# Sample size confounded with factors (ANOVA)

What do you suggest doing when sample size is confounded with factors in an ANOVA?

"For example, in a two-way ANOVA, let’s say that your two independent variables (factors) are Age (young vs. old) and Marital Status (married vs. not).

Let’s say there are twice as many young people as old. So unequal sample sizes.

And say the younger group has a much larger percentage of singles than the older group. In other words, the two factors are not independent of each other. The effect of marital status cannot be distinguished from the effect of age.

So you may get a big mean difference between the marital statuses, but it’s really being driven by age." (Thank you to the analysis factor for this example)

If I am interested specifically in the interaction (age*marital status), how can I address the issue of an uneven distribution of marital status across age?

• I'm not sure I would say that the sample size is confounded with other factors. Presumably the sample size itself is always independent of other factors, right? You're really asking about how to handle two supposedly independent variables that you suspect have a causal relationship, right? – Adrian Keister Jun 11 at 15:58
• I would recommend drawing a causal diagram. What are the fundamental cause and effect you're investigating? What other variables are important? How are they causally related? Here you draw $A\to B$ if you think of $A$ as causing $B.$ – Adrian Keister Jun 11 at 15:59

I notice that you research Alzheimer's disease, including its factors. So let's suppose you want to know the causal influence of age on a particular factor. You've mentioned that you think age influences marital status (surely not the reverse, except perhaps figuratively!), and perhaps you're also thinking that marital status influences the Alzheimer's factor. In that case, you would have a causal diagram like this:

This situation is known as a mediator: marital status is a mediator of the causal effect of age on the Alzheimer's factor. In this scenario, marital status is NOT a confounder, because it does not set up a backdoor path from Age to the Alzheimer's factor. If you were to regress the Alzheimer's Factor on Age, and you were to include the marital status, you'd be conditioning on marital status and you would get an incorrect causal effect of age on the Alzheimer's factor. Conclusion: regress the Alzheimer's factor solely on age, without including marital status.

On the other hand, you might be interested in investigating the causal effect of marital status on the Alzheimer's factor. Using the same causal diagram, but re-arranging it a bit yields this quite different picture:

Now age sets up a backdoor path from marital status to the Alzheimer's factor. If you want the correct causal effect of marital status on the Alzheimer's factor, you must condition on age (include in your regression).

I've made some assumptions in this answer which might be quite different from what you're really trying to accomplish, but I wanted to show you how the New Causal Revolution can really help clarify what to do when, particularly when it comes to confounding variables. Indeed, I view understanding confounding variables as one of the most important advances of the New Causal Revolution.

• Thank you very much for the answer! I appreciate you looking up my background for your examples, and the effort in drawing causal diagrams. I think I expect a moderation effect, where the Age -> Alzheimer's factor path is moderated by Marital Status – Annalise Jun 17 at 15:29
• For the second diagram, could you clarify what you mean by 'condition on age' using a regression equation? Do you mean Alzheimer's Factor ~ Age + Age x Marital Status? Is it possible to have only the interaction with no main effects, i.e., Alzheimer's Factor ~ Age x Marital Status – Annalise Jun 17 at 15:33
• @Annalise You're very welcome! Glad I could be of help. In the second diagram, the term "condition on age" in a regression setting means including it on the RHS. That's how you typically condition on a variable in the linear regression setting. Other ways of conditioning include stratifying your analysis based on age values, backdoor adjustment, frontdoor adjustment, instrumental variables, and probably other approaches as well. For more details, I highly recommend Causal Inference in Statistics: A Primer. – Adrian Keister Jun 17 at 15:38
• I see you edited your comment. You can try the regression with the interaction effect, but that's not going to be the causal effect of age on Alzheimer's Factor. As you preferred the mediation scenario, I would recommend regressing $\text{Alzheimer's Factor}\sim\text{Age}.$ – Adrian Keister Jun 17 at 15:41