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Phil. Trans. R. Soc. A (2011) 369, 4149–4161
doi:10.1098/rsta.2011.0227
A tentative taxonomy for predictive models in
relation to their falsifiability
BY MARCO VICECONTI*
Laboratorio di Tecnologia Medica, Istituto Ortopedico Rizzoli,
Via di Barbiano 1/10, 40136 Bologna, Italy
The growing importance of predictive models in biomedical research raises some concerns
on the correct methodological approach to the falsification of such models, as they are
developed in interdisciplinary research contexts between physics, biology and medicine.
In each of these research sectors, there are established methods to develop cause–effect
explanations for observed phenomena, which can be used to predict: epidemiological
models, biochemical models, biophysical models, Bayesian models, neural networks,
etc. Each research sector has accepted processes to verify how correct these models
are (falsification). But interdisciplinary research imposes a broader perspective, which
encompasses all possible models in a general methodological framework of falsification.
The present paper proposes a general definition of ‘scientific model’ that makes it possible
to categorize predictive models into broad categories. For each of these categories, generic
falsification strategies are proposed, except for the so-called ‘abductive’ models. For
this category, which includes artificial neural networks, Bayesian models and integrative
models, the definition of a generic falsification strategy requires further investigation by
researchers and philosophers of science.
Keywords: biomedicine; predictive models; falsification; integrative models
1. Introduction
A model is an invention, not a discovery [1, p. 277].
As biomedical researchers realize that a purely reductionist approach is
not sufficient to investigate some of the most challenging processes of human
physiopathology [2–7], the importance of predictive models within biomedical
research grows exponentially (see the articles included in Phil. Trans. R. Soc.
A [8–13]). This is mostly due to the fact that any integrative approach requires
a way to capture the fragments of reductionist knowledge (i.e. knowledge on a
phenomenon limited to a single characteristic scale, or to a subset of interactions)
into artefacts that make it practically possible to compose these fragments
of knowledge into a systemic understanding of physiopathology processes, and
predictive models are the most effective way to capture knowledge in a reusable
and composable way [5,14,15].
*[email protected]
One contribution of 11 to a Theme Issue ‘Towards the virtual physiological human: mathematical
and computational case studies’.
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But also this poses a number of new challenges. One relevant to this paper is
that these predictive models must be developed and investigated in terms of their
correctness in a territory of science that is at the crossroads between biological,
medical, physical and engineering sciences. Each of these domains is mature
enough to have its own well-established epistemology, in particular with respect
to the formulation of theories and the assessment of their truth content; however,
these epistemologies are significantly different from each other. Without entering
into a difficult and potentially controversial discussion, we can recognize that
medicine and engineering are applied sciences, whose scope is to solve problems,
whereas biology and physics are fundamental sciences, whose scope is to establish
new scientific knowledge about the world. In addition, engineering and physics
tend to embrace the scientific method in full, aiming towards a mechanistic
explanation of a phenomenon whenever possible, whereas biology and medicine
tend to amplify the empirical/observational component of research.
Developing predictive models in this interdisciplinary region of science is
dangerous. Is it possible to develop physics-based predictive models when the
observations we start from in order to develop the model are subject to severe
uncertainty? If, following the proposal by Popper [16], theories can only be
falsified but never confirmed, what does it mean to validate a predictive model? Is
there an inherent difference in the potential truth content of a mechanistic model
with respect to a phenomenological model based exclusively on observations?
In the following pages, I shall try to answer these questions, by first providing
a definition of ‘scientific model’ that, while general in nature, provides those
prescriptive elements to make them potentially useful for scientists.
2. What is a model?
Before I try to provide a tentative answer to this apparently simple question,
it is important to point out that with the term ‘scientific model’ I intend
to indicate any artefact that we use to idealize a portion of reality for the
purpose of scientific investigation. In this sense what follows is applicable not
only to mathematical/computer models, but also to any other ‘idealization
vehicle’ that we use in the practice of science. For example, when we investigate
an animal under the assumption that it will provide information on certain
aspects of human physiopathology, that animal is a model. When we use some
tissue-engineering product to investigate properties of real human tissues, that
tissue-engineering product is a model. Thus, what is discussed here is potentially
of much broader interest.
At the present time we do not have a precise definition of what a model is in
science, shared and accepted by most scientists. In such cases, we need to resort
to those who look to science from outside, the philosophers of science. Most
of the philosophical literature debates the role of scientific models with respect
to theories. In their relevant extensive review, Frigg & Hartmann [17] argued
that authors advocating the so-called syntactic approach to theories recognize a
limited role for models, while the advocates of the semantic approach to theories,
which reject the concept of calculus as a unique representation of theories, see
theories as families of models. While this net interpretation of previous works is
being questioned [18], the point remains that most authors until recently have
considered models only in the context of scientific theorization.
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More recently alternative perspectives have been proposed, wherein models
are recognized as autonomous agents possibly independent from theories (i.e.
[19]). In these papers, models are recognized as complements, replacements
or preliminaries of theories, but much of the recent philosophical debate
focuses on the representational purpose of models and whether this has unique
characteristics, or if it can be conducted to the more general philosophical
discussion on the representation of reality [20–24].
While this debate is of great interest and relevance, it lacks, so far, those
prescriptive elements that scientists require in order to improve their work. In
addition, in the current philosophical debate, there is an implicit assumption that
models are a product of science. However, if we look at the recent neuroscience
literature, and in particular at how long-term memory works, we are forced to
recognize that idealizing reality is a fundamental cognitive process [25–28]. From
this neuroscience perspective, we can define models, in a general sense, as:
finalized cognitive constructs of finite complexity that idealize an infinitely
complex portion of reality.
Here I pose, as an a priori concept, the assumption that every portion of reality,
no matter how small or limited it is, is always infinitely complex to ‘understand’.1
On the contrary, any idealization of that portion of reality is of finite complexity;
this is made possible by finalizing the cognitive elaboration of the perceptual
information. So to remember Mary’s face, we idealize certain elements of our
perceptions, and associate them with certain memories, whereas to recognize
that Mary does not feel well, we idealize other elements of our perceptions, and
associate them with other memories.
A careful reader can spot an apparent contradiction here: before I stressed how
scientific models are also physical things, such as tissue-engineering products or
research animals. But now I define models as products of our minds. But the
contradiction is only apparent. In reality, the animal is not the model; the model
is the idealization we apply in interpreting the observations that we make on that
animal as representative of human physiology. Knuuttila [32] defines a model
as ‘a two-fold phenomenon, comprised of both a material sign-vehicle and an
intentional relation of representation that connects the sign-vehicle to whatever
it is that is being represented’. I prefer to use the term model to indicate the
idealization, and the term analogue to indicate the ‘device that can be used for
the purpose of making a model’ [1].
3. What is a scientific model?
If we accept that modelling is an essential process of the human mind, then what
is special about scientific models? I propose that scientific models are discernible
from the other idealizations of the human mind because of the finality that informs
the choice of idealization.
1 The
infinite complexity of reality descends, for example, from the so-called ‘many worlds
interpretation’ of quantum theory [29]; it is also advocated by various sociologists, such as
Durkheim [30] or Weber [31]. If we limit this concept by considering only the disparity between the
complexity of reality and the capacity of the human brain, simple considerations across dimensional
scales down to sub-atomic particles would support this statement.
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But ‘science’ is a very broad term. It is used to indicate formal sciences, such
as mathematics, and empirical sciences, such as physics or sociology. In formal
sciences, the role of models is not necessarily linked to the concept of reality, and
thus our definition appears inappropriate.
Among empirical sciences, we must distinguish between natural sciences and
social and behavioural sciences. Hereinafter I shall refer with the term ‘science’ to
all areas that use the scientific method. If we define science as ‘a systematic
enterprise of gathering knowledge based on the scientific method’ and the
scientific method as ‘the typical modality by which science proceeds to reach
a knowledge of reality that is objective, reliable, verifiable and shareable’, then
we can define scientific models as:
finalized cognitive constructs of finite complexity that idealize an infinitely
complex portion of reality through idealizations that contribute to the
achievement of knowledge on that portion of reality that is objective,
shareable, reliable and verifiable.
This definition applies to all natural sciences. Whether it might also be
useful in social sciences such as economics is less clear. The adoption of the
scientific method in social sciences is debated, in particular with reference to
the requirement of objectivity. But in social sciences such as economics, where
modelling is a standard practice, I believe the definition above can be relevant
and prescriptive.
4. Attributes of scientific models
As I declared in §1, I do not pretend that this definition has any particular
philosophical value; however, as I shall show in the following pages, it has the
considerable advantage of yielding a number of prescriptive elements that are
potentially useful in dealing with the epistemological confusion that the use of
predictive models in biomedicine is posing.
A first set of prescriptive observations emerge directly from the definition,
by noticing that every scientific model should present the four fundamental
attributes the definition provides. Alternatively, following the approach of Popper,
we should consider ranking scientific models according to the degree to which they
are capable of achieving these fundamental attributes. Now, let us take a close
look to these four attributes.
— Objective is the opposite of subjective: agreeable by many peers, mostly
unbiased by personal views. In empirical sciences, no model will ever be
entirely objective; this is an abstract concept, unachievable in full, but to
which nevertheless we all should aim.2 If we recognize that the search for
2 It
should be noted that the search for objectivity does not imply the existence of a unique
perspective. To cite one of the reviewers of this paper: ‘Instead of searching for the meta-viewpoint,
i.e. the point from which the condition of objectivity as correspondence (with the structure of
reality) is achieved, multiple viewpoints could be employed in describing or explaining a given
phenomenon. These might complement each other and allow one to acquire a more complete
understanding of it’. See also the concept of complementarity applied to biology [33].
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scientific truths3 is a collective endeavour, then the search for objectivity
is strongly linked with the second attribute, shareability.
— Shareable means that it can be shared with our peers. Again, from a
prescriptive perspective, objectivity and shareability are strongly related
to the need for reproducibility. This opens up a whole new argument, which
I cannot discuss in full here. However, it should be pointed out that while,
in principle, reproducibility of experiments is possible through a textual
description (materials and methods) as the artefacts they involve are
generally available, the reproducibility of numerical studies too frequently
relies on digital artefacts that are not readily available (i.e. in-housedeveloped software), or the fine details about parameters and settings
that are necessary to reproduce the study are not provided. This is why
many peers are now advocating the need for mechanisms for model sharing
attached to peer-review journals that host numerical studies of this kind.4
— Reliable means ‘that can be trusted’. In this context, I define this property
as the ability of a model to provide comparable quality of performance
within a given range of conditions. The quality of performance depends
on the finality of the model, which thus must always be stated explicitly.
We also recognize that almost no model remains reliable for every possible
range of conditions (or more correctly to every possible state the portion
of reality under examination can assume). This range is the defined limit
of validity of the model, and it also must be stated explicitly every time
we develop a scientific model.
— Verifiable means ‘conceived and presented so that the quality of its
performance can be assessed’. This brings us directly to one of the problems
I listed at the beginning of this paper: if, according to Popper, no theory
can be confirmed, what is the sense of validating a predictive model? This
apparent contradiction derives from the fact that the definition of science I
provided above is somehow incomplete. In fact, science pursues two distinct
objectives: problem-solving and knowledge production.
Traditionally, each scientific domain is aimed at one or the other of these
objectives: typically medicine and engineering aim to solve problems, whereas
physics and biology aim to understand why some problems can be solved in
a certain way, i.e. to form new knowledge about how nature works. However,
in integrative biomedical research, this separation is much more blurred, and
the risk of mixing up these two scopes is greater. This point is relevant here
because, when the scope of a model is to solve a given problem, that model can
be validated. Indeed, we can imagine one or more experiments that confirm that
the model is capable of predicting the phenomenon of interest with a level of
3 This
is the only point where I use the word truth, whereas in the rest of the paper I use truth
content: the ‘truth content’ of a theory is the class of true propositions that may be derived from
it, and the ‘falsity content’ of a theory is the class of the theory’s false consequences [34]. Here, the
accent is on the search for truth, rather than on the existence of an absolute scientific truth, which
in agreement with Popper I do not believe will ever exist. In the rest of the paper, the word true
is used only in the context of logic.
4 See, for example, the petition for model reproducibility promoted by the Virtual Physiological
Human Network of Excellence (http://www.biomedtown.org/VPH/petitions/reproducibility/).
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accuracy sufficient to solve the problem. On the contrary, if a predictive model
aims to provide some new knowledge on a certain phenomenon, then we can
only try to falsify this model, but it is not possible to develop a finite number of
experiments that conclusively confirm that model to be valid.
5. Idealization and truth content
In order to be objective and shareable, the idealizations of scientific models should
always be formulated through logical reasoning. This consideration makes it
possible to elaborate a tentative taxonomy for scientific models that makes
possible a general discussion on the truth content and the falsification strategies
for each family of models.
There are three recognized kinds of logical reasoning:
— deduction, deriving a as a consequence of b;
— induction, inferring a from multiple instantiations of b; and
— abduction, inferring a as an explanation of b.
While deduction and induction should be familiar to most readers, it might be
worth providing a bit more detail on the third type of logical reasoning, abduction.
First proposed by Charles S. Peirce [35], this type of reasoning can be summarized
as follows:
— the surprising fact b is observed;
— but if a was true, b would be a matter of course; and
— hence, there is a reason to suspect that a is true.
Every scientific model contains multiple idealizations, which, in general, are
formulated according to different types of logical reasoning. However, it is usually
possible to identify one particular type of reasoning as predominant in the
construction of the model. We can thus taxonomize scientific models according
to the predominant type of logical reasoning used to build its idealizations, and
use this partitioning to discuss in a general context the potential truth content
and the possible falsification strategies that can be used for each category.
(a) Deductive models
To this family belong all physics-based mechanistic models. In these models,
we idealize the observed phenomenon according to a given set of physics laws,
capable of capturing the aspects of the phenomenon that the model scope regards
as relevant. All the rest of the model is deduced from these general laws. Typical
examples are the solid and fluid mechanics models used in biomechanics (i.e.
[36,37]), or the electro-chemo-mechanical models used in cardiac physiology (see
Bassingthwaighte’s review article on the cardiac physiome [38]).
The potential truth content of deduction is totally dependent on the truth
content of the initial assumptions: if the precedents are true, deduction ensures
that the consequents are also true. If the set of laws we choose to represent the
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phenomenon of interest (i) have an extensive truth content and (ii) are accurate in
capturing the relevant aspects of the phenomenon of interest, everything deduced
from them has the same truth content.
Another important property of deductive models is potential generality. If the
laws I used as antecedents are sufficiently general, I might find that the same
model is effective in predicting a very large set of similar phenomena.
Because in a deductive model, the consequents’ truth content derives from that
of the antecedents, deductive models are usually challenged by questioning the
laws of physics or of biology that we used to build our deductive model. As these
laws frequently have already been challenged for a long time, the point is not to
prove the laws false in themselves, but rather to challenge the assumption that
the phenomenon under observation can be accurately modelled in its aspects
relevant to the modelling scope by that choice of laws. So, for example, if we
idealize the biomechanical behaviour of the mineralized extracellular matrix of
bones with the theory of elasticity, in order to falsify this idealization, we should
conduct experiments to see if, within the range of physiological skeletal loading,
bone tissue shows a significant deviation from proportionality in the relationship
between stress and strain.
The other falsification strategy is to investigate phenomena that, while
independent from the one we modelled, logically descend from the idealization
we made. So staying with our bone example, if bone behaviour in physiological
conditions is linear elastic, then, in the range of deformation rates that is
physiologically possible, bones should exhibit no significant viscous dissipation,
which can be verified with isothermal experiments [39].
(b) Inductive models
All models belong to this category, where the prediction is generated by
induction, i.e. models based on the interpolation or extrapolation of observational
data or measurements. Typical examples are predictive models based on
epidemiological studies (e.g. population-based models to predict the risk of bone
fractures in osteoporotic patients [40]).
In models where the idealizations are predominantly formulated by induction,
within a deterministic framework the potential truth content is null. In fact, if an
inductive model accurately predicts a phenomenon n times, nothing guarantees
that for the n + 1 set of parameters the model will also be accurate. In a
probabilistic framework, which postulates that all physical phenomena show a
certain degree of regularity, the potential truth content of an inductive model
depends on statistical indicators such as level of significance, power of the test, etc.
The first approach to the falsification of inductive models should also be
inductive. We frequently forget that, if we want to use a data regression to
make predictions, it is not sufficient to verify that regression is accurate ‘on
average’. In this sense, attempts to falsify an inductive model by verifying if the
predicted values and the measured values simply correlate should be considered
largely inadequate. Of course, a predictive model must make predictions that are
correlated with the reality they intend to model. But predicting is much more
than correlating; it implies an explanatory nature that simple correlation lacks.
What we must do is to verify if the predictive error remains for every prediction
with the acceptance limits over the entire range of validity of the model.
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Another way to see this problem is to recognize that the peak predictive error
we observe over a large set of validation experiments should be considered as the
interval of uncertainty of our model’s predictions. If we run our model with two
different sets of parameters, and the two predictions differ by less than the peak
error we observed during the validation experiments, nothing can be concluded,
because we cannot positively confirm that the difference really exists and it is
not due to the inadequacy of our model.
But this is not sufficient. Inductive models must also be challenged on the
statistical side. If the model assumes the data are normally distributed, are
they? What is the level of statistical significance for our predictor? What is
the statistical power? These tests should be done not only on the data that
we use to inform our inductive model, but also on the predictions produced by
our model. A typical example is the following. Say that we want to model the
relationship between the electric charge on and the concentration of a chemical
species in an electrophysiology experiment. The concentration is measured for a
wide range of positive and negative charges, and then the entire set of data pairs
is fitted with a linear regression that shows a regression coefficient R2 > 0.9. We
thus use the regression equation as a predictive model of species concentration
as a function of the electric charge. If the process is polarized, i.e. positive
charges produce opposite effects to negative charges, such a model might be easily
falsified by repeating the regression separately on the positive and the negative
charge measurements. We might discover that the two resulting regressions are
significantly different in slope and/or intercept from the one obtained by pooling
all data together. This is an example of a regression that is statistically significant,
but that has a very low statistical power.
(c) Abductive models
This is not only the most difficult but also the most interesting family of
models. What are the models that change as we add new observations? So far I
have identified three examples:
— Neural networks are models that change according to the training set we
provide them, i.e. new observations.
— Bayesian models are models that change the probability associated with
the model as we add new observations.
— Integrative models are models made of component idealizations and
of relational idealizations defining the relations between components.
Integrative models are abductive: by adding a new component idealization
into our model, it predicts a surprising fact b; if we observe something close
to b under the same conditions, the compositional idealization that forms
the model is the fact a that by abduction is suspected to be true.
Neural network models are being used in a myriad of biomedical research
domains, such as analytical chemistry [41], diagnosis of pancreatic diseases [42],
pharmacokinetics [43] and drug discovery [44], to predict renal colic in emergency
settings [45], in the recognition of the activity type in accelerometer signals [46], in
neuromotor control modelling [47], etc. In fact, ‘artificial neural network’ returns
2736 papers in PubMed.
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Similarly, Bayesian models are used in myriad applications, including the
interpretation of magnetic resonance imaging of the human brain [48], the study
of metabolic systems [49], etc. ‘Bayesian model’ returns 6346 papers in PubMed.
The term ‘integrative model’ is quite recent and not yet universally
adopted. It was probably first used extensively in psychology, to indicate
approaches that combined psychology and other scientific disciplines such
as psychoneuroimmunology [50]. With respect to physiology and pathology
modelling, the idea of integrative modelling largely remains a conceptual
manifesto, although under the umbrella of the Virtual Physiological Human
initiative a considerable amount of research is being conducted that should
produce a large number of examples in the near future. Some early examples are
the multi-scale study of skeletal strength [14], a multi-scale model of gas exchange
in fruit [51], a multi-scale model of gastric electrophysiology [52], a multiscale model of bone remodelling [53] and a multi-scale model of cardiovascular
regulation [54].
This last category, in particular, is central to our reflection; as I already
mentioned in §1, various authors are suggesting that, to overcome the drift
in biomedical research to frequently transform the necessary methodological
reductionism5 into the very dangerous causal reductionism,6 and to properly
account for complex systemic emergence [57] and the issue of complementarity
[33], a possible approach is to capture reductionist knowledge into component
idealizations, and then add them into integrative models together with the
appropriate relational idealizations.
However, at the current state of our analysis very little can be said on the
potential truth content of abductive models, and on the best strategies to falsify
them. The best advice for integrative models is first to attempt the falsification
of each component idealization and of each relational idealization separately, and
then to repeat it for the integrative model now considered as a whole.
However, this strategy does not apply to Bayesian models or to neural
networks; thus a more general falsification framework, valid for all abductive
models, needs to be elaborated. A possible direction of research is that of the
so-called bootstrap methods [58]: the repetition of the construction of the model
through the abduction cycle using multiple sets of independent observations, all
relative to the phenomenon to be modelled, should yield the same model.
6. Discussion
The potential truth content of deductive models is very high. Using the Popperian
approach, the falsifiability of deductive models is, in general, excellent. By
formulating our idealizations in terms of a set of fundamental principles from
which we deduce, we provide an excellent target for the falsification of the model.
5 Methodological reductionism is defined as ‘an approach to understanding the nature of complex
things by reducing them to the interactions of their parts, or to simpler or more fundamental
things’ [55].
6 Causal reductionism implies that the causes acting on the whole are simply the sum of the effects
of the individual causalities of the parts [55,56]. In other words, macro-level causal relations are
seen as reducible to micro-level causal relations.
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On the contrary, the potential truth content of inductive models is low. With
the exception of the most trivial cases, inductive models can be falsified only
on a statistical basis, which in the end transfers the problem of model validity
to that more general one of the regularity of the predicted phenomenon. In this
sense, while we recognize the usefulness of inductive models in a number of cases,
everything else being the same (ceteris paribus clause), deductive models should
always be preferred to inductive models for their superior potential truth content,
i.e. their higher falsifiability.
The discussion for abductive models is more complex; we are tempted to say
that their recursive nature perfectly matches the process of scientific research,
and because of this they promise to embody the scientific method better than any
other modelling approach. However, the current incompleteness of the falsification
framework for this family of models casts some concerns on their use. Because of
this I strongly recommend that the scientific community, possibly with the help
of the philosophy of science community, adopts the problem of the falsification of
abductive models as a central goal for future research.
This paper results from the collective reflection on the use of integrative models in biomedicine
that is permeating the research community that participates in the Virtual Physiological Human
initiative.7 In particular, I would like to acknowledge the contributions of various individuals,
including Denis Noble, Peter Hunter, Ralph Müller and Peter Kohl, who found the time and the
patience to discuss with me these fairly convoluted arguments.
This study was supported by the European Commission through the projects ‘The Osteoporotic
Virtual Physiological Human—VPHOP’ grant FP7-ICT2008-223865 and ‘Virtual Physiological
Human Network of Excellence’ grant FP7-2007-IST-223920. The author declares that he does not
have any financial or personal relationships with other people or organizations that could have
inappropriately influenced this study.
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