Sub-Balanced Data and the Mixed Analysis of Variance
研究了混合模型中数据平衡性条件，提出了次平衡随机效应的概念，并指出当数据次平衡时，传统ANOVA方法可能不适用，需用协方差矩阵谱分解。
Abstract Abstract In the mixed model, it is well known that balanced data are a sufficient condition for an ANOVA with terms that are independent multiples of chi-squared variables (called a proper ANOVA). In this article a weaker condition, which is both necessary and sufficient, is determined and shown to be equivalent to what is defined here as sub-balanced random effects, a condition easily checked in practice. The usual method of generating an ANOVA by fitting sums of squares for a given ordering of the effects in the model is sometimes inadequate with sub-balanced data, requiring instead a spectral decomposition of the covariance matrix. An example is given. Key Words: Variance componentsBalanced dataSub-balanced dataAnalysis of varianceMixed model