# The myth and fallacy of simple extrapolation in medicine

## Abstract

Simple extrapolation is the orthodox approach to extrapolating from clinical trials in evidence-based medicine: extrapolate the relative effect size (e.g. the relative risk) from the trial unless there is a compelling reason not to do so. I argue that this method relies on a myth and a fallacy. The myth of simple extrapolation is the idea that the relative risk is a ‘golden ratio’ that is usually transportable due to some special mathematical or theoretical property. The fallacy of simple extrapolation is an unjustified argument from ignorance: we conclude that the relative effect size is transportable in the absence of evidence to the contrary. In short, simple extrapolation is a deeply problematic solution to the problem of extrapolation.

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## Notes

1. 1.

An even less complicated strategy—simplest extrapolation—would let the assumption of generalizability go completely unchecked. Broadbent (2013) uses ‘simple extrapolation’ to connote this simplest approach.

2. 2.

Not all members of the EBM community recommend simple extrapolation. Howick et al. (2013) argue that several historical examples of failed extrapolations are enough to reject it as a robust strategy.

3. 3.

In math, two numbers (A, B) are ‘in the golden ratio’ if the ratio of the larger number (A) to the smaller number (B) is equal to the ratio of their sum to the larger number, or A/B = (A + B)/A = φ. The golden ratio (φ)—variously known as the ‘golden section’ or ‘divine proportion’—is a constant, approximately equal to 1.618. It is an important ratio in mathematics, but also creeps up frequently in art, architecture and nature. Analogously, in medicine we often proceed as if relative risks are golden constants.

4. 4.

In comparison, some clinical researchers assume that quantitative effects typically differ across patient subgroups inside or outside of a trial (Lubsen and Tijssen 1989; Bailey 1994). Salim Yusuf et al. suggest that “‘quantitative’ differences in the size of the treatment effect in different subgroups are quite likely to exist” (1984, p. 413). Meanwhile, in the first edition of Evidence-Based Medicine: How to Practice and Teach EBM, David Sackett et al. note: “this (constancy of RR) is a big assumption” (1997, p. 170).

5. 5.

If we construe RRs and ARRs as quantifying a change in probability of the outcome (as a ratio or as a difference, respectively), the problem does not go away because negative probabilities and probabilities greater than 100% are incoherent.

6. 6.

While the framework assumes determinism, we could adapt this formula to allow for indeterminism by inserting coefficients representing the probability of the outcome for each unique type of individual in the population. The RR would then represent the effect size as a change in probability of the outcome.

7. 7.

Kravitz et al. (2004) provide several examples of treatment effect heterogeneity produced by particular genetic, behavioural and environmental variations.

8. 8.

One reviewer suggests that my criticism of GG relies on an argument from ignorance: I do not know of any justification for GG; therefore, GG is not justified. My argument is rather that GG is unjustified because it must itself make unjustified assumptions: either specious assumptions about the mathematical constancy of the RR or specious theoretical assumptions about homogeneity of individual treatment effects.

9. 9.

Elliott Sober (2009) provides a formal probabilistic justification for these ‘absence of evidence’ arguments.

10. 10.

It turns out that this RR is not generalizable to the 30–49% stenosis population (Rothwell et al. 2003).

11. 11.

Steel (2008) as well as Elias Bareinboim and Judea Pearl (2013) offer theories of extrapolation using structural models. Cartwright formulates sufficient conditions for external validity using the probabilistic theory of causality (2010) as well as causal equations (2012).

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## Acknowledgements

Thanks to Nancy Cartwright, Iain Chalmers, Luis Flores, Nicholas Howell, Ayelet Kuper, Mathew Mercuri, Jacob Stegenga, David Teira, Paul Thompson, and Ross Upshur for helpful and insightful feedback on earlier drafts of this paper. I am especially grateful to Mathew Mercuri for suggesting the analogy between unknown differences in extrapolation and unknown confounders in causal inference. Thanks also to the audience at the “Science and Certainty” workshop at the University of California, San Diego (2015) as well as the “Too Much Medicine” conference at Oxford University (2017) for comments and discussion of these ideas. I am thankful for funding support from the Canadian Institutes of Health Research and the McLaughlin Centre. I have no conflicts of interest to declare.

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## Appendix

### Appendix

We can model the question of whether or not to extrapolate the effect size using decision theory, by comparing the expected utility of extrapolating and intervening (EUI) with the expected utility of not extrapolating and not intervening (EU¬I). If we intervene with I, there are two possible scenarios: the intervention’s major benefit is generalizable (G), or the intervention’s major benefit is not generalizable (¬G). There is a partial expected utility associated with G (EUG), as well as a partial expected utility associated with ¬G (EU¬G). The EUG depends on the benefits and harms of the intervention, as well as their probabilities. We can predict the probability of a beneficial outcome (B) or harmful outcome (H) using the effect size and the untreated outcome probability (e.g. RR = p(B|I)/p(B|¬I), thus: *p(B|I) = RR × p(B|¬I)). If we assume one major benefit B (the primary outcome targeted by the intervention) and one major harm H (a major side effect or adverse event), then *EUG = p(B|I)u(B) + p(H|I)u(H), where p(B|I) and p(H|I) are the probabilities and u(B) and u(H) are the utilities (u(B) is positive and u(H) is negative). We can further assume that EU¬G < EUG; otherwise, there is no point in even considering intervention. Meanwhile, the total expected utility of intervening is: EUI = p(G)EUG + p(¬G)EU¬G. Substituting the previous equations for *EUG and *p(B|I):

$${\text{EU}}_{\text{I}} = {\text{p}}\left( {\text{G}} \right)\left[ {\left( {{\text{RR x p}}\left( {{\text{B}}|\neg {\text{I}}} \right)} \right){\text{u}}\left( {\text{B}} \right) + {\text{p}}\left( {{\text{H}}|{\text{I}}} \right){\text{u}}\left( {\text{H}} \right)} \right] + {\text{p}}\left( {\neg {\text{G}}} \right){\text{EU}}_{{\neg {\text{G}}}}$$

According to decision theory, we should implement the intervention if EUI > EU¬I. Simple extrapolation implies that this will usually be the case unless compelling evidence substantially raises the p(¬G) (and thus substantially lowers EUI). However, EUI also depends on EUG and EU¬G, and thus on the benefits and harms of intervention and their effect sizes. Therefore, even if p(¬G) is low to begin with (a risky assumption), EUI might only be slightly greater than EU¬I to begin with, and raising p(¬G) by just a little might tip the balance in favour of EU¬I.

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Fuller, J. The myth and fallacy of simple extrapolation in medicine. Synthese (2019). https://doi.org/10.1007/s11229-019-02255-0