Partial Least Squares (PLS)

Authors: Golia Shafiei

This tutorial explains the concept of Partial Least Squares (PLS) analysis in general, which was first introduced to the neuroimaging community in 1996 (McIntosh et al., 1996). In addition, we illustrate how to use PLS process on a sample data in Brainstorm.


PLS is a free toolbox that is a part of Baycrest, University of Toronto. The PLS code is written entirely in MATLAB (Mathworks Inc) and can be downloaded from To cite PLS Toolbox, please see the “References” section of this tutorial.


Partial Least Squares (PLS) analysis is a multivariate statistical technique that is used to find the relationship between two blocks of variables. PLS has various applications and types (Krishnan et al., 2011); however, the focus of this tutorial is on Mean-Centered PLS analysis, which is a common type of PLS while working with neuroimaging data. In this type of PLS analysis, one data block is neural activity (e.g. MEG measurements/source data here) while the other one is the experiment design (e.g. different groups/conditions).

PLS analysis is based on extracting the common information between the two data blocks by finding a correlation matrix and linear combinations of variables in both data blocks that have maximum covariance with one another. In the example provided here, we find a contrast between different conditions as well as patterns of brain activity that maximally covary with that specific contrast.

For this purpose, we take the neural activity as one data block, matrix X, where the rows of matrix X are observations (participants/trials) nested in conditions or groups, and the columns of X are variables (time scales nested within sources). The other data block, matrix Y, is a matrix of dummy coding that corresponds to experimental design (different groups or conditions) (Krishnan et al., 2011).

PLS analysis first calculates a mean-centered matrix using matrices X and Y. Then, singular value decomposition (SVD) is applied on the mean-centered matrix. The outcome of PLS analysis is a set of latent variables that are in fact linear combinations of initial variables of the two data blocks that maximally covary with the corresponding contrasts (Krishnan et al., 2011, Mišić et al., 2016). More specifically, each latent variable consists of a set of singular values that describe the effect size, as well as a set of singular vectors, or weights, that define the contribution of each initial variable to the latent variables (Mišić et al., 2016).

Finally, the statistical significance of a latent variable is defined by a p-value calculated from permutation test. In addition, bootstrapping is used to assess the reliability of each original variable (e.g. a source at a time point) that contributes to the latent variable. Bootstrap ratios are calculated for each original variable for this purpose. Bootstrap ratio is the ratio of the weights to the standard errors estimated from bootstrapping. Therefore, the larger the magnitude of a bootstrap ratio, the larger the weight (i.e. contribution to the latent variable) and the smaller the standard error (i.e. higher stability) (McIntosh and Lobaugh, 2004, Mišić et al., 2016). Bootstrap ratio can be equivalent to a z-score if we have an approximately normal bootstrap distribution (Efron and Tibshirani, 1986).

PLS analysis was explained in general in this section. However, this tutorial assumes that the users are already familiar with basics of PLS analysis. If PLS is new to you or if you want to read more about PLS and its applications in details, please refer to the articles introduced in “References” section.

Download and installation

In order to run PLS process in Brainstorm, the PLS Toolbox must be downloaded from here and added to MATLAB pathway.

Data, Pre-Processing and Source Analysis

The data processed here is the same dataset that is used in MEG visual tutorial: Single subject and MEG visual tutorial: Group analysis. This dataset consists in simultaneous MEG/EEG recordings of 19 subjects performing a simple visual task on a large number of famous, unfamiliar and scrambled faces. The detailed presentation of experiment is available in the MEG visual tutorial: Single Subject.

You can follow this tutorial after processing the data as illustrated in MEG visual tutorial: Single Subject. Then:

Running PLS

There are two PLS processes available in Brainstorm:

Both of these two processes work in a similar way:

We will continue with explaining the PLS process for only two conditions from now and we will leave PLS process for more conditions for the Advanced section of this tutorial.

Select files


Run the process when the options are set. The process will take some time depending on the number of files, number of permutations and number of bootstraps. The results are then saved in Group Analysis > Intra-Subject folder.

PLS Results





Advanced Option

You can run PLS for more than 2 conditions through Process1 tab. However, you should be careful when arranging the files in this option. We will use the same dataset to illustrate how to use this option in Brainstorm.


Run process Test > Partial Least Squares (PLS) – More than Two Condition.


  1. Efron B, Tibshirani R (1986). "Bootstrap methods for standard errors, confidence intervals and other measures of statistical accuracy." Stat. Sci. 1, 54– 77.

  2. Krishnan A, Williams LJ, McIntosh AR, Abdi H (2011). "Partial Least Squares (PLS) methods for neuroimaging: a tutorial and review." Neuroimage, 56(2), 455-475.

  3. McIntosh AR, Bookstein F, Haxby J, Grady C (1996). "Spatial pattern analysis of functional brain images using partial least squares." Neuroimage 3, 143–157.

  4. McIntosh AR, Lobaugh NJ (2004). "Partial least squares analysis of neuroimaging data: applications and advances." Neuroimage 23, S250–S263.

  5. Mišić B, Dunkley BT, Sedge PA, Da Costa L, Fatima Z, Berman MG, ... & Pang EW (2016). "Post-traumatic stress constrains the dynamic repertoire of neural activity." The Journal of Neuroscience, 36(2), 419-431.

GoliaShafiei (last edited 2017-01-10 19:25:25 by GoliaShafiei)