Measurement 42 (2009) 1270–1277
Contents lists available at ScienceDirect
Measurement
journal homepage: www.elsevier.com/locate/measurement
Widely-defined measurement – An analysis of challenges
Ludwik Finkelstein
Measurement and Instrumentation Centre, City University London, London, UK
a r t i c l e
i n f o
Article history:
Received 26 November 2008
Received in revised form 26 February 2009
Accepted 9 March 2009
Available online 28 March 2009
Keywords:
Measurement theory
Measurement philosophy
Widely-defined measurement
a b s t r a c t
The paper examines fundamental problems of widely-defined measurement that lie outside the representation concerns of measurement theory. It is intended as a starting point
of a research agenda. It shows that measurement is applied in a wide range of diverse
domains of knowledge and enquiry for which a wide-sense definition of measurement is
necessary. It examines philosophical objections to the application of measurement. It considers in particular problems of measurand concept formation, validity, verifiability and of
theories for the measurand. It concludes that Measurement Science should address the
whole range of applications of measurement and should endeavour to provide a universal
framework of concepts and principles to address all applications of measurement.
Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction
Measurement, that is the descriptive representation of
the attributes of objects and events of the real world by
symbols on the basis of an objective empirical process, is
a basic tool of modern human thought. It is the way in
which we describe and reason about the world.
Measurement has been developed through the physical
sciences, which serve as a paradigm. From this basis its
application has been extended to virtually all domains of
human knowledge and discourse. However, the concepts
and methods of measurement in this wider and more diverse range of disciplines offer significant conceptual problems, compared with measurement in the physical
sciences that is the normative view of much metrological
discourse.
Some of these problems will be outlined in the present
paper as a starting point of discussion and an agenda for
research.
2. Historical development
The present concepts and principles of measurement
are the product of a long historical development. An under-
E-mail address: l.fi[email protected]
0263-2241/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.measurement.2009.03.009
standing of this process of development is very necessary
to help the extension of the application of measurement
to new domains, or to areas where measurement is still
problematic. The literature of the history of mathematics
and science does not, in general, treat the development
of measurement explicitly in its general account. Some
critical historical philosophical studies of the development
of the concepts of measurement have, however, also been
published. The subject should be an important item on
the research agenda.
Only a brief outline of the historical development of
measurement can be presented here. The references are
illustrative rather than exhaustive [1–8].
Measurement originated in counting, at the very dawn
of human culture. It developed in antiquity, on an intuitive
basis, through applications in crafts, trade, surveying and
calendar determination.
The rise of modern science was promoted by the advance of methods of measurement and in turn drove them
forward. The 19th century saw the development of concepts and methods of measuring intangible physical variables, such as those of thermal and electromagnetic
phenomena. There was established for physical phenomena an arsenal of measurement techniques, and a system
of scales and units, based on comprehensive theories of
the relevant domains of physics. A theory of measurement,
based on the concepts of the physical sciences, was devel-
L. Finkelstein / Measurement 42 (2009) 1270–1277
oped by Helmholtz and Hoelder and presented in detail in
the works of Campbell.
The descriptive and explanatory power of the physical
sciences made them a model for endeavours to extend
the same concepts and methods to psychological and social
domains of knowledge. The classical view of measurement
was inadequate for the purpose and a wider concept of
measurement was developed.
This historical development is well described in the literature. The present author discussed it in outline in [9].
More detailed analytical discussions are presented by Diez
[10,11] and Michell [12].
3. Measurement theory
Modern logical and philosophical understanding of the
fundamental concepts of measurement is based on the representational theory. The theory is based on the viewing of
the real world as empirical relational systems and measurement as a process of mapping them into symbolic relational systems.
The theory has been extensively presented in the literature [13–18]. An outline has been presented in [19]. The
theory has been extensively considered and important
contributions have been made by Mari and Rossi [20–23].
The theoretical the basis of the arguments in this paper
is outlined below informally for convenience. The arguments have been presented in detail in [9,24].
Measurement will be defined in the wide sense as a
process of empirical, objective assignment of symbols to
attributes of objects and events of the real world in such
a way as to represent them, or to describe them.
Description, or representation, means that when a symbol, or measure, is assigned by measurement to the property of an object, and other symbols are assigned by the
same process to other manifestations of the property, then
the relations between the symbols or measures imply, and
are implied, by empirical relations between the property
manifestations.
What is meant by an objective process in the above definition of measurement is that the symbols assigned to a
property manifestation by the measurement process must,
within the limits of acceptable uncertainty, be independent of the observer.
An empirical process in the definition of measurement
presented above means, first, that it must be the result of
observation and not, for example, of a thought experiment.
Further, the concept of the property measured must be
based on empirically determinable relations and not, say,
on convention.
This wide definition of measurement is often disputed.
Some consider the paradigm of measurement in the physical sciences as normative. Others require that, at least,
measurement be a numerical representation that reflects
an order. For this reason, it is convenient to distinguish between strongly and weakly defined measurements.
Strongly defined measurement is defined as a class of
widely-defined measurement that follows the paradigm of
the physical sciences. In particular, it has precisely defined
1271
empirical operations, representation by numbers and
well-formed theories for broad domains of knowledge.
Measurement that constitutes representation by symbols of properties of entities of the real world based on
an objective empirical process, but lacks some, or all, of
the above distinctive characteristics of strong measurement, may be termed weakly defined.
4. Properties of measurement
The properties of measurement arising from the above
wide-sense definition will now be analysed and discussed
to provide an explanation of the usefulness of its wide and
diverse applications.
Measurement provides an objective description of the
measurand. The description is not merely a matter of opinion or feeling. It is invariant in rational discourse.
Measurement is verifiable. Given a specification of the
measurement process the same symbolic description of a
measurand should in principle be obtainable by any
observer.
Measurement is based on a well-defined empirical process of observation. It is thus a basis of justified, true belief;
in other words it is the basis of true knowledge.
Measurement is not naming. It provides descriptions of
relations of the property manifestation measures to other
manifestations of the same property.
The value of a measurement process depends upon the
richness of the relations it can represent.
Measures are descriptions of great conciseness. A single
number tells us what it would take many words to express.
Measurement gives, further, a description that is precise, pinpointing by a single number a particular entity,
where a verbal description indicates a range of similar
but differing things.
Measurement is description by a well-defined symbolism. A measure of a property gives us an ability to express
facts and conventions about it in a formal symbolic language. Without the convenient notation of such a language, the complex chains of induction and deduction by
which we describe and explain the universe would be
too cumbersome to express.
It follows from what has been said that description by
symbols is not good in itself. The only value of measurement lies in the use to which the information is put. Science is not just the amassing of numerical data; it
depends upon the way in which the data are interpreted,
analysed and organised.
Finally, measurement describes measurands by symbols, which can be realised as signals and can be acquired,
processed and effectuated by information machines.
5. Applications of measurement
As was discussed above, measurement is extensively applied outside the physical sciences, in domains where
widely-defined concepts of measurement are used. This outline of the range and diversity of applications illustrates the
significance of measurement outside the physical sciences.
1272
L. Finkelstein / Measurement 42 (2009) 1270–1277
Measurement in psychology is the first example to be
cited. It is the endeavour to apply measurement in psychology that first challenged the restrictive view of measurement of the physical sciences and led to the present
wider concepts. An account of the historical and philosophical aspects of measurement in psychology is given
by Michell [12]. Measurement in psychology embraces
measurement of such attributes as intelligence, attitude
and the like [24,25]. It also concerns measurement of the
subjective perception of physical stimuli such as loudness,
colour, odour and the like [26]. There are at present major
programmes of research in the field of the measurement of
these sensory variables, as well as in the measurement of
emotions.
Closely related to measurements in psychology are educational measurements. Tests of knowledge, both declarative and procedural, are extensively carried out on
individuals in modern societies. Equally, tests of aptitude
and like personal characteristics are widely used. The tests
result in the assignment of numbers, or more general symbols. Aiming at objectivity, they are measurements in the
wide sense. They are immensely influential in the present
world. The literature of educational measurement is
immense and cannot be conveniently summarised in this
paper. A good survey, with extensive bibliographic information, is presented in [27].
Measurement is applied extensively in sociology. It is
concerned with investigation of class, status, segregation,
attitudes, poverty, literacy and the like. It requires widelydefined concepts of measurement. The theoretical concepts
are discussed in [28–30]. A recently published encyclopaedia gives a very wide view of theory and applications of measurement in the social sciences [31].
Related to sociological measurement is its application in
history, as discussed in [32]. There is also application to
political analysis [33].
A major area of application of measurement is economics. The important work brought together by Boumans [34]
provides an overview of the theory and philosophy of those
applications.
Accounting is concerned with the measurement of
economic activity of enterprises. It is the basis of the management of all enterprises. Accountancy is the key measurement activity of modern economies. It presents,
however, many problems of measurement objectivity and
validity [35]. Accounts are essentially models of an enterprise. While some quantities are measured, others are
based on conventions, assumptions and judgement. These
must be explicitly stated for the model to be valid. Valuation of intangible assets such as knowledge, intellectual
property, goodwill and the like is of great importance
and offers significant challenges [36–38].
Recent economic developments have shown the importance of credit risk valuation, which purports to be an
objective, empirical description. It is of vital importance,
especially in the case of the financial services industry. It
appears to have systemic faults [39,40].
In recent decades there has been a significant application of measurement in Linguistics, with the establishment
of a discipline of Quantitative Linguistics. It is concerned
with measurement of phonological, lexical, grammatical
and other attributes of natural language communication.
An important aspect of measurement as applied to natural
language is content, or text, analysis, which can be considered the measurement of meaning of text. Refs. [41–43]
provide a view of the field.
A significant area of the application of numerical
description is the determination of utility. Utility is a very
important concept in human affairs. In Economics utility is
the degree of satisfaction obtained from the consumption
of some goods or services. In Decision Theory it is the relative desirability of an outcome of a decision. It is a key
concept of Operational Research, which can be described
as the conceptual planning of effective action. In Utilitarianism it was proposed as a criterion of moral action with
a slogan of the greatest good of the greatest number and
the idea of a felicic calculus.
There have been developed representational scales that
assign numbers to utility. Utilities can be represented on
ordinal, or ratio scales. Utility functions can be formulated
relating attributes of candidates to values of utility. Utility
functions for multi-attribute candidate choices have been
developed. Representations of utility by numbers, or like
symbols, have the formal characteristics of measurement.
Determination of utility was at the very centre of the
development of modern measurement theory and the treatise of Roberts [13] provides a very good statement of the
issues. The encyclopaedia edited by Sage is an excellent
introduction to utility and decision making [44]. Utility
type calculations are extensively used in modern management [45–48]. They are, of course related to accountancy
information.
The status of utility determination as measurement is
doubtful. It is not in any way a measurement of the value
of the entity under consideration, but only of the value assigned to it by the formulator of the utility criterion.
It may be considered to be the measurement of the value judgement of the decision maker, as it is objective and
based on an observation of the actor. However, the basis of
the utility criterion is an arbitrary judgement and is not
based on empirical laws. Models of preference are generally normative rather than based on observation. Social
psychology provides ample evidence that practical decision making does not necessarily follow a rational model.
For such considerations see [49]. While the usefulness of
utility determination as a tool is undoubted, it does not
constitute true measurement as defined here.
Quality and organisational performance measures represent essentially utility measurements.
Utility-like measurements are also extensively used in
clinical management and public health, namely in the
measurement of what is termed as health status [50].
A major area of application of measurement has been
the use of so called software metrics. This is a measure of
attributes of a software system [51].
Classification based on objective empirical operations
constitutes measurement in the wide sense. Such measurement, therefore, embraces objective, empirical taxonomies such as biological systematics [52]. However, there
are other taxonomies that, while they are based on objective principles and based on empirical information about
the classified objects, are not measurement, because they
L. Finkelstein / Measurement 42 (2009) 1270–1277
are based on a principle of classification that is based on
convention rather than empirical observable properties.
An example is the international classification of economic
activities [53]. Such taxonomies have close similarities to
measurement in the wide sense.
The above outline has demonstrated the range and
diversity of measurement applications. There is an urgent
need for their systematic analysis.
6. Philosophical considerations
The wide application of measurement is driven by
philosophical considerations. It is based on a belief that
empirical observation represents the only secure basis of
knowledge and that objectivity is possible.
In this outline it is not possible to review the relevant
philosophical discussions adequately. Relevant entries in
[54–56] are recommended for further reading. A good
summary discussion is presented in [27].
There is a wide adoption of the programme of Galileo to
measure what is measurable and to make measurable that
which is not. Knowledge that is not based on an expression
in numbers is often viewed, with Kelvin, as of a meagre and
unsatisfactory kind.
One may view much of the accounts of the construction
of measurement scales in the present theory of measurement as having its roots in the philosophy of logical positivism that attempted to base all knowledge on direct
observational statements [57]. More strongly, much practical approach to measurement is based on operationalism
which defines concepts in science by the operations by
which we measure them [58].
This programme of basing all knowledge on objective
observations and measurement statements has been significantly challenged. Quine has stressed the importance
of holism and theory in interpretation of observations
[59]. Popper has put forward the view that all scientific
knowledge is conjectural [60] and that falsifiability, rather
than verifiability, is the test of scientific validity.
Further the scientific Kuhn questioned the nature of scientific theories and objectivity from the point of view of
history and sociology of science [61]. Feyerabend advocated methodological anarchy [62].
With these and like developments measurement science
must place philosophical considerations on its agenda.
In the application of measurement to systems with human actors, there are strong views that there is no unity of
science and that human sciences demand different methods of investigation from the natural sciences. There is
rejection of the belief that human behaviour is governed
by general universal laws and characterised by underlying
regularities. The human world can only be understood in
terms of the behaviour of individuals within it. They have
will, freedom, irrationality and intelligence. The detached,
objective observer required for measurement, cannot give
an account of this world. It requires involvement and
empathy.
It is specially important in the human sciences that
observations are theory laden. The theories are commonly
not objective, but reflect the interests of their formulators.
1273
They are, for instance, influenced by the gender, or class
interests of the observer.
There are objections that the aim of measurement is
control and that this leads to manipulation of society in
the interests of those with knowledge and power.
Ethical objections have been made that measurement
treats human beings as objects, rather than as the unique
persons they are. Objective descriptions, in terms of general laws are argued to be essentially dehumanising.
Thus, on the one hand the scientific method and the
universal application of measurement have a sound record
of success, on the other hand its basic assumptions are
challenged. The study and development of the relevant
philosophical problems is an important item for the research agenda of measurement science.
7. Measurand concept formation
The establishment of a process and scale of measurement
requires the formation of the concept of the measurand. This
has been a central problem of the establishment of measurement concepts of such attributes as temperature, or electrical current, which unlike length and weight are not simply
perceived by the human senses.
The process of measurand concept formation has been
discussed in the literature of logical positivism [63] and
considered in [64].
The consideration of widely-defined measurement requires further study of the process of measurand concept
formation in the light of the history of science, to understand how measurement was developed for intangible
physical phenomena. The study of Chang on the measurement of temperature points the way [65].
It may be useful to consider here the measurement of
intelligence as an example of the case where it is easier
to devise tests, than to establish what it is that they are
measuring [66]. Another significant similar problem is
the measurement of poverty, where substantive conceptual issues arise about what is measured [67,68].
Finally, we may consider the problem of what educational examinations measure [27]. A useful examination
must be a valid test of some well-defined attribute of the
examinee, other than the ability to pass similar examinations. It is not clear that this attribute is always clearly
defined and explained.
8. Experiment and observation
Measurement theory relies on an ability to perform
experiments for the purpose of establishing the existence
of empirical relational systems.
In many cases direct observation of the measurand is
not possible and it must be indirectly observed. Such indirect experiments consist of exciting the system under test
and observing the response. In many cases of widelydefined measurement, such experiments are not possible,
because the system under observation cannot be significantly disturbed.
This is done by formulating a model of the system,
based on theory, and estimating the measurands on the ba-
1274
L. Finkelstein / Measurement 42 (2009) 1270–1277
sis of measurements of the observables, by treating the
measurands as parameters or internal variables of the
model.
The problem is that it is difficult to test adequately the
theoretical model used, so that the measurement process is
theory laden. It is also usually necessary to adjust the model to permit the application of appropriate statistical techniques so that the model is not an adequate representation
of reality.
Problems may arise when the models have many variables and only limited observations are possible. Further
there are difficulties in the estimation of internal variables
and parameters of non-linear systems. In systems under
test that are not specially designed for the purpose of
experiment, the structure and parameters of the systems
may vary during the observation affecting the result. Also
observations to determine regularities in the behaviour of
systems under observation depend on the maintenance
of constancy of all conditions, other than the excitation
variable. This condition may be impossible to achieve in
natural systems. Further chaos theory has shown that even
simple systems may behave chaotically and not show replicable relations between variables [69].
We may illustrate these problems with the example of
economic measurements. Economies cannot, in general,
be disturbed for the purpose of measuring economic quantities. It is necessary to measure such measurands under
conditions of normal functioning of the economy. The subject is discussed in the literature of econometrics [70].
A similar problem occurs in biological and biomedical
measurements. Measurements of some attributes can only
be performed on intact, functioning, living organisms. The
possibility of disturbance of such organisms for experimental purposes is limited. Measurements are performed
on the basis of theoretical models of the system, by estimation of parameters and internal variables on the basis of
measurements of observables with minimal disturbance
[71].
9. Replicable relations
The establishment of a scale of measurement for a quality is based on the existence among the quality manifestations of empirical relations that can be observed to be
replicable. Such relations are termed nomological, or law
like.
In many systems, however, there are no replicable relations for the measurand.
As has been discussed in the section dealing with philosophical considerations humans are autonomous intelligent actors and may behave differently under apparently
identical conditions. They are also not passive objects of
investigation, but are aware of the tests performed on
them. Idiographic methods of investigation, that take accounts of their uniqueness, may be more appropriate for
many such systems. This is an objection to measurement
in many aspects of the social sciences.
Nevertheless, there is empirical evidence in measurement in the psychological and social science of regularities,
which when statistically treated, represent an adequate
basis for measurement in the wide sense. This is the basis
of measurement in the social sciences.
The other reason for the difficulty in determining of
replicable relations involving the measurand is the fact
that the measurand may be embedded in a complex system. The difficulties in measurement in such systems have
been discussed in the section above.
10. Reliability, validity and generalisability
The concepts of reliability, validity and generalisability
are central to widely-defined measurement. A good summary account is given in [27].
Reliability is the term used to describe the extent to
which a measurement procedure yields the same result
on repeated trials. It is thus the same concept as that of
uncertainty used in strictly defined measurement. This is
well discussed in the literature [72]. Reliability may be difficult to achieve in practice in many applications of measurement, but it does not present conceptual difficulties.
Validity in the social and psychological sciences is a
very comprehensive concept. It is used for the general fitness for purpose of an investigation and its results.
In the context of the present discussion the validity of a
measurement procedure is the totality of evidence about
whether a particular measurement procedure adequately
represents what is intended by theoretical concept of the
measurand. Thus, the symbols assigned by the procedure
should bear to each other the same relation in the symbolic
relational system, as the measurand manifestations of
which they are images, bear to each other in the measurand empirical relational system. They should correlate
with other operationalisations of the same concept. They
should not correlate with measurements with which theoretically they should not be related to.
The term generalisability is used to describe extent to
which research tests and observations from a study conducted on a sample population can be applied to the population at large [73]. In this connection one may contrast
wide sense measurement with strict sense measurement
in the physical sciences in which there are well-established
theories for broad domains of knowledge and the above
problems do not arise to a significant extent.
11. Theories
The aim of scientific inquiry is, in general, to establish a
well-defined theory for a domain of knowledge. Such a theory consists of a system of quantities and relations between them. It provides descriptive, explanatory and
predictive power. In the physical sciences such a system
is well established.
Such theories are not well developed in the psychological and social sciences and, as been discussed above, the
possibility of such theories for systems incorporating human actors is widely denied. However, it is recognized in
the application of measurement in these domains that
the establishment of networks of relations connecting different quantities is an important task. They are termed
nomological networks [74].
L. Finkelstein / Measurement 42 (2009) 1270–1277
It is necessary to recognise that the value of the concept
of a measurand depends upon the number of relations with
other variables into which it enters. Thus, for example, the
discussion of the definition of a measure of intelligence depends is concerned with the absence of adequate theories,
or nomological networks [75].
Nevertheless there are wide applications of economic
models that are essentially such theories. While they are
limited in predictive power they are deemed to have practical use [76].
Further the techniques of systems dynamics are extensively applied to model social and like systems [77,78].
Even in physical measurements there are examples of
physical attributes for which adequate theories are not
available. Hardness is one of them. It is defined as resistance of solid matter to local deformation, Scratch hardness, penetration hardness and rebound hardness are
recognized, Measurement tests are available, but there is
no adequate theory to relate them [79].
12. Verifiability
One of the principal pragmatic strengths of measurement is that measurement statements are verifiable. A
measurement statement consists of a symbol assigned
to the measurand, together with the specification of the
scale of measurement employed. Given such a statement
the measurement can be replicated by any other
observer.
In the strongly defined measurement in the physical
sciences a measurement result consists of a number and
the specification of the unit.
In physical and chemical measurements the system of
units is defined by international agreement and there is
an international organisation, together with a network of
national organisations to maintain it [80]. Valid measurements should always be traceable to the international
standards. They are thus verifiable.
In some domains such standards do not exist, or are
only partially developed. Fisher discusses some of these
difficulties [81].
Consider, for example, measurements in economics and
accountancy. They are expressed in units of a currency.
However, the value of a unit of a currency changes over
time. In such measurements it is significant to specify precisely how the change of value of the currency is taken into
account [82,83].
Similar problems arise in economic and accountancy
measurements involving international comparisons and
multiple currencies. The exchange rates between currencies vary and measurements must specify exactly how
the exchange rates are taken into account. There may not
necessarily be a satisfactory objective basis for the exchange rate used [84].
International Financial Reporting Standards (IFRS), the
standards and interpretations adopted by the International
Accounting Standards Board (IASB) deal with the above
and other issues. They are thus a step towards verifiability.
However they are less prescriptive and strict than is the
case with strictly defined measurement [85].
1275
Another case that may be examined is that of educational standards. Of the large number of examples that
may be given, consider the standards of educational attainment set out in the OECD Programme for International Student Assessment. The desired competences of the student
are set out in a set of statements in natural language. They
are translated into a set of tests. The results of the tests are
statistically treated for the cohort of students. This is typical for an educational attainment test. The question arises
with such tests as to how the test relates to the stated
objectives and what is the uncertainty involved [86].
13. Conclusions
Measurement is applied in a wide range of human inquiry and discourse. Measurement in the physical sciences
is the dominant paradigm. However, in significant areas of
application of measurement that paradigm is inapplicable
and a wide-sense definition of measurement is necessary.
Measurement science should address the whole range
of applications of measurement. It should endeavour to
provide a universal framework of concepts and principles
to address all applications of measurement.
Measurement theory, on representational principles
provides a basis for a universally applicable measurement
science.
There are, however, a range of problems of widely-defined measurement that require addressing. They constitute a research agenda. Among them are the need to
engage in the history and philosophy of science and the
methodology of the sciences in which measurement is
applied.
Acknowledgements
The author thanks his colleagues Professors K.T.V. Grattan and S.H. Khan for their support. He is also grateful to
Professors D. Hofmann, R. Morawski and L. Mari for creating the environment in which fundamental measurement
science ideas can be discussed.
References
[1] R. Porter, The Cambridge History of Science, Cambridge University
Press, Cambridge, New York, 2003.
[2] C.B. Boyer, U.C. Merzbach, A History of Mathematics, second ed.,
Wiley, 1989.
[3] J.L. Heilbron, The Oxford Companion to the History of Modern
Science, Oxford University Press, Oxford, New York, 2003.
[4] J.L. Heilbron, The Oxford Guide to the History of Physics and
Astronomy, Oxford University Press, Oxford, New York, 2005.
[5] A.W. Crosby, The Measure of Reality: Quantification and Western
Society, 1250–1600, Cambridge University Press, Cambridge, 1997.
[6] J.L. Heilbron, Weighing imponderables and other quantitative
science around 1800, University of California Press, Berkeley,
California, 1993.
[7] M.S. Morgan, M. Morrison, Models as Mediators: Perspectives on
Natural and Social Science, Cambridge University Press, Cambridge,
1999.
[8] J.J. Roche, The Mathematics of Measurement: A Critical History,
Athlone, London, 1998.
[9] L. Finkelstein, Widely, strongly and weakly defined measurement,
Measurement 34 (2003) 39–48.
[10] J. Diez, A hundred years of numbers. An historical introduction to
measurement theory 1887–1990: Part I: The formation period. Two
lines of research: axiomatics and real morphisms, scales and
1276
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
L. Finkelstein / Measurement 42 (2009) 1270–1277
invariance, Studies in History and Philosophy of Science Part A 28
(1997) 167–185.
J. Diez, A hundred years of numbers. An historical introduction to
measurement theory 1887–1990: Part II: Suppes and the mature
theory. Representation and uniqueness, Studies in History and
Philosophy of Science Part A 28 (1997) 237–265.
J. Michell, Measurement in Psychology: Critical History of a
Methodological Concept, Cambridge University Press, Cambridge,
New York, 2005.
F.S. Roberts, Measurement Theory: With Applications to Decision
Making, Utility, and the Social Sciences, Cambridge University Press,
Cambridge, 1984.
J. Pfanzagl, V. Baumann, H. Huber, Theory of Measurement, second
ed., Physica-Verlag, Würzburg, 1971.
D.H. Krantz, R.D. Luce, A. Tversky, P. Suppes, Foundations of
Measurement Volume I: Additive and Polynomial Representations,
Academic Press, New York, 1971.
D.H. Krantz, R.D. Luce, A. Tversky, P. Suppes, Foundations of
Measurement Volume II: Geometrical, Threshold, and Probabilistic
Representations, Academic Press, New York, 1989.
P. Suppes, D.H. Krantz, R.D. Luce, A. Tversky, Foundations of
Measurement Volume III: Representation Axiomatization, and
Invariance, Academic Press, New York, 1990.
L. Narens, Abstract Measurement Theory, MIT Press, Cambridge, MA,
1985.
L. Finkelstein, Foundational problems of measurement, in: K. Kariya,
L. Finkelstein (Eds.), Measurement Science – A Discussion, Ohmsha
Press, Tokyo, Washington, DC, 2000, pp. 13–21.
G.B. Rossi, Measurability, Measurement 40 (2007) 545–562.
L. Mari, Epistemology of measurement, Measurement 34 (2003) 17–
30.
L. Mari, The problem of foundations of measurement, Measurement
38 (2005) 259–266.
G.B. Rossi, An attempt to interpret some problems in measurement
science on the basis of Kuhn’s theory of paradigms, Measurement 39
(2006) 512–521.
A. Anastasi, S. Urbina, Psychological Testing, seventh ed., Prentice
Hall, Upper Saddle River, NJ, 1997.
J. Rust, S. Golombok, Modern Psychometrics: The Science of
Psychological Assessment, second ed., Routledge, London, 1999.
G.A. Gescheider, Psychophysics: The Fundamentals, third ed., L.
Erlbaum Associates, Mahwah, NJ, 1997.
L. Cohen, L. Manion, K. Morrison, Research Methods in Education,
sixth ed., Routledge, London, 2007.
P. Abell, Model Building in Sociology, Weidenfeld and Nicolson,
London, 1971.
H.M. Blalock, Measurement in the Social Sciences: Theories and
Strategies, Macmillan, London, 1975.
R.A. Zeller, E.G. Carmines, Measurement in the Social Sciences: The
Link Between Theory and Data, Cambridge University Press,
Cambridge, 1980.
K. Kempf Leonard, Encyclopedia of Social Measurement, Elsevier/
Academic Press, Amsterdam/London, 2005.
P. Hudson, History by Numbers: An Introduction to Quantitative
Approaches, Arnold/co-published in the USA by Oxford University
Press, London/New York, 2000.
D.E. McNabb, Research Methods for Political Science: Quantitative
and Qualitative Methods, Eurospan/M.E. Sharpe, London/Armonk,
New York, 2004.
M. Boumans, How Economists Model the World into Numbers,
Routledge, London, New York, 2005.
B.E. Needles, M. Powers, Principles of Financial Accounting, tenth ed.,
Houghton Mifflin Co., Boston, 2008.
M.J. Mard, J.R. Hitchner, S.D. Hyden, Valuation for Financial
Reporting: The Determination of Fair Value for Audited Intangible
Assets, second ed., John Wiley, Hoboken, NJ, 2007.
J. Berry, Tangible Strategies for Intangible Assets: How to Manage
and Measure Your Company’s Brand, Patents, Intellectual Property,
and Other Sources of Value, McGraw-Hill, New York, 2005.
J.R. Hitchner, Financial Valuation: Applications and Models, John
Wiley & Sons, Hoboken, NJ, 2006.
T.R. Bielecki, M. Rutkowski, Credit Risk: Modeling, Valuation and
Hedging, Springer, Berlin, New York, 2002.
J.F. Dean, The Art & Science of Financial Risk Analysis: The Analyst’s
Guide to Determining a Company’s Financial Strength and
Creditworthiness, third ed., National Association of Credit
Management, Columbia, MD, 2006.
R. Franzosi, Content Analysis, Sage, Los Angeles, London, 2008.
[42] K. Johnson, Quantitative Methods in Linguistics, Blackwell, Oxford,
2008.
[43] R. Köhler, G. Altmann, R.D.G. Piotrovskin012D, Quantitative
Linguistik: ein internationales Handbuch = Quantitative Linguistics:
an International Handbook, M. de Gruyter, Berlin, New York, 2005.
[44] A.P. Sage, Concise Encyclopedia of Information Processing in
Systems & Organizations, first ed., Pergamon Press, Oxford,
England, New York, 1990.
[45] A. Neely, Business Performance Measurement: Unifying Theories
and Integrating Practice, second ed., Cambridge University Press,
Cambridge, 2007.
[46] R.B. Carton, C.W. Hofer, Measuring Organizational Performance:
Metrics for Entrepreneurship and Strategic Management Research,
Cheltenham/Edward Elgar, UK/Northampton, MA, 2006.
[47] W.D. Cook, J. Zhu, Modeling Performance Measurement: Applications
and Implementation Issues in DEA, Springer, New York, 2005.
[48] A.D. Neely, Measuring business performance, The Economist in
association with Profile Books, London, 1998.
[49] D. Kahneman, A. Tversky, Choices, Values, and Frames, Cambridge
University Press/Russell sage Foundation, New York, Cambridge/UK,
2000.
[50] I. McDowell, Measuring Health: A Guide to Rating Scales and
Questionnaires, third ed., Oxford University Press, New York,
Oxford, 2006.
[51] N.E. Fenton, S.L. Pfleeger, Software metrics: a rigorous and practical
approach, second ed., Boston/International Thomson Computer
Press, PWS/London, 1996.
[52] R.T. Schuh, Biological Systematics: Principles and Applications,
Cornell University Press, Ithaca, New York, 2000.
[53] United Nations, Statistical Division, International Standard Industrial
Classification of all Economic Activities (ISIC), Revision 3.1 ed.,
United Nations, New York, 2004.
[54] N. Abercrombie, S. Hill, B.S. Turner, The Penguin Dictionary of
Sociology, fifth ed., Penguin, London, New York, 2006.
[55] R. Audi, The Cambridge Dictionary of Philosophy, second ed.,
Cambridge University Press, Cambridge, New York, 1999.
[56] T. Honderich, The Oxford Companion to Philosophy, second ed.,
Oxford University Press, Oxford, New York, 2005.
[57] O. Hanfling, Logical Positivism, B. Blackwell, Oxford, 1981.
[58] P.W. Bridgman, The Logic of Modern Physics, The Macmillan
Company, New York, 1927.
[59] W.V. Quine, R.F. Gibson, Quintessence: Basic Readings from the
Philosophy of W.V. Quine, Belknap Press of Harvard University Press,
Cambridge, MA, 2004.
[60] K.R. Popper, The Logic of Scientific Discovery, Basic Books, New York,
1959.
[61] T.S. Kuhn, The Structure of Scientific Revolutions, third ed.,
University of Chicago Press, Chicago, IL, 1996.
[62] P. Feyerabend, Against Method, third ed., Verso, London, New York,
1993.
[63] C.G. Hempel, Fundamentals of Concept Formation in Empirical
Science, University of Chicago Press, Chicago, 1952.
[64] L. Finkelstein, Theory and philosophy of measurement, in: P.H.
Sydenham (Ed.), Handbook of Measurement Science, vol. 1, Wiley,
Chichester, 1982, pp. 1–30.
[65] H. Chang, Inventing Temperature: Measurement and Scientific
Progress, Oxford University Press, Oxford, New York, 2004.
[66] R.J. Sternberg, Handbook of Intelligence, Cambridge University Press,
Cambridge, New York, 2000.
[67] A.B. Atkinson, On the Measurement of Poverty, London School of
Economics and Political Science, London, 2000.
[68] M. Jäntti, On the Measurement of Poverty: Conceptual Issues and
Estimation Problems, Åbo Akademi, Åbo, 1990.
[69] R.L. Devaney, An Introduction to Chaotic Dynamical Systems, second
ed., Westview/Colo., Boulder/Oxford, 2003.
[70] P. Kennedy, A Guide to Econometrics, sixth ed., Blackwell Pub.,
Malden, MA, 2008.
[71] E.R. Carson, C. Cobelli, Modelling Methodology for Physiology and
Medicine, Academic Press, San Diego, 2001.
[72] D.J. Hand, Measurement Theory and Practice: The World through
Quantification, Arnold, 2004.
[73] R.L. Brennan, Generalizability Theory, Springer, New York, 2001.
[74] L. Cronbach, P. Meehl, Construct validity in psychological tests,
Psychological Bulletin 52 (1955) 281–302.
[75] R.L. Gregory, The Oxford Companion to the Mind, second ed., Oxford
University Press, Oxford, 2004.
[76] B.S. Ferguson, G.-C. Lim, Introduction to Dynamic Economic Models,
Manchester University Press, Manchester, 1998.
L. Finkelstein / Measurement 42 (2009) 1270–1277
[77] J.W. Forrester, Industrial Dynamics, The MIT Press, Cambridge, MA,
1962.
[78] J. Sterman, Business Dynamics: Systems Thinking and Modeling for a
Complex World, Irwin/McGraw-Hill, Boston/London, 2000.
[79] J. Malzbender, Comment on hardness definitions, Journal of the
European Ceramic Society 23 (2003) 1355–1359.
[80] Le Système International d’Unités (SI) = The International System of
Units (SI), eighth ed., Bureau International des Poids et Mesures,
Sèvres, 2006.
[81] W.P. Fisher, New metrological horizons: invariant reference
standards for instruments measuring human, social, and natural
capital, in: E. Benoit (Ed.), 12th IMEKO TC1 & TC7 Joint Symposium
[82]
[83]
[84]
[85]
[86]
1277
on ‘‘Man Science & Measurement”, Universite de Savoie, Annecy,
France, 2008, pp. 51–58.
I. Fisher, The Money Illusion, Adelphi Company, New York, 1928.
J.A. Kay, C.P. Mayer, Inflation Accounting, Institute for Fiscal Studies,
London, 1984.
D.W. Pearce, Macmillan Dictionary of Modern Economics, fourth ed.,
Macmillan, London, 1992.
D. Alexander, C. Nobes, International Financial Reporting Standards:
Context, Analysis and Comment, Routledge, London, New York,
2008.
OECD, Sample Tasks from the PISA 2000 Assessment Reading,
Mathematical and Scientific Literacy, OECD Publishing, 2002.