{"id":631,"date":"2011-03-24T16:47:10","date_gmt":"2011-03-24T20:47:10","guid":{"rendered":"https:\/\/cyrussamii.com\/?p=631"},"modified":"2011-03-24T16:49:45","modified_gmt":"2011-03-24T20:49:45","slug":"technical-imputation-ipw-causal-inference-snowden-et-al-2011-with-disc","status":"publish","type":"post","link":"https:\/\/cyrussamii.com\/?p=631","title":{"rendered":"(technical) imputation, ipw, causal inference (Snowden et al, 2011, with disc)"},"content":{"rendered":"

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<\/a>). 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. <\/p>\n

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<\/a>). 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<\/em> 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. <\/p>\n

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<\/em>-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<\/a>).)<\/p>\n

Stijn Vansteelandt and Niels Keiding provide an invited commentary (gated link<\/a>) 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:<\/p>\n

The IPTW [inverse probability of treatment] approach is not commonly
\nused in practice because of the traditional reliance on out-
\ncome-regression-based analyses, which tend to give more
\nprecise estimates. Its main virtue comes when the con-
\nfounder distribution is very different for the exposed and
\nunexposed subjects (i.e., when there is near violation of the
\nassumption of the experimental treatment assignment), for
\nthen the predictions made by the G-computation approach
\nmay be prone to extrapolate the association between out-
\ncome and confounders from exposed to unexposed
\nsubjects, and vice versa. The ensuing extrapolation uncer-
\ntainty is typically not re\ufb02ected in con\ufb01dence intervals for
\nmodel-based standardized effect measures based on tradi-
\ntional outcome regression models, and thus the IPTW
\napproach may give a more honest re\ufb02ection of the overall
\nuncertainty (provided that the uncertainty resulting from
\nestimation of the weights is acknowledged) (19). A further
\nadvantage of the IPTW approach is that it does not re-
\nquire modeling exposure effect modi\ufb01cation by covariates
\nand may thus ensure a valid analysis, even when effect
\nmodi\ufb01cation is ignored.<\/p><\/blockquote>\n

\nI 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:<\/p>\n

We here propose a compromise that combines the bene\ufb01ts of G-computation\/
\nmodel-based standardization and of the IPTW approach. Its
\nimplementation is not more dif\ufb01cult than the implementa-
\ntion of these other approaches. As in the IPTW approach,
\nthe \ufb01rst step involves \ufb01tting a model of the exposure on
\nrelevant covariates; this would typically be a logistic re-
\ngression model. The \ufb01tted values from this model express
\nthe probability of being exposed and are commonly called
\n\u2018\u2018propensity scores.\u2019\u2019 They are used to construct a weight
\nfor each subject, which is 1 divided by the propensity score
\nif the subject is exposed and 1 divided by 1 minus the pro-
\npensity score if the subject is unexposed. The second step
\ninvolves \ufb01tting a model, the Q-model, for the outcome on
\nthe exposure and relevant covariates but using the afore-
\nmentioned weights in the \ufb01tting procedure (e.g., using
\nweighted least squares regression). Once estimated, the
\nimplementation detailed in the article by Snowden et al.
\n(4) is followed; that is, counterfactual outcomes are pre-
\ndicted for each observation under each exposure regimen
\nby plugging a 1 and then subsequently a 0 into the
\n\ufb01tted regression model to obtain predicted counterfactual
\noutcomes. Finally, differences (or ratios) between the aver-
\nage predicted counterfactual outcomes corresponding to
\ndifferent exposure regimens are calculated to arrive at a stan-
\ndardized mean difference (or ratio) (see reference 19 for
\na similar implementation in the context of attributable
\nfractions). We refer to this compromise approach as doubly robust
\nstandardization. Here, the name doubly robust expresses
\nthat doubly robust standardized effect measures have 2
\nways to give the right answer: when either the Q-model
\nor the propensity score model is correctly speci\ufb01ed, but
\nnot necessarily both. <\/p><\/blockquote>\n

\nThis approach has been demonstrated elsewhere—e.g. a recent paper by Vansteelandt and co-authors in the journal Methodology<\/em> (ungated version<\/a>, gated published<\/a>). 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<\/a>). I wonder if there is a difference in practice.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 … <\/p>\n

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