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The statistical area of methods of variance component estimation (VCE) has seen numerous changes, improvements, and advancements in the last 50 years. Thus, a complete history of the evolution of VCE methods would be very interesting especially if comments about the discoverers of methods were included, and about how various methods came into existence. Some of that history is included in the supplemental notes.
Because of the evolution of methods, the teaching of VCE methods for animal breeders can be very cumbersome if all of the historical developments are covered in great detail. On one hand, an historical perspective is needed so that history does not repeat itself and it does provide a good overview of methods that do not work correctly in animal breeding situations. On the other hand, a complete historical coverage of VCE methods would require an entire course of its own, and in the end only a couple of methods would be of immediate importance for a future researcher. Thus, in this course only the simple basics of VCE estimation will be reviewed and the details for two methods will be given. Those methods are REML (restricted maximum likelihood) and the Bayesian approach using Gibbs Sampling as the computational tool for obtaining Bayesian estimates. Another current method, known as Method R, will not be covered and is, in general, not advocated for use in animal breeding (my opinion, at least) because it has been shown by Flavio Schenkel that Method R can give very biased estimates of variances when relationship matrices are not complete and selection has been practiced (which is the usual situation in animal breeding).
By definition, variances are positive, non-zero quantities. Variances are used to derive heritabilities, repeatabilities, prediction error variances or reliabilities of genetic evaluations. They assist in design of experiments to determine the necessary sample size to detect significant differences. They are useful in predicting expected genetic change. Thus, in order to evaluate livestock, animal breeders must firstly estimate the genetic variances and covariances to be used.
Variances are commonly estimated from quadratic forms, which
are simply weighted sums of squares of the observations. The different
methods of VCE that exist merely define how those quadratic forms
are to be calculated and what to do with them after you have
A quadratic form is a scalar
quantity of the form
where Q is assumed to be
symmetric. If V =
Variance of Quadratic Forms
The variance of a quadratic form is given by
The best way to describe unbiased methods of estimation is to
give a small example with only three observations. Let
In this example, there are 3 unknown variances to be estimated,
and consequently, at least three quadratic forms are needed in
order to estimate the variances.
The -matrices are the 'weights' of the observations in the
quadratic forms. These matrices differ depending on the method of
estimation that is chosen. Below are three arbitrary -matrices
that were chosen such that
not necessarily correspond to any known method of estimation, but
are for illustration of the calculations.
The expectations of the quadratic forms are
Unbiased methods of estimation required that the values of the
quadratic forms be equated to their corresponding expectations,
which gives a system of equations to be solved, such as
In this case, the equations
Normally, the variance-covariance matrix of the estimates, commonly
known as the sampling variances of the estimates, were never
actually computed during the days of unbiased methods due to their
computational complexity. However, with today's computers their
calculation can still be very challenging and usually impossible.
For small examples, the calculations can be easily demonstrated.
In this case,
To get numeric values for these variances, the true components need
to be known. Assume that the true values are
then the variance
of w1 is
Variance of Heritability
Often estimates of ratios of functions of the variances are needed
for animal breeding work, such as heritabilities, repeatabilities,
and variance ratios. Let such a ratio be denoted as a/c where
From Osborne and Patterson (1952) and Rao (1968) an approximation to
the variance of the ratio is given by
Another approximation method assumes that the denominator has been
estimated fairly accurately, so that it is considered to be a
Properties of Estimators
Methods of estimation of variance components differ in the properties of their estimates. Due to the complexity of estimating non-zero scalar quantities from quadratic forms, there is no method that can possibly include all of the desirable properties that animal breeders would like to have. Below is a description of the properties that animal breeders would like to see in a VCE method.
Generally, researchers tend to use software that is readily available to them. Sometimes the software does not cover the precise model that the researcher would like to employ, or there may be a limit to the amount of data that can be included. I have found it best to write specific software for a specific model and dataset, but not all researchers have the time or experience to do this. Also, writing one's own software there is always the possibility for errors in the code which may or may not be detected.
This LaTeX document is available as postscript or asAdobe PDF.Larry Schaeffer