Canonical LMM reductions

These reductions connect specialized APIs to the shared primitives in the model catalog. They are used to detect duplicate proposals: a named model that is already an exact composition or a qualified reduction is not a missing production engine.

Representation reductions

General representation	Reduced case	Established route
Incomplete multitrait pedigree	one trait	Single-trait pedigree MME/REML
Irregular pedigree random regression	complete common grid and matching basis	Balanced pedigree longitudinal representation
Generic record-longitudinal	complete common grid	Structured balanced longitudinal representation
Component LMM pedigree component with one intercept channel	one component and homogeneous residual	Single-trait animal model
Component LMM identity component with one intercept channel	one component	Ordinary grouped random intercept
Pedigree plus identity intercept components	additive plus repeated-record identity domain	Repeatability/permanent-environment model
Shared pedigree component with direct and maternal channels	individual incidence plus dam incidence	Correlated direct/maternal model; covariance is within one channel block
Shared relationship component with intercept and basis channels	weighted channel incidence	Random regression or reaction-norm component
Scalar identity components	domain names changed	Litter, herd, year, herd-year, block, site, family, or common environment

Residual reductions

• Residual blocks of size one reduce exactly to the independent grouped- variance path.
• On integer occasions and positive rho, continuous exponential correlation with \(\phi=-\log(\rho)\) equals discrete AR(1).
• Missing longitudinal occasions are absent records in both correlated and independent residual representations; no balanced rows are synthesized.
• Patterned multitrait residual covariance uses the principal block for the traits actually observed in a residual unit. It is not temporal residual correlation.

Active-set reductions

• A zero scalar component is omitted from the active MME and restored as zero coefficients and zero prior/PEV variance in external results.
• A rank-one two-channel covariance reduces to one active relationship-based latent direction while preserving an estimable orientation. Results are transformed back to the full two-channel coordinates.
• Positive-definite component blocks and strictly positive residual parameters retain the positive-interior representation. Blocks larger than two do not gain a semidefinite boundary through these reductions.

Inference reductions and non-reductions

• Complete-MME PEV is the random-effect block of the inverse of the complete MME. It does not reduce to inversion of the random-only coefficient submatrix when fixed effects are estimated.
• CHOLMOD selected-inverse trace and uncertainty requests reuse one retained sparse factor and the requested inverse-entry closure.
• PARDISO coordinate traces reduce trace contractions to bounded batches of exact coordinate systems, but do not provide selected inversion.
• PARDISO selected_solve PEV reduces requested complete-MME inverse blocks to bounded retained-factor solves \(CX=E_T\) followed by target-row extraction. It never reduces to selected inversion or a full dense inverse.
• PCG coefficient solutions do not imply an exact inverse diagonal because no retained direct factor exists.

API consequence

Specialized pedigree, multitrait, longitudinal, repeatability, and formula interfaces may remain useful adapters for their validated input contracts. Identity/group effects, permanent or common environment, direct/maternal covariance, and ordinary random slopes are composition recipes whenever the component-LMM scalar-record contract is sufficient. Future work should target a precise missing adapter, backend contract, or qualification cell instead of reimplementing these reductions.