Looking for an explanation to determine the conceptual and statistical difference between one way with four groups vs. two way ANOVA.

(I am conjuring up this situation). Situation – first IV is alcohol use, levels: Yes, No. Second IV is Migraine headaches, Levels: Yes, No

Dependent variable is a memory test, continuous variable, # of words recall.

What is the difference if I do this:

One way ANOVA (combine the levels of each group and create four groups): four groups – group 1 is alcohol use "yes", Migraine "Yes"; group 2 is alcohol use "yes", Migraine "No"; group 3 is alcohol use "no", Migraine "Yes"; group 4 is alcohol use "no", Migraine "no".

Two way ANOVA: first IV – alcohol use, yes or no, second IV – migraine use, yes or no; then interaction between the two IV's.

When I conduct my own version of these tests (made up the data)...for the one way ANOVA, the F is 2.022, p = .118. for the two way anova, first IV F is 1.94, p = .16, second IV F is .67, p = .41, interaction F is 1.32, p = .25.

So the interaction F value for two IV's is slightly different than the F value for the one way ANOVA with four groups.

Conceptually, I am wondering what the interaction is saying. With the four groups, each group is treated as if they are independently formed and unique groups. Each group is unique in contributing something to the variability in the DV. But with an interaction, we are saying that the level of the DV for the first IV group will change depending on the status of the second IV group. So for the IV, each category in the group will contribute to the level of the IV, but then if we add an interaction, conceptually we are saying that the relationship between the first IV and the DV will change depending on which group you are in.

As I reflect, it would seem that "collapsing" two IV's with two levels each into one group with four levels and running a one-way ANOVA might be conceptually incorrect if we are truly interested in the combination, i.e., interaction between the two IV's.

I think the issue of "collapsing" two IV's with two levels into a One-way ANOVA has to do with obtaining the post-hoc tests in order to pinpoint where the interaction has an effect. In SPSS, I've seen it where a one-way ANOVA is conducted with two IV's with two levels, but collapsed into four groups for a one-way ANOVA only for the purpose of obtaining post-hoc tests across all four "groups". The purpose of "converting" an ANOVA with two IV's into a one-way ANOVA with one IV with four groups is for the convenience of obtaining the post-hoc tests (can't run post-hoc tests on the interaction). But for a two-way ANOVA, you can use an EMmeans statement with a Compare option to obtain a pairwise comparison between the means to determine which aspects of the interaction are significant. So that would eliminate that need to "collapse" two Iv's with two levels into four groups.

Interested in receiving comments or additional reading on this issue.

Thanks in advance.

Peter Ji