Analyzing Multi-Block Data: A Tutorial of Multiple Factor Analysis in R

Abstract

In multivariate analyses, data are typically structured with observations (i.e., participants) on the rows and measurements (i.e., variables) on the columns. These data structure types are commonly analyzed with methods such as principal components analysis (PCA) or factor analyses (FA). However, PCA, FA, and related techniques are not always suitable for data structures such as multi-table, multi-block, or even multi-modal data sets that often include complex multifactorial structures. Usually, these more complex structures include data comprised of, for example, (1) distinct sets of rows and columns per observation (i.e., a matrix per participant), (2) groups of measurements across the columns, or (3) combinations of the two. To address these issues, we require a technique particularly suited for data like these: Multiple factor analysis (MFA). MFA is a robust multivariate extension of PCA designed specifically for complex data sets with multifactorial structures, wherein at least one dimension (e.g., observations) is shared across many distinct tables (or blocks). Some examples of these data structures include (1) ratings data where each individual (a block of columns) provides ratings for items of interest (e.g., consumer products), or (2) large batteries comprised of cognitive, neuropsychological, and behavioral variables (where each set of variables is a block of columns) are measured for all individuals (rows). Briefly, MFA works by first normalizing each block so that each block contributes equal variance (akin to Z-scoring a matrix). Next, each block is concatenated (usually column-wise) and then a PCA is applied to the concatenated table. MFA also creates a consensus model that helps illustrate how each block contributes to the overall model. While these types of data structures are very common, MFA is a rarely used technique in the psychological sciences, and thus we have created this workshop to explicate this method. Our workshop focuses on MFA and its application to a variety of data structures using the R statistical environment. In this workshop, participants will gain a basic understanding of MFA, how it can be applied to multi-table data structures, and how to interpret and report results from MFA. In addition to descriptive (fixed-effects) MFA, we also show how to utilize inferential (random-effects) techniques, such as bootstrapping and permutation. We will illustrate MFA with two data sets similar to those previously discussed: (1) ratings of beer (rows) across many sensory categories (columns) by 8 participants (blocks), and (2) a large battery of variables (columns) grouped by instrument (blocks) for a large number of participants (rows); specifically, cognitive, neuropsychological, and clinical variables for an Alzheimer’s disease population. We will provide R code and example data sets for participants of the workshop. Finally, we conclude with a brief discussion on the extensions and variants of MFA (e.g., discriminant analyses) for various problems (e.g., longitudinal studies, multi-site studies, brain imaging, genetics). Overall, participants will leave with a basic understanding of MFA (and extensions), and how to perform MFA on their own data