I'm fitting a linear model with outcome $Y$. I have measurements for variables $X_1$ and $X_2$.
- I hypothesize that $X_1$ and $Y$ are linearly related. I want to know the slope and significance of this relationship.
- I know from world knowledge that $X_1$ and $X_2$ are related.
Specifically, regardless of whether the hypothesis in (1) is true or not, I expect that there is a relationship in the underlying generative model such that $X_1$ is a multiple of $X_2$. There's also noise during this process, but it's not clear whether (a) the multiplier itself is noisy, or (b) $X_1$ is a fixed multiple of $X_2$, and then some noise is added.
- From world knowledge, I don't expect $X_2$ and $Y$ to be related except via $X_1$. It's theoretically possible for $X_2$ and $Y$ to have an independent relationship, but it doesn't matter to me either way.
How should I fit this model so that I get the best estimate for $Y \sim X_1$? Should I divide $X_1$ by $X_2$ before entering it into the model? Should I include Z as a covariate ($Y \sim X_1 + X_2$ possibly with an interaction)?
I'm willing to do something more complicated to infer the estimate if necessary, but I'd like to keep it simple if possible since the actual model predicting $Y$ is mixed-effects and includes a number of other predictors that I have independent hypotheses about.