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Canonical correlation analysis (CCA) and partial least squares (PLS) are powerful multivariate methods for capturing associations across 2 modalities of data (e.g., brain and behavior). However, when the sample size is similar to or smaller than the number of variables in the data, standard CCA and PLS models may overfit, i.e., find spurious associations that generalize poorly to new data. Dimensionality reduction and regularized extensions of CCA and PLS have been proposed to address this problem, yet most studies using these approaches have some limitations. This work gives a theoretical and practical introduction into the most common CCA/PLS models and their regularized variants. We examine the limitations of standard CCA and PLS when the sample size is similar to or smaller than the number of variables. We discuss how dimensionality reduction and regularization techniques address this problem and explain their main advantages and disadvantages. We highlight crucial aspects of the CCA/PLS analysis framework, including optimizing the hyperparameters of the model and testing the identified associations for statistical significance. We apply the described CCA/PLS models to simulated data and real data from the Human Connectome Project and Alzheimer's Disease Neuroimaging Initiative (both of n500). We use both low- and high-dimensionality versions of these data (i.e., ratios between sample size and variables in the range of ∼1-10 and ∼0.1-0.01, respectively) to demonstrate the impact of data dimensionality on the models. Finally, we summarize the key lessons of the tutorial.

Canonical Correlation Analysis (CCA) and its regularised versions have been widely used in the neuroimaging community to uncover multivariate associations between two data modalities (e.g., brain imaging and behaviour). However, these methods have inherent limitations: (1) statistical inferences about the associations are often not robust; (2) the associations within each data modality are not modelled; (3) missing values need to be imputed or removed. Group Factor Analysis (GFA) is a hierarchical model that addresses the first two limitations by providing Bayesian inference and modelling modality-specific associations. Here, we propose an extension of GFA that handles missing data, and highlight that GFA can be used as a predictive model. We applied GFA to synthetic and real data consisting of brain connectivity and non-imaging measures from the Human Connectome Project (HCP). In synthetic data, GFA uncovered the underlying shared and specific factors and predicted correctly the non-observed data modalities in complete and incomplete data sets. In the HCP data, we identified four relevant shared factors, capturing associations between mood, alcohol and drug use, cognition, demographics and psychopathological measures and the default mode, frontoparietal control, dorsal and ventral networks and insula, as well as two factors describing associations within brain connectivity. In addition, GFA predicted a set of non-imaging measures from brain connectivity. These findings were consistent in complete and incomplete data sets, and replicated previous findings in the literature. GFA is a promising tool that can be used to uncover associations between and within multiple data modalities in benchmark datasets (such as, HCP), and easily extended to more complex models to solve more challenging tasks., Comment: 52 pages, 18 figures (including supplementary material)