Is it safe to run a Factorial ANOVA if you cannot collect data for some cells?
Scenario: There are two independent variables, and one dependent variable.
All independent variables are nominal:
D (data structure) has 4 levels: $d_1$ to $d_4$.
F (distance function) has 6 levels: $f_1$ to $f_6$.
The dependent variable is interval: Y (outcome) has domain $[0, 1] \subset \mathbb{R}$.
Data looks like this:
(subject, D, F, Y) Example: (user001, $d_1$, $f_1$, 0.3).
Goal: To determine the effect of D and F on Y.
Problem: there is incompatibility between some levels of D and F.
Example: $d_1$ is incompatible with $f_6$. This means that Y cannot be measure in this cell.
There are other incompatibilities:
$d_2$ is incompatible with $f_6$.
$d_3$ is incompatible with $f_3$ to $f_6$.
$d_4$ is incompatible with $f_1$ to $f_5$. This means that two objects represented as data structures of type $d_4$ can only be compared by using the function $f_6$.