# Multiple linear regression with dependent as a dummy predictor

I have a model $$Y = \alpha + \beta_1X_1 + \beta_2X_2$$.
$$Y$$ has a bimodal normal(ish) distribution, so I'm looking to see if the relationship between the predictors and the response is different for low values of $$Y$$ compared with high values of $$Y$$.

Thus far I've recoded $$Y$$ as a dummy variable $$D$$ (0 if Y < median(Y), else 1) and added it as a predictor yielding $$Y = \alpha + \beta_1X_1 + \beta_2X_2 + \beta_3D$$ (plus interactions).
$$D$$ is significant as both a main effect (obviously) and in interaction with $$X_1$$. This suggests to me that, indeed, the relationship between $$X_1$$ and $$Y$$ differs for high values of $$Y$$ compared with low values of $$Y$$.

But somehow this method feels off.
Adding the dummy increased $$R^2$$ from 20% to 80% (caused by the fact that the dummy is derived from $$Y$$). I'm getting a large amount of multicollinearity between my dummy and the dummy in interaction with $$X_1$$ and $$X_2$$, which, again makes sense, but signals to me this method is not what I'm looking for.

I can't find any useful information on recoding a response variable as a dummy and adding it as a predictor (again, for me, that's a cause for concern), so I'm hoping someone here will point out the flaw and explain why this method is not cricket.

EDIT: For clarification, I'm trying to avoid splitting my data and running the analysis separately for each mode.

• " bimodal normal(ish)": no such thing. If bimodality is strong enough to deserve emphasis; your distribution is not normal. What you mean by "(ish)" can't be clear to anyone else. Why not show a graph (e.g. a histogram or quantile plot of $Y)$? – Nick Cox May 5 at 9:45
• More to the point, the dummy idea is interesting but -- as you sense -- fatally flawed given the element of circularity. I'd see your problem as natural for quantile regression in which you look at predicting selected quantiles of $Y$ from the $X_j$. How many quantiles you should be looking at depends partly on your sample size and partly on the shape of the distribution. – Nick Cox May 5 at 9:48
• @NickCox Bimosal normalish was poorly phrased, please read as: "looks like 2 approximately normal distributions stuck together". Am definitely going to check out quantile regression, thanks or your suggestion! – user156060 May 5 at 11:16
• Since you have started down this path by introducing $D,$ why stop there? Your definition of $D$ loses information. Use $Y$ itself! Your model becomes $$Y=\alpha+\beta_1X_1+\beta_2X_2+Y$$ with the (obvious) solution $(\alpha,\beta_1,\beta_2)=(0,0,0)$ giving a perfect fit. – whuber May 5 at 15:32
• @ whuber Yes, as Nick stated, circularity is indeed the problem here, and as you correctly point out, I should have seen it. When you're not an expert at something, and you look at that something for a few hours, sometimes you miss the blatantly obvious :-). – user156060 May 6 at 6:58