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Statistical Methods

This page documents methodological choices in plsdo that differ from or extend common PLS implementations. For a general introduction to interpreting the output, see Interpreting output.

Design matrix encoding (discriminatory PLS)

Discriminatory PLS constructs a design matrix from categorical group labels. plsdo uses full-rank dummy coding: a factor with k levels produces k indicator columns (one per level), not k − 1.

The rows of this matrix sum to 1, so the design has rank k − 1 and the SVD returns k − 1 non-trivial latent variables. Group contrasts are implicitly referenced to the grand mean.

This follows the convention in [@mcintosh2004] and differs from packages that use mean-centred deviation coding (e.g. plscmd in Matlab), where the reference level is explicit. The resulting singular values and loadings are mathematically equivalent; the difference is interpretive, not computational.

Permutation p-values

Permutation p-values use the corrected formula from [@phipson2010]:

p = (c + 1) / (n_perms + 1)

where c is the count of permuted singular values ≥ the observed value. This guarantees exact control of the Type I error rate at any number of permutations and avoids p-values of exactly zero.

The same correction is applied to the cross-validation permutation test.

Latent variable filtering

plsdo applies a two-stage filter before reporting latent variables as significant:

  1. The permutation test p-value must be < 0.05.
  2. At least one feature must have |bootstrap ratio| > 1.96 on both the X and Y sides.

Most PLS implementations check only the Y side. Requiring reliability on both sides is a conservative choice that reduces false positives where a significant singular value is driven by noise on the X side. The threshold of 1.96 corresponds to a 95% confidence interval under the standard-normal approximation to the bootstrap distribution.

Bootstrap alignment

Each bootstrap resample produces a new SVD, whose singular vectors may be rotated or sign-flipped relative to the reference. plsdo aligns bootstrap replicates using Procrustes rotation on the Y-side loadings (Vt), then applies a sign correction based on the dot product with the reference loadings.

Aligning in a single space (Vt) is sufficient because the SVD guarantees a shared coordinate system and the rotation matrix Q is orthogonal. This follows the approach in [@mcintosh2004].

References

BibTeX entries are in references.bib.

  • mcintosh2004 — McIntosh AR, Lobaugh NJ (2004). Partial least squares analysis of neuroimaging data: applications and advances. NeuroImage, 23, S250–S263. doi:10.1016/j.neuroimage.2004.07.020
  • krishnan2011 — 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. doi:10.1016/j.neuroimage.2010.07.034
  • mcintosh2013 — McIntosh AR, Mišić B (2013). Multivariate statistical analyses for neuroimaging data. Annual Review of Psychology, 64, 499–525. doi:10.1146/annurev-psych-113011-143804
  • phipson2010 — Phipson B, Smyth GK (2010). Permutation P-values should never be zero: calculating exact P-values when permutations are randomly drawn. Stat Appl Genet Mol Biol, 9(1). doi:10.2202/1544-6115.1585