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Basis set of independence claims for a distributional SEM

Source: R/dsep.R
basis_set.Rd

The basis set is the collection of non-adjacent variable pairs (X, Y) where Y is endogenous and X is causally no later than Y. Each claim asserts that X has no effect on any modelled distributional component of Y, conditional on Y's existing parents. This any-component reading is drmSEM's definition of a missing arrow (see docs/design/03-dsep.md).

Usage

basis_set(object, ...)

# S3 method for class 'drm_sem'
basis_set(object, ...)

Arguments

object

A drm_sem object.

...

Unused.

Value

A data frame with columns claim, x, y, given (comma-separated conditioning set). On a DAG, given is Y's parents. On a MAG (latent = supplied), it is ant({X, Y}) \\ {X, Y}.

Details

A covariance edge declared with covary() (a residual rho12 or higher-level corpair arc) is an allowance that the two responses stay associated, so the y1 _||_ y2 claim is dropped from the basis set (Shipley's bidirected-edge rule; OQ-14). A declared feedback motif (drm_cycle(), 0.5) likewise drops independence claims among its nodes — DAG d-separation does not hold across a cycle, and the goodness-of-fit test is scoped to the acyclic part until sigma-separation lands.

When latent = names marginalised latents (L_M), claims are generated on the implied MAG by Richardson & Spirtes (2002) Corollary 5.3: each non-adjacent observed pair is conditioned on the anteriors of the pair (not on Shipley & Douma's observed parents). Bidirected spouses are not anteriors. Pairwise ⇒ global is licensed when the independence model is a compositional graphoid (Sadeghi & Lauritzen 2014 Thm 3; Lauritzen & Sadeghi 2018 Thm 4) — automatic for homoscedastic all-Gaussian nodes, otherwise assumed via faithfulness (a cli_inform() fires for non-Gaussian or sigma ~ nodes). Selection / conditioned latents are not supported.

The any-component reading of an independence claim is a drmSEM choice on top of the local-likelihood d-separation framework of Shipley; the d-separation graphical criterion itself is due to Pearl, and the bidirected-edge handling of declared residual correlations follows Shipley's path-analysis treatment.

References

Shipley B (2000). “A New Inferential Test for Path Models Based on Directed Acyclic Graphs.” Structural Equation Modeling, 7(2), 206–218. doi:10.1207/S15328007SEM0702_4 .

Shipley B (2009). “Confirmatory Path Analysis in a Generalized Multilevel Context.” Ecology, 90(2), 363–368. doi:10.1890/08-1034.1 .

Shipley B (2016). Cause and Correlation in Biology: A User's Guide to Path Analysis, Structural Equations and Causal Inference with R, 2nd edition. Cambridge University Press, Cambridge.

Pearl J (2009). Causality: Models, Reasoning, and Inference, 2nd edition. Cambridge University Press, Cambridge.

Lefcheck JS (2016). “piecewiseSEM: Piecewise Structural Equation Modelling in R for Ecology, Evolution, and Systematics.” Methods in Ecology and Evolution, 7(5), 573–579. doi:10.1111/2041-210X.12512 .

Richardson T, Spirtes P (2002). “Ancestral Graph Markov Models.” The Annals of Statistics, 30(4), 962–1030. doi:10.1214/aos/1031689015 .

Sadeghi K, Lauritzen S (2014). “Markov Properties for Mixed Graphs.” Bernoulli, 20(2), 676–696. doi:10.3150/12-BEJ502 .

Lauritzen SL, Sadeghi K (2018). “Unifying Markov Properties for Graphical Models.” The Annals of Statistics, 46(5), 2251–2278. doi:10.1214/17-AOS1618 .

Examples

if (FALSE) { # \dontrun{
sem <- drm_sem(
  size = drm_node(drmTMB::bf(size ~ temp + habitat, sigma ~ temp),
                  family = stats::gaussian()),
  abundance = drm_node(drmTMB::bf(abundance ~ size + temp, zi ~ habitat),
                       family = drmTMB::nbinom2()),
  data = dat)
basis_set(sem)
} # }