# S1. 方差分析的方法

Venkateswarlu K., 1997 Estimation of Variance Components Based on a Diallel Model Involving Maternal and Maternal Interaction Effects. Biometrical J. 39: 287–295.

There are several methods available in the literature for the estimation of variance components. These are:

1. Analysis of Variance (ANOVA),
2. Maximum Likelihood (ML) (HARTLEY and RAO, 1967),
3. Restricted Maximum Likelihood (REML) (PATTERSON and THOMPSON, 1971; CORBEIL and SEARLE, 1976),
4. Minimum Norm Quadratic Unbiased Estimation (MINQUE) (RAo,1971a, 1971b),
5. Henderson’s method III (HENDERSON, 1953),
6. Quadratic Least Squares Estimation (QLSE) (SEELY,1969, 1970, 1971; SEELY and ZYSKIND, 1971; YUAN, 1977) and
7. Symmetric Sums (SS)(KOCH,1967,1968) methods.

When-the data is balanced ANOVA, MINQUE, REML(ignoring non negativity), SSA and QLSE will give same estimators and they areknown to have properties (under normality) like:

1. unbiasedness;
2. minimum variance among all unbiased estimators which are quadratic functionsof observations;
3. known sampling variances under normality.

When the data is balanced **ANOVA, SSA **and QLSE **is **easy to compute and also does not require distributional assumptions but ML and REML requires distribution alassumptions and involves inverting large matrices. In case of MINQUE and Henderson method 3 distributional assumptions are not required but still have aproblem of inverting a large matrices. QLSE and SS estimators reduces to **ANOVA **estimators in the case ofbalanced data.
However often an experimenter has unbalanced data (i.e. data withunequal number of observations in the sub classes). More often the datacollection cannot be controlled to ensure balance, thus it is frequently necessary to estimate variance components with unbalanced data. For estimationof variance components with unbalanced data one can use methods like ML, REML, Hendersonmethod3, MINQUE, SSA and QLSE. But the complexity of using methods like ML, REMLand MINQUE will be unimaginable when considering Triallel and Double cross matingdesigns with unbalanced data.

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