Two-way analysis of variance

In statistics, the two-way analysis of variance (ANOVA) is an extension of the one-way ANOVA that examines the influence of two different categorical independent variables on one continuous dependent variable. The two-way ANOVA not only aims at assessing the main effect of each independent variable but also if there is any interaction between them.

In 1925, Ronald Fisher mentions the two-way ANOVA in his celebrated book from 1925, Statistical Methods for Research Workers (chapters 7 and 8). In 1934, Frank Yates published procedures for the unbalanced case. Since then, an extensive literature has been produced. The topic was reviewed in 1993 by Yasunori Fujikoshi. In 2005, Andrew Gelman proposed a different approach of ANOVA, viewed as a multilevel model.

Let us imagine a data set for which a dependent variable may be influenced by two factors which are potential sources of variation. The first factor has ${\displaystyle I}$ levels (${\displaystyle i\in \{1,\ldots ,I\}}$) and the second has ${\displaystyle J}$ levels (${\displaystyle j\in \{1,\ldots ,J\}}$). Each combination ${\displaystyle (i,j)}$ defines a treatment, for a total of ${\displaystyle I\times J}$ treatments. We represent the number of replicates for treatment ${\displaystyle (i,j)}$ by ${\displaystyle n_{ij}}$, and let ${\displaystyle k}$ be the index of the replicate in this treatment (${\displaystyle k\in \{1,\ldots ,n_{ij}\}}$).

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