Type of analysis¶

Type 1: AI-REML estimation of co-variance component (previous DMUAI)¶

AI-REML for estimation of (co)variance components using Average Information REstricted Maximum Likelihood
The implementation is based on Jensen et al. 1997.

The algorithm use the Average Information (AI) as second differentials of the likelihood
function. The AI matrix is obtained by averaging the information matrices based on observed and expected
information. The implementation can also use Expectation Maximization (EM) to maximize the restricted likelihood
function.

It is only the part of Likelihood that depends on random effect that is maximized. The full Likelihood also
contains a factor (constant) that depends on data and fixed effects. Therefore, the reported “-2LogL +
constant” value can only be used to compare models using the exact same data and fixed effects.

Asymptotic standard errors of estimated (co)variance components are obtained from the Average Information
matrix.  If parameters in the form of interclass correlations and correlations between random effects
(e.g. genetic correlations) the standard errors of these parameter estimates are computed based on a Taylor
series approximation.

AI-REML can yield updates of the parameter vector outside the parameter space. To overcome this problem
different methods are implemented:
     1   AI, but combining AI and EM if an update goes outside the parameter space.
     2   EM based on a algorithm by Robin Thompson
     3   EM based on an algorithm by Esa Mäntysaari
     4   AI, but with step halving if an update goes outside the parameter space.

AI-REML can use sparse computation based on FSPAK subroutines or dens computation based on LAPACK subroutines.

As for the in-core BLUP, the dens computation can use parallel computations on SMP (multi core/CPU) computer.
The default is to parallelize over all over all available CPU’s/cores, but can be limited by setting the
environment variable MKL_NUM_THREADS=n, where n is the number of CPU’s/cores to use.

Type 2: MCMC sampling of dispersion and location parameters (previous RJMC)¶

Module for single and multiple trait Markov Chain Monte Carlo (MCMC) based estimation of location and
dispersion parameters. The implementation is based on iteration on data techniques and can handle
Gaussian, binary, ordered categorical, 2-component mixture, zero inﬂated sequential binary traits and
reaction norm models with unknown environmental gradient.
Sampling of genetic dispersion parameters can be restricted so only information on parental animals is used.
This approach is especially useful for cross-sectional binary traits (Ødegård et al. (2010)).

Type 11: BLUP in core (previous DMU4)¶

This module can be used to predict future outcomes of random effects (e.g. breeding values) and to estimate
fixed effects. The multiple trait mixed model equations are set up and solved using techniques, that requires
that all non-zeros of the whole system is stored in memory of the computer.
The multiple trait mixed model equations can be solved by seven different iterative methods using various
forms of adaptive relaxation techniques from the subroutine package ITPACK (Kicaid et al., 1982), or by
direct methods based on FSPAK (Perez-Enciso et al., 1994) or LAPACK subroutines.

The optimum solver depends on the model used, the amount of data, and the data structure. For sparse system,
FSPAK and ITPACK solvers are the most efficient. The ITPACK solver requires less memory then FSPAK solvers.
Among the FSPAK solvers, method 8 has the smallest  and method 9 the largest memory requirement.
Time requirements for the FSPAK solvers are less for method 9, followed by 10 and 8. If a direct method
cannot be used due to memory requirement,ITPAK  method 1 (JCG) is a good choice.

For dens system as occurs in SNP- and G-BLUP models, the dens solver is the most efficient in terms of
computer time. If run on a SMP (multi core/CPU) computer, the solving step is parallelized over all available
CPU’s/cores, or on the number of CPU’s/cores specified by setting the environment variable MKL_NUM_THREADS=n,
where n is the number of CPU’s/cores to use.

If solutions are obtained by FSPAK or for dense systems by LAPACK, standard error of estimate for fixed
effects and standard error of prediction for random effects in the model are also computed. Irrespectively
of the solver used, standard error of estimate/prediction and correlation among estimates/predictions of
selected fixed or random effects in the model(s) can be obtained. Such information can be used to test
various hypotheses.

Type 12: BLUP-PCG iteration on data (previous DMU5)¶

BLUP-PCG can be used to solve the multiple trait mixed model equations based on iteration on data, and
can to solve much larger systems than BLUP-IC.
BLUP-PCG solves the model based on processing/reading the data file in each round of iteration.
the data file in each round of iteration. The iterative solver is based on the Preconditioned Conjugate
Gradient method.