nmf.ffb.test: Test the feed-forward null of an NMF-FFB fit

In nmfkc: Non-Negative Matrix Factorization with Kernel Covariates

Test the feed-forward null of an NMF-FFB fit

Description

Calibrated test of the hypothesis that the outcomes need no indicator-level feedback: the feed-forward model Y_1 = X(\Theta_2 Y_2 + U) + \mathcal{E} with a full random-effect covariance against the feedback model with \Theta_1 \ne 0. This is the only inferential step of the procedure (see the workflow below); nmf.ffb.inference is for the uncertainty of the coefficients after the null has been rejected, and nmf.ffb.diagnostics for what the selected feedback looks like.

Usage

nmf.ffb.test(
  object,
  Y1,
  Y2,
  B = 1000L,
  calibration = c("procedure", "conditional"),
  seed = 123L,
  ...
)

Arguments

object	A fit from nmf.ffb with method = "fiml".
Y1, Y2	The endogenous and exogenous matrices the fit was computed from (variables in rows, units in columns).
B	Number of bootstrap replicates (default 1000). The smallest reportable p-value is 1/(1+B).
calibration	What is re-applied to each null replicate; see above. "conditional" is selected automatically, with a warning, when the fit used a basis or an exclusion restriction supplied by the caller, since there is then nothing to re-estimate.
seed	Base seed; replicate b uses seed + b. The caller's random stream is restored on exit.
...	Passed to the bootstrap: cores (or ncores) to run the replicates in parallel, defaulting to getOption("mc.cores", 1L) as elsewhere in the package; fiml.maxit and factr to control the optimizer in the replicates; print.trace = TRUE for progress.

Details

The likelihood ratio does not have a \chi^2 null distribution: the constraint \Theta_1 \ge 0 puts the null on the boundary of the parameter space, and the support of \Theta_1 is chosen from the same data. It is therefore calibrated by a parametric bootstrap from the fitted feed-forward null,

Y_1^{*} = (\hat X(\hat\Theta_2 Y_2 + U^{*}) + \mathcal{E}^{*})_{+}, \quad U^{*} \sim N(0, \hat\Phi), \; \mathcal{E}^{*} \sim N(0, \hat\Psi),

with negative draws clipped at zero, and the reported p-value is (1 + \#\{LR^{*} \ge LR\}) / (1 + B_{ok}).

What is re-applied to each replicate

calibration = "procedure" (the default) re-applies the whole procedure: stage 1 is re-run on each Y_1^{*}, the basis is re-estimated, the exclusion restriction is re-derived from that basis, and stage 2 follows. The p-value is then the operating characteristic of the complete exploratory procedure, which is what is applied to a fresh data set. This is the calibration to use.

"conditional" holds \hat X and the exclusion restriction at their fitted values and re-runs stage 2 only. It is valid only if basis and restriction are independent of the Y_1 being tested, which they are not when all three come from the same sample; it is provided for comparison with that practice, and it is anti-conservative.

Sample splitting – basis and restriction from one half of the units, test on other – also removes the dependence, and was offered here in 0.9.8. It was withdrawn in 0.9.8: measured against "procedure" it has the same size and lower power, because the test uses N/2 units, and it cannot be run at all when the halves are too small for stage 1.

Reading the result

Report LR.p.boot together with prob.select.null and, for calibration = "procedure", C1.restriction.change.rate. The last two say whether the exclusion restriction is determined well enough for the test to mean anything: if the dominant factor of an outcome moves from one null replicate to the next, a previously blocked entry becomes free and a feedback coefficient can absorb loading structure that the feed-forward model attributes to X. A large prob.select.null with a large null quantile is the signature of a data set in which the procedure finds feedback whether or not there is any.

Value

An object of class "nmf.ffb.test":

LR, LR.df

Observed c(full, selected) likelihood ratios and the number of free feedback entries left by the exclusion restriction.

LR.p.boot

Calibrated p-values of the two statistics.

LR.null.quantile

The 0.95 quantiles of the null distributions.

LR.boot

B \times 2 matrix of the null replicates.

prob.select.null

Null false-selection rate: the probability that BIC retains at least one feedback entry when the feed-forward model is true.

C1.restriction.change.rate

("procedure" only) share of null replicates in which the exclusion restriction moved.

LR.boot.df

Free entries per null replicate (varies under "procedure").

LR.boot.n.ok, LR.boot.n.nonconv

Usable replicates, and how many did not meet the optimizer tolerance.

Workflow

ecv <- nmf.ffb.ecv(Y1, Y2, rank = 1:5)   # 1. choose Q
fit <- nmf.ffb(Y1, Y2, rank = ecv$rank)  # 2. estimate; BIC selects the support
tst <- nmf.ffb.test(fit, Y1, Y2)         # 3. test the feed-forward null  <- only inference
dgn <- nmf.ffb.diagnostics(fit)          # 4. cycles, spectral radius, identifiability
inf <- nmf.ffb.inference(fit, Y1, Y2)    # 5. intervals for the retained entries

See Also

nmf.ffb, nmf.ffb.ecv, nmf.ffb.inference, nmf.ffb.diagnostics

nmfkc documentation built on Oct. 30, 2026, 9:06 a.m.