Measurability invariance, continuity and a portfolio representation
Highlights
► The unwritten law of the sciences is to measure that which has not been measured. ► A measure should be invariant across observers and instruments. ► A measure should satisfy a continuity property across time and attributes. ► Measurement is a process of convergence to a preferred measure. ► This convergence involves anchoring.
Introduction
Galileo’s maxim “count what is countable, measure what is measurable and what is not measurable make measurable” (Rossi [1, p. 545]) appears now to underscore an unwritten law of both the sciences and the social sciences that it is better to measure than not to measure. But, the assumption of measurability has rarely been examined. The modern theory of measurability began with the analytical foundations laid in a paper by Helmholtz [2]. As noted by Rossi [1], Helmholtz’s principal idea was that measurement entailed the measurement of characteristics, analogous to counting, requiring measured values to satisfy two conditions:
- (i)
An order relation (>, < and =) so that for any couplet (a and b) of measured values, either a > b or a < b, or a = b.
- (ii)
An addition operation (+) so that the measured value for the combined characteristic (a and b) is a + b.
Many characteristics, however, are not naturally ordered or invariant to the observer, and, in multi-dimensional data, the measurement problem is complicated by the need to reduce the measurement of many dimensions to a single value. While there have been a few notable attempts at measuring multivariate data, for example, Chernoff [3], there has been little formal consideration of multivariate measurement.
The International Vocabulary of Metrology [4] defines measurement in the physical sciences as a process which presupposes a measurement procedure, a calibrating system, a measured value and usually a targeted uncertainty. It also emphasises the requirement that measurement be repeatable and reproducible, a condition formalised in this paper. Metrology tends to diverge between the physical and social sciences (Finkelstein [5, p. 269]). The soft metrology of the social sciences is confounded by three factors; measurement by humans of human behaviour, the inability to replicate observations, and the theory dependence of data. Soft metrology is then more subjective and less temporally consistent than physical measurement, and has objectives often related to the pre-disposition of the observer. As Mari [6] has surmised, measurement is then an evaluation designed to satisfy certain objectives, and not necessarily to elicit true values.
Uncertainty in measurement is often as important as the measured value itself. The International Vocabulary of Metrology [4] prescribes measurement error as the difference between the measured value and a typically unknown reference value, and dichotomises the error as systematic and random. Measurement error is particularly important in the social sciences, where estimation uncertainty is larger and less predictable due to behavioural biases. One theoretical response posited to the problem of measurement uncertainty is to widen the definition of measurement to include strong and weak forms of measurement (Finkelstein [7]), recognising that not all measurement is of the same standard. Another response to uncertainty is anchoring which occurs when individuals who are asked to estimate an unknown quantity anchor their answers to an unrelated attribute (Tversky and Kahneman [8]). This paper discusses how anchoring is being used in social science measurement.
In this paper, we formally examine measurability and, in particular, a set of conditions that any measurement process should satisfy. These conditions are consistent with existing concepts of measurement, such as repeatability and reproducibility in the International Vocabulary of Metrology [4]. This is the contribution of Section 2 of the paper. In Section 3 we consider measurement in multi-dimensional data, a common problem in the social sciences. The approach here is to borrow an analogy from the portfolio theory of finance to revisit the question of measurability in characteristics that have many dimensions.
The discussion is illustrated by several measurement problems including the measurement of the hardness of minerals, the measurement of the national product of an economy and the measurement of the ranking of universities. These measurement problems span both the physical and social sciences, have different exposures to measurement uncertainty, include both one-dimensional and multi-dimensional problems, and exemplify the difference in objectives in measurement. In sum, they encompass the many problems which constitute measurability.
Section snippets
Measurability
To understand measurement theory, it is necessary to revisit the theory of integration and, particular, Lebesgue measure theory (Bartle [9]). A measure is a real-valued function F defined on a σ-algebra X of subsets of a measurable space X such that
- (i)
F(ϕ = 0).
- (ii)
F(x) ⩾ 0 for all × ϵ X.
- (iii)
F is countably additive so that if xi is a sequence of disjoint sets in X, then F(∪xi) = ∑F(xi).
Measurement as a portfolio
In the social sciences, when measuring objects indexed by an unknown characteristic τ, it is common to construct measures based on a set of component characteristics τ(1), … , τ(N). The implicit assumption is that the unknown characteristic is a weighted combination of componentswhere w(1), … , w(N) are a set of weights and w(i) the relative importance of component i. In some cases, the set of weights is standardised so that ∑ w(i) = 1. For example, in the Times Higher Education ranking of
Conclusion
Measurement imparts certainty or at least a reduction in uncertainty. The minimisation of uncertainty has underwritten the unbounded expansion of measurement, consistent with Galileo’s view that we should measure the measurable and make that which is unmeasured measurable. Never has measurement been more important, and never has it been more important to define measurability.
In two significant papers, Rossi [1] and Rossi [15], established a framework to assess measurability in both a
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