A short comment on statistical versus mathematical modelling

From Nature Communications

Andrea Saltelli

Nature Communications volume 10, Article number: 3870 (2019) | Download Citation

While the crisis of statistics has made it to the headlines, that of mathematical modelling hasn’t. Something can be learned comparing the two, and looking at other instances of production of numbers.Sociology of quantification and post-normal science can help.

While statistical and mathematical modelling share important features, they don’t seem to share the same sense of crisis. Statisticians appear mired in an academic and mediatic debate where even the concept of significance appears challenged, while more sedate tones prevail in the various communities of mathematical modelling. This is perhaps because, unlike statistics, mathematical modelling is not a discipline. It cannot discuss possible fixes in disciplinary fora under the supervision of recognised leaders. It cannot issue authoritative statements of concern from relevant institutions such as e.g., the American Statistical Association or the columns of Nature.

Additionally the practice of modelling is spread among different fields, each characterised by its own quality assurance procedures (see1 for references and discussion). Finally, being the coalface of research, statistics is often blamed for the larger reproducibility crisis affecting scientific production2.

Yet if statistics is coming to terms with methodological abuse and wicked incentives, it appears legitimate to ask if something of the sort might be happening in the multiverse of mathematical modelling. A recent work in this journal reviews common critiques of modelling practices, and suggests—for model validation, to complement a data-driven with a participatory-based approach, thus tackling the dichotomy of model representativeness—model usefulness3. We offer here a commentary which takes statistics as a point of departure and comparison.

For a start, modelling is less amenable than statistics to structured remedies. A statistical experiment in medicine or psychology can be pre-registered, to prevent changing the hypothesis after the results are known. The preregistration of a modelling exercise before the model is coded is unheard of, although without assessing model purpose one cannot judge its quality. For this reason, while a rhetorical or ritual use of methods is lamented in statistics2, it is perhaps even more frequent in modelling1. What is meant here by ritual is the going through the motions of a scientific process of quantification while in fact producing vacuous numbers1.

All model-knowing is conditional on assumptions4. Techniques for model sensitivity and uncertainty quantification can answer the question of what inference is conditional on what assumption, helping users to understand the true worth of a model. This understanding is identified in ref. 3 as a key ingredient of validation. Unfortunately, most modelling studies don’t bother with a sensitivity analysis—or perform a poor one5. A possible reason is that a proper appreciation of uncertainty may locate an output on the right side of Fig. 1, which is a reminder of the important trade-off between model complexity and model error. Equivalent formulations of Fig. 1 can be seen in many fields of modelling and data analysis, and if the recommendations of the present comment should be limited to one, it would be that a poster of Fig. 1 hangs in every office where modelling takes place.

Fig. 1

figure1figure1

Model error as ideally resulting from the superposition of two curves: (i) model inadequacy error, due to using too simple a model for the problem at hand. This term goes down by making the model more complex; (ii) error propagation, which results from the uncertainty in the input variables propagating to the model output. This term grows with model complexity. Whenever the system being modelled in not elementary, overlooking important processes leaves us on the left-hand side of the plot, while modelling hubris can take us to the right-hand side

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October 26, 2019 at 12:56AM

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