Author: Max MattessichJoaquin ReynaEdel AronAnna Konstorum

# Analysis of MCIA Decomposition

# Introduction

## Motivation

Multiple co-inertia analysis (MCIA) is a member of the family of joint dimensionality reduction (jDR) methods that extend unsupervised dimension reduction techniques such as Principal Components Analysis (PCA) and Non-negative Matrix Factorization (NMF) to datasets with multiple data blocks (alternatively called views) (Cantini, 2021).

Here, we present a new implementation in R of MCIA, `nipalsMCIA`, that uses an extension of Non-linear Iterative Partial Least Squares (NIPALS) to solve the MCIA optimization problem (Hanafi, 2011). This implementation has several features, including speed-up over approaches that employ the Singular Value Decomposition (SVD), several options for pre-processing and deflation to customize algorithm performance, methodology to perform out-of-sample global embedding, and analysis and visualization capabilities to maximize result interpretation.

While there exist additional implementations of MCIA (e.g. mogsa, omicade4), ours is unique in providing a pipeline that incorporates pre-processing data options including those present in the original development of MCIA (including a theoretically grounded calculation of inertia, or total variance) with an iterative solver that shows speed-up for larger datasets, and is explicitly designed for simultaneous ease of use as a tool for multi-view data decomposition as well as a foundation for theoretical and computational development of MCIA and related methodology. A manuscript detailing our implementation is forthcoming.

## Overview

In this vignette, we will cover the most important functions within the `nipalsMCIA` package as well as downstream analyses that can help interpret the MCIA decomposition using a cancer data set from Meng et al., 2016 that includes 21 subjects with three data blocks. The data blocks include mRNA levels (12895 features), microRNA levels (537 features) and protein levels (7016 features).

The `nipals_multiblock` function performs MCIA using the NIPALS algorithm. `nipals_multiblock` outputs a decomposition that includes a low-dimensional embedding of the data in the form of global scores, and the contributions of the data blocks (block score weights) and features (global loadings) to these same global scores. `nipalsMCIA` provides several additional functions to visualize, analyze, and interpret these results.

The `nipals_multiblock` function accepts as input a MultiAssayExperiment (MAE) object. Such objects represent a modern classed-based approach to organizing multi-omics data in which each assay can be stored as an individual experiment alongside relevant metadata for samples and experiments. If users have a list of data blocks with matching sample names (and optional sample-level metadata), we provide a simple conversion function (`simple_mae.R`) to generate an MAE object. For more sophisticated MAE object construction, please consult the MAE documentation.

In the context of the NCI-60 data set and this vignette, we will show you the power of MCIA to find important relationships between mRNA, microRNA and proteins. More specifically, we will show you how to interpret the global factor scores in Part 1: Interpreting Global Factor Scores and global loadings in Part 2: Interpreting Global Loadings.

## Installation

    # devel version

    # install.packages("devtools")
    devtools::install_github("Muunraker/nipalsMCIA", ref = "devel",
                             force = TRUE, build_vignettes = TRUE) # devel version

    # release version
    if (!require("BiocManager", quietly = TRUE))
      install.packages("BiocManager")

    BiocManager::install("nipalsMCIA")

    library(ComplexHeatmap)
    library(dplyr)
    library(fgsea)
    library(ggplot2)
    library(ggpubr)
    library(nipalsMCIA)
    library(stringr)

    # NIPALS starts with a random vector
    set.seed(42)

## Preview of the NCI-60 dataset

The NCI-60 data set has been included with the `nipalsMCIA` package and is easily available as shown below:

    # load the dataset which uses the name data_blocks
    data(NCI60)

    # examine the contents
    data_blocks$miRNA[1:3, 1:3]

    ##            MI0000060_miRNA MI0000061_miRNA MI0000061.1_miRNA
    ## CNS.SF_268           11.91            6.71             13.11
    ## CNS.SF_295           11.94            7.13             12.86
    ## CNS.SF_539           11.50            5.79             12.10

    data_blocks$mrna[1:3, 1:3]

    ##            5-HT3C2_1_mrna A1BG-AS1_2_mrna A2LD1_3_mrna
    ## CNS.SF_268           0.53            0.35        -0.05
    ## CNS.SF_295          -0.42            0.54        -1.04
    ## CNS.SF_539           0.00            0.80         0.85

    data_blocks$prot[1:3, 1:3]

    ##            STAU1_1_prot NRAS_2_prot HRAS_3_prot
    ## CNS.SF_268     5.712331    7.385177    5.758845
    ## CNS.SF_295     0.000000    6.327175    0.000000
    ## CNS.SF_539     0.000000    6.597432    0.000000

To convert data_blocks into an MAE object we provide the `simple_mae()` function:

    data_blocks_mae <- simple_mae(data_blocks, row_format = "sample")

# Running and reviewing the MCIA output

We can compute the MCIA decomposition for \(r\) global factors. For our example, we take \(r=10\).

    set.seed(42)
    mcia_results <- nipals_multiblock(data_blocks_mae,
                                      col_preproc_method = "colprofile",
                                      num_PCs = 10, tol = 1e-12, plots = "none")

The result is an NipalsResult object containing several outputs from the decomposition:

    slotNames(mcia_results)

    ##  [1] "global_scores"        "global_loadings"      "block_score_weights"
    ##  [4] "block_scores"         "block_loadings"       "eigvals"
    ##  [7] "col_preproc_method"   "block_preproc_method" "block_variances"
    ## [10] "metadata"

We describe the first two in more detail below, and will discuss several others in the remainder of the vignette. For additional details on the decomposition, see (Hanafi, 2011, Mattesich, 2022).

## Brief overview of the Global Scores Matrix (\(F\))

The `global_scores` matrix is represented by \(\mathbf{F}\) with dimensions \(n \times r\), where \(n\) is the number of samples and \(r\) is the number of factors chosen by using the `num_PCs = r` argument. Each column \(j\) of this matrix represents the global scores for factor \(j\),

\[ \mathbf{F} = \begin{pmatrix} | & |& & |\\ \mathbf{f}^{(1)} &\mathbf{f}^{(2)} & \dots & \mathbf{f}^{(r)}\\ | & |& & | \end{pmatrix} \in \mathbb{R}^{n \times r} \]

This matrix encodes a low-dimensional representation of the data set, with the \(i\)-th row representing a set of \(r\)-dimensional coordinates for the \(i\)-th sample.

## Brief overview of the Global Loadings Matrix (\(A\))

The `global_loadings` matrix is represented by \(\mathbf{A}\) that is \(p \times r\), where \(p\) is the number of features across all omics and \(r\) is as before. Each column \(j\) of this matrix represents the global loadings for factor \(j\), i.e.

\[ \mathbf{A} = \begin{pmatrix} | & |& & |\\ \mathbf{a}^{(1)} &\mathbf{a}^{(2)} & \dots & \mathbf{a}^{(r)}\\ | & |& & | \end{pmatrix} \in \mathbb{R}^{p \times r} \]

This matrix encodes the contribution (`loading`) of each feature to the global score.

The remainder of this vignette will be broken down into two sections, Part 1: Interpreting Global Factor Scores and Part 2: Interpreting Global Loadings where we show how to interpret \(\mathbf{F}\) and \(\mathbf{A}\), respectively.

# Part 1: Interpreting Global Factor Scores

## nipals_multiblock() Generates Basic Visualizations

In the introduction we showed how to calculate the MCIA decomposition using `nipals_multiblock()` but used the parameter `plots = "none"` to avoid the default plotting behavior of this function. By default, this function will generate two plots which help establish an initial intuition for the MCIA decomposition. Here we will re-run `nipals_multiblock()` with the default `plots` parameter (`all`):

    set.seed(42)
    mcia_results <- nipals_multiblock(data_blocks_mae,
                                      col_preproc_method = "colprofile",
                                      num_PCs = 10, tol = 1e-12)

The first plot visualizes each sample using factor 1 and 2 as a lower dimensional representation (factor plot).

- Each sample is represented by 4 points, a center point (solid bla’ck dot) which represents the global factor score, a mRNA factor score (square), a miRNA factor score (circle), and a protein factor score (triangle).
- The last three omic-specific block factor scores are connected to the global factor score. If a block factor scores is plotted far from its corresponding global factor score, then this is an indication that the block does not agree with/contribute to the trend found by the global decomposition.
- As an example, we can take a look at the global factor score at \((-1.1, 0.4)\). The block factor scores are all quite near which suggests all three omics are contributing somewhat equally. This is in contrast to the global factor score at \((-0.3, 0.9)\) where the mRNA factor score is close, but the mRNA and miRNA are far from their respective global factor score.

The second plot is a scree plot of normalized singular values corresponding to the variance explained by each factor.

## Visualizing a Factor Plot with Only Global Factor Scores

For clustering, it is useful to look at global factor scores without block factor scores. The `projection_plot()` function can be used to generate such a plot using `projection = "global"`.

    projection_plot(mcia_results, projection = "global", orders = c(1, 2))

In addition, scores can be colored by a meaningful label such as cancer type which is highly relevant to NCI-60. To do so, the `colData` slot of the associated MAE object must be loaded with sample-level metadata prior to invoking `projection_plot()`. The sample metadata is composed of row names corresponding to the primary sample names of the MAE object, and columns contain different metadata (e.g. age, disease status, etc). For instance, each of the 21 samples in the NCI-60 dataset represents a cell line with one of three cancer types: CNS, Leukemia, or Melanoma. We have provided this metadata as part of the `data(NCI60)` dataset and we next show how it can be included in the resulting MAE object using the `colData` parameter in `simple_mae()`:

    # preview of metadata
    head(metadata_NCI60)

    ##            cancerType
    ## CNS.SF_268        CNS
    ## CNS.SF_295        CNS
    ## CNS.SF_539        CNS
    ## CNS.SNB_19        CNS
    ## CNS.SNB_75        CNS
    ## CNS.U251          CNS

    # loading of mae with metadata
    data_blocks_mae <- simple_mae(data_blocks, row_format = "sample",
                                  colData = metadata_NCI60)

We now rerun `nipals_multiblock()` using the updated MAE object, where the `colData` is passed to the `metadata` slot of the `NipalsResult` instance, and then visualize the global factor scores using the `projection_plot()` function.

    # adding metadata as part of the nipals_multiblock() function
    set.seed(42)
    mcia_results <- nipals_multiblock(data_blocks_mae,
                                      col_preproc_method = "colprofile",
                                      plots = "none",
                                      num_PCs = 10, tol = 1e-12)

The `color_col` argument of `projection_plot()` can then be used to determine which column of `metadata` is used for coloring the individual data points, in this case `cancerType`. `color_pal` is used to assign a color palette and requires a vector of colors (i.e. `c("blue", "red", "green")`). To help create this vector we also provide `get_metadata_colors()`, a helper function (used below) that can be used with a `scales::<function>` to return an appropriate vector of colors. Note: colors are applied by lexicographically sorting the list of unique metadata values then assigning the first color to the first value, second with second and so on.

    # meta_colors = c("blue", "grey", "yellow") can use color names
    # meta_colors = c("#00204DFF", "#7C7B78FF", "#FFEA46FF") can use hex codes
    meta_colors <- get_metadata_colors(mcia_results, color_col = 1,
                                       color_pal_params = list(option = "E"))

    projection_plot(mcia_results, projection = "global", orders = c(1, 2),
                    color_col = "cancerType", color_pal = meta_colors,
                    legend_loc = "bottomleft")

Using this plot one can observe that global factor scores for factor 1 and 2 can separate samples into their cancer types.

## Visualizing the Clustering of Samples by Factor Scores

A heatmap can be used to cluster samples based on global scores across all factors using `global_scores_heatmap()`. The samples can be colored by the associated metadata using `color_cor` + `color_pal`as shown below.

    global_scores_heatmap(mcia_results = mcia_results,
                          color_col = "cancerType", color_pal = meta_colors)

Like the projection plot, one can observe that the clustering can differentiate between cancer type quite effectively.

# Part 2: Interpreting Global Loadings

In addition to the global scores matrix, MCIA also calculates a global loadings matrix \(A\) that is \(p\times r\).

## Pseudoeigenvalues Representing the Contribution of Each Omic to the Global Factor Score

The global loadings matrix is calculated using a weighted sum of the block loadings. These weights can be interpreted as the contribution of each omic to the global factor score (i.e. pseudoeigenvalue), and can be visualized using the function `block_weights_heatmap()`.

    block_weights_heatmap(mcia_results)

For example, we can observe that factors 7-9 have the strongest contribution from the protein data.

## Visualize All Feature Loadings on Two Axes

MCIA returns feature loadings for each factor that can be used to understand important contributions. To get a sense of which omics features show high contribution to each factor, one can plot feature loadings for any two omics using the `vis_load_plot` function. The features that take the most extreme values will be the strongest contributors to that factor.

    # colors_omics = c("red", "blue", "green")
    # colors_omics <- get_colors(mcia_results, color_pal = colors_omics)
    colors_omics <- get_colors(mcia_results)

    vis_load_plot(mcia_results, axes = c(1, 4), color_pal = colors_omics)

    ## Warning: Use of `gl_f[[colnames(gl_f)[1]]]` is discouraged.
    ## ℹ Use `.data[[colnames(gl_f)[1]]]` instead.

    ## Warning: Use of `gl_f[[colnames(gl_f)[2]]]` is discouraged.
    ## ℹ Use `.data[[colnames(gl_f)[2]]]` instead.

From this plot, we can deduce that individual miRNA features tend to have the most extreme values in factors 1 and 4, thereby contributing most strongly to the global scores for those factors.

## Scree Plot: Visualizing the Top Features per Factor

To dive into each factor users can generate a scree plot of top loading values using the following two steps/functions: 1) `ord_loadings()` which calculates a list of ranked feature loadings, and 2) `vis_load_ord()` which takes as input ranked values from step (1) to generate a scree plot. Features can be ranked based on ascending or descending values and/or absolute value. Users can also plot features from all omics together or focus on a specific omic by using a corresponding block name (i.e. mrna).

### Factor 1

Here, we visualize two ranked lists for Factor 1: using all omics, or just mrna.

    # generate the visualizations
    all_pos_1_vis <- vis_load_ord(mcia_results, omic = "all", factor = 1,
                                  absolute = FALSE, descending = TRUE)
    mrna_pos_1_vis <- vis_load_ord(mcia_results, omic = "mrna", factor = 1,
                                   absolute = FALSE, descending = TRUE)

    ggpubr::ggarrange(all_pos_1_vis, mrna_pos_1_vis)

The ordering itself can be easily obtained using the `ord_loadings(...)` function.

    # obtain the loadings as plotted above
    all_pos_1 <- ord_loadings(mcia_results, omic = "all", factor = 1,
                              absolute = FALSE, descending = TRUE)
    mrna_pos_1 <- ord_loadings(mcia_results, omic = "mrna", factor = 1,
                               absolute = FALSE, descending = TRUE)

    # visualize the tabular data
    all_pos_1[1:3, ]

    ##                    loading  omic       omic_name factor
    ## MI0000284_miRNA 0.09759126 miRNA MI0000284_miRNA      1
    ## MI0000266_miRNA 0.09732188 miRNA MI0000266_miRNA      1
    ## MI0000267_miRNA 0.08809128 miRNA MI0000267_miRNA      1

    mrna_pos_1[1:3, ]

    ##                      loading omic         omic_name factor
    ## GNS_3797_mrna     0.01854570 mrna     GNS_3797_mrna      1
    ## WASL_12215_mrna   0.01850120 mrna   WASL_12215_mrna      1
    ## LAPTM4A_4988_mrna 0.01749899 mrna LAPTM4A_4988_mrna      1

The first plot shows that that miRNA has the overall top positive feature loadings for factor 1, whereas the second plot allows us to focus on only mRNA signals, where genes like GNS and WASL come up as top contributors.

### Factor 2

Here, we return and visualize two ranked lists for Factor 2: using all omics, or just protein, this time ranking features by magnitude (absolute value).

    # define the loadings
    all_abs_2 <- vis_load_ord(mcia_results, omic = "all", factor = 2,
                              absolute = TRUE, descending = TRUE)

    prot_abs_2 <- vis_load_ord(mcia_results, omic = "prot", factor = 2,
                               absolute = TRUE, descending = TRUE)

    ggpubr::ggarrange(all_abs_2, prot_abs_2)

    # obtain the loadings as plotted above
    all_abs_2_vis <- ord_loadings(mcia_results, omic = "all", factor = 2,
                              absolute = TRUE, descending = TRUE)
    prot_abs_2_vis <- ord_loadings(mcia_results, omic = "prot", factor = 2,
                              absolute = TRUE, descending = TRUE)

    # visualize the tabular data
    all_abs_2_vis[1:3, ]

    ##                    loading  omic       abs       omic_name factor
    ## MI0001735_miRNA -0.1505451 miRNA 0.1505451 MI0001735_miRNA      2
    ## MI0003132_miRNA -0.1326578 miRNA 0.1326578 MI0003132_miRNA      2
    ## MI0000744_miRNA -0.1254875 miRNA 0.1254875 MI0000744_miRNA      2

    prot_abs_2_vis[1:3, ]

    ##                      loading omic        abs        omic_name factor
    ## THY1_1801_prot   -0.02671046 prot 0.02671046   THY1_1801_prot      2
    ## IGFBP7_1337_prot -0.02612204 prot 0.02612204 IGFBP7_1337_prot      2
    ## COL5A1_6730_prot -0.02450671 prot 0.02450671 COL5A1_6730_prot      2

For factor 2, we again see that across all omics, miRNA is playing the strongest role. One important difference (relative to factor 1) is made by ranking the values according to the absolute value. Doing so allows us to identify 9 (out of 15) features with a strong negative loading value. The right plot shows that the negatively values protein features have the largest absolute values, indicating that they may play a distinct role from the miRNA in global scores for this factor.

### Factor 4

We return and visualize a ranked list for Factor 4 with the top 60 features.

    # visualization
    all_4_plot <- vis_load_ord(mcia_results, omic = "all", factor = 4,
                               absolute = FALSE, descending = TRUE, n_feat = 60)
    all_4_plot

    # visualize the tabular data
    all_4 <- ord_loadings(mcia_results, omic = "all", factor = 4,
                          absolute = FALSE, descending = TRUE)

One can observe that while miRNA dominate the top features, proteins are also represented.

## Pathway Analysis for the Top Factors using Data from Gene-Centric Omics Blocks

The NCI-60 data set includes gene expression data, and its corresponding global loading matrix is a gene by factor matrix. We can learn more about the pathways over-represented in a given factor by running the global loadings for the factor through a gene set enrichment analysis (GSEA). Here, we look at factors 1 and 3. We run `gsea_report()`, which reports on the p-value of the most significant pathway as well as the total number of significant pathways for each factor.

### Gather Data and Generate the Report

    # extract mRNA global loadings
    mrna_gfscores <- nmb_get_gl(mcia_results)
    mrna_rows <- str_detect(row.names(mrna_gfscores), "_mrna")
    mrna_gfscores <- mrna_gfscores[mrna_rows, ]

    # rename rows to contain HUGO based gene symbols
    row.names(mrna_gfscores) <- str_remove(rownames(mrna_gfscores), "_[0-9]*_.*")

    # load pathway data
    path.database <- "https://data.broadinstitute.org/gsea-msigdb/msigdb/release/6.2/c2.cp.reactome.v6.2.symbols.gmt"
    pathways <- fgsea::gmtPathways(gmt.file = path.database)

    # generate the GSEA report
    geneset_report <- gsea_report(metagenes = mrna_gfscores, path.database,
                                  factors = seq(1, 3), pval.thr = 0.05, nproc = 2)

### Investigating the GSEA Summary Table

The report comes in the form of a list where the first element is a data frame with summary level of the GSEA analysis per factor. Ideally, each factor is capturing a select number of pathways with a high significance. From this report (below) we can see that the most significant pathway is associated with Factor 3 and that there is a large variation in the number of total (significant) pathways ranging from 7 (Factor 8) to 143 (Factor 4).

    geneset_report[[1]]

    ##         min_pval total_pathways
    ## Factor1 4.81e-09            119
    ## Factor2 2.19e-08             95
    ## Factor3 1.92e-21             88

As just mentioned, Factor 3 contains the most enriched gene set so we can re-run GSEA for this factor in order to get a full list of enrichment scores across all gene sets:

    # re-running GSEA
    factor3_paths <- fgseaMultilevel(pathways, stats = mrna_gfscores[, 3],
                                     nPermSimple = 10000, minSize = 15,
                                     nproc = 2, maxSize = 500)

    head(factor3_paths[, c("pathway", "padj")])

    ##                                                  pathway     padj
    ##                                                   <char>   <char>
    ## 1:                                   REACTOME CELL CYCLE 6.16e-21
    ## 2:                           REACTOME CELL CYCLE MITOTIC 7.29e-19
    ## 3:                REACTOME GENERIC TRANSCRIPTION PATHWAY 1.22e-13
    ## 4:                        REACTOME MITOTIC M M G1 PHASES 2.34e-09
    ## 5:                              REACTOME DNA REPLICATION 3.22e-09
    ## 6: REACTOME PROCESSING OF CAPPED INTRON CONTAINING PR... 8.37e-09

We observe that the most significant gene set is the REACTOME_CELL_CYCLE gene set. Below we can take a look at the list of genes from REACTOME_CELL_CYCLE that likely contribute to factor 3. This analysis can be repeated as necessary to make sense of other gene based factor loadings.

    # extracting REACTOME_CELL_CYCLE, the most significant gene set
    sig_path3 <- factor3_paths[min(factor3_paths$padj) == factor3_paths$padj, ][1, ]
    sig_path3$leadingEdge[[1]][1:10]

    ##  [1] "XRN2"  "FBXW7" "FBXL5" "USP11" "FBXW4" "NOP56" "FBXL3" "FBXO4" NA
    ## [10] NA

# Session Info

Session Info

    sessionInfo()

    ## R version 4.6.0 (2026-04-24)
    ## Platform: x86_64-pc-linux-gnu
    ## Running under: Ubuntu 24.04.4 LTS
    ##
    ## Matrix products: default
    ## BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
    ## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
    ##
    ## locale:
    ##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    ##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
    ##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    ##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    ##  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    ## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    ##
    ## time zone: Etc/UTC
    ## tzcode source: system (glibc)
    ##
    ## attached base packages:
    ## [1] grid      stats     graphics  grDevices utils     datasets  methods
    ## [8] base
    ##
    ## other attached packages:
    ## [1] stringr_1.6.0         nipalsMCIA_1.11.0     ggpubr_0.6.3
    ## [4] ggplot2_4.0.3         fgsea_1.39.0          dplyr_1.2.1
    ## [7] ComplexHeatmap_2.29.0 BiocStyle_2.41.0
    ##
    ## loaded via a namespace (and not attached):
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    ##  [3] farver_2.1.2                S7_0.2.2
    ##  [5] fastmap_1.2.0               pracma_2.4.6
    ##  [7] digest_0.6.39               lifecycle_1.0.5
    ##  [9] cluster_2.1.8.2             magrittr_2.0.5
    ## [11] compiler_4.6.0              rlang_1.2.0
    ## [13] sass_0.4.10                 tools_4.6.0
    ## [15] yaml_2.3.12                 data.table_1.18.4
    ## [17] knitr_1.51                  ggsignif_0.6.4
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    ## [21] DelayedArray_0.39.3         RColorBrewer_1.1-3
    ## [23] abind_1.4-8                 BiocParallel_1.47.0
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    ## [29] stats4_4.6.0                colorspace_2.1-2
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    ## [33] MultiAssayExperiment_1.39.0 SummarizedExperiment_1.43.0
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    ## [81] backports_1.5.1             broom_1.0.13
    ## [83] bslib_0.11.0                Rcpp_1.1.1-1.1
    ## [85] fastmatch_1.1-8             SparseArray_1.13.2
    ## [87] xfun_0.57                   MatrixGenerics_1.25.0
    ## [89] buildtools_1.0.0            pkgconfig_2.0.3
    ## [91] GlobalOptions_0.1.4