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positive While definite our estimates are unbiassed case) as simple functions is correct ofx, of our observed of z, is if the model definite, and the estimate if the Error MS is positive both mean the estimate matrices will often not be positive with error even if our definite. Vol13 calculate k, the number of progeny per sire in the balanced matrices. Then we can estimate 416 these component matrices (given with we can Proc.
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We might suppose the Error MS matrix is a matrix sampled from a distribution expected with expected value 2 E and that the Sire MS matrix is sampled from a distribution value I: E + kz. Assuming these mean square matrices are positive definite, the problem arises when we impose an expectation model on them.
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missing data may lead to non-positive definite mean square matrices. However, an uneven distribution and amount of necessarily but usually will be positive definite. Both matrices will always be non-negative semidefinite and will be positive definite unless all the residuals/effects are zero for one trait or the residuals/effects for the When data is unbalanced, these mean square matrices are not traits are perfectly correlated. Sire Mean Square (Sire MS) matrix can be calculated from the sums of squares and cross-products (weighted) of the sire effects. The Error Mean Square (Error MS) Similarly, the matrix can be calculated from sums of squares and cross products of the residuals. HOW NON-POSITIVE DEFINITE MATRICES ARISE Consider the simple sire model with 2 traits and balanced data. I will consider first why this problem occurs and then how this problem is addressed in ASREML. It is matrix for some for the proposed space. Thus, strategies are required to produce estimates which are in the parameter with estimating the estimate. the estimated matrix is outside the theoretical parameter space model. Assuming we have an appropriate variance model, there is a problem associated variance component matrices which is independent of the algorithm used to obtain that sampling variation in the data often supports a non-positive definite components, ie. 1995) matrix and is generally quite efficient. ASREML uses a quadratic convergence method based on the Average Information (Gilmour et al. 1971) is now the preferred method and is implemented in many programs including ASREML (Gilmour et al. These include genetic and residual matrices between traits and between times which are used to calculate Restricted Maximum Likelihood (Patterson and Thompson heritabilities and genetic correlations. Keywords: Genetic correlation estimation, average information, REML, variance components INTRODUCTION Animal breeders are often interested in estimating variance/co-variance matrices from data. Difficulty arises when the assumed model is not correct and because of sampling variation in the matrices. Gilmour NSW Agriculture, Orange Agricultural Institute, Forest Road, Orange, NSW 2800 SUMMARY The paper explains why it is often difficult to estimate co-variance matrices from data and describes the variance structures available in ASREML to investigate the problems. Vol13 VARIANCE STRUCTURES AVAILABLE IN ASREML A.