(technical) imputation, ipw, causal inference (Snowden et al, 2011, with disc)

In the advance access pages of the American Journal of Epidemiology, Jonathan M. Snowden, Sherri Rose, and Kathleen M. Mortimer have a nice tutorial on what they refer to as “G-computation” for causal inference with observational data (ungated link). An average causal effect for a binary treatment can be defined as the average of individual level differences between the outcome that obtains when one is in treatment versus in control. Because people are either in treatment or control, one of these two “potential” outcomes is unobserved, or missing (within subjects designs do not overcome this, because the ordering of treatment assignment is itself another dimension of treatment). Given this “missing data” problem, G-computation refers to fitting models to available data that allow you to impute (i.e., predict) unobserved counterfactual values. You can then use this complete set of counterfactual values to estimate various types of causal effects. The idea isn’t so new or groundbreaking, but many theoretical insights have been elucidated only recently. Snowden et al’s presentation focuses on effects that average over the entire population.

These authors don’t cite it in their paper, but I think the most sophisticated application of this approach for a cross-sectional study is Jennifer Hill’s “Bayesian Nonparametric Modeling for Causal Inference” study (gated link). Hill uses the magical BART algorithm to fit a response surface and generate the imputations, from which various flavors of causal effect might be constructed (okay, full disclosure—Hill was one of my grad school teachers, but hey, BART is pretty cool). I understand that there has been a fair amount of application, at least beta-testing, of such counter-factual imputation methods in longitudinal studies as well, although I don’t have references handy.

This approach is especially appealing when you anticipate lots of measurable effect modification that you want to average-over in order to get average treatment effects. Actually, I think Snowden et al’s article does a good job of demonstrating how in some cases, it’s not classic confounding and omitted variable bias per se that is the major concern, but rather effect modification and effect heterogeneity (i.e., interaction effects) associated with variables that also affect treatment assignment. Traditional regression is clumsy in dealing with that. As far as I know conventional social science teaching, rooted as it is in constant effects models, does not have a catchy name for this kind of bias; maybe we can call it “heterogeneity bias.” Another thing that makes this kind of bias special relative to the usual kinds of confounding is that, as far as I understand, imputation-based strategies (like g-computation) that try to correct for it may in fact take advantage of measured heterogeneity associated with post-treatment variables. That is one of the reasons that these methods have appeal for longitudinal studies. (On this point, I’ll refer you to a little tutorial that I’ve written on a related set of methods—augmented inverse propensity weighting for attrition and missing data problems (link).)

Stijn Vansteelandt and Niels Keiding provide an invited commentary (gated link) on Snowden et al’s paper, and they make some really interesting points that I wanted to highlight. First, they note that imputation based strategies such as g-computation have a long history in association with the concept of “standardization.” More importantly are two points that they make later in their commentary. First is a point that Vansteelandt has made elsewhere, discussing the similarities and differences between imputation/standardization and inverse probability weighting:

The IPTW [inverse probability of treatment] approach is not commonly used in practice because of the traditional reliance on out- come-regression-based analyses, which tend to give more precise estimates. Its main virtue comes when the con- founder distribution is very different for the exposed and unexposed subjects (i.e., when there is near violation of the assumption of the experimental treatment assignment), for then the predictions made by the G-computation approach may be prone to extrapolate the association between out- come and confounders from exposed to unexposed subjects, and vice versa. The ensuing extrapolation uncer- tainty is typically not reflected in confidence intervals for model-based standardized effect measures based on tradi- tional outcome regression models, and thus the IPTW approach may give a more honest reflection of the overall uncertainty (provided that the uncertainty resulting from estimation of the weights is acknowledged) (19). A further advantage of the IPTW approach is that it does not re- quire modeling exposure effect modification by covariates and may thus ensure a valid analysis, even when effect modification is ignored.

I think this is an exceptionally important point, making clear that the apparent “inefficiency” of IP(T)W relative to imputation based methods is, in some sense, illusory. Vansteelandt and Keiding also discuss one approach to combining imputation and IPW in order to get the best of both worlds:

We here propose a compromise that combines the benefits of G-computation/ model-based standardization and of the IPTW approach. Its implementation is not more difficult than the implementa- tion of these other approaches. As in the IPTW approach, the first step involves fitting a model of the exposure on relevant covariates; this would typically be a logistic re- gression model. The fitted values from this model express the probability of being exposed and are commonly called ‘‘propensity scores.’’ They are used to construct a weight for each subject, which is 1 divided by the propensity score if the subject is exposed and 1 divided by 1 minus the pro- pensity score if the subject is unexposed. The second step involves fitting a model, the Q-model, for the outcome on the exposure and relevant covariates but using the afore- mentioned weights in the fitting procedure (e.g., using weighted least squares regression). Once estimated, the implementation detailed in the article by Snowden et al. (4) is followed; that is, counterfactual outcomes are pre- dicted for each observation under each exposure regimen by plugging a 1 and then subsequently a 0 into the fitted regression model to obtain predicted counterfactual outcomes. Finally, differences (or ratios) between the aver- age predicted counterfactual outcomes corresponding to different exposure regimens are calculated to arrive at a stan- dardized mean difference (or ratio) (see reference 19 for a similar implementation in the context of attributable fractions). We refer to this compromise approach as doubly robust standardization. Here, the name doubly robust expresses that doubly robust standardized effect measures have 2 ways to give the right answer: when either the Q-model or the propensity score model is correctly specified, but not necessarily both.

This approach has been demonstrated elsewhere—e.g. a recent paper by Vansteelandt and co-authors in the journal Methodology (ungated version, gated published). I am intrigued by this because it differs from the manner in which I have implemented doubly robust estimators that combine weighting and imputation (again, see link). I wonder if there is a difference in practice.