Weighted sampling as a way to eliminate specific source of variation?

Question

I am facing a problem of predicting probability of an event given two correlated predictors where only one of them is of interest. Thus, I’m trying to eliminate one of them from the model while making corrections in the training procedure to account for that.

Data is of e-commerce web tracking, where each individual datapoint is a display of an item in catalog. Goal is to predict probability of a purchase of a specific item given some item in the catalog will be purchased. Available features are rank of an item in the catalog (its # position on the page), its price as well as some other variables. All items on the page are of the same type. Probability of purchase follows exponential distribution w.r.t. to both rank and price. The goal is to eliminate item’s rank from the resulting model, since it is a variable we wish to control. However, there is a complication: almost all historic data was generated by displays where items were sorted by the price to begin with, so any effect of rank and price are most often entangled to indistinguishability.

Two approaches come to mind:

1. Fit polynomial regression with interaction variables between price and rank, then simply drop terms related to rank. Since we generate predictions for the full page and normalise all predicted probabilities this should not be a problem.

2. Try to eliminate the variation induced by item’s rank from training data. To do so, I calculate P(purchase|rank) and then sample each training data point according to the inverse of it, 1/P(purchase|rank). With this data I train a model with all relevant variables except rank as predictors.

Approach 2 has some intuitive appeal, however it is hard to validate and hard to prove that it is doing what is required.

Hence, my question is: does this approach makes any sense, and is it maybe related to some existing statistical procedure?

Answer 1

Approach two is called "inverse probability of treatment weighting." It is a valid technique and it is popular in the "causal inference" literature. By "causal" I mean causal in the sense of Angrist and Pischke (2009) in Mostly Harmless Econometrics

I actually made an attempt for while to extend IPTW to high dimensional control variables and didn't meet much success. In fact I asked several questions about it on this site this past summer (~July 2014) that might be interesting.