# Supported features

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### Supported operations[edit]

Currently **lmt** support the following operations on linear mixed models:

- Solving for BLUP and BLUE solutions conditional on supplied variances for random and fixed factor, respectively;
- Gibbs sampling of variance components in single pass and blocked mode;
- MC-EM-REML estimation of variance components
- Sampling (block)diagonal elements of the inverse of the mixed model coefficient matrix
- Solving for (block)diagonal elements of the inverse of the mixed model coefficient matrix

### Supported factors and variables[edit]

**lmt** supports

- fixed
- random factors
- classification variables
- continuous co-variables, which can be nested. For continuous co-variables
**lmt**support user-defined polynomials and hard coded Legendre polynomials up to order 6. - genetic group co-variables

All classification and co-variables can be associated to a fixed or random factor.

### Supported variance structures[edit]

For random factor **lmt** supports variance structures of

- structure $$\Gamma\otimes\Sigma$$, where $$\Sigma$$ is an dense symmetric positive definite matrix, and
- $$\Theta_L(\Gamma\otimes I_{\Sigma})\Theta_L^{'}$$, where $$\Theta$$ is symmetric positive definite block-diagonal matrix of $$n$$ symmetric positive definite martices $$\Sigma_i, i=1,..,n$$, $$\Theta_L$$ is the lower Cholesky factor of $$\Theta$$ and $$I_{\Sigma}$$ is an identity matrix of dimension $$\Sigma_i$$.

When solving linear mixed models $$\Sigma$$ and $$\Gamma$$ are user determined constants, whereas when estimating variances $$\Gamma$$ is a user determined constant and $$\Sigma$$ is a function of the data.

Supported type for $$\Gamma$$ are

- an identity matrix
- an arbitrary positive definite diagonal matrix
- a pedigree-based numerator relationship matrix $$A$$ which may contain meta-founders
- a pedigree- and genotype-based relationship matrix $$H$$ which may contain meta-founders
- a user-defined(u.d.) symmetric, positive definite matrix of which inverse is supplied
- as a sparse upper-triangular matrix stored in csr format
- as a dense matrix

- a co-variance matrix of a selected auto-regressive process

Further **lmt** supports special variance structures which are not covered by the above description

- SNP-BLUP variance for the model of Liu and Goddard 2014 with the option to model marker co-variances as above.

### Supported linear mixed model solvers[edit]

**lmt** supports

- a direct solver requiring to explicitly build the linear mixed model equations left-hand-side coefficient matrix($$C$$)
- an iteration-on-data pre-conditioned gradient solver which
**does not**require $$C$$

### [edit]

- direct use of genomic marker data
- building of genomic relationship matrices($$G$$) from supplied genomic data
- uploading of a u.d. $$G$$
- adjustment of $$G$$ to $$A_{gg}$$ in SS-G-BLUP and SS-SNP-BLUP
- solving Single-Step-G-BLUP models
- Variance component estimation for Single-Step-G-BLUP models
- solving Single-Step-T-BLUP models
- solving Single-Step-SNP-BLUP models
- calculation of true H matrix diagonal elements
- all Single-Step models can be run from "bottom-up", that is the user supplies the genotypes and all necessary ingredients(e.g. $$G$$) are built on the fly.

### Supported pedigree types[edit]

- ordinary pedigrees
- probabilistic pedigrees with an unlimited number of parent pairs per individual
- genetic group pedigrees
- meta-founder pedigrees
- ignoring of inbreeding
- iterative derivation of inbreeding coefficients