Causa Nostra: The Potentially Legitimate Business of Drawing Causal Inferences From Observational Data

I was delighted to participate as a speaker and panel discussant at the 2018 American Conference on Pharmacometrics in a session devoted to dialogue between “pharmacometric modelers” and “statisticians.” In the current perspective, I reprise the major points of my talk, which addressed the possibility of drawing valid causal inferences from observational data. I augment this reprise with brief remarks comparing diagrammatic conventions in pharmacometrics to those used in formal causal diagrams. The 2018 American Conference on Pharmacometrics was held on the narrow sand spit that connects Coronado Island to Imperial Beach on the mainland; in other words, on a causeway. I want to begin by expressing the hope that the burgeoning field of causal inference will come to provide a metaphorical causeway (or “cause-way”!), bridging fundamental perspectival gaps between the disciplines of pharmacometrics and statistics. In my talk I suggested, albeit with caveats, that a treatment effect estimate derived from pharmacometric population simulation will generally have a causal interpretation that is rigorously defensible, even when the treatments being compared (e.g., high exposure vs. low exposure) have not been randomized. I supported this claim by showing that pharmacometric population simulation generally relies on the same logic as the causal inference procedure known as G-computation,1 and the latter procedure is known to produce valid causal estimates under assumptions that can be fully formalized and articulated. I developed my argument with reference to kidney stone surgery data published by Charig et al.2 The comparison of open surgery to percutaneous nephrolithotomy based on those data has become a classic known in statistical discussions as “Simpson's paradox.”3 The “paradox” manifests in that case as a conflict between a misleading overall analysis, which seems to favor percutaneous nephrolithotomy and subset analyses that clearly favor open surgery. This phenomenon, which is exceedingly unlikely in randomized experiments, is readily explained in terms of confounding when treatment is not randomized. The data therefore provide a simple and standard test case for causal inference procedures: if an analysis methodology avoids the misleading inference associated with the naive “overall” analysis of those data, that is a good sign. In my talk, I proposed that a normative pharmacometric approach to these data would rely on population simulation, proceeding as follows: An R script to recreate the relevant subset of the Charig et al. data and to implement the analysis described previously is provided as Supplemental Material. These two differences are motivated by the same consideration, namely, that pharmacometric response models are typically nonlinear mixed-effects models for which predicted values are not directly available as a function of model parameters. In that context, for every covariate combination present in the original data, one wants to generate multiple draws from the random effect distribution to arrive at simulation-based estimates of the predicted response that properly average over the random-effect distribution. Notwithstanding these relatively minor differences in procedures, G-computation and pharmacometric population simulation share an implicit invocation of what Pearl et al.5 simply call “the adjustment formula.”5 From a purely probabilistic perspective, this is nothing more than applying the law of total probability to average conditional response distributions with respect to a covariate distribution. As the previous example is intended to illustrate, it would hardly be novel to propose that the law of total probability be applied in the context of pharmacometric simulation to properly average over covariate distributions. Moreover, to do so would be to merely advance various probabilistic and statistical methods, which misses the point of formal causal reasoning entirely. What is ultimately on offer from the causal inference literature is not merely a new arsenal of statistical techniques but rather a language and a set of formalisms to represent concepts that cannot be reduced to probabilistic and statistical concepts.5, 6 Formalisms associated with causal diagrams provide a major case in point. At the probabilistic level of analysis, the simple graph “oral dose → plasma concentration ← intravenous dose” implies that the joint distribution of the three variables can be specified in terms of the joint marginal distribution of the dose variables and the conditional distribution of plasma concentration given the levels of the two dose variables. Of course, that much could be said of any three random variables; it is trivially true, so the graph fails to convey any nontrivial semantic content as long as it is interpreted through that exclusively probabilistic lens. When interpreted through that lens it does not tell us what would happen to oral dose if we exogenously intervened to manipulate intravenous dose. By contrast, we associate causal semantics with the graph if and only if we stake out positions regarding the effects of (perhaps hypothetical) interventions on the represented variables. One may object that such graphs are also nothing new in pharmacometrics: typical graphical representations of pharmacometric models (for example, the graphical representations presented by Mager et al.7) are clearly meant to convey implications about interventions and are interpreted as such. This is only partly true, and for pharmacometrics to harvest the fruits of causal diagram research it remains to systematically compare the (seemingly unarticulated) “rules” for interpretation of pharmacometric model diagrams to the rigorously articulated rules for interpretation of causal diagrams. As an example of the differing semantics, note that in a typical pharmacometric diagram such as Figure 1a, the arrows associated with clearance of the drug (labeled CL and CLE) do not denote causal influence on graphical descendants (in fact the descendants are not even articulated) but rather, the longitudinal progression of the nodes from which they emanate. In contrast, the strict application of causal diagram principles would dictate that arrows only be used to represent causal influence, in which case temporal evolution would need to be represented in some other fashion (e.g., by indexing variables by time in a discrete-time approximation, as in Figure 1b). Indeed, at present, a substantial barrier to the application of causal diagram methodologies in pharmacometrics is the underdevelopment of causal diagram techniques for continuous-time dynamic systems. The development of a hybrid graphical system that would support the formal operations that pertain to causal diagrams without losing the representational economy of conventional pharmacometric diagrams would appear to be a worthwhile objective for future research. Although it is not yet clear how to marry the two graphical ecosystems, the potential progeny of such a marriage seem worth fighting for. In my talk, I reviewed a simplified adaptation of the gamma-glutamyl transpeptidase system example discussed by Daniel et al.,8 as shown in Figure 2a. In the context of this example, I illustrated the value of what Pearl et al.5 call “graph surgery” as a means of formalizing the causal questions of interest and simultaneously clarifying the simulation logic that should be implemented to address such questions (see Figure 2b,c). Both my podium presentation and my summary here have borrowed heavily from Pearl and Mackenzie's The Book of Why.6 In closing, I would like to briefly call attention to one important aspect of that book that I did not broach in my talk: the historical provenance of statisticians’ beliefs and attitudes with regard to causality. In reading the narrative of that history by Pearl and Mackenzie, I was struck by how many of the familiar antagonisms between the “pharmacometric modeler” community and the “statistics” community are explicable in terms of the contentious battles over the meaning of causality in the last two and a half centuries, battles that can be traced back (at least) to the pivots in thinking that occurred during the Age of Enlightenment. For anyone seeking to advance the field of pharmacometrics via dialogue with the statistical community, this history is no mere curiosity. The more we understand the historical forces that have shaped our differing perspectives, the more we are likely to find a causeway of common ground. That would be cause to celebrate! No funding was received for this work. The author declared no competing interests for this work. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.