dsep-basis

name	dsep-basis
description	Build basis sets and run any-component d-separation / Fisher's C correctly — adjacency = direct parent on ANY component, causal ordering, conditioning on parents, LRT df bookkeeping, honest framing as drmSEM's definition.

d-separation & basis sets (any-component)

drmSEM's d-separation is the ANY-COMPONENT definition (DECISIONS.md #2, docs/design/03-dsep.md): a missing arrow X → Y asserts X has NO effect on ANY modelled distributional component of Y. This is drmSEM's definition of a missing arrow — frame it that way, not as a generic graphical-SEM claim.

Adjacency = direct parent on ANY component

• X is adjacent to Y if X is a direct parent of Y on mu, sigma, nu, zi, hu, sd(group), or rho12 — any one suffices (drm_is_parent, scanning the full typed edge table from drm_build_edges).
• A predictor that appears only in Y's sigma formula still makes X adjacent to Y. Do not restrict adjacency to mu.

Basis set construction (basis_set.drm_sem)

For each endogenous Y and each variable X (endogenous or exogenous):

1. Skip X == Y.
2. Skip if X is causally LATER than Y (drm_order_index: topological position; exogenous variables sort at index 0, before all nodes).
3. Skip if X is already a parent of Y (adjacent → not a missing arrow).
4. Skip if Y is a parent of X (orientation guard).
5. Otherwise emit claim X _||_ Y | {parents(Y)}. The conditioning set is Y's existing parents (drm_parents), so the LRT controls for them.

Testing a claim (dsep.drm_sem)

• Refit Y's node adding X to EVERY modelled component sub-formula (drm_refit_augmented, with se = TRUE). This realizes the any-component alternative hypothesis.
• LRT: LR = 2*(logLik_aug - logLik_base), df = df_aug - df_base.
• df bookkeeping: df is the number of extra coefficients across ALL augmented components (one per component sub-formula in the simplest case, more with factors/interactions). Read both df values from drm_fit_logLik — never hard-code df = 1.
• Guard degenerate cases: status no_data_column, refit_failed, degenerate (df <= 0 or NA LR) are recorded and EXCLUDED from Fisher's C.

Fisher's C (fisher_c, drm_fisher_c_from_p)

• C = -2 * sum(log p) over the ok claims with p > 0; df = 2k where k is the number of contributing claims; p = pchisq(C, df, lower.tail = FALSE).
• A SMALL Fisher p-value means the DAG omits a needed path (reject the model's independence claims). Report C, df = 2k, k, and p together.

Do / Don't

• DO: state results as "X carries information about some component of Y beyond Y's parents" when a claim's p is small — not "X affects the mean of Y".
• DO: require nodes fitted with se = TRUE (declarative drm_sem() does this) so refits converge and vcov/LRT work; warn via check_sem() if not.
• DON'T: build claims only from mu edges, or condition on non-parents.
• DON'T: feed failed/degenerate claims into Fisher's C.
• DON'T: add or change a d-sep rule without a recovery test (a DGP where the true missing arrow is known) — see validation-harness.