Transformation Algorithms in Analysis of Single Trait and of Multitrait Models with Equal Design Matrices and One Random Factor per Trait: A Review

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Abstract
Transformation algorithms for models with two variance components per trait are reviewed and illustrated with a numerical example. The emphasis is on multiple trait models with equal design matrices. Algorithms of canonical, “Cholesky,” and Householder transformations are discussed. The series of transformations offers an alternative that drastically reduces the amount of computation per round of the iterative expectation maximization algorithm for estimating (co)variance components by the restricted maximum likelihood method. After all the transformations are carried out, no matrices need to be inverted and the computations in each round of the iteration process can be evaluated in linear time. Thus, in practice, after the initial computing work is done, any number of iterations can be performed with ease. This allows the use of conservative stopping criteria. The stopping criteria often need to be conservative because considerable changes in parameter estimates can occur during later rounds of the iteration process, even though the change per round is very small.

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