Hello! Hoping I could find some answers here. I have this experimental setup shown in this image:

I was wondering using the formula design = ~ batch + condition would be sufficient enough to control for batch effect considering the fact that the control group is not represented in the other batch. How should I proceed with further analysis? Surprisingly, Limma was able to somewhat correct the data to cluster broadly by condition. I am stuck on what to do.

I will appreciate any answer. Thank you so much!

Nope. Batch is entirely confounded with day. You will have no way at all of knowing what day 4 changes are not batch effects.

What does that date represent exactly? Because date of illumina run will add no artifacts. But if that date represents anything else, then you are in trouble.