Imputing missing values for linear regression model, using linear regression

Question

I scraped a real estate website and would like to impute missing data on total area (about 40% missing) using linear regression. I achieve the best results using price, number of rooms, bedrooms, bathrooms, and powder rooms.

Adding price to the room information makes a significant difference. This makes sense, since the number of rooms alone don't give you any information on how large those rooms may be. Price can reduce some of that uncertainty. There is a 20 point difference between the R^2 scores of the model that includes and the one that excludes price (0.62 vs 0.82).

The problem that I see, is that my final model would likely also be a liner regression with price as the target. With this, it seems wrong to include price in predicting total area for imputation. In essence, I'm using the target to predict a feature and then use that feature to predict the target again. That's circular and seems problematic to me but I could be wrong. My final model will look better as a consequence but I will have engineered a synthetic correlation. This seems especially critical since about 40% of values need to be replaced.

Does anyone disagree with this? Should I keep price as a predictor to impute missing values even though it will be the target of my final model?

Answer 1

As Robert Long says in a comment, "With single imputation, you shouldn't be doing it in the first place."

There is a well respected way to deal with missing data, by doing multiple imputations. The idea is to acknowledge and incorporate variability in the imputation process by producing probabilistically several different imputed data sets. You then perform your analysis separately on each of the imputed sets, and combine the information in a way that takes both within-set and between-set variability into account. That gets around the inherent circularity that you rightly fear with the single-imputation approach.

Functions implementing the imputation and analysis process are available in many statistical software systems. This online book is a helpful introduction. And yes, including the outcomes (prices in your case) is an important part of the multiple imputation process.