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, Statistical Methods for Research Workers (chapters 7 and 8).
In 1934, Frank Yates published procedures for the unbalanced case.
The topic was reviewed in 1993 by Yasunori Fujikoshi.
[2] In 2005, Andrew Gelman proposed a different approach of ANOVA, viewed as a multilevel model.
[3] Let us imagine a data set for which a dependent variable may be influenced by two factors which are potential sources of variation.
We represent the number of replicates for treatment
be the index of the replicate in this treatment (
From these data, we can build a contingency table, where
, and the total number of replicates is equal to
The experimental design is balanced if each treatment has the same number of replicates,
In such a case, the design is also said to be orthogonal, allowing to fully distinguish the effects of both factors.
data points, for instance via a histogram, "probability may be used to describe such variation".
The two-way ANOVA models all these variables as varying independently and normally around a mean,
Specifically, the mean of the response variable is modeled as a linear combination of the explanatory variables:
is the additive main effect of level
from the first factor (i-th row in the contingency table),
is the additive main effect of level
from the second factor (j-th column in the contingency table) and
is the non-additive interaction effect of treatment
from both factors (cell at row i and column j in the contingency table).
Another equivalent way of describing the two-way ANOVA is by mentioning that, besides the variation explained by the factors, there remains some statistical noise.
This amount of unexplained variation is handled via the introduction of one random variable per data point,
random variables are seen as deviations from the means, and are assumed to be independent and normally distributed:
Following Gelman and Hill, the assumptions of the ANOVA, and more generally the general linear model, are, in decreasing order of importance:[5] To ensure identifiability of parameters, we can add the following "sum-to-zero" constraints:
In the classical approach, testing null hypotheses (that the factors have no effect) is achieved via their significance which requires calculating sums of squares.
Testing if the interaction term is significant can be difficult because of the potentially-large number of degrees of freedom.
[6] The following hypothetical example gives the yields of 15 plants subject to two different environmental variations, and three different fertilisers.
Five sums of squares are calculated: Finally, the sums of squared deviations required for the analysis of variance can be calculated.