# Factorial experiment design with incompatible levels in IVs

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 $f_6$.