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Estimation of Components of Variance and Covariance by Symmetric Differences Squared and Minimum Norm Quadratic Unbiased Estimation: a Comparison¹

The Ohio State University, Columbus 43210

Abstract

Standard errors were obtained of four different estimators of components of variance and covariance applied to five specific data set designs. The designs were for estimating the genetic and environmental components of variance and covariance of a maternal model. The four estimators were symmetric differences squared (SDS), minimum norm quadratic unbiased estimation (MINQUE) with all prior values for the components of variance and covariance set to zero except for the environmental variance (MINQUE(0)), MINQUE with all prior values for the components of variance and covariance set to 1 (MINQUE(1)), and minimum variance quadratic unbiased estimation (MIVQUE). The standard errors of SDS, MINQUE(0), MINQUE(1), and MIVQUE estimates of variances due to direct and maternal additive effects of genes, covariance between direct and maternal additive effects of genes, variance due to environmental effects common to maternal half- and full-siblings and variance due to temporary environmental effects were computed for data sets consisting of 200 sets of Thompson's A or B design (1976), or 200 sets of Eisen's 1, 2, or 3 design (1967). Efficiency was computed as standard error of MIVQUE divided by standard error of SDS, MINQUE(0), or MINQUE(1). Symmetric differences squared weighted by the inverse of the environmental variance-covariance matrix (WSDS) is equivalent to MINQUE(0). Over all parameter sets, components estimated, and designs, the efficiency of MINQUE(1) ranged from .80 to .99, the efficiency of SDS ranged from .30 to .95 and the efficiency of MINQUE(0) ranged from .33 to .99. On the average, MINQUE(0) was more efficient than SDS and MINQUE(1) was more efficient than MINQUE(0). The efficiency of MINQUE(1) was insensitive to differences between the true components of variance and covariance and their prior values.

¹ Salaries and research support provided by state and federal funds appropriated to the Ohio Agric. Res. and Development Center, The Ohio State Univ. Journal Article No. 125-88.

² Present address: Roman L. Hruska Meat Anim. Res. Center, ARS, USDA, Clay Center, NE 68933.

³ Dairy Sci. Dept., The Ohio State Univ.