Algorithms

Solving Linear Mixed model Equations

lmt supports two types of solver for solving MME's: a direct solver and an iterative solver

Iterative solver

The iterative solver uses the preconditioned conjugate gradient method and is lmt's default solver. It does not require the explicit construction of any mixed model equation, and is therefore less resource demanding than the direct solver. That is, many models which cannot be solved using the direct solver can still be solved using the iterative solver. Even for small models the iterative solver usually outperforms the direct solver in terms of total processing time.

The iterative solver has converged to a stable solution if $$log_e\left(\sqrt{\frac{(Cx-b)'(Cx-b)}{b'b}}\right)<t$$, where $$C$$ is the mixed-model coefficient matrix, $$x$$ is the solution vector, $$b$$ is the right-hand side and $$t$$ is the convergence threshold. The default convergence threshold is -18.42, which is equivalent to $$ \sqrt{\frac{\sum_{i=1}^n ((Ax)_i-b_i)^2}{n}/\frac{\sum_{i=1}^n b_i^2}{n}}<10^{-8} $$

Direct solver

The direct solver requires the mixed model coefficient matrix to be build and all Kronecker products to be resolved. This can be quite memory demanding and should therefore be used carefully. The direct solver uses a Cholesky decomposition and forward-backward-substitution to solve the mixed model equation system, where especially the decomposition step can be very resource demanding and time consuming.

Gibbs sampling of variance components

Single pass Gibbs sampling

lmt's single pass Gibbs sampling algorithm is described in . In short, all location parameters are drawn from their joint conditional posterior distribution. Note that this requires solving the mixed model equation system once per iteration which usually leads to a substantial increase in processing time.

Blocked Gibbs sampling

For random factors lmt's blocked Gibbs sampler draws correlated location parameters within factor level from their joint conditional posterior distribution. Location parameters of fixed factors are drawn in scalar mode from their fully conditional posterior.