C7 A list of the possible effects in designs with two, three, or four IVs

Designs involving more than one IV are commonly analysed using the statistical technique known as Analysis of Variance (ANOVA) – see Section B4.2 of this Web site. The standard approach is to test all the possible sources of variation – interactions as well as main effects – for statistical significance. Table C5 lists the main effects and interactions that you can get for experiments with two, three and four IVs. Note that you do not need to know whether the IVs use unrelated or related samples in order to be able to predict how many main effects and interactions there can be. (You do, however, need to know this in order to know how to analyse them properly.) Nor do you need to know how many levels there are on each IV. Note also that whereas there is potentially one main effect for every IV in the design, after two IVs, each further IV increases the number of potential interactions by more than one, and the more IVs that you have, the more interactions are added each time you add another IV. The complexity this generates for designing your study and interpreting your findings is one reason why I cautioned you in the book against designing experiments involving more than three IVs. Another reason – and the main one – is the difficulty of interpreting three- or four-way interactions.

Summary

Table C5
The Potential Main Effects and Interactions in Designs Employing Two to Four Independent Variables

Experiment with 2 IVs	Potential main effects:	IV1
		IV2
	Potential two-way interaction:	IV1 x IV2
Experiment with 3 IVs	Potential main effects:	IV1
		IV2
		IV3
	Potential two-way interactions:	IV1 x IV2
		IV1 x IV3
		IV2 x IV3
	Potential three-way interaction:	IV1 x IV2 x IV3
Experiment with 4 IVs	Potential main effects:	IV1
		IV2
		IV3
		IV4
	Potential two-way interactions:	IV1 x IV2
		IV1 x IV3
		IV1 x IV4
		IV2 x IV3
		IV2 x IV4
		IV3 x IV4
	Potential three-way interactions:	IV1 x IV2 x IV3
		IV1 x IV2 x IV4
		IV1 x IV3 x IV4
		IV2 x IV3 x IV4
	Potential four-way interaction:	IV1 x IV2 x IV3 x IV4

You can use Table C5 to check against your analyses when you run experiments involving the simultaneous manipulation of two, three or four IVs. It gives you the maximum number of main effects and interactions possible with such designs. If you find that Table C5 mentions an effect that does not appear in your output – and you are sure that you are not simply misreading your output – likely explanations for this are that either you do not have data for all the possible combinations of the levels of your IVs or something is wrong with the way that you specified the analysis.

When you analyse your findings you can find any or all of the effects displayed in Table C5 are statistically significant. That is, the main effect plus any or all of the interactions involving the same IV may be statistically significant, or that only the main effect, but not any of the interactions involving that IV, are statistically significant, or there may be no statistically significant main effect, but some or all of the interactions involving that IV may be statistically significant.