and imaging phenotypic traits) and discrete [e.g. genotyping values for single nucleotide polymorphisms (SNPs)] datatypes simultaneously.
Our work bridges the gap that there are few multi-omics studies applied to Alzheimer’s disease. We applied our framework to extract biologically meaningful information shared by multiple modalities in the ADNI database. We performed the subsequent machine learning analyses to predict the Alzheimer’s disease-related cognitive score using the extracted latent factors. Our structural Bayesian factor analysis model achieves the best prediction accuracy for the subsequent machine learning prediction problems compared with the other state-of-the-art factor-analysis-based methods.
Acknowledgements
This study was partly supported by NIH grants RF1AG063481, R01AG071174, R01 AG071470, U01 AG066833 and U01 AG068057. Data used in this study were obtained from the ADNI database (adni.loni.usc.edu), which was funded by NIH U01AG024904.
Jingxuan Bao is a Ph.D. student co-advised by Dr. Li Shen and Dr. Qi Long at the University of Pennsylvania Perelman School of Medicine.
Changgee Chang is an assistant professor at Indiana University School of Medicine.
Qiyiwen Zhang is a postdoctoral fellow in Dr. Qi Long’s lab at the University of Pennsylvania Perelman School of Medicine.
Andrew J. Saykin is the director of the Center for Neuroimaging and Indiana Alzheimer's Disease Research Center. He is a professor at Indiana University School of Medicine.
Li Shen is a Professor of Informatics at the University of Pennsylvania Perelman School of Medicine.
Qi Long is a Professor of Biostatistics at the University of Pennsylvania.
Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (adni.loni.ucla.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: {http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf}.
Contributor Information
Jingxuan Bao, Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania Perelman School of Medicine, Philadelphia, 19104, PA, USA.
Changgee Chang, Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania Perelman School of Medicine, Philadelphia, 19104, PA, USA.
Qiyiwen Zhang, Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania Perelman School of Medicine, Philadelphia, 19104, PA, USA.
Andrew J Saykin, Department of Radiology and Imaging Sciences, Indiana University, Indianapolis, 46202, IN, USA.
Li Shen, Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania Perelman School of Medicine, Philadelphia, 19104, PA, USA.
Qi Long, Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania Perelman School of Medicine, Philadelphia, 19104, PA, USA.
Data availability
The source code is available through GitHub (https://github.com/JingxuanBao/SBFA). The genotyping data, gene expression data, demographic data, and imaging data used in the preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. The data are available after application through the ADNI website (http://adni.loni.usc.edu).
Author contributions statement
Jingxuan Bao and Dr. Changgee Chang take full responsibility for the integrity of the data and the accuracy of the data analysis.
Study concept and design: Jingxuan Bao, Dr. Changgee Chang and Dr. Qi Long.
Acquisition, analysis or interpretation of data: Jingxuan Bao, Dr. Changgee Chang, Dr. Andrew J. Saykin, Dr. Li Shen and Dr. Qi Long.
Drafting of the manuscript: Jingxuan Bao, Dr. Changgee Chang and Dr. Qi Long.
Critical revision of the manuscript for important intellectual content: All authors.
Study supervision: Dr. Changgee Chang and Dr. Qi Long.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The source code is available through GitHub (https://github.com/JingxuanBao/SBFA). The genotyping data, gene expression data, demographic data, and imaging data used in the preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. The data are available after application through the ADNI website (http://adni.loni.usc.edu).