On the Principle of marginality

Question

If we include only the square of a covariable x as an explanatory variable but not x in a glm (glmm) do we violate the Principle of marginality?

Answer 1

This is going to come down to semantics. The principle of marginality, as I am used to seeing it discussed (and as it is discussed on the Wikipedia page), pertains to interactions between two (or more) different variables. In your case, you don't have interactions, so that specific term does not apply. However, I have also seen using power terms to represent curvature as metaphorically described as 'a variable interacting with itself'. Indeed, this seems almost like a literal truth to me, rather than just a metaphor. What's important to note here is that the same underlying issues apply, whether or not we want to call it the "principle of marginality". To see this, it may help to read, Does it make sense to add a quadratic term but not the linear term to a model?, especially, um, @whuber's answer. Moreover, the underlying issues apply when other methods of trying to capture curvature (e.g., including $\log(X)$ as a variable) where the metaphor of a variable interacting with itself is less obvious.