Reading Identification off the Diagram

Hsiu-Ting Yu

2026-09-28

Because a well-formed UMG determines a likelihood, several necessary conditions for identification can be read from the picture before any estimation is attempted. The functions in this vignette operationalise that idea. None of them replaces a formal identification analysis; each catches a routine error at the drawing board, where errors are cheapest to fix.

Scaling marks for latent variables

Every latent continuous random vertex must have its scale fixed. umg_check_scaling() reports the mechanism detected for each.

umg_check_scaling(umg_factor("F", paste0("y", 1:4)))   # marker loading
#>   vertex scaled            via location_fixed
#> 1      F   TRUE marker loading          FALSE
umg_check_scaling(umg_esem("F1", paste0("y", 1:4)))    # fixed variance
#>   vertex scaled            via location_fixed
#> 1     F1   TRUE fixed variance          FALSE

A latent vertex reported as "none" is the single most common specification error in latent variable modelling, surfaced as a missing mark rather than a nonconvergence.

The counting rule

umg_count_parameters() applies the classical t-rule, comparing free parameters against the distinct pieces of information the observed variables supply. A non-negative df is necessary, not sufficient.

umg_count_parameters(umg_factor("F", paste0("y", 1:6)))
#> $data_information
#> [1] 21
#> 
#> $free
#> $free$loadings_regressions
#> [1] 5
#> 
#> $free$covariances
#> [1] 0
#> 
#> $free$variances
#> [1] 7
#> 
#> $free$means
#> [1] 0
#> 
#> 
#> $free_total
#> [1] 12
#> 
#> $df
#> [1] 9
#> 
#> $applicable
#> [1] TRUE

Label switching

Mixture and latent class models are identified only up to a permutation of the classes.

umg_labelswitching(umg_mixture(umg_growth(4), c("I", "S")))
#> [1] "c"

Conditional independence and d-separation

umg_dsep() reads a conditional-independence claim off the diagram. Covariance edges are treated as latent common causes (the projection of an acyclic directed mixed graph onto a DAG).

med <- umg_mediation(direct = FALSE)
umg_dsep(med, "X", "Y")               # FALSE: connected through M
#> [1] FALSE
umg_dsep(med, "X", "Y", given = "M")  # TRUE: blocked by the mediator
#> [1] TRUE

The implied independencies of the model (its testable implications) are enumerated under the local Markov property:

umg_implied_ci(umg_mediation(direct = FALSE))
#>   x y given
#> 1 X Y     M

A single summary

umg_identify() collects these readings into one printable object.

umg_identify(umg_factor("F", paste0("y", 1:6)))
#> UMG identification summary
#> --------------------------
#> Latent scaling: 1 latent continuous vertex(es); all scaled
#> Counting rule: 21 data moments, 12 free parameters, df = 9
#> Implied conditional independencies (observed): 0
#> Note: necessary conditions only; not a formal identification proof.