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I have a problem with an unbalanced ANOVA design and wasn't able to solve it through research on the Internet yet. In a lot of questions, when people are talking about their unbalanced designs, it's about different sample sizes or similar. My problem, however, is in the factors itself:

In theory, I have a 3x3 design but on two factors, there are only two repeats possible. So, let's say we have a factor1 (=f1) and factor2 (=f2) that have three levels each, then only the following combinations are possible:

f1-level1 x f2-level1;

f1-level1 x f2-level2;

f1-level1 x f2-levek3;


f1-level2 x f2-level1;

f1-level2 x f2-level2;


f1-level3 x f2-level2;

f1-level3 x f2-level3.

Unfortunately, this is nothing, I can change or avoid by design. Is it possible to reasonably analyse this in ANOVA style or mixed effect models? How do I approach implementing these in SPSS or R?

I am very grateful for every single response. I'm really stuck and after reading a lot on other people's questions, also slightly confused.

Thank you very much in advance!

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This is called a fractional factorial design. You might find this link helpful. In your design, you only have two factors and one interaction and cannot assess the simple interaction between f1 and f2 (3 levels by 3 levels). However, you can test two specific interactions (f1 levels 1-2 and f2 levels 1-2; and f1 levels 2-3 and f3 levels 2-3). My recommendation is to first run a model with just the two main effects but no interaction. Then run a full model (main effects and interaction) with just the 4 cells for the first possible interaction (f1 levels 1-2 and f2 levels 1-2) and then another with the 4 cells of the second interaction.

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  • $\begingroup$ Thank you so much for this great answer! This makes things really clear, especially with the link. Now I was just wondering what the best software to do this is. I guess you want to use something that allows you to compare model fits for the different possible models? $\endgroup$ – MrSomething Jul 23 '18 at 8:15
  • $\begingroup$ You don't need to compare model fits (each has a different sample size). SPSS, Stata, SAS, R are all fine for this. What you are interested in from each model is different. From the first, you are interested in the two main effects, from the second and third, you are only interested in the interaction effect. $\endgroup$ – dbwilson Jul 23 '18 at 11:26
  • $\begingroup$ Oh, I got it. Thank you so much for your time and effort! $\endgroup$ – MrSomething Jul 23 '18 at 12:10

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