Editing Analysis of variance (section)

==For multiple factors==
{{Main|Two-way analysis of variance}}
ANOVA generalizes to the study of the effects of multiple factors. When the experiment includes observations at all combinations of levels of each factor, it is termed [[Factorial experiment|factorial]]. Factorial experiments are more efficient than a series of single factor experiments and the efficiency grows as the number of factors increases.<ref name="Montgomery">Montgomery 
(2001, Section 5-2: Introduction to factorial designs; The advantages of factorials)</ref> Consequently, factorial designs are heavily used.

The use of ANOVA to study the effects of multiple factors has a complication. In a 3-way ANOVA with factors x, y and z, the ANOVA model includes terms for the main effects (x, y, z) and terms for [[Interaction (statistics)|interactions]] (xy, xz, yz, xyz). 
All terms require hypothesis tests. The proliferation of interaction terms increases the risk that some hypothesis test will produce a false positive by chance. Fortunately, experience says that high order interactions are rare.<ref>Belle (2008, Section 8.4: High-order interactions occur rarely)</ref> {{verify source|date=December 2014}}
The ability to detect interactions is a major advantage of multiple factor ANOVA. Testing one factor at a time hides interactions, but produces apparently inconsistent experimental results.<ref name="Montgomery" />

Caution is advised when encountering interactions; Test interaction terms first and expand the analysis beyond ANOVA if interactions are found. Texts vary in their recommendations regarding the continuation of the ANOVA procedure after encountering an interaction. Interactions complicate the interpretation of experimental data. Neither the calculations of significance nor the estimated treatment effects can be taken at face value. "A significant interaction will often mask the significance of main effects."<ref>Montgomery (2001, Section 5-1: Introduction to factorial designs; Basic definitions and principles)</ref> Graphical methods are recommended to enhance understanding. Regression is often useful. A lengthy discussion of interactions is available in Cox (1958).<ref>Cox (1958, Chapter 6: Basic ideas about factorial experiments)</ref> Some interactions can be removed (by transformations) while others cannot.

A variety of techniques are used with multiple factor ANOVA to reduce expense. One technique used in factorial designs is to minimize replication (possibly no replication with support of [[Tukey's test of additivity|analytical trickery]]) and to combine groups when effects are found to be statistically (or practically) insignificant. An experiment with many insignificant factors may collapse into one with a few factors supported by many replications.<ref>Montgomery (2001, Section 5-3.7: Introduction to factorial designs; The two-factor factorial design; One observation per cell)</ref>