The present note should be read with patience and tenacity because some technical notions, simple but abstract, are required for a minute conceptualisation. If the need arises, the reader should not hesitate to scroll the Internet to get information on specific notions (function, function composition, increasing function, Cartesian product…).
The conceptual difference between psychotechnical scoring and genuine psychotechnical measurement can be revealed by describing both of them as the composition of two functions, namely, a ○ x (scoring) vs. f ○ q (ordinal measurement). It suffices to show that the two function compositions are not identical to show that scoring and measurement are not interchangeable concepts.
To determine the score of a given person at a given moment (by using a psychological test), one has to determine then to score a psychotechnical description of this person at this moment. To measure supposes that the psychotechnical description results from a measurement process. If the supposition is false and if one endorses the scientific attitude, one cannot consider that the description results from a measurement process (hence, it is tempting to score the description as a non-scientific way of compressing the psychotechnical description into a numerical description).
1. The psychotechnical description x
The psychotechnical description is a function denoted by x: Ω → D. Its domain, Ω, is a statistical population. An observation unit from Ω consists usually of a person at a given moment, where “moment” means the temporal interval required by the test administration. Any observation unit is denoted by ω. The codomain of the description function x is denoted by D (for description). The elements from D ares m-tuples, where m indicates the number of the test items. A 3-tuple is for example (0, 1, 1), where 0 and 1 denote incorrect and correct responses, respectively. The bold type of the notations x and D indicates that the description is multivariate (as opposed to onevariate)–this is a m-tuple; a score is a 1-tuple.
For example, let us consider the observation unit ω = (Paul, t). The psychotechnical description of Paul when he is tested at moment t is given by his m responses to the test. This m-tuple is denoted by x(ω). Depending on the context, the notation x may denote any description, that is, x(ω) where ω is any observation unit, or the psychotechnical description viewed as a function, that is, x: Ω → D.
The function x is the conjoint function
x1x2…xm: Ω → D1×D2×…×Dm = D
(which shows that D is a Cartesian product), where any xi, i = 1, 2, …, m, assigns one and only one image to any observation unit in the scale of each item i–the scale of the item is the set of its descriptive values.
2. The scoring rule a
A psychotechnical scoring rule is a function denoted by a: D → S. Its domain is the codomain of x, that is, the set of the m-tuples that are possible given the test design. The codomain of a is denoted by S; S is a set of scores–the test scale. The score of any m-tuple from D is determined by “transformation rules”, or aggregation rules (hence the notation a). For example, the score of (0, 1, 1) is 2 points if one uses the common rule, which consists in counting the correct responses in the m-tuple.
3. The psychotechnical scoring a ○ x
To test a person at a given moment by using a given test consists in determining her responses to the m test items–x(ω)–then to assign a score to this description– a[x(ω)]. This function composition is denoted by a ○ x. The score of the observation unit ω is
(a ○ x)(ω) = a[x(ω)] = x(ω).
The notation x denotes a psychotechnical score but it can also denote the function a ○ x.
4. The quantitative hypothesis q
Measurement rests on the fundamental hypothesis that a quantity does exist (one mesures a quantity or one does not measure; saying that one measures a process is an example of uncareless speaking). This hypothesis can be stated as the hypothesis of a function
q: Ω → [0, max],
which assigns one and only one amount from the segment [0, max], where “max” denotes the highest possible amount of the quantity, to any unit observation from Ω.
The statistical notion of an observation unit deserves a remark in the present context. To suppose that the person possesses an instantaneous amount of the quantity is one thing, to suppose that this amount does not vary within the temporal interval the test administration requires is another thing. The invariance convention according to which the amount to be measured during the test administration does not vary is mandatory to test the hypothesis that this amount is measurable.
5. The measurement function f
The measurement funtion
f: [0, max] → D
is defined on the domain [0, max] and takes its values in the codomain D. Let us consider a m-item test. For any item i of the test, one supposes that there exists an increasing function fi: [0, max] → Di, where Di is the descriptive scale of the item i (see Measuring with Item Responses). The function f is the conjoint function
f1f2…fm: [0, max] → D1×D2×…×Dm = D
(which shows, again, that D is a Cartesian product). The measurement value of the amount q by f is a m-tuple of values to be denoted by y.
A critical property of f‘s values is that they are simply ordered because any function fi is increasing. This point is detailed in Section 6. This property of simple order, or, in other words, of comparability, implies that the proposition, according to which two specific descriptions x(ω1) and x(ω2) result from the measurement of the amounts q(ω1) and q(ω2), is falsifiable. If x(ω1) and x(ω2) are incomparable, and one writes
x(ω1) <> x(ω2),
then at least one of these descriptions is not a measurement.
6. Psychotechnical measurement f ○ q
A test enables one to measure the amount of a psychological quantity within an observation unit ω if this amount, q(ω), determines the psychotechnical description f[q(ω)]. The psychotechnical measurement is the function composition f ○ q.
Let ω1 and ω2 be two observation units. Their measurements are respectively (f ○ q)(ω1) = f[q(ω1)] = y(ω1), and (f ○ q)(ω2) = f[q(ω2)] = y(ω2), where the letter y denotes the resulting description or the function f ○ q.
Direct product order (definition). One says that f[q(ω1)] ≥ f[q(ω2)] if and only if for all is, fi[q(ω1)] ≥ fi[q(ω2)]. This definition applies also to x.
Theorem. f is increasing.
Let ω1 and ω2 be such that q(ω1) ≠ q(ω2):
- If q(ω1) > q(ω2), then for all i = 1, 2, …, m, fi[q(ω1)] ≥ fi[q(ω2)], hence f[q(ω1)] ≥ f[q(ω2)].
- If q(ω1) < q(ω2), invert the subscripts of ω and go back to the preceding case.
Comparability (simple order). The increasing property of f implies that two measurements y1 and y2 are comparable.
7. Scoring vs. measurement
It can be seen formally that scoring and measurement are distinct operations, since x = a ○ x is not the same symbol as y = f ○ q. Put in other words, the availability of the score x does not necessarily means that there is an increasing function f such that x = f ○ q (f is onevariate as x is onevariate).
Let Ωn be a set of n observation units. If there is one function f such that the image of Ωn by x is the image of Ωn by f ○ q, that is, x(Ωn) = (f ○ q)(Ωn), then one can suppose that the psychotechnical descriptions x(Ωn) are ordinal measurements of the amounts q(Ωn). In other terms, these descriptions are simply ordered, that is, the descriptions of two distinct observation units are either equal or strictly ordered. Then the score x expresses this ordinal property of comparability.
If such a function f does not exist, one ignores how to measure the amounts q(Ωn). It is possible that our ignorance results from the fact that these amounts do not exist–in which case we suppose them wrongly–, or from our inability to determine the experimental conditions that would allow us to suppose not against the empirical evidence that f exists.
If two observations units ω1 et ω2 exist in such a way that their descriptions are incomparable, i.e., i and j exist such that xi(ω1) > xi(ω2) and xj(ω1) < xj(ω2), then f does not exist and hence one cannot assert that the descriptions result from the measurement by f of the supposed amounts.