Similarly:

> Lastly, I = AC means that we can’t tell the difference between doing nothing at all, compared to adding an egg and adding nuts on top.

But test 0 and test 2 are doing nothing and adding an egg & nuts respectively. What are we missing here?

On further reading, it seems like "doing all three options" should be interpreted to mean "the interaction of all three options"? So we aren't able to tell if an improvement of taste comes from B alone or from the interaction of A, B, and C. I'm mostly guessing though.

As best I can tell from reading https://www.stat.purdue.edu/~yuzhu/stat514s2006/Lecnot/fracf... the unstated assumption here is that we're going to do linear regression where (critically) the "-" case for each condition is -1, and the "+" is +1. This has the surprising-to-me effect of making "I", which looks like it might be the control group based on notation, actually a positive exemplar of AC interaction (and any even-order interaction). You can think of this as a change in basis in how you parse out the effects, where we're talking about

1 -1

-1 1

instead of

0 0

0 1

for an interaction.

I have a gut feeling it's done this way mostly because the tools being used expect things to be expressed this way rather than any conscious choice by experimenters. Through this lens, testing I, B, AC, and ABC, every experiment has a positive effect from AC interaction, and taking B-I, which we might think of as the effect from B, is in this paradigm also sensitive to the ABC interaction and the AB and BC interactions. The "real" effect from B would be approximated as (B + ABC - AC - I)/2, which is exactly the same as the effect from ABC interaction (which is positive when an odd number of its constituents are positive...).

Thanks for the explanation and the link.

So the goal is to find real numbers f_I, f_A, ...f_ABC such that

result = f_I + f_A*v_A + f_B*v_B + f_AB*v_A*v_B + ... + f_ABC*v_A*v_B*v_C

where v_A is 1 or -1 depending on whether A is present or absent in the experiment. f_AB is being abbreviated to AB, which is causing some confusion, since when heading a column, AB means v_A*v_B. The article should say that we can't tell the difference between the effect associated with B and the effect associated the 3-way interaction (for this definition of the effect associated with the 3-way interaction).