# How to choose the right analysis for a partially nested design?

I am not even sure if it is appropriate to call this scenario a partially nested design, but I settled for it for lack of a better term.

My design is similar to the following in terms of its overall structure: Suppose we are interested in the effects of (i) caffeine intake (IV 1) and (ii) room temperature (iv 2) on how well people do on a test (DV).

Life would be easy, of course, if this was a 2x2 Factorial design where all 4 possible combinations of the two IVs were available (e.g., +Caffeine/Cold room, -Caffeine/Cold room, +Caffeine/Warm room, -Caffeine/Warm room). But for a reason which does not apply well to this toy example, I could not fully cross the two IVs I am working with. So I have +Caffeine/Cold room, -Caffeine/Cold room, and +Caffeine/Warm room, but not the final combination of -Caffeine/Warm room. So not all combinations are present. If you are in the cold room, you may be either +caffeine or -caffeine, but if you are in the warm room, this necessarily means you are also +caffeine. Similarly, if you are +caffeine, you can be either in a cold room or a warm room, but -caffeine necessarily means you are in a cold room, because this is the only pairing that contains that.

Now, the descriptions of nested designs say that we have a nested design if ALL levels of a factor are fully contained within another factor. This is obviously not the case here, which is why I chose to call it a "partially nested design."

But I would appreciate if someone told me about the right terminology.

Also, what design would be appropriate here? If I had all 4 combinations, I'd use a 2x2 factorial ANOVA. But for the current design, I am puzzled as to what would be the correct choice.

• What is the goal of the experiment? This is still an (uncomplete) anova design, what to be done depends on which contrasts you want to test. Please tell us! Commented Oct 25, 2021 at 15:38
• This experiment wants to test the effects of room temperature (cold vs. warm room) and caffeine intake (caffeine vs. no caffeine) on test performance (as measured by a score on a test people take in a room). As you say, this design is "incomplete" in the sense that not all combinations of this 2x2 design are present. I am asking how the data would be analyzed under these circumstances. Only 3 of the 4 theoretically possible conditions are available here. Commented Oct 25, 2021 at 16:38