# 2x2 design - two separate hypotheses and one interaction hypothesis. how many tests?

I am planning to conduct a study with three hypotheses in a 2×2 between subjects design.

Both independent variables are categorical and the dependent variable is metric. Let's say the levels of my first independent variable is A1 and A2 and for my second independent variable B1 and B2.

I have three hypotheses stating that

• (1) A2 > A1,
• (2) B2 > B1 and the interaction
• (3) (A2-A1) > (B2-B1).

How would you recommend testing it? Would you test (1) and (2) in separate one-way ANOVAs and then the (3) in a two-way ANOVA? Or would you rather recommend testing it all at once with one two-way ANOVA? Is there any disadvantage with splitting it into three tests?

EDIT: to be more clear:

The experiment is related to task performance. One variable is a goal level (high goal vs. low goal) and the second variable is the dimension of the goal (speed vs. accuracy). So it is predicted that both goal dimensions will lead to higher task performance (1) & (2), however the effect of the speed goal will be higher than the accuracy goal (3). There are 40 participants per cell, which means a total sample size of n=160. Each participant experiences each condition only once!

• Could you please elaborate on the subject of "2x2 between subject design" ? May 28, 2020 at 17:02
• @Rodolphe Thank you for your quick answer. It is related to task performance. One variable is a goal level (high goal vs. low goal) and the second variable is the dimension of the goal (speed vs. accuracy). so it is predicted that both goal dimensions will lead to higher task performance (1) & (2) , however the effect of the speed goal will be higher than the accuracy goal (3). May 28, 2020 at 17:10
• Thanks for your clarification @FS110. So each subject will experience each of the 4 possible combinations ? Or did you assign randomly each subject to one, and only one, combination ? May 28, 2020 at 17:16
• @Rodolphe each participant randomized only in one condition! (n=40) per cell, so n=160 in total May 28, 2020 at 17:19
• Do you mean that each participant has been assigned to one unique combination and that he/she will experience it 4 times ? (4 repeated measurements by participant in same combination of the two factors) May 28, 2020 at 17:23

ANOVA / linear modelling can include both factors with their simple effects as well as their interactions. No need to split.

In r that would be summary(lm(y~A*B))

That's equivalent to summary(lm(y~1+A+B+A:B))

and you get everything you need to answer all hypotheses at once, more or less.

But before interpretation of the individual effects' significance, the main question : Is my model better nothing - significant ?

And then ( if the answer to the main question is a big yes) :

• Is the effect of A significant ?

• Is the effect of B significant ?

• Is the effect of the interaction of A with B significant ?

All with one line of code... But great care.