American Journal of Epidemiology Advance Access originally published online on August 24, 2005
American Journal of Epidemiology 2005 162(7):621-622; doi:10.1093/aje/kwi256
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ORIGINAL CONTRIBUTIONS |
van der Laan et al. Respond to "Hypothetical Interventions to Define Causal Effects"
1 Department of Statistics, University of California, Berkeley, CA
2 Division of Biostatistics, School of Public Health, University of California, Berkeley, CA
3 Division of Epidemiology, School of Public Health, University of California, Berkeley, CA
Correspondence to Thaddeus J. Haight, Division of Epidemiology, School of Public Health, University of California, Berkeley, 140 Warren Hall, #7360, Berkeley, CA 94720-7360 (e-mail: tad{at}stat.berkeley.edu).
Received for publication June 2, 2005. Accepted for publication June 7, 2005.
Abbreviations: L/F, ratio of lean body mass to fat mass; MSM, marginal structural model
Dr. Hernán (1
) wrote a very insightful and educational response to our paper (2). He distinguishes between two types of epidemiologic analyses that apply marginal structural models (MSMs) or any other causal model: one in which the assignment of treatment (exposure) is uniquely defined (e.g., administration of a drug) and one in which it is unclear what route a subject followed to the reported level of exposure (e.g., lean-to-fat ratio (L/F)). One could represent L/F as having two components (A1, A2), A1 being the actual level of the exposure and A2 being the manner in which the exposure level is achieved, resulting in corresponding counterfactuals 
One could make the observation from our paper that we are only concerned with estimating the distribution of the counterfactual that sets the exposure level but leaves the assignment mechanism random to follow the course it follows in the population, that is, 
Since the counterfactual population distribution of 
can, in principle, be reproduced by using a hypothetical experiment, we can address the question raised by Dr. Hernán (1) of how to design a randomized experiment to replicate the causal parameter for L/F as defined by an MSM. He inquires about different interventions (e.g., exercise, surgery) that one would use in such an experiment. We realize that none of these interventions would result in a causal parameter comparable with the one learned from the observational data. However, we have that the marginal expectation of 
equals the weighted average across the interventions a2 of the population mean of for the subpopulation of all subjects who actually took A2 = a2, weighted by the population proportion of subjects taking A2 = a2: 
Thus, by carrying out experiments on subpopulations intervening on the completely defined exposure (A1, A2), one can actually reproduce the causal effect learned in the observational study. However, note that doing so requires not only measuring but also knowing the population distribution P (A2 = a2) of interventions of A2.
It is worth noting that one cannot account for all possible interventions that would contribute to the causal effect for a given treatment. This applies not only to an exposure such as L/F but also to the effect of a single intervention based on a randomized controlled drug trial. Indeed, the effect of the intervention would be the average of the effects of that intervention based on different subpopulations given that different people could respond differently to the same drug.
Regardless of how treatment is defined (e.g., a1 or (a1, A2)), counterfactuals exist as random variables whose distribution is a treatment-specific modification of the population distribution of the observed data, leaving all other factors affecting this random variable untouched (3, 4). So, formally, the existence of the counterfactuals is a nonissue; consequently, the formal definition of an MSM is also a nonissue. The issue that Dr. Hernán (1) raises concerns the actual interpretation of these counterfactual distributions, and thereby the MSM parameter 
and how this is affected by the type of treatment variable. To address this issue, one simply needs to understand the interpretation of the causal parameter targeted by the MSM fit as a function of the population distribution of the observed data.
Even without the notion of counterfactuals, this parameter of the population distribution of the observed data can be interpreted meaningfully. For example, given observational data under conditions where treatment is exchangeable within strata of baseline covariates, the parameter one targets by fitting a point-treatment MSM is actually a very interesting one of the population without concern as to well-defined outcomes for the given levels of treatment.
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ACKNOWLEDGMENTS |
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Conflict of interest: none declared.
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References |
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 TOP References |
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- Hernán MA. Invited commentary: Hypothetical interventions to define causal effects—afterthought or prerequisite? Am J Epidemiol 2005;162:618–20.
[Free Full Text] - Haight T, Tager I, Sternfeld B, et al. Effects of body composition and leisure-time physical activity on transitions in physical functioning in the elderly. Am J Epidemiol 2005;162:607–17.
[Abstract/Free Full Text] - Gill RD, Robins JM. Causal inference for complex longitudinal data: the continuous case. Ann Stat 2001;29:1785–811.[CrossRef]
- Yu Z, van der Laan M. Construction of counterfactuals and the G-computation formula. Berkeley, CA: Berkeley Electronic Press/UCB Biostatistics Division, 2002:122.
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