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.


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.


# 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))


# NIPALS starts with a random vector

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

# 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 \(r=10\).

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:

##  [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):

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 block 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
# 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
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_palas shown below.

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