What is a good reference that discusses the problem of common support?

Question

Asking this for a colleague:

They have two groups of people with various characteristics and to address the issue of comparability of outcomes they would like to reference a paper or book that discusses the concept and problems of common support.

Answer 1

Matching and Balance

The general issue falls under 'matching', which is the process of sampling units to create subsamples that are maximally 'balanced' i.e as similar as possible on uninteresting (e.g. non-treatment) covariates while differing on an interesting one, with an eye to making causal claims about the latter. Ideally, matched samples have the same distribution of uninteresting covariates. Less stringently, their covariates have overlapping support.

Propensity scores are one approach to the balance problem, but there are a more, not all of which are so dependent on building another model to make a matching 'score' and implicitly relying on its assumptions. For example, a recent alternative would be 'coarsened exact matching'.

Reference

This field is getting pretty big, but here is an unsystematic sample from my shelf and hard disk, that might prove helpful.

Gelman and Hill ch. 9 and 10 are very good, I think, assuming that you're operating in a regression framework to start with. For just a reference to the issue and its possible solutions, this might be a good one. You might also find Imai et al. (2008) helpful, mostly for the error decomposition.

More generally, there are a bunch of article length reviews available from Stuart's papers page or from King's causal inference page. Morgan and Winship is a thorough book length treatment of causal inference that puts the common support issue in a bigger picture.

Answer 2

If I understand, you have a classification problem where you observe $(X_1,Y_1),\dots,(X_n,Y_n)$. Denoting by $\mathcal{X}$ the space where $X$ takes values, you assume the existance of $\mathcal{X}_{10}$ the smallest closed subset of $\mathcal{X}$ such that:

$\mathcal{X}\setminus \mathcal{X}_{10}$ can be partinioned into $\mathcal{X}_{1\setminus 0}\cup \mathcal{X}_{0\setminus 1}$ with $P_{0}(X\in \mathcal{X}_{1\setminus 0})=0$, $P_{1}(X\in \mathcal{X}_{0\setminus 1})=0$ (for $i=0,1$ $P_i$ is the distribution of $(X|Y=i)$)

I think there are two problems that may arise from your question:

1. How do you justify a decomposition of the distribution to be able to solve the problem only on $\mathcal{X}_{10}$ (Theoretical problem)
2. How do you estimate $\mathcal{X}_{1\setminus 0}$, $\mathcal{X}_{0\setminus 1}$ and$\mathcal{X}_{0 1}$ from data (practical estimation problem)

Theoretical problem In this case (and under suitables conditions) you can decompose $P_{1}$ and $P_0$ using the Radon-Nikodym theorem (This version) into:

$$P_{1}=P_{1,\mathcal{X}_{1\setminus 0}}+P_{1,\mathcal{X}_{0 1}}$$

$$P_{0}=P_{0,\mathcal{X}_{0\setminus 1}}+P_{0,\mathcal{X}_{0 1}}$$

with $P_{0,\mathcal{X}_{0 1}}\sim P_{1,\mathcal{X}_{0 1}}$ (mutually absolutly continuous)

$P_{1,\mathcal{X}_{0 1}} \bot P_{1,\mathcal{X}_{1\setminus 0}} \bot P_{0,\mathcal{X}_{0\setminus 1}}$ (mutually singular) Note that distributions in the decomposition are not probabilities, and you need to renormalize (i.e. compute $P_{i}(X\in A)$ when $A$ is one of the three above mentionned sets). This renormalization is also required from a practical point of view.

Once you have done the renormalization you can construct a classification rule on $\mathcal{X}_{0 1}$ using renormalized version of $P_1$ and $P_0$ 's restrictions. You can extend this classification rule in the obvious way to the whole $\mathcal{X}$.

Estimation problem If you want to "learn" the above mentionned rule from the data, you need to estimate the corresponding sets I guess there is a large litterature in "support" estimation but in view of the classification problem at the end, it is not a big deal since the places where you might be wrong in terms of support estimation are does where you don't have much "mass" and hence does that are not really important for the classification errors at the end... at least from a bayesian perspective of which error is important... (see my answer here)