Identify Significant Components via RMT and Inter-block Coherence Tests

This function determines which components from a multi-block analysis (e.g., penalized MFA) are statistically significant. It combines two complementary tests: a Random Matrix Theory (RMT) test based on the Marchenko-Pastur distribution, and an Inter-block Coherence (ICC) test that assesses whether loadings are consistent across blocks.

Usage

significant_components(
  fit,
  n,
  k_vec = NULL,
  alpha = 0.05,
  check_rmt = TRUE,
  tail_frac = 0.3
)

Arguments

fit

A fitted multi-block model object containing at minimum a `V_list` attribute (list of loading matrices) and optionally `sdev` (singular values).

n

Integer; the number of observations (rows) used to fit the model.

k_vec

Optional integer vector of block dimensions (number of columns per block). If NULL, inferred from `fit` via `block_indices` or `V_list`.

alpha

Numeric; significance level for hypothesis tests (default 0.05).

check_rmt

Logical; if TRUE (default), performs the Marchenko-Pastur edge test to check if eigenvalues exceed the noise threshold.

tail_frac

Numeric; fraction of smallest eigenvalues used to estimate noise variance for the RMT test (default 0.3).

Value

A list with the following elements:

keep

Integer vector of component indices that pass both tests.

rmt_pass

Logical vector indicating which components pass the RMT test.

icc_pass

Logical vector indicating which components pass the ICC test.

icc

Numeric vector of inter-block coherence values per component.

icc_pvalue

Numeric vector of p-values for the ICC test.

mp_edge

The Marchenko-Pastur edge threshold (NA if RMT skipped).

sigma2_est

Estimated noise variance (NA if RMT skipped).

lambda

Eigenvalues (squared singular values).

n, k_vec, alpha

Input parameters echoed back.

Details

The RMT test uses the Marchenko-Pastur distribution to identify eigenvalues that exceed the expected bulk edge under the null hypothesis of pure noise. Noise variance is estimated robustly from the tail of the eigenvalue distribution.

The ICC test measures the squared cosine similarity of loading vectors across all pairs of blocks. Under the null hypothesis of random loadings, the expected value is 1/k (where k is the harmonic mean of block dimensions). A z-test with Bonferroni correction is applied.