Monte Carlo Comparison of ANOVA, MIVQUE, REML, and ML Estimators of Variance Components

Abstract

For the one-way classification random model with unbalanced data, we compare five estimators of cr.' and cf, the among-and within-treatments variance components: analysis of variance (ANOVA), maximum likelihood (ML), restricted maximum likelihood (REML), and two minimum variance quadratic unbiased (MIVQUE) estimators. MIVQUE (0) is MIVQUE with a priori values 62 = 0 and 5: = 1; MIVQUE(A) is MIVQUE with the ANOVA estimates used as a priori's, We enforce nonnegativity for all estimators, setting any negative estimate to zero in accord with usual practice. The estimators are compared through their biases and MSE's, estimated by Monte Carlo simulation. Our results indicate that the ANOVA estimators perform well, except with seriously unbalanced data when u.'/u: > 1; ML is excellent when u:/c: < 0.5, and MIVQUE(A) is adequate; further iteration to the REML estimates is unnecessary. When u:/u: 2 1, MIVQUE(0) (the default for SASS PROCEDURE VAR-COMP) is poor for estimating 0,' and very poor for u:, even for just mildly unbalanced data.

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“…Especially when using badly unbalanced data, one should consider using instead maximum likelihood (ML) or restricted maximum likelihood (REML) estimators . For a more detailed discussion and comparison of these variance components estimators, see, for example, Swallow & Monahan (1984) .…”

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“…However, the ML estimation of the variance components has been criticized for its failure to take into account the loss of degrees of freedom resulting from estimating the coefficients on the explanatory variables. As a consequence, the ML estimates of variance components are generally biased downward (Swallow and Monahan 1984). These "deficiencies" are eliminated in the Restricted Maximum Likelihood (REML) method set forth in general form by Patterson and Thomson (1971).…”