Two-Way Analysis of Variance
We have examined the one way ANOVA but we have only considered one factor at a time. Remember, a factor is an Independent Variable (IV), so we have only been considering experiments in which one Independent Variable was being manipulated.
We are now going to move up a level in complexity and consider two factors (i.e., two Independent Variables) simultaneously. Now these two IVs can either be both between groups designs, both repeated measures designs, or a mixed design. The mixed design obviously has one between groups IV and one repeated measures IV. Each IV can also be a true experimental manipulation or a quasi experimental grouping (i.e., one in which there was no random assignment and only pre-existing groups are compared).
If a significant F-value is found for one IV, then this is referred to as a significant main effect. However, when two or more IVs are considered simultaneously, there is also always an interaction between the IVs - which may or may not be significant.
An interaction may be defined as:
There is an interaction between two factors if the effect of one factor depends on the levels of the second factor. When the two factors are identified as A and B, the interaction is identified as the A X B interaction.
Often the best way of interpreting and understanding an interaction is by a graph. A two factor ANOVA with a nonsignificant interaction can be represented by two approximately parallel lines, whereas a significant interaction results in a graph with non parallel lines. Because two lines will rarely be exactly parallel, the significance test on the interaction is also a test of whether the two lines diverge significantly from being parallel.
If only two IVs (A and B, say) are being tested in a Factorial ANOVA, then there is only one interaction (A X B). If there are three IVs being tested (A, B and C, say), then this would be a three-way ANOVA, and there would be three two-way interactions (A X B, A X C, and B X C), and one three-way interaction (A X B X C). The complexity of the analysis increases markedly as the number of IVs increases beyond three. Only rarely will you come across Factorial ANOVAs with more than 4 IVs.
A word on interpreting interactions and main effects in ANOVA. Many texts including Ray (p. 198) stipulate that you should interpret the interaction first. If the interaction is not significant, you can then examine the main effects without needing to qualify the main effects because of the interaction. If the interaction is significant, you cannot examine the main effects because the main effects do not tell the complete story. Most statistics texts follow this line. But I will explain my pet grievance against this! It seems to me that it makes more sense to tell the simple story first and then the more complex story. The explanation of the results ends at the level of complexity which you wish to convey to the reader. In the two-way case, I prefer to examine each of the main effects first and then the interaction. If the interaction is not significant, the most complete story is told by the main effects. If the interaction is significant, then the most complete story is told by the interaction. In a two-way ANOVA this is the story you would most use to describe the results (because a two-way interaction is not too difficult to understand). One consequence of the difference in the two approaches is if, for example, you did run a four-way ANOVA and the four-way interaction (i.e., A X B X C D) was significant, you would not be able to examine any of the lower order interactions even if you wanted to! The most complex significant interaction would tell the most complete story and so this is the one you have to describe. Describing a four-way interaction is exceedingly difficult and would most likely not represent the relationships you were intending to examine and would not hold the reader's attention for very long. With the other approach, you would describe the main effects first, then the first order interactions (i.e., A X B, A X C, A X D, B X C, B X D, C X D) and then the higher order interactions only if you were interested in them! You can stop at the level of complexity you wish to convey to the reader.
Another exception to the rule of always describing the most complex relationship first is if you have a specific research question about the main effects. In your analysis and discussion you need to address the particular hypotheses you made about the research scenario, and if these are main effects, then so be it! However, not all texts appear to agree with this approach either!
For the sake of your peace of mind, and assessment, in this unit, there will be no examination questions or assignment marks riding on whether you interpret the interaction first or the main effects first. You do need to realise that in a two-way ANOVA, if there is a significant interaction, then this is the story most representative of the research results (i.e., tells the most complete story and is not too complex to understand).