An overview of variance component estimation

Variance components estimation originated with estimating error variance in analysis of variance by equating error mean square to its expected value. This equating procedure was then extended to random effects models, first for balanced data (for which minimum variance properties were subsequently established) and later for unbalanced data. Unfortunately, this ANOVA methodology yields no optimum properties (other than unbiasedness) for estimation from unbalanced data. Today it is being replaced by maximum likelihood and restricted maximum likelihood (REML) based on normality assumptions and involving nonlinear equations that have to be solved numerically. There is also minimum norm quadratic unbiased estimation which is closely related to REML but with fewer advantages.