Scientific model recipes

These definitions preserve useful scientific distinctions from the earlier A-C planning catalog. They are mathematical recipes, not installed syntax, implementation priorities, or qualification claims. Consult the model catalog for each public route and the reductions for established equivalences.

For \(y = X\beta + \sum_k Z_k u_k + e\), each component must declare its external coefficient domain, record incidence and weights, relationship, covariance layout, missingness, and prediction targets. The notation Z describes the mathematics; it does not require a production incidence matrix.

Animal, grouped, and correlated-channel recipes

Recipe	Scientific meaning and interpretation
Animal	\(a \sim \mathcal{N}(0,\sigma_a^2 A)\); records reference animal coefficients, including pedigree animals without observations
Multitrait animal	Relationship A shared across declared traits with covariance G; observed residual units use R[S,S]; different fixed designs and missing traits require explicit mapping
Repeatability	Independent animal and subject-level identity terms, plus residual; \(h^2 = v_a/(v_a+v_p+v_e)\) and repeatability \(\frac{v_a+v_p}{v_a+v_p+v_e}\)
Common environment	Independent identity component over litter, pen, household, herd, site, or another declared domain; domains need not be nested in pedigree order
Sire	Records reference sire coefficients with covariance \(\sigma_s^2 A_s\); report sire variance explicitly, without silently converting it to additive genetic variance
Direct/maternal	Animal and dam incidence channels share one pedigree relationship and a joint 2-by-2 genetic covariance; all dam references must resolve
Maternal permanent environment	Independent identity component indexed by dam; combining it with direct/maternal genetics gives the full maternal recipe
Trait-specific random effects	Components declare their acting traits and records; absent incidence is structural, distinct from a missing response
Correlated random coefficients	Several declared incidence channels share one relationship K and covariance Q; matching grouping labels alone do not imply correlation
Direct/social	Direct and social genetic channels may correlate, while a social channel has several weighted contributing group mates; self-exclusion, duplicates, weight scale, and member order need explicit contracts
Discrete pedigree GxE	Environment-specific breeding values with covariance \(A\otimes G_E\); a multitrait reduction applies only when incidence and fixed-design mappings match
Random regression/reaction norm	u_i has relationship-by-basis covariance and contributes \(b(t)^{\mathsf T}u_i\) or \(b(E)^{\mathsf T}u_i\); common-grid and irregular records use supplied bases, with explicit prediction rows and extrapolation interpretation

Repeatability and other variance ratios require a valid positive denominator and stated units; their uncertainty support must be assessed separately. For a trajectory, covariance between supplied basis rows is \(b(t)^{\mathsf T}Qb(s)\). Basis rank and external axis order are part of the model, not solver choices.

Covariance and residual definitions

These families may apply to different domains. A family available in a generic longitudinal API is not thereby available in the component or formula API.

Family	Definition or required distinction
Homogeneous / heterogeneous diagonal	\(\sigma^2 I\) / \(\operatorname{diag}(\sigma_j^2)\); distinguish trait, residual-group, and level-domain variances
Unstructured SPD	\(LL^{\mathsf T}\) with positive diagonal L; log-Cholesky is a parameterization, not another covariance family
Compound symmetry	\(\sigma^2((1-\rho)I+\rho J)\); admissible rho depends on dimension
Heterogeneous compound symmetry	\(D((1-\rho)I+\rho J)D\), where D contains marginal standard deviations
Banded	Ordered levels and a declared bandwidth; specify omitted zeros and a positive-definite parameterization
Antedependence	Structured precision or modified Cholesky, with innovation variances and dependence coefficients; zero dependence reduces to diagonal covariance
Discrete AR(1)	\(\sigma^2\rho^{\lvert t_i-t_j\rvert}\) on integer positions, \(\lvert\rho\rvert<1\); heterogeneous scaling is \(DC_{\mathrm{AR1}}D\)
AR/MA/ARMA	Each order requires its own stationarity/invertibility, ordering, and derivative contract
Continuous exponential	\(\sigma^2\exp(-\phi\lvert t_i-t_j\rvert)\), \(\phi>0\), on irregular finite times; duplicate-time handling must be explicit
Unstructured within-subject residual	A bounded occasion covariance with observed principal submatrices; distinct from a permanent-environment random effect
Factor analytic FA(k)	\(\Lambda\Lambda^{\mathsf T}+\Psi\), positive diagonal specifics; declared rank and loading identification are required; compare derived covariances under permitted sign conventions
Reduced rank	\(\Lambda\Lambda^{\mathsf T}\) is singular when rank is below dimension; it requires an explicit singular precision/determinant/boundary contract
Spatial AR1 by AR1	\(\sigma^2 C_{\mathrm{row}}\otimes C_{\mathrm{col}}\); declare grid order and missing-cell behavior, separately from random row/column effects
General separable covariance	\(S_1\otimes S_2\); both factors need validated ordering and derivatives; \(\log\det(S_1\otimes S_2)=d_2\log\det(S_1)+d_1\log\det(S_2)\)

Support for one positive-interior family does not establish its zero-variance, singular, mixed-family, or optimized boundary extension. Component omission and fixed-parameter reductions remain useful when a chosen route excludes boundaries.