Overview of Multiple Factor Analysis

Multiple factor analysis (MFA) represents an extension of principal component analysis for the case where multiple quantitative data tables are to be simultaneously analyzed. In particular, the MFA approach weights each table individually to ensure that tables with more features or those on a different scale do not dominate the analysis; all features within a given table are given the same weight. These weights are chosen such that the first eigenvalue of a PCA performed on each weighted table is equal to 1, ensuring that all tables play an equal role in the global multi-table analysis. According to the desired focus of the analysis, data can be structured either with molecular assays (e.g., RNA-seq, methylation, miRNA-seq, copy number alterations) as tables (and genes as features within omics), or with genes as tables (and molecular assays as features within genes). The MFA weights balance the contributions of each omic or of each gene, respectively. In padma, we focus on the latter strategy in order to allow different omics to contribute to a varying degree depending on the chosen pathway.

Figure 1 from Rau et al. (2019), illustrating the padma approach.

More precisely, consider a pathway or gene set composed of p genes (Figure 1A), each of which is measured using up to k molecular assays (e.g., RNA-seq, methylation, miRNA-seq, copy number alterations), contained in the set of gene-specific matrices X₁, …, X_p that have the same n matched individuals (rows) and j₁, …, j_p potentially unmatched variables (columns) in each, where j_g ∈ {1, …, k} for each gene g = 1, …, p. Because only the observations and not the variables are matched across data tables, genes may be represented by potentially different subset of omics data (e.g., only expression data for one gene, and expression and methylation data for another).

In the first step, these data tables are generally standardized (i.e., centered and scaled). Next, an individual PCA is performed using singular value decomposition for each gene table X_g, and its largest singular value λ_g¹ is calculated (Figure 1B). Then, all features in each gene table X_g are weighted by 1/λ_g¹, and a global PCA is performed using a singular value decomposition on the concatenated set of weighted standardized tables, $X^\ast = \left[ \frac{X_1}{\lambda_1^1}, \ldots, \frac{X_p}{\lambda_p^1}\right]$ (Figure 1C). This yields a matrix of components (i.e., latent variables) in the observation and variable space.

The MFA thus provides a consensus across-gene representation of the individuals for a given pathway, and the global PCA performed on the weighted gene tables decomposes the consensus variance into orthogonal variables (i.e., principal components) that are ordered by the proportion of variance explained by each. The coordinates of each individual on these components, also referred to as factor scores, can be used to produce factor maps to represent individuals in this consensus space such that smaller distances reflect greater similarities among individuals. In addition, partial factor scores, which represent the position of individuals in the consensus for a given gene, can also be represented in the consensus factor map; the average of partial factor scores across all dimensions and genes for a given individual corresponds to the factor score (Figure 1D).

Calculation of individualized pathway deviation scores

In the consensus space obtained from the MFA, the origin represents the ``average” pathway behavior across genes, omics, and individuals; individuals that are projected to increasingly distant points in the factor map represent those with increasingly aberrant values, with respect to this average, for one or more of the omics measures for one or more genes in the pathway. To quantify these aberrant individuals, we propose an individualized pathway deviation score d_i based on the multidimensional Euclidean distance of the MFA component loadings for each individual to the origin: where f_i, ℓ corresponds to the MFA factor score of individual i in component ℓ, and L corresponds to the rank of X^*. Note that this corresponds to the weighted Euclidean distance of the scaled multi-omic data (for the genes in a given pathway) of each individual to the origin. These individualized pathway deviation scores are thus nonnegative, where smaller values represent individuals for whom the average multi-omic pathway variation is close to the average, while larger scores represent individuals with increasingly aberrant multi-omic pathway variation with respect to the average. An individual with a large pathway deviation score is thus characterized by one or more genes, with one or more omic measures, that explain a large proportion of the global correlated information across the full pathway.

Calculation of individualized per-gene deviation scores

In order to quantify the role played by each gene for each individual, we decompose the individualized pathway deviation scores into gene-level contributions. Recall that the average of partial factor scores across all MFA dimensions corresponds to each individual’s factor score. We define the gene-level deviation for a given individual as follows: where as before f_i, ℓ corresponds to the MFA factor score of individual i in component ℓ, L corresponds to the rank of X^*, and f_{i, ℓ, g} corresponds to the MFA partial factor score of individual i in gene g in component ℓ. Note that by construction, the contributions of all pathway genes to the overall deviation score sum to 0. In particular, per-gene contributions can take on both negative and positive values according to the extent to which the gene influences the deviation of the overall pathway score from the origin (i.e., the global center of gravity across individuals); large positive values correspond to tables with a large influence on the overall deviation of an individual, while large negative values correspond to genes that tend to be most similar to the global average.

LUAD_subset: D4-GDI signaling pathway in the TCGA-LUAD multi-omic data

In this vignette, we focus on the example of multi-omic (RNA-seq, methylation, copy number alterations, and miRNA-seq) data for the D4-GDI signaling pathway from individuals with lung adenocarcinoma (LUAD) from The Cancer Genome Atlas (TCGA) database. The multi-omic TCGA data were downloaded and processed as previously described, including batch correction for the plate effect. The total number of individuals considered here is n = 144. See Rau et al. (2019) for additional details about data processing and mapping of miRNAs to genes.

The D4-GDP dissociation inhibitor (GDI) signaling pathway is made up of 13 genes; it was chosen for follow-up here as it was identified as having individualized pathway deviation scores that are significantly positively correlated with smaller progression-free intervals. RNA-seq, methylation, and CNA measures are available for all 13 genes, with the exception of CYCS and PARP1, for which no methylation probes were measured the promoter region. In addition, miRNA-seq data were included for one predicted target pair: hsa-mir-421 → CASP3.

The available multi-omic data for the D4-GDI signaling pathway in TCGA LUAD patients is included within the padma package as the LUAD_subset object, which is a named list of data.frames each containing the relevant data (clinical parameters, RNA-seq, methylation, miRNA-seq, CNA) for the p = 13 genes of the D4-GDI pathway in the n = 144 individuals.

names(LUAD_subset)
#> [1] "clinical" "rnaseq"   "methyl"   "mirna"    "cna"

lapply(LUAD_subset, class)
#> $clinical
#> [1] "data.frame"
#> 
#> $rnaseq
#> [1] "data.frame"
#> 
#> $methyl
#> [1] "data.frame"
#> 
#> $mirna
#> [1] "data.frame"
#> 
#> $cna
#> [1] "data.frame"

lapply(LUAD_subset, dim)
#> $clinical
#> [1] 144  55
#> 
#> $rnaseq
#> [1]  13 144
#> 
#> $methyl
#> [1]  13 144
#> 
#> $mirna
#> [1]   1 144
#> 
#> $cna
#> [1]  13 144

There are some important data formatting issues that are worth mentioning here:

• The clinical data (LUAD_subset$clinical) have individuals as rows and clinical variables as columns; all other datasets contain assay data with biological entities (e.g., genes, miRNAs) as rows and individuals as columns, with appropriate row and column names. Sample (column) names for assay data correspond to the patient barcodes that are provided in the bcr_patient_barcode column of LUAD_subset$clinical. For LUAD_subset$mirna, the first two columns represent the miRNA ID (miRNA_lc) and corresponding targeted gene symbol (gene).

• A gene_map data.frame can be optionally included to map biological entities (e.g. miRNAs) to gene symbols as needed. By default, a set of 10,754 predicted miRNA gene targets with “Functional MTI” support type from miRTarBase (version 7.0) are used for the gene_map.

• All omics data have already been appropriately pre-processed before subsetting to this pathway (e.g., RNA-seq and miRNA-seq data have been normalized and log-transformed, all datasets were batch-corrected for the plate effect).

• Although padma does require that all datasets have the same number of individuals, and that all datasets are sorted so that individuals are in the same order, each dataset does not need to represent every considered gene. Note that in these data, a single gene (CASP3) is predicted to be targeted by a miRNA (hsa-mir-421).

• Finally, these toy data have been subsetted to include only values relevant to the D4-GDI signaling pathway solely in the interest of space. In practice, users do not need to subset their data prior to input in padma or match miRNAs to genes, as both of these steps are performed automatically for a ser-provided pathway name or set of genes.

msigdb: the MSigDB canonical pathway gene sets

In our work, we considered the pathways included in the MSigDB canonical
pathways curated gene set catalog, which includes genes whose products are involved in metabolic and signaling pathways reported in curated public databases. We specifically used the 1322 “C2 curated gene sets” catalog from 601MSigDB v5.2 available at http://bioinf.wehi.edu.au/software/MSigDB/as described in the limma Bioconductor package. For convenience, the gene sets corresponding to each pathway have been included in the msigdb data frame:

head(msigdb)
#>                         geneset
#> 1   c2_cp_BIOCARTA_41BB_PATHWAY
#> 2   c2_cp_BIOCARTA_ACE2_PATHWAY
#> 3    c2_cp_BIOCARTA_ACH_PATHWAY
#> 4 c2_cp_BIOCARTA_ACTINY_PATHWAY
#> 5  c2_cp_BIOCARTA_AGPCR_PATHWAY
#> 6    c2_cp_BIOCARTA_AGR_PATHWAY
#>                                                                                                                                                                                                                            symbol
#> 1                                                                                                              JUN, MAP3K5, MAPK8, CHUK, MAPK14, MAP3K1, RELA, TNFRSF9, IKBKB, MAP4K5, IFNG, TRAF2, ATF2, IL4, NFKB1, NFKBIA, IL2
#> 2                                                                                                                                         ACE2, COL4A1, COL4A2, REN, COL4A3, COL4A4, CMA1, AGT, AGTR1, ACE, AGTR2, COL4A6, COL4A5
#> 3                                                                                                                      SRC, PTK2, TERT, CHRNG, FOXO3, RAPSN, AKT1, PIK3CA, MUSK, FASLG, CHRNB1, YWHAH, BAD, PTK2B, PIK3CG, PIK3R1
#> 4                                                                                         ABI2, WASF2, WASL, ACTA1, PIR, WASF1, ARPC4, PSMA7, ARPC5, ARPC1A, ACTR2, ACTR3, ARPC1B, ARPC3, NCKAP1, WASF3, ARPC2, RAC1, NTRK1, NCK1
#> 5                                                                                                                                PRKCB, PRKCA, PRKAR2B, PRKACB, PRKACG, GNAS, ARRB1, GNGT1, GNB1, PRKAR1A, PRKAR2A, PRKAR1B, GRK4
#> 6 JUN, MAPK8, PAK3, ACTA1, EGFR, NRG1, RAPSN, DAG1, CHRM1, PAK2, LAMA3, LAMA2, PAK4, UTRN, PXN, NRG3, PAK7, DMD, CTTN, SRC, PTK2, ITGA1, CHRNA1, PAK1, SP1, GIT2, DVL1, NRG2, MUSK, ITGB1, MAPK3, LAMA4, RAC1, PAK6, CDC42, MAPK1

In practice, to use padma a user can either directly provide a vector of gene symbols, or the geneset name as defined in this data frame (e.g., c2_cp_BIOCARTA_41BB_PATHWAY).

mirtarbase: predicted gene targets of miRNAs

padma integrates multi-omic data by mapping omics measures to genes in a given pathway. Although this assignment of values to genes is relatively straightforward for RNA-seq, CNA, and methylation data, a definitive mapping of miRNA-to-gene relationships does not exist, as miRNAs can each potentially target multiple genes. Many methods and databases based on text-mining or bioinformatics-driven approaches exist to predict miRNA-target pairs. Here, we make use of the curated miR-target interaction (MTI) predictions in miRTarBase (version 7.0),
using only exact matches for miRNA IDs and target gene symbols and predictions with the “Functional MTI” support type. As a reference, we have included these predictions in the built-in mirtarbase data frame:

head(mirtarbase)
#>             miRNA Target Gene
#> 1  hsa-miR-20a-5p       HIF1A
#> 2  hsa-miR-20a-5p       HIF1A
#> 3 hsa-miR-146a-5p       CXCR4
#> 4 hsa-miR-146a-5p       CXCR4
#> 5  hsa-miR-122-5p      CYP7A1
#> 6  hsa-miR-222-3p      STAT5A

In practice, unless an alternative data frame is provided for the gene_map argument of padma, the mirtarbase data frame is used to match miRNA IDs to gene symbols.

Numerical outputs

The numerical outputs of padma include the following (which can all be accessed using the matching accessor functions):

There are two primary numerical outputs from padma:

• pathway deviation scores: accessed via assay(...),
  a data.frame providing the individualized pathway deviation scores for each individual. The group column indicates whether an individual was included as part of the reference population ("Base") or as a supplementary individual projected onto the reference ("Supp"), as indicated by the user in the base_ids and supp_ids arguments of padma.
• pathway gene-specific deviation scores: accessed via pathway_gene_deviation(...), a matrix of dimension n (nuumber of individuals) x p (number of genes in the pathway) containing the individualized per-gene deviation scores

There are also several outputs related to the MFA itself, and obtained via the FactorMineR R package implementation of the MFA:

• MFA_results: accessed via the MFA_results(...), a named list containing detailed MFA output. When padma is run with full_results = TRUE, these include eigenvalues (eig), the percent contributions of each individual, gene, and omic to the MFA components (ind_contrib_MFA, gene_contrib_MFA, omics_contrib_MFA), the formatted gene tables used as input for the MFA (gene_tables), the pairwise Lg coefficients between genes (gene_Lf_MFA), and the full FactoMineR MFA output (total_MFA), which is an object of class "MFA". See the FactoMineR reference manual for a full description of this full MFA output. When padma is run with full_results = FALSE, these include eigenvalues (eig), and a summary of the cumulative contributions of each individual, gene, and omic to the first ten and full set of MFA components (ind_contrib_MFA_summary, gene_contrib_MFA_summary, omics_contrib_MFA_summary). In both cases, eig is a matrix of dimension c (number of MFA components) x 3 providing the respective eigenvalues, percentage of variance explained, and cumulative percentage of variance explained by each MFA component.

Finally, there are additional outputs related to the pathway itself that can be accessed via appropriate accessor functions:

• pathway_name: accessed via pathway_name(...), the user-provided name of pathway analyzed by padma. If a vector of gene symbols was provided rather than a built-in name, "custom" is returned.
  
• ngenes: accessed via ngenes(...), number of genes with data available in the pathway
  
• imputed_genes: accessed via imputed_genes(...), names of entities for which values were automatically imputed by padma
  
• removed_genes: accessed via removed_genes(...), names of entities which were automatically removed by padma due to small variance (< 10e-5) or all missing values across samples.

Graphical outputs

padma includes three primary plotting functions, all of which can be produced using ggplot2 (default) or base graphics (using argument ggplot = FALSE). Note that when ggplot2 is used, ggrepel is additionally used to better position sample labels.

• factorMap: plot the MFA components for each individual for a selected pair of components. This would typically be done for the first few components, and serves as a visualization of the structuring of indviduals for the pathway; for example, below the factor map suggests that individual TCGA-78-7536 is likely to have a larger overall pathway deviation score given that it is positioned quite far from the origin (for these two axes).

factorMap(run_padma, dim_x = 1, dim_y = 2)

The base version of this plot may be produced as follows.

factorMap(run_padma, dim_x = 1, dim_y = 2, ggplot = FALSE)
#> Loading required namespace: car

• factorMap (partial mode): plot the MFA partial components for one specific individual for a pair of components. This function would be used to zero in on one or several indivduals (for example, TCGA-78-7536) to visualize which genes appear to be pulling the overall score away from the origin. Here we note the seeming large influence of CASP1 and CASP3. Note that the individual’s overall pathway score, which is represented as the large black point, is the same as that from the previous plot.

factorMap(run_padma,
           partial_id = "TCGA-78-7536",
           dim_x = 1, dim_y = 2)

As before, ggplot2 plotting can be optionally turned off, if desired.

factorMap(run_padma,
           partial_id = "TCGA-78-7536",
           dim_x = 1, dim_y = 2, ggplot = FALSE)

• omicsContrib: visualize the contribution of each omics to the variation of a subset of MFA components, and summarize this contribution as a weighted overall average. In this plot, transparency of the bars reflects the percentage variance explained for each component. Note that if ggplot2 is used, the cowplot package is used to combine graphics.

omicsContrib(run_padma, max_dim = 10)
#> Loading required namespace: cowplot

Once again, ggplot2 plotting can be turned off if desired.

omicsContrib(run_padma, max_dim = 10, ggplot = FALSE)

Additional options

run_padma_supp <- 
  padma(mae, pathway_name = D4GDI, verbose = FALSE,
        base_ids = sampleMap(mae)$primary[1:10],
        supp_ids = sampleMap(mae)$primary[15:20])

run_padma_impute <- 
  padma(mae, pathway_name = D4GDI, 
        impute = TRUE, verbose = FALSE)

run_padma_concise <- 
  padma(mae, pathway_name = D4GDI, full_results = FALSE)