Chapter 1
Introduction

Purpose

WOMBAT is a program to facilitate analyses fitting a linear, mixed model via restricted maximum likelihood (REML). It is assumed that traits analysed are continuous and have a multivariate normal distribution.

WOMBAT is set up with quantitative genetic analyses in mind, but is readily applicable in other areas. Its main purpose is the estimation of (co)variance components and the resulting genetic parameters. It is particularly suited to analyses of moderately large to large data sets from livestock improvement programmes, fitting relatively simple models. It can, however, also be used as simple generalised least-squares program, to obtain estimates of fixed and predictions (BLUP) of random effects. In addition, it provides the facilities to simulate data for a given data and pedigree structure, invert a matrix or combine estimates from different analyses.

WOMBAT replaces DfReml [29, 30] which has been withdrawn from distribution at the end of 2005.

Features

WOMBAT consists of a single program. All information on the model of analysis, input files and their layout, and (starting) values for (co)variance components is specified in a parameter file. A large number of run options are available to choose between (partial) analyses steps, REML algorithms to locate the maximum of the likelihood function, strategies to re-order the mixed model equations, and parameterisations of the model.

• WOMBAT accommodates standard uni- and multivariate analyses, as well as random regression (RR) analyses, allowing a wide range of common models to be fitted and offering a choice between full and reduced rank estimation of covariance matrices.
• WOMBAT incorporates the so-called ‘average information’ (AI) algorithm, and the standard (EM) as well as the ‘parameter expanded’ (PX) variant of the expectation-maximisation algorithm. In addition, derivative-free maximisation via Powell’s method of conjugate directions or the Simplex procedure is available. By default, WOMBAT carries out a small number of PX-EM iterates to begin with, then switches to an AI REML algorithm.
• Computational efficiency and memory requirements during estimation depend strongly on the amount of ‘fill-in’ created during the factorisation of the coefficient matrix. This can be reduced by judicious ordering of the equations in the mixed model, so that rows and columns with few elements are processed first. Several ordering procedures aimed at minimising fill-in are available in WOMBAT, including MeTis [22], a multilevel nested dissection procedure which has been found to perform well for large data sets from livestock improvement programmes.
• WOMBAT allows for analyses to be broken up into individual steps. In particular, carrying out the ‘set-up’ steps separately facilitates thorough checking and allows memory requirements in the estimation step to be minimised.
• Generally, WOMBAT assumes covariance matrices to be estimated to be unstructured. Estimation can be carried out on the ’original’ scale, i.e. by estimating the covariance components directly, or by reparameterising to the matrices to be estimated elements of the Cholesky factors of the covariance matrices. The latter guarantees estimates within the parameter space, in particular when combined with a transformation of the diagonal elements to logarithmic scale. Reduced rank estimation, equivalent to estimating the leading principal components only, is readily carried out by estimating the corresponding columns of the respective Cholesky factor(s) only.

WOMBAT offers few features related to data editing, such as selection of subsets of records, transformation of variables or tabulation of data features. There are a number of general statistical packages to choose from which perform these tasks admirably, including free software such as the R package [21].

Remarks

This document is a manual with instructions how to use WOMBAT. It does not endeavour to explain restricted maximum likelihood estimation and related issues, such as approximation of sampling errors, likelihood ratio test, use of information criteria, or how to assess significance of fixed effects.

Throughout the manual, it is assumed that users have a thorough knowledge of mixed linear models and a sound understanding of maximum likelihood inference in general. Numerous textbooks covering these topics are available. To get the best from WOMBAT, users should also be familiar with some of the technical aspects of REML estimation, in particular properties of various algorithms to maximise the likelihood, ordering strategies and parameterisations – otherwise many of the (advanced) options provided will not make a great deal of sense.