Supported features

Supported operations

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

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

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
• genetic groups absorbed into $$A$$ or $$H$$
• 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

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 $$C$$

Supported features related to genomic data

• 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

• 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

Supported features related to meta-founders and genetic groups

• meta-founders can be modeled for all $$\Gamma$$ which contain $$A$$(.e.g. $$A$$, $$H$$ for ssGBLUP, ssGTBLUP and ssSNPBLUP)
• genetic groups can be modeled as an extra factor or can be absorbed into all $$\Gamma$$ which contain $$A$$