SECTION 1: Theory | 1) Why reduce the cross-environment covariance | 2) The factor-analytic (FA) model | 3) The reduced-rank (RR) model | 4) Choosing k and comparing to other structures | SECTION 2: Fitting factor-analytic and reduced-rank models | 1) Data | 2) Factor-analytic model with fam() | 3) Reduced-rank model with rrm() | 4) Comparing FA and RR | SECTION 3: Extracting loadings, scores, and diagnostic plots | 1) Loadings and specific variances | 2) Genotype scores | 3) Why and how to rotate loadings and scores | 4) Percentage of variance explained | 5) Loadings plot | 6) Genotype scores plot | 7) Latent regression plots | 8) Heatmap of the fitted genetic correlation matrix | Literature

Chapter 1: The linear mixed model and Henderson's equations | 1.1 The model | 1.2 Henderson's Mixed Model Equations (MME) | $$C\begin | 1.3 Where covariance structures come in | 1.4 Preview of the key distinction (developed in Chapter 4) | Chapter 2: Parameterization theory, explained from first principles | 2.1 Why we can't just let the computer search freely | 2.2 The working-parameter vector | 2.3 Transform family 1 — variances: the exponential map | 2.4 Transform family 2 — bounded quantities: logit and tanh | 2.5 Transform family 3 — many correlated quantities at once (factorizations) | 2.6 Why the reference level matters: avoiding scale confounding | Chapter 3: Catalog of covariance structures | 3.1 ism(x) — identity | 3.2 dsm(x, values, fixed, theta) / atm(x, levs, values, fixed) — heterogeneous diagonal variances | 3.3 csm(x, rho, fixed, variance, values) — compound symmetry | 3.4 ar1m/ar2m/ar3m/toeplitzm — autoregressive & general Toeplitz correlation | 3.5 mam/ma1m/ma2m — moving-average covariance | 3.6 corgm(x, theta, fixed) — general correlation matrix | 3.7 usm(x, theta, fixed) — general unstructured covariance | 3.8 fam(x, k, loadings, specific, fixed, variance) / rrm(x, k, loadings, fixed) — factor-analytic covariance | 3.9 antem(x, order, beta, innovations, fixed) — antedependence | 3.10 maternm(x, range, nu, fixed, distance) — Matérn spatial correlation | 3.11 sar(x, W, rho, fixed) — simultaneous autoregressive spatial covariance | 3.12 car(x, W, rho, fixed) — proper conditional autoregressive covariance | 3.13 ownm(x, K, fun, par, fixed, dfun) — user-defined shape | 3.14 covm(ran1, ran2, thetaC, theta, ...) — legacy two-random-effect helper | 3.15 Summary table | Chapter 4: REML, AI-REML, and ai_mme_sp2() | 4.1 The REML log-likelihood sommer maximizes | 4.2 The AI-REML update | 4.3 The core object: precision and its derivative | 4.4 Variance-type parameters: sparse/diagonal $B_k$ | 4.5 Correlation-type parameters: dense $B_k$ | 4.6 trust_cap: a second REML-stability mechanism tied to parameter type | 4.7 Summary: the full chain per parameter type | Chapter 5: Worked examples, one per structure | 5.1 ism() | 5.2 dsm() | 5.3 csm(), homogeneous | 5.4 csm(), heterogeneous | 5.5 ar1m() | 5.6 ar2m() / ar3m() / toeplitzm() | 5.7 mam() / ma1m() / ma2m() | 5.8 usm() | 5.9 corgm() | 5.10 fam() / rrm() | 5.11 antem() | 5.12 maternm() | 5.13 sar() | 5.14 car() | 5.15 ownm() | Chapter 6: End-to-end numerical trace, from $\eta$ to the REML objective | 6.1 A 4-observation, 2-line x 2-trait toy data set | 6.2 Step 1 — working parameters to the shape and its precision (Chapter 2-3) | 6.3 Step 2 — precision to Henderson's $C$ matrix (Chapter 1) | 6.4 Step 3 — solving the MME (BLUEs/BLUPs depend on $\eta$) | 6.5 Step 4 — from the MME solution to the REML ingredients (Chapter 4.1) | 6.6 Step 5 — perturbing $\eta$: where the score/AI-matrix machinery lives | Where to go from here

SECTION 1: Covariance structures in vsm() | 1) The product-level variance scale | 2) Covariance-shaping factors | 3) Homogeneous and heterogeneous marginal variances | SECTION 2: Fitting structured models | 1) Compound symmetry | 2) Autoregressive covariance | 3) Fixing covariance parameters | SECTION 3: Complete covariance-structure catalog | 2) Combining two simple effects with covm() | 3) Covariance across several terms with strm() | 4) Equality and scaling constraints with vcc | SECTION 4: CovarianceFactor descriptors | 1) Descriptor fields | 2) Working and reported parameters | Literature

What the error means | Identify the R installation and BLAS | Install the appropriate R packages | Route A: R uses FlexiBLAS | Route B: R uses a recognized OpenBLAS or MKL library directly | If R still uses a reference BLAS | Verify control with a small model | Retry your model | Cluster allocations and startup configuration | Troubleshooting and interpreting usage | References

Overview | Simulated population | Pedigree and relationship inverse | Breeding values and phenotypes | Step 1: REML on a subset of herds | Step 2: evaluation of the whole population | Accuracy of the predicted breeding values | Other ways to give the variance components | Practical notes

SECTION 1: Introduction | Backgrounds in linear algebra | SECTION 2: Topics in quantitative genetics | 1) Marker and non-marker based heritability calculation | 2) Specifying heterogeneous variances in univariate models | 3) Using the vpredict calculator | 3.1) Standar error for heritability | 4) Half and full diallel designs (use of the overlay) | 5) Genomic selection: predicting mendelian sampling | 6) Indirect genetic effects | 7) Genomic selection: single cross prediction | 8) Multivariate genetic models and genetic correlations | 9) Bivariate model with repeated records | SECTION 3: Special topics in Quantitative genetics | 1) Partitioned model | 2) UDU' decomposition | 3) Mating designs | North Carolina Design I (Nested design) | North Carolina Design II (Factorial design) | 4) GWAS by GBLUP | Final remarks | Literature

1) Random slopes | 2) Random slopes and random intercepts (without correlation) | 3) Random slopes and random intercepts (with correlation) | 4) Random slopes with a different intercept | 4) Other models available in sommer but not in lme4 | Literature

1. Where does the spatial part go? | 2. The building block: AR1 correlation | 3. A single trial | 3.1 A field where we know the truth | 3.2 Five candidate models | 3.3 Comparing the models | 3.4 Reading the estimated parameters | 3.6 Reproducing SpATS with P-spline ANOVA | 4. Several trials at once | 4.1 Three simulated trials | 4.2 Shared correlations: a Kronecker product with vsm() | 4.3 Trial-specific correlations: a direct sum with dsumm() | 4.4 Which model should I use? | 4.5 Trials without spatial information: levels= | 4.6 Spatial effects on the G side for several trials | 5. Practical advice | Translating from ASReml-R | Literature

1. The problem the engine solves | 1.1 A structured linear mixed model | 1.2 What is and is not estimated here | 2. Henderson equations and the restricted likelihood | 2.1 The coefficient matrix | 2.2 The REML objective in Henderson form | 2.3 A useful connection to the textbook score | 3. Covariance parameters: legal coordinates for optimization | 3.1 Natural values and working values | 3.2 Derivatives belong to the covariance descriptor | 3.3 Fixed parameters and covariance validity | 4. How AI-REML proposes and accepts updates | 4.1 Score, information, and an update | 4.2 Why blend AI with EM information? | 4.3 Safeguards around the proposed step | 5. Three solvers for the MME system | 5.1 solver="ldlt": sparse direct factorization | 5.2 solver="cholmod": supernodal direct factorization | 5.3 solver="pcg": iterative solves and stochastic traces | 5.4 At-a-glance comparison | 6. Structure-aware shortcuts and where the speed comes from | 6.1 Kronecker products: invert the factors, not the product | 6.2 Diagonal and repeated-block residuals | 6.3 Selected inverse instead of full inverse | 6.4 Reuse what does not change across iterations | 6.5 Matrix-free PCG for compatible REML models | 6.6 Why dense relationship matrices are a different regime | 7. The C++ numerical libraries | 8. A complete small Henderson solve: two environments and three entries | 8.1 Data, ordering, and design matrices | 8.2 The entry relationship and the supplied Ai | 8.3 The environment covariance and the full random covariance | 8.4 Assemble Henderson's equations | 8.5 What the CHOLMOD solve returns | 8.6 Quadratic form and one likelihood evaluation | 8.7 How the four covariance coordinates affect this example | 8.8 The converged fit is a different point | 8.9 What this six-observation example teaches | 9. Choosing settings and interpreting results

1) Single environment model | 2) MET: main effect model | 3) MET: diagonal model (DG) | 4) MET: compund symmetry model (CS) | 5) MET: unstructured model (US) | 6) MET: random regression model | 7) Other GxE covariance structures | 8) Finlay-Wilkinson regression | 9) Factor analytic and reduced rank models | 10) Two stage analysis | Literature

Overview | How the fit works | PQL and IRLS | Dispersion and covariance structures | Offsets and precision weights | Families and examples | Gaussian | Binomial | Poisson | Gamma | Inverse Gaussian | Quasi families | Extracting results and controlling PQL | Practical guidance

1) Optimizing the selection of one feature with a constraint in another feature | 2) Obtaining subsample of a population to maximize a feature while constraining the relationship between individuals in the population | 2a) Best parents for the next generation | 2b) Obtaining optimal N crosses from a population for a given trait/feature | 3) Optimizing a subsample of size N to be representative | 3a) Optimizing a subsample of size N to be representative of its own | 3b) Optimizing a subsample of size N to be representative of another population | 6) How to specify constraints | Gender in breeding | Number of times a parent should be used | 6) Customizing a fitness function (linear regression example) | 7) Travel salesman problem | 8) How to force the initial solutions or founders | 9) How to optimize the number of progeny to produce per cross | Literature