A Note on Multiple Solutions in Multivariate Restricted Maximum Likelihood Covariance Component Estimation

¹ Department of Animal Sciences, University of Illinois, Urbana 61801

Restricted maximum likelihood has gained wide acceptance for estimating variance components by animal breeders. Converged solutions from maximization algorithms are usually considered REML solutions. However, neither the expectation maximization nor the derivative-free downhill simplex algorithm guarantees convergence to the global maximum. To investigate the occurrence of multiple solutions, residual and additive genetic covariance matrices for two traits were estimated by the derivative-free approach using the downhill simplex method and an EM algorithm for a multivariate animal model. Expectation maximization yielded three different converged solutions for the data set used, while downhill simplex identified two of the three expectation maximization solutions. Multiple solutions from both expectation maximization and downhill simplex suggest the existence of local maxima. Visualization of the log-likelihood surface however, did not support this view. Instead, points of inflection in some of the dimensions could be the cause for multiple solutions. The results prove that multiple solutions can exist, casting doubt on the merit of algorithms—like expectation maximization and downhill simplex—that do not guarantee global maximization. Downhill simplex was superior to expectation maximization. In terms of central processing unit time it was faster by a factor of 22 misidentifying only one solution instead of 2 as expectation maximization did.

Key Words: covariance components • restricted maximum likelihood • derivative free

Submitted on October 27, 1989
Accepted on February 20, 1990