We will propose a new unifying framework for statistical models, which we call Regression by Composition (RBC). RBC includes as special cases all generalized linear and generalized additive models, as well as many other models. Among the advantages of RBC is that it allows link functions based on the switch relative risk. The RBC framework also facilitates conceptual insight about central issues in statistical modelling, including collapsibility and the nature of homogeneity assumptions.

When we started this project, the primary motivation was to generalize regression models to allow link functions that are consistent with Sheps’ preferred variant of the relative risk. However, the solution we came up with took on a life of its own, and turned into a unified modelling framework that has advantages which go significantly beyond the switch relative risk.

We are very excited about the novelty and applicability of these ideas, and certainly hope to see all our statistically oriented readers in the audience. The BEMC presentation is going to be more technical than my talk at LHSTM, but I believe applied epidemiologists and other data scientists will be able to follow the argument.

Ps. A recording of my talk at LHSTM is now available (scroll to talk 4)

Then, on Feb 2nd at 3pm UTC, I will give a joint talk with Rhian Daniel and Daniel Farewell at the Berlin Epidemiology Methods Colloquium, to launch our joint work (with Mats Julius Stensrud) about Regression by Composition (RBC), a flexible generalization of general linear models. Rhian has previously talked about some aspects of this work at her keynote presentation at the World Congress of Epidemiology, but this will be the first complete explanation of how these models work. One of the many advantages of RBC models, is that this framework admits some models that have no GLM form, including models based on the Switch Relative Risk. This therefore allows regressions models that are consistent with Sheps’ advice. A link to the stream will be posted at https://bemcolloquium.com/meetings-calendar/

In a very short paper from 1925, W.S. Abbott, an entomologist working at the US Department of Agriculture, proposed measuring the effect of insect sprays using what is now known as Abbott’s Formula. Abbott’s formula, which is still used by entomologists today, is mathematically equivalent to Sheps’ suggestion for the case where the intervention increases risk of the outcome. Abbott does not consider the situation where exposure reduces the risk of the outcome.

In 1939, another entomologist, C.I. Bliss from the Institute for Plant Protection in Leningrad, extended Abbott’s formula to the setting where exposure reduces incidence. To do so, he developed the Joint Independent Action model, which has become central to how toxicologists think about interaction between poisons. Toxicologists have made attempts to convince epidemiologists about the utility of this framework for interaction, which according to Howard and Webster has “firm biological foundations” in contrast with epidemiological models that consider interaction in terms of departures from risk additivity.

(I will note that while Bliss’ works appears to precede Sheps’ , Sheps was certainly the first to recognize the implications for medical statistics and epidemiology, and appears to have done so independently of the earlier work).

So if these models are good enough to be standard tools of the trade for toxicologists and entomologists, why are they still not being used by epidemiologists, medical statisticians or clinicians? Every time I find a new researcher who has rediscovered part of this idea, I get more and more puzzled over this.

It is not because people haven’t tried. In 1986, Clarice Weinberg, suggested using the Joint Independent Action model in epidemiological research, leading to recommendations that are identical to Sheps’. Weinberg is a leading epidemiologist publishing in a leading journal, and the journal did not publish any convincing counterarguments.

Yet Sheps recommendations are still not used in practice. There seems to be a deep resistance to these ideas, and it is unclear where it is coming from. Hopefully, our manuscript can contribute to clarifying the scope and limitations of this line of reasoning, and bring together closely related insights that have been scattered across the literature in fields as diverse as entomology, psychology, philosophy and computer science!

This post originally contained an animated video to explain the idea. The video contained a segment which stated that a popular textbook had “dismissed Sheps’ ideas”. While the relevant paragraph from the textbook has certainly been used by reviewers to dismiss our work on Sheps’ ideas, this may not have been the intention of the textbook authors. We have temporarily removed the video from this newsletter and delisted it from YouTube, until a correction version can be made available.

Ten years ago, when I was a doctoral student in Epidemiology at the Harvard School of Public Health, I had what I then believed was a truly original and important idea about how to choose the scale for measuring the effects of a medication (for example, in randomized trials).

It quickly became apparent that my idea had been "scooped" more than half a century earlier by Mindel Cherniack Sheps (1913-1973). In her 1958 paper "Shall we count the living or the dead", Sheps relied on the same intuition to reach the same conclusion: That when an intervention reduces the risk of an outcome, the effect should be summarized using the standard risk ratio (which “counts the dead”, i.e. considers the relative probability of the outcome event), whereas when the intervention increases risk, the effect should instead be summarized using the survival ratio (which “counts the living”, i.e. considers the relative probability of the complement of the outcome event).

Sheps’ ideas have never been implemented in practice, and most medical statisticians and epidemiologists are unaware of her work. I therefore concluded that I had two important task ahead of me:

• Formalize Sheps’ intuition in terms of a fully specified counterfactual causal model (these models did not exist in 1958)

• Convince methodologists and applied statisticians that this wasn't just a cute idea, but the correct solution to a well-recognized problem with real implications for medical decision making.

I spent the next several years working on this. This research program was not fundable, so I recruited personal friends as collaborators and coauthors; these superstars of Epidemiology, Medicine and Statistics donated their time and efforts towards sharpening the idea, identifying where the argument was unclear and how it could be improved, coming up with examples and improving the structure and flow of the manuscripts. Despite a lot of resistance from reviewers, we were able to publish several papers, yet we are still seeing little progress in terms of getting statisticians and applied medical researchers to actually implement Sheps' solution.

This latest manuscript adds the following:

• We discuss the relevant considerations for choice of effect measure, and clarify why we consider stability to be paramount. We show why the typical decision-theoretic argument for using the risk difference fails.

• The causal model that supports Sheps’ conclusions is presented in terms of Sufficient-Component Cause Models (“Causal Pie Models”), a close variation of Mackie’s INUS framework for causation. This allows us to demonstrate that the argument for stability of the survival ratio is actually an improved version of an argument that is used in many epidemiology courses in favor of the risk difference.

• Instead of focusing exclusively on the situation where the effect is exactly homogeneous between groups, we clarify the advantages of defining effect heterogeneity in terms of deviations from Sheps' preferred variant of the relative risk, rather than deviations from the other variant of the relative risk.

• We provide evolutionary reasons for stability of Sheps’ preferred variant of the relative risk. The conclusion depends on a biological asymmetry between levels of the exposure variable, which will generally only be viable when there was a “default state” in evolutionary history (for example: not being treated with Penicillin was the default state for our ancestors, whereas there was no default state of some other exposure variables, including gender).

• In the appendix, we provide an impossibility proof for the odds ratio. Specifically, we show that if scientists choose the conditioning set (set of effect modifiers) by reasoning about the distribution of covariates which determine whether an individual will respond to treatment, conditional stability of the odds ratio will only be obtained if the conditioning set is large enough to imply stability of all other measures of effect.

While we were working on this paper, we gradually became aware that variations of the same basic idea have been rediscovered several times across different academic fields:

• Patricia Cheng, a psychology professor at UCLA, published the “Power-PC” model in 1997. This model relies objects called “causal generative and preventive power”. Philosopher Clark Glymour from CMU has referred to these causal powers as "a brilliant piece of mathematical metaphysics", and a substantial literature has developed in those fields in support of this approach to extrapolation of causal effects. The Power-PC model is very closely related to our independently developed causal models, and its recommendations are identical to Sheps’.

• Andre Bouckaert and Michel Mouchart, statisticians from Universite Catholique de Louvain, developed the “Sure outcomes of random events” model in 2001; this model contains mathematical objects that are identical to the objects used in our justification for Sheps' conclusion.

• Mark van der Laan, Alan Hubbard and Nicholas Jewell, biostatisticians at UC Berkeley, proposed a measure of effect called “the causal switch relative risk”, which automatically selects the variant of the risk ratio recommended by Sheps.

• Rose Baker, an Emeritus Professor of Applied Statistics at Salford Business School, and Dan Jackson, a biostatistician at AstroZeneca, developed a new measure of effect for meta-analysis of binary outcomes which they called the “generalized relative risk reduction (GRRR)”. GRRR can be understood as a convenient representation of the causal switch relative risk, and their justification for the effect measure hints at the same underlying understanding of underlying causal mechanisms.

• Les Irwig and Paul Glasziou, leading thinkers in Evidence-Based Medicine, suggested in BMJ in 1995 that effects of interventions should be summarized using “relative benefits and absolute harms”. While this is technically a different proposal, it will lead to identical predictions if the outcome event is rare.

• Sainyam Galhotra, Romila Pradhan and Babak Salimi are computer scientists working on algorithmic fairness, who recently posted a pre-print on arXiv which argues that objects called “sufficiency scores” and “necessity scores” can improve on state-of-the-art approaches to algorithm fairness and interpretability. These objects are mathematically very close variations of objects used in models that justify Sheps’ conclusions

• In response to some of my earlier work, Carlos Cinelli and Judea Pearl, computer scientists at UCLA, developed a variation of causal Directed Acyclic Graphs which uses identical assumptions, and leads to identical conclusions, as the causal models that support Sheps' conclusion.

Almost all these scientists were working without knowledge of the others. This convergence of ideas from very different academic traditions hints that there is something about the underlying concepts which is appealing and intuitive to scientists who spend time thinking seriously about models for binary outcomes. We believe this ideas constitutes an attractor in idea space, and that it will continue to be “rediscovered” until it is either explicitly refuted, or routinely implemented in statistical practice.

Today, in addition to the arXiv preprint, I am also announcing plans for a documentary about all the researchers who have rediscovered variations of Sheps’ insight. When travel becomes possible again, a small film crew will follow me across North America and Europe while I have conversations with some of the scientists discussed in this blog post, as well as some of the detractor, and possibly some people who knew Sheps or who have written about other aspects of her life.

I am personally committing significant financial resources to this project, and will also accept crowdsourced backing via GoFundMe and Dogecoin (DPZNaH8zCiVAm7irRvAL9gX9ij2uuw3RGk). All contributions (no matter how small) will be acknowledged in the credits; all contributions above $15 will receive a digital copy of the final version of the documentary.

This newsletter will be used for weekly updates on my progress in convincing medical statisticians that not only was Sheps right, her ideas are fundamentally important for clinical decision making. Each newsletter will include questions and comments from readers, and my response to those comments. These comments can be submitted either via email at anders@huitfeldt, or anonymously at admonymous.co

All posts will be open-access for at least the first year. Voluntary contributions to a documentary movie about this project are welcome via GoFundMe or using Dogecoin (DPZNaH8zCiVAm7irRvAL9gX9ij2uuw3RGk)

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