Abstract
Inference to the Best Explanation (IBE) is usually employed in the Scientific Realism debates. As far as particular scientific theories are concerned, its most ready usage seems to be that of a theory of confirmation. There are however more uses of IBE, namely as an epistemological theory of testimony and as a means of categorising and justifying the sources of evidence. In this paper, I will present, develop and exemplify IBE as a theory of the quality of evidence - taking examples from medicine and showing that IBE can thereby provide the epistemological underpinning and justify the criteria of grading quality of mechanistic evidence that have been recently provided in the Clarke et al. (2014) paper on how evidence of medical mechanisms is to be construed alongside population studies.
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Notes
Lipton 2004, pp. 103–126. As it is well known, Lipton distinguishes between what he calls ‘Inference to the Likeliest Explanation’ and ‘Inference to the Loveliest Explanation’. Roughly speaking, ‘Inference to the Likeliest Explanation’ is concerned with confirmation, and, although connected with IBE, Lipton tends to identify it with the Bayesian theory. On the other hand, ‘Inference to the Loveliest Explanation’ constitutes the proper use of IBE in finding the relevant evidence, in pointing to the most fruitful and promising hypotheses, in a preliminary stage, before one arrives at, or takes into account, the level of confirmation (hence, IBE as Inference to the Loveliest Explanation could be used, in collaboration with a Bayesian approach, to approximate the prior probabilities of hypotheses to be confirmed; see Lipton 2001, p. 22, and more recently, McCain and Poston 2014). In a sense, my enquiry in the present paper could be portrayed as an attempt to refine the use the Inference to the Loveliest Explanation as a theory for relevant evidence, and for grading the quality of relevant evidence. However, for the purpose of clarity of exposition, I will rather pursue the way opened by Lipton’s work on testimony, since it offers a more direct access to the level of evidence and makes it easier to keep apart the concepts of confirmation and that of the quality of evidence.
For the original contributions urging that in medicine mechanisms should be treated alongside population studies, see Russo and Williamson (2007), Russo and Williamson (2011). Note that I will be using in this paper the neutral definition of mechanism provided in Illari and Williamson (2012) ‘a mechanism for a phenomenon consists of entities and activities organized in such a way that they are responsible for the phenomenon’ (Illari and Williamson 2012, p. 125).
A theory of the quality of evidence should be able to provide such ‘rules of thumb’ even when (or especially in the circumstances in which) we do not have numerical expressions of the probabilistic dependence between the evidence and the hypotheses at stake, or the probability attached to evidence itself. It is for this reason that I think IBE is an interesting path to investigate, even if, given the tremendous popularity of Bayesianism (as a theory of confirmation) one would prima facie consider it also as a candidate for assessing the quality of evidence. We have in fact at disposal sophisticated Bayesian accounts of testimony, (e.g. Bovens and Hartmann 2003). As I said, however, I am focusing here on IBE given that it is more amenable to such ‘rules of thumb’ - which does not mean that I ignore the ‘friendly companionship’ between IBE and Bayesianism (to use the expression employed by Lipton to describe the relation between the two theories on the level of theory confirmation). Such a ‘friendly companionship’ could as well hold for the level of assessing quality of evidence, and I will mention in footnotes the conceptual support that my approach to IBE could gather from the Bayesian perspective.
In Clarke et al.‘s own words “In the paper up to this point, we have made the case that evidence of mechanisms can usefully supplement evidence of correlation. Of course, all evidence and all conclusions reached in medicine are fallible. Evidence of correlation is fallible; evidence of mechanisms is fallible; and conclusions drawn from that evidence are fallible. That is the nature of any science. We focus here on the important point that we can get varying quality of evidence of mechanisms, just as we can get varying quality of evidence of correlation. We have been pressing the point that this kind of variation in quality of evidence of mechanisms needs a great deal more attention—indeed, it needs just as much attention as quality of evidence of correlation. Here we make a very preliminary attempt to lay out some ways in which evidence of mechanisms may be graded. We acknowledge that much more work will need to be done in this regard.” (Clarke et al. 2014, p. 358)
See for instance, the Oxford criteria of the levels of medical evidence, in which the evidence of mechanisms is placed on the bottom of the scale. http://www.cebm.net/oxford-centre-evidence-based-medicine-levels-evidence-march-2009/
Of course, one can discuss the latter question without going into the former.
Psillos 2002, pp. 612–613, Lipton 2004, pp. 67, 82–83. Psillos also discusses other advantages of IBE over the hypothetico-deductive method, like the fact that, when an instance appears to disconfirm a theory, the hypothetico-deductive method does not have an easy life in distinguishing whether it is the core-theory or the auxiliary assumptions that should be dismissed.
Lipton 2007.
Psillos 2002, pp. 615–616,
Lipton 1993, pp. 39–40, 42–43, Lipton 2004, pp. 41–43, Mill [1843] 2002. Mill’s criteria amount to simple but extremely powerful causal intuitions that can be found in the background of most contemporary sophisticated methodologies of causal discovery (see for instance Cartwright 1989). Note that there are various ways in which one can conceptualise IBE. The one which is adopted here follows Peter Lipton’s (and Gilbert Harman’s) footsteps, and lays stress on the explanatory values, and Mill’s criteria. An alternative view lays more stress on the similarities between IBE and Pierce’s abduction, privileges the context of discovery, is amenable to a more formal expression, and tends to leave in the background the role of explanatory values, or the use of Mill’s methods (see Schurz 2008 for an excellent, detailed discussion). I prefer to use here Lipton’s way because the causal vein in his account makes its application to realms like medicine easier, and also because my rationale is derived from Lipton’s work on the epistemology of testimony, as it will be discussed below.
Psillos 2002, p, 621.
See Dragulinescu 2016a, where I discuss the use of IBE as a theory of confirmation in a medical context and show how quality of evidence is taken into account when adjudicating among hypotheses on explanatory grounds. For a general argumentation that IBE reckons with quality of evidence when inferring explanatory conclusions, see Psillos 2002, p, 620, Harman 1965, pp. 90–91. The general idea is that one simply cannot infer the best explanation for a phenomenon based on unreliable, biased or falsified evidence.
Which is, in fact, a (crude) form of the Humean reductionist approach to testimony, as we shall see in the next section.
Gelfert 2010, Lipton 2007. From a different perspective than Lipton’s, Lackey (2006) and Lackey (2008) also advocates a middle way between the reductionist and non-reductionist approaches; as the title of her (2006) article colorfully says ‘It Takes Two to Tango’. Graham (2006) accepts that reductionism and non-reductionism are not incompatible positions, but does not go on the way of seeing a fruitful collaboration between assumptions belonging to both parties, arguing instead that justification of testimonial beliefs can be overdetermined.
Hume 1977 [1748], 74. There are important differences within the camp of reductionism (e.g., general vs. local reductionism), and the positions of proponents of this view should of course not be assimilated to mere pastiches of Hume.
“What is to be explained is often not just that the speaker said what she said, but that she said it, and that she said it in the way that she did, for example in a way of exaggerated earnestness.” Lipton 2007, p. 245.
“There will also be evidence that has nothing to do with either the speaker or her present utterance. Thus the fact that she has been so reliable on these matters in the past encourages me to trust her this time.” Lipton 2007, p. 245.
‘These sorts of account [non-reductionist ones] make credibility remarkably independent of the content of the testimony. What counts is who says it, not what is said.[...] In any event, it is clear that the decision whether to believe someone depends not just on who they are but also on what they say and how what they say fits with what the audience already accepts. The central question about testimony is not just whom to trust, but what to believe.’ Lipton 1998, p. 14, italics added. Obviously, a good example for instances where the content of the testimony should take central stage are the testimonial reports in special sciences, including medicine. More on this will follow below.
Harman 1965, p. 89. One of Lackey’s most important contributions is that testimonial inference could well go direct to (what I have called) first order truth, without going through (what I have called) second order truth; this is, I think, consistent with an explanatory framework; see Lackey (2008). However, I am inclined to doubt that in the scientific contexts with which this paper is concerned, one could really dispense with the second-order truth as to the author of testimony believing the facts s/he is reporting.
Cardiomiopathy being a condition in which the heart’s capacity to contract is reduced, due to damage in the myocardium produced by toxic, metabolic, or infectious agents. It could also be idiopathic. The most common form of cardiomopathy is dilated cardiomiopathy, in which the cavities of the heart become enlarged.
Waagstein et al. 1975. It is not a purely mechanistic report but is embedded in mechanistic evidence - see all the investigations pointed out in the report, as well as the relation to animal experimentation, described below.
cf. Waagstein 2002.
We would not want our theory of testimony to rule out conceding the truth of a report which bears witness to unusual findings, especially in a scientific context, in which such unusual findings may trigger important new discoveries or pathways of research. One would want however that these unusual findings have a minimal coherence with the background knowledge of the science in question. In our example, as I said above, one could concede the truth of Waagstein’s report, even if it concerned a new path of research in the domain of cardiac failure, since the findings of Waagstein were correlations that could have meant either that this new paths of research is promising, or that the correlations were accidental from the point of view of the direct efficacy of beta-blockers.
See Darsee 1983.
Incidentally, the idea that truth can be a cause might sound unusual but it has been propounded before - famously by Aristotle, who declares, when reviewing in the first chapters of his Metaphysics the philosophical positions of his predecessors, that, in spite of their overall errors, some had to acknowledge certain proper metaphysical statements, ‘being forced by the truth -ὑπ᾽ αὑτῆς τῆς ἀληθείας ἀναγκαζόμενοι; Met A, 984b9–10″). Truth means here for Aristote certain states of affairs that his predecessors could not ignore.
A causal interpretation of testimony is provided, from a different, Bayesian perspective, in Bovens and Hartmann (2003). It should not be forgotten that Hume formulated the position that stands in the background of the reductionist approach as a particular case of his general reasoning concerning causation ‘The reason, why we place any credit in witnesses and historians, is not derived from any connexion, which we perceive a priori, between testimony and reality, but because we are accustomed to find a conformity between them’. (Hume 1977 [1748], 75)
“And clearly, the decision whether to believe what one is told will have some dependence on the prior probability one assigned to what is asserted, which can itself be based on all sorts of evidence”(Lipton 2007, p. 245). Coherence figures also at the center of the Bayesian account developed in Bovens and Hartmann (2003), even though the authors argue that there can be no definitive measure of it.
Again, Lipton hinted towards the use of the method of agreement, for different testimonial sources; the method of agreement applies also, from a certain point of view, the explanatory virtue of coherence “[…] incompatibility between what the speaker says and some of the hearer’s deeply held beliefs is often a reason for rejecting the testimony. Yet, here there is no obvious explanatory link to the fact of utterance. Similar remarks apply to cases [...] of contradictions between different speakers’ testimony. If our speaker is contradicted by another speaker’s testimony, this provides reason not to believe, but the second speaker’s testimony may bear no explanatory relation to the first speaker’s testimony. But here to the defender of TIBE has some kind of reply, since she can say that negative evidence, whether from background or from contradictory testimony, will be registered by making a truth-entailing explanation less attractive and so less likely to be inferred” Lipton (2007), p. 251. The same point is made in Lipton (1998), p. 27.
The explanatory values could also apply to inferences which take into account features other than the content of the testimony itself. Take the Humean track-histories, according to which the truth of a testimony is to be inferred by induction from previous correspondences between similar testimonial reports and actual states of affaires (or, when it comes to a single speaker, from previous testimonial reports, on any subject, show to correspond to facts). If Harman and Psillos are right and the successful or warranted inductions are limit cases of inferences to the best explanation guided by the explanatory values hinted at by Harman and developed by Psillos, then track-history justifications, which are compatible with TIBE, can also be shown to rest on such explanatory values.
‘But Inference to the Best Explanation cannot then be understood as inference to the best actual explanation. Such a model would make us too good at inference, since it would make all our inferences true’ (Lipton 2004, p. 58). Among others, this was Lipton’s way of responding to van Fraassen’s charge of ‘the bad lot’. Gelfert has already made the move from truth to probable truth in his own, slightly twisted version of TIBE, in which the default stance of testimonial acceptance is justified on abductive grounds, and inferences to the best explanation are used to reject testimonies that are probably false. ‘On the one hand, the coherence and success of our testimony-based projects provides general abductive support for a default stance of testimonial acceptance; on the other hand, we are justified in rejecting specific testimonial claims whenever the best explanation of the instances of testimony we encounter entails, or makes probable, the falsity or unreliability of the testimony in question’(Gelfert 2010, p. 386).
Again, Lipton hinted towards the use of the method of agreement, for different testimonial sources; the method of agreement applies also, from a certain point of view, the explanatory virtue of coherence “[…] incompatibility between what the speaker says and some of the hearer’s deeply held beliefs is often a reason for rejecting the testimony. Yet, here there is no obvious explanatory link to the fact of utterance. Similar remarks apply to cases [...] of contradictions between different speakers’ testimony. If our speaker is contradicted by another speaker’s testimony, this provides reason not to believe, but the second speaker’s testimony may bear no explanatory relation to the first speaker’s testimony. But here to the defender of TIBE has some kind of reply, since she can say that negative evidence, whether from background or from contradictory testimony, will be registered by making a truth-entailing explanation less attractive and so less likely to be inferred” Lipton (2007), p. 251. The same point is made in Lipton (1998), p. 27.
The explanatory values could also apply to inferences which take into account features other than the content of the testimony itself. Take the Humean track-histories, according to which the truth of a testimony is to be inferred by induction from previous correspondences between similar testimonial reports and actual states of affaires (or, when it comes to a single speaker, from previous testimonial reports, on any subject, show to correspond to facts). If Harman and Psillos are right and the successful or warranted inductions are limit cases of inferences to the best explanation guided by the explanatory values hinted at by Harman and developed by Psillos, then track-history justifications, which are compatible with TIBE, can also be shown to rest on such explanatory values.
Moreover, in the accounts of IBE framed in terms closer to the abductive framework, the reasoning by analogy is one main feature of explanatory inferences (see Schurz 2008)
Waagstein, 1975.
Waagstein 2002, pp. 215–216, 218.
Mann and Bristow 2005.
Feldman et al. 1987
Swedberg 1993.
Mann and Bristow 2005.
“I kick myself now for not trying beta-blockers. We were afraid. We were concerned that if we blocked the body’s response to heart failure, the heart failure would get worse. In fact, we did studies that showed that beta-blockers could worsen heart failure. But the Swedes showed that if they started at a low dose and slowly increased it, beta blockers could be used safely in many patients. And the benefit could be enormous, because the high sympathetic tone that we demonstrated wasn’t a protective response – it was actually part of the problem.” (Lee 2013, pp. 162–163).
The mechanism of cardiac failure and its treatment are still subject to debate. In a recent interview, Braunwald was confessing that after several decades of research into the mechanisms of cardiac failure, the issues that were confronting him in the beginning of his career are still on the table for the medical community (Landau 2012). One can say that, on the level of confirmation, one final resolution has not yet been reached. However, these decades of research have brought about quality evidence about numerous sub-mechanisms of cardiac failure, detailing the neuro-hormonal consequences of the adrenergic influences in cardiac failure. Incidentally, this touches upon a point I have been repeated throughout this paper - that aside from the studies of confirmation, one needs a theory that does justice to the quality of evidence.
Illari and Williamson 2012, p. 125
See for instance Hernandez et al. 2009, who point out a potential lack of effectiveness for elderly patients.
References
Adler, J. (2012). Epistemological problems of testimony. Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/testimony-episprob/.
Audi, R. (1997). The place of testimony in the fabric of knowledge and justification. American Philosophical Quarterly, 34, 405–422.
Bird, A. (2010). Eliminative abduction—examples from medicine. Studies in History and Philosophy of Science, 4, 345–352.
Bird, A. (2011). The epistemological function of Hill’s criteria. Preventive Medicine, 53, 85–96.
Bovens, L., & Hartmann, S. (2003). Bayesian epistemology. Oxford: Clarendon Press.
Braunwald, E., & Chidsey, C. A. (1965). The adrenergic nervous system in the control of the normal and failing heart. Proceedings of the Royal Society of Medicine, 58(12), 1063–1066.
Cartwright, N. (1989). Nature’s capacities and their measurement. Oxford: Oxford University Press.
Clarke, B., Gillies, D., Illari, P., Russo, F., & Williamson, J. (2014). Mechanisms and the evidence hierarchy. Topoi, 33(2), 339–360.
Darsee, J. (1983). A retraction of two papers on cardiomyopathy. The New England Journal of Medicine, 308, 1419.
Douven, I. (2016). Inference to the best explanation: What Is It? And Why Should We Care? In K. McCain & T. Poston (Eds.), Best explanations: new essays on inference to the best explanation. Oxford: Oxford University Press.
Douven, I., & Schupbach, J. (2015a). Probabilistic alternatives to Bayesianism: the case of explanationism. Frontiers in Psychology, 6, 459.
Douven, I., & Schupbach, J. (2015b). The role of explanatory considerations in updating. Cognition, 142, 299–311.
Douven, I., and S. Wenmackers. (2015). Inference to the best explanation versus Bayes’s rule in a social setting. British Journal for the Philosophy of Science, Online first. http://bjps.oxfordjournals.org/content/early/2015/07/31/bjps.axv025.abstract. Accessed 8 March 2016.
Dragulinescu, S. (2012). On ‘Stabilising’ medical mechanisms, truth-makers and epistemic causality: a critique to Williamson and Russo’s approach. Synthese, 187(2), 785–800.
Dragulinescu, S. (2016a). Inference to the best explanation and mechanisms in medicine. Theoretical Medicine and Bioethics, 37(3), 211–232.
Dragulinescu, S. (2016b). Mechanisms and difference-making. Acta Analytica. doi:10.1007/s12136-016-0292-1.
Epstein, S., Robinson, B. F., Kahler, R. L., & Braunwald, E. (1965). Effects of beta-adrenergic blockade on the cardiac response to maximal and submaximal exercise in man. The Journal of Clinical Investigation, 44(11), 1745–1753.
Feldman, M. D., Copelas, L., Gwathmey, J. K., Phillips, P., Warren, S. E., Schoen, F., Grossman, W., & Morgan, J. P. (1987). Deficient production of cyclic AMP: pharmacologic evidence of an important cause of contractile dysfunction in patients with end-stage heart failure. Circulation, 75, 331.
Gaffney, T., & Braunwald, E. (1963). Importance of the adrenergic nervous system in the support of circulatory function in patients with congestive heart failure. The American Journal of Medicine, 34, 320–324.
Gelfert, A. (2010). Reconsidering the role of inference to the best explanation in the epistemology of testimony. Studies in History and Philosophy of Science Part A, 41(4), 386–396.
Glass, D. (2012). Inference to the best explanation: does it track truth? Synthese, 185, 411–427.
Graham, P. J. (2006). Testimonial justification: inferential or non-inferential. Philological Quarterly, 56, 84–95.
Harman, G. (1965). The inference to the best explanation. The Philosophical Review, 74, 88–95.
Hempel, C. G. (1970). On the ‘Standard Conception’ of scientific theories. In M. Radner & S. Winokur (Eds.), Minnesota studies in the philosophy of science (Vol. Vol. IV, pp. 142–163). Minneapolis: University of Minnesota Press.
Hempel, C. G., & Oppenheim, P. (1948). Studies in the logic of explanation. Philosophy of Science, 15, 135–175.
Hernandez, et al. (2009). Clinical effectiveness of Beta-blockers in heart failure. Journal of the American College of Cardiology, 53(2), 184–192.
Hume, D. (1977) [1748]. An Enquiry concerning human understanding. Eric Steinberg (ed.). Indianapolis: Hackett Publishing Company.
Illari, P., & Williamson, J. (2012). What is a mechanism: thinking about mechanisms across the sciences. European Journal for Philosophy of Science, 2, 119–135.
Lackey, J. (2006). It takes two to tango: beyond reductionism and non-reductionism in the epistemology of testimony. In J. Lackey & E. Sosa (Eds.), The epistemology of testimony. Oxford University Press.
Lackey, J. (2008). Learning from words. Oxford: Oxford University Press.
Landau, M. (2012). An Interview with Dr. Eugene Braunwald. Clinical Chemistry, 58(1), 11–20.
Lee, T. (2013). Eugene Braunwald and the rise of modern medicine. Harvard University Press
Lipton, P. (1993). Making a difference. Philosophica, 51, 39–54.
Lipton, P. (1994). Truth, existence, and the best explanation. In A. A. Derksen (ed.), The scientific realism of rom Harré. Tilburg University Press
Lipton, P. (1998). The epistemology of testimony. Studies in History and Philosophy of Science, 29, 1–31.
Lipton, P. (2000). Inference to the best explanation. In W.H. Newton-Smith (Ed.), A companion to the philosophy of science (pp. 184–193). Maalden: Blackwell.
Lipton, P. (2001). Is explanation a guide to inference? A reply to Wesley C. Salmon. In G. Hon & S. Rakover (Eds.), Explanation, theoretical approaches and applications (pp. 93–120). Dordrecht: Springer.
Lipton, P. (2004). Inference to the best explanation (second ed.). London: Routledge.
Lipton, P. (2007). Alien abduction: Inference to the best explanation and the management of testimony. Episteme, 4(3), 238–251.
Mann, D., & Bristow, M. (2005). Mechanisms and models in heart failure. The biomechanical model and beyond. Circulation, 111(21), 2837–2849.
McCain, K., & Poston, T. (2014). Why explanatoriness is evidentially relevant. Thought, 3(2), 145–153.
McMullin, E. (1992). The inference that makes science. Milwaukee: Marquette University Press.
Mill, JS [1843] (2002). A system of logic. Honolulu: University Press of the Pacific.
Niiniluoto, I. (1999). Defending abduction. Philosophy of Science, 66, S436–S451.
Niiniluoto, I. (2007). Abduction and scientific realism. In F. Keskin (Ed.), The proceedings of the twenty-first world congress of philosophy, vol. 12: Philosophical trends in the XXth Century (pp. 137–142). Ankara: Philosophical Society of Turkey.
Perrine, T. (2014). In defense of non-reductionism in the epistemology of testimony. Synthese, 191(14), 3227–3237.
Psillos, S. (1999). Scientific realism: how science tracks truth. London: Routledge.
Psillos, S. (2000). Abduction: between conceptual richness and computational complexity. In A. C. Kakas & P. Flach (Eds.), Abduction and induction: essays in their relation and integration (pp. 59–74). Dordrecht: Kluwer.
Psillos, S. (2002). Simply the best: a case for abduction. In A. C. Kakas & F. Sadri (Eds.), Computational logic: from logic programming into the future (pp. 605–625). Dordrecht: Springer.
Relman, A. (1983). Lessons from the Darsee affair. The New England Journal of Medicine, 308(23), 1415–1417.
Russo, F., & Williamson, J. (2007). Interpreting causality in the health sciences. International Studies in the Philosophy of Science, 21(2), 157–170.
Russo, F., & Williamson, J. (2011). Epistemic causality and evidence-based medicine. History and Philosophy of the Life Sciences, 33(4), 563–582.
Schurz, G. (2008). Patterns of abduction. Synthese, 164(2), 201–234.
Swedberg, K. (1993). Initial experience with Beta blockers in dilated cardiomyopathy. The American Journal of Cardiology, 71(9), 30C–38C.
Waagstein, F. (2002). Beta-blockers in congestive heart failure: the evolution of a new treatment concept – mechanisms of action and clinical implications. Journal of Clinical and Basic Cardiology, 5(3), 215–223.
Waagstein, F., Hjalmarson, A., Varnauskas, E., & Wallentin, I. (1975). Effect of chronic beta-adrenergic receptor blockade in congestive cardiomyopathy. British Heart Journal, 37(10), 1022–1036.
Wilmshurst, P. (2007). Dishonesty in medical research. The Medico-Legal Journal, 75(1), 3–12.
Woodward. (2002). What is a Mechanism: A Counterfactual Account. Philosophy of Science 69, supplement. Proceedings of the 2000 Biennial meeting of the philosophy of science association. Part II: Symposia Papers: S366–S377.
Woodward (2011). Mechanisms revisited. Synthese, 83, 409–427.
Acknowledgments
This article was written as part of the project ‘Grading Evidence of Mechanisms in Physics and Biology’, funded by the Leverh ulme Trust (http://blogs.kent.ac.uk/jonw/projects/gradingevidence-of-mechanisms-in-physics-and-biology/). I have received useful comments from Kristoffer Ahlstrom-Vij, Rachel Cooper, Veli-Pekka Parkkinen, Jon Williamson, and three anonymous referees for this journal.
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Dragulinescu, S. Inference to the best explanation as a theory for the quality of mechanistic evidence in medicine. Euro Jnl Phil Sci 7, 353–372 (2017). https://doi.org/10.1007/s13194-016-0165-x
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DOI: https://doi.org/10.1007/s13194-016-0165-x