Traditional methods for covariate adjustment of treatment means in designed experiments are inherently conditional on the observed covariate values. In order to develop a coherent general methodology for analysis of covariance, we propose a multivariate variance components model for the joint distribution of the response and covariates. It is shown that, if the design is orthogonal with respect to (random) blocking factors, then appropriate adjustments to treatment means can be made using the univariate variance components model obtained by conditioning on the observed covariate values. However, it is revealed that some widely used models are incorrectly specified, leading to biased estimates and incorrect standard errors. The approach clarifies some issues that have been the source of ongoing confusion in the statistics literature.
"A Multivariate Variance Components Model for Analysis of Covariance in Designed Experiments." Statist. Sci. 24 (2) 223 - 237, May 2009. https://doi.org/10.1214/09-STS294