Statistical inference for the exogenous parameter matrix C2

nmf.sem.inference performs statistical inference on the exogenous parameter matrix \(C_2\) from a fitted nmf.sem model, conditional on the estimated basis matrix \(\hat{X}\) and the endogenous parameter matrix \(\hat{C}_1\).

Under the working model \(R = Y_1 - X C_1 Y_1 \approx X C_2 Y_2 + \varepsilon\), inference on \(C_2\) is conducted via sandwich covariance estimation and one-step wild bootstrap with non-negative projection.

Usage

nmf.sem.inference(object, Y1, Y2, wild.bootstrap = TRUE, ...)

Arguments

object

A list returned by nmf.sem, containing at least X, C1, and C2.

Y1

Endogenous variable matrix (P1 x N). Must match the data used in nmf.sem().

Y2

Exogenous variable matrix (P2 x N). Must match the data used in nmf.sem().

wild.bootstrap

Logical. If TRUE (default), performs wild bootstrap for confidence intervals and bootstrap standard errors.

...

Additional arguments:

wild.B: Number of bootstrap replicates. Default is 1000.
wild.seed: Seed for bootstrap. Default is 42.
wild.level: Confidence level for bootstrap CI. Default is 0.95.
sandwich: Logical. Use sandwich covariance. Default is TRUE.
C.p.side: P-value type: "one.sided" (default) or "two.sided".
cov.ridge: Ridge stabilization for information matrix inversion. Default is 1e-8.
print.trace: Logical. If TRUE, prints progress. Default is FALSE.

Value

The input object with additional inference components:

sigma2.used: Estimated \(\sigma^2\) used for inference.
C2.se: Sandwich standard errors for \(C_2\) (Q x P2 matrix).
C2.se.boot: Bootstrap standard errors for \(C_2\) (Q x P2 matrix).
C2.ci.lower: Lower CI bounds for \(C_2\) (Q x P2 matrix).
C2.ci.upper: Upper CI bounds for \(C_2\) (Q x P2 matrix).
coefficients: Data frame with Estimate, SE, BSE, z, p-value for each element of \(C_2\).
C2.p.side: P-value type used.

References

Satoh, K. (2025). Applying non-negative matrix factorization with covariates to structural equation modeling for blind input-output analysis. arXiv:2512.18250. https://arxiv.org/abs/2512.18250

Examples

Y <- t(iris[, -5])
Y1 <- Y[1:2, ]; Y2 <- Y[3:4, ]
res <- nmf.sem(Y1, Y2, rank = 2)
res2 <- nmf.sem.inference(res, Y1, Y2)
res2$coefficients
#>     Basis    Covariate     Estimate           SE          BSE      z_value
#> 1 Factor1 Petal.Length 9.157274e-03 0.0007674188 0.0007462585 1.193256e+01
#> 2 Factor2 Petal.Length 2.602520e-09 0.0006634281 0.0003779676 3.922836e-06
#> 3 Factor1  Petal.Width 2.300680e-06 0.0023326828 0.0012550356 9.862806e-04
#> 4 Factor2  Petal.Width 2.134356e-10 0.0018436918 0.0009884945 1.157654e-07
#>        p_value      CI_low     CI_high
#> 1 4.003418e-33 0.007747119 0.010681964
#> 2 4.999984e-01 0.000000000 0.001239286
#> 3 4.996065e-01 0.000000000 0.004173533
#> 4 5.000000e-01 0.000000000 0.003146949