不平衡数据的方差分析：生态学和进化学的更新。

Analysis of variance with unbalanced data: an update for ecology & evolution.

Institute of Environmental Sciences, University of Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland.

J Anim Ecol. 2010 Mar;79(2):308-16. doi: 10.1111/j.1365-2656.2009.01634.x. Epub 2009 Dec 3.

Abstract

Factorial analysis of variance (anova) with unbalanced (non-orthogonal) data is a commonplace but controversial and poorly understood topic in applied statistics. 2. We explain that anova calculates the sum of squares for each term in the model formula sequentially (type I sums of squares) and show how anova tables of adjusted sums of squares are composite tables assembled from multiple sequential analyses. A different anova is performed for each explanatory variable or interaction so that each term is placed last in the model formula in turn and adjusted for the others. 3. The sum of squares for each term in the analysis can be calculated after adjusting only for the main effects of other explanatory variables (type II sums of squares) or, controversially, for both main effects and interactions (type III sums of squares). 4. We summarize the main recent developments and emphasize the shift away from the search for the 'right'anova table in favour of presenting one or more models that best suit the objectives of the analysis.

摘要

非平衡（非正交）数据的析因方差分析（ANOVA）在应用统计学中是一个常见但有争议且理解不深的话题。
我们解释了 ANOVA 按顺序（I 型平方和）计算模型公式中每个项的平方和，并展示了调整后的平方和 ANOVA 表如何由多个顺序分析组合而成的综合表。对于每个解释变量或交互作用，都会进行不同的 ANOVA，因此每个项在模型公式中依次放在最后，并根据其他项进行调整。
可以仅根据其他解释变量的主效应（II 型平方和）或有争议地根据主效应和交互作用（III 型平方和）来计算分析中每个项的平方和。
我们总结了主要的最新发展，并强调了从寻找“正确”的 ANOVA 表转向支持呈现最适合分析目标的一个或多个模型。