I am analysing a variable (DV) as a function of an experimental manipulation (X), presented in different visual contexts.The DV varies by subject and context. I have repeated measures per subject and context at the different levels of X.
To capture by-subject variation in the DV, I add a random intercept/slope for each partcipant (using lme4 in R):
$DV$ ~ $X + (X|Subject)$
The context really describes an interaction between three stimulus factors (say: color (2) $*$ direction (2) $*$ form (2)).
Separately, these factors cannot be treated as random effects, having only two levels each. Instead, including their interaction (8 levels) - I am afraid I am ignoring something (like the factor-levels being related or not "random").
Thus, my question: Is it valid to use the interaction between (these) factors as a random effect (such as: $DV$ ~ $X + (X|Sub)+(X|C)$ )?