Logistic Regression Models Without Main Effects?

Question

I am building logistic regression models measuring human behaviour, which consist of categorical variables: demographics, conditions, and interactions between the demographics and the condition variable.

The big issue is that as per my theory tested I am not interested at all in the main effects of demographics but only in the interactions with the condition variable. Moreover, there are extreme collinearity forces among the demographics e.g. income and education etc.

Hence, I would intend to present log models without main effects in my thesis. Do you think this can be defended? If so, could you please give me ideas or statistical arguments of how to argue in favour of that?

Answer 1

Short answer: No that cannot be defended.

Longer answer: With an interaction effect you show how the effect of one variable ($x_1$) changes when another variable ($x_2$) changes. So you cannot estimate the interaction term without also saying something about the effects of $x_1$ and $x_2$ (the main effects). To be more precise if you leave out the main effect of $x_1$ then you constrain the effect of $x_1$ to be $0$ when $x_2$ is $0$. Such a constraint will normally hugely bias any interaction effect.

In all the datasets I know income and education are not extremely collinear. Correlation has to be in the order of .8 or .9 before I speak of extreme collinearity, while I would expect to see a correlation of about .3 or .4 between education and income. The largest (cross sectional) genuine correlation in human behaviour is between the education of two partners, which is about .6. So even that is far far away from extreme colinearity.

Answer 2

A point that's worth perhaps more than just a comment:

Moreover, there are extreme collinearity forces among the demographics e.g. income and education etc.

@MaartenBuis's answer is helpful in saying these are probably not that collinear. But it is more helpful to step back and say, "What is collinearity doing?" New statisticians are taught about collinearity with little regard to the modeling and causal inference. Put simply, there's no reason to adjust for variables if there is not some extent of collinearity: it is the price we pay [power] for precise/applicable measurement of effects.

The set of adjustment variables to a model determines the hypothesis which is being tested. Omitting a strong confounding variable on the basis of collinearity is simply unacceptable. This approach reveals that the analyst and investigator previously affirmed the variable's role as a possible confounder, then made a decision based on the structure of the data to change their hypothesis. Changing a hypothesis based on data seriously compromises the analysis. Another way to see this is that analyses which omit confounders are biased.

There may be other reasons to omit a variable which was previously believed to be a confounder from analyses. If the causal model is such that only an indirect path connects a sequence of confounders from an exposure to an outcome, then adjusting for the confounder most proximal to the outcome satisfies the backdoor criterion. An excellent SE discussion from @CarlosCinelli addresses that here.

Collinearity can make power a problem. Or it can improve power: we rarely can predict the impact of adjusting for highly correlated predictors even in large sample sizes. It is acceptable for sensitivity analyses to consider adding or omitting certain variables: however one must not then favor a particular model and report it as a final or main analysis. Omitting a confounding variable may not change the term of interest, but it may shrink the confidence interval substantially, perhaps to the point of achieving statistical significance which wasn't seen before. In this case, it is always advisable to conduct the underpowered analysis and comment on its relevance.