Constructs Inferences and Mental Measurement Critica a La Medida en Psicologia

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Joel Michell School of Psychology, University of Sydney, Sydney NSW 2006, Australia

Available online 22 March 2011. http://dx.doi.org/10.1016/j.newideapsych.2011.02.004,How to Cite or Link Using DOI Permissions & Reprints

Abstract

The construct concept occupies a significant place in psychology and, yet its role is

misunderstood. Psychologists think that theorising in the area of psychological testing involves

conjuring constructs, which are operationally defined and measured via psychometric tests.

However, the construct concept is unworkable and laden with confused philosophical baggage

accrued under the hegemony of logical empiricism, and its real function in psychology is

obscured. Via an analysis of its history and logic, I expose its flawed conception of the relation

between theoretical and observable concepts and the way in which it serves the myth of mental

measurement. Finally, it is shown how the actual logic of theorising in science, which entails

that theories are best inferredfrom relevant phenomena, not imaginatively constructed,

oppugns this myth and promises to coordinate theoretical concepts with the phenomena to be

explained.

Keywords

Construct; Theoretical concept; Psychometrics; Measurement; Continuous quantity; Order

The construct concept plays a prominent role in psychology, shaping theories about the

attributes mental tests assess. As a result, its use raises issues concerning definition and

discovery of theoretical concepts, any discussion of which presumes a philosophical framework.

The conclusions reached here presume anempirical realistposition of the kind expounded in,

for example,Anderson (2007)andArmstrong (2010). Furthermore, these conclusions build

upon earlier work on the realist theory of measurement and its implications for psychology

(e.g.,[Michell, 1990],[Michell, 1999],[Michell, 2005]and[Michell, 2007a], &2009a). There is not

space here to defend this framework or theory and, so, I simply signal their relevance.

In many areas, scientists have uncovered otherwise hidden causes, unobserved entities and

processes behind phenomena seen with the naked eye. So, for example, chemical phenomena
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are understood to happen because of processes involving atoms, and biological phenomena,

because of processes involving genes. Few scientists doubt the existence of atoms or genes

and, yet, these remain theoretical in the sense that they are never directly observed. In

psychology, the search continues for theoretical concepts able to perform similar explanatory

wonders, and in psychometrics this search has led to the postulation of constructs.

Interestingly, the construct concept, as such, does not figure in physics, chemistry, or biology.

Scientists there talk of theoretical concepts, but not constructs. Have psychologists simply

adopted a different word for the same thing? An adequate answer requires investigating the

history and logic of the construct concept.

1. The construct concept, part I: philosophical trajectory

The view that some concepts1are composed of others is traceable toLockes

(1690/1959)doctrine that all complex ideas, say, that ofapple, are composed of simple ideas,

like red, roundand sweet. The term, construct, was not used for such concepts untilPearson

(1892), but it wasRussell (1914)who popularised it, prescribing, as he later put it, wherever

possible, substitute constructions out of known entities for inferences to unknown entities

(Russell, 1924/1988, p. 164). Whitehead and he applied it in Principia Mathematica(1910

1913), trying to secure mathematics by constructing its concepts of those of logic. For

example, they attempted to construct the concept ofnatural numberout of those

ofclassand equivalence. Later, Russell thought he could similarly secure knowledge of theexternal world by constructing material objects of sense data (Russell, 1914). While few now

think that he succeeded in either enterprise (Bostock, 2009; &Linsky, 2003), his constructivist

aspirations inspired Carnap, who confessed that Russells attempts made an especially vivid

impression on me because they formulated clearly and explicitly a view of the aim and method

of philosophy which I had implicitly held for some time (Carnap, 1963, p. 13). Consequently,

the dazzling sequel toRussell (1914)was Carnaps (1928)Der Logische Aufbau der Welt

(Quine, 1967, p. 313), which is generallyand rightlyregarded as one of the most important

classics of twentieth-century positivist thought(Friedman, 1999, p. 89).

Later, Carnap applied the method of construction to theoretical concepts in science. This

encounters an insurmountable logical obstacle. To the logical positivists, the method of

construction seemed to provide the only route whereby meaning could adhere to concepts that

are not directly observed, for it was presumed that meaning must flow from the observed to the

unobserved. However, if theoretical concepts are constructed entirely of observed concepts,

then, asHospers (1956, p. 205)noted, to say something about a particular theoretical concept

is only to say something about (in principle) observable phenomena and, so, it is not to say

anything about things existing in a way that is logically independent of those observable

phenomena. But this is incompatible with the empirical realist understanding of the causalrelation, viz., that it can only hold between logically independent events (Maze, 1954/2009).
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Take any theoretical, scientific concept, such as, for example, that ofdinosaur. A dinosaur was

a kind of entity, which although never observed by us, had its own characteristics and, while

causal connections to observable concepts, like those of various fossils, exist, saying something

about dinosaurs is not itself equivalent to saying something about fossils, for the connection

between dinosaurs and fossils is contingent and never logically necessary. When it is said, for

example, that a particular kind of dinosaur was a carnivore, something about these past

creatures is asserted, not something about present fossils, although fossils may provide

evidence for such claims. So while propositions involving theoretical concepts may be inferred

from propositions involving observed concepts, the meaning of theoretical concepts cannot be

constructed out of the meaning of these observed concepts. In successful sciences, theoretical

concepts get meaning via inferences, as, for example, the meaning of the dinosaur concept

developed as more and more was inferred about them from fossil features and other evidence,

scientists often reasoning by analogy with the relationship between presently living creatures

and features of their skeletons.

Over time, the logical positivists diluted their constructivist program, but not enough. Carnap,

with Feigl and Hempel, disseminated logical positivism (re-branded as logical empiricism) in

the United States from the 1930s and during that decade Carnap made a fundamental shift

( [Carnap, 1936]and[Carnap, 1937]; andMormann, 2007): he still held that theoretical

constructs absorb content from observational terms, via what he called correspondence rules

but their meaning, he now thought, is never completely reducible to such terms. This became

the standard, logical empiricist view, whichHempel (1952)famously described using the

analogy of a complex spatial network (p. 36), which, in turn influencedCronbach and Meehls

(1955)concept of a nomological network. In Hempels analogy, theoretical concepts are

likened to knots, definitions and hypotheses to threads connecting them and the entire network

is thought of as anchored to observed concepts by strings, which correspond to rules of

interpretation. The theoretical apparatus Hempel insisted cannot, in general be formulated in

terms of observables alone (p. 36) and, so,

Guided by his knowledge of observable data, the scientist has to invent a set of concepts

theoretical constructs, which lack immediate experiential significance, a system of hypotheses

couched in terms of them, and an interpretation of the resulting theoretical network (Hempel,

1952, p. 37).

While theoretical constructs were thought to absorb meaning from the plane of observation,

their meaning was said to remain open to some degree because the connections to that plane

may alter, depending upon the predictive and explanatory successes or failures of the theory

over time. This picture liberalised Russell s model, but the idea that meaning flows from

privileged concepts (i.e., observables) remained.

Feigl, in fact, was most eager to reconcile logical empiricism with some sort of realism

(Mormann, 2007, p. 159) and proposed supplementing the interpretation of constructs via

existential hypotheses (Feigl, 1950), that is, as partly involving real entities or processes.
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Constructs in Russells sense, composed entirely of directly experienced concepts, go no

further than those same concepts because no surplus content is added. Such concepts are

the intervening variablesofMacCorquodale and Meehl (1948). At best, intervening variables

simply say how observable concepts are interrelated and, so, it is a mistake to regard them as

something intervening betweenobservable concepts (Maze, 1954/2009). On the other hand,

when constructivism is liberalised and it is thought that the meaning of constructs is

underdetermined by observable concepts, the way appears open to the possibility of constructs

possessing surplus content. This suited Feigl who wanted theoretical constructs to also

incorporate real entities (such as, in psychology, neural states).

What Feigl failed to see, however, is that realism and constructivism are logically incompatible.

Theoretical concepts serve causal explanations of observable phenomena, and cause and

effect are always logically independent of each other. Therefore, the one thing that theoretical

concepts must not be is constructs, if byconstructsis meant concepts constructed of, even

partly, the observable concepts to which they are causally connected.

This crucial logical point concerning the independence of theoretical and observable concepts

was not just ignored byRussell (1913, p. 1), who saw causality as a relic of a bygone age,

surviving like the monarchy, only because it is erroneously supposed to do no harm, it was

never fully appreciated by the logical empiricists, albeit for a different reason: they wanted to

model scientific theories upon axiomatic theories in mathematics. They were in awe of the

axiomatic method as applied by[Hilbert, 1899]and[Whitehead and Russell, 19101913], and

others. However, while mathematicians may define higher-order concepts in terms of lower-

order ones for their own special reasons, scientists, because their concepts are causally

interconnected, require concepts all on the same level, coexisting in mutual logical

independence.

In erring this way, the logical empiricists embraced a debilitating logical fallacy, viz., that of

confusing epistemic with ontological priority. Just because observable concepts are known prior

tounobservable ones, it does not follow that observable concepts are grounded any more

securely in reality thanunobservable concepts or that the latter, like ontological leeches, survive

only by sucking meaning from observable ones. While we may depend upon facts about

observable things for knowledge of unobserved entities or processes, the latter remain logically

independent of the former.

Thus, it is clear that the trajectory of the construct concept up to the point where it penetrated

psychology shed very little light on the character of theoretical concepts. Beset by logical flaws,

its difficulties resulted from attempts to impose an inappropriate model of meaning; a model

ignoring the logical independence of causally connected concepts. Despite this, and because of

the hegemony of logical empiricism, American psychologists embraced the construct concept.

However, in order to make it serve their special interests, they also transformed it.

2. The construct concept, part II: psychological trajectory
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The construct concept entered psychology beforeMacCorquodale and Meehl (1948)(e.g.,

seeBoring et al., 1945). During the 1930s, opinion leaders in psychology sought to boost the

credentials of the discipline by promoting philosophies of science that had associated

themselves most closely with recent revolutions in physics (viz., operationalism and positivism).

The prevailing rejection of mentalism, which was thought to involve not just unobservable but

intrinsically private concepts, created a gap in the array of potential theoretical concepts, which

the construct concept filled. Psychologists of that era wanted to base theories upon publicly

observable concepts and to confine their terms to the behaviourists lexicon. The result was

identification of an array of concepts, operationally defined in terms of behaviour and, following

the positivists, designated as constructs.

Even within the philosophy of science, use of the construct concept was not entirely consistent

(Beck, 1950) and in psychology it was at first assimilated to operationalism. By the 1940s, the

opinion leaders followed the logical empiricists in regardingBridgmans (1927)operationalism

as too narrow and favoured a more liberal version (Boring et al., 1945), according to which

operational definitions were used not so much to define constructs, as to identify observable

criteria for their application (Feest, 2005). The epoch-making papers of Meehl and associates

advocated consistency of usage by promoting Feigls more realist-oriented version of logical

empiricism.2In defining their concept,MacCorquodale and Meehl (1948)utilized Feigls concept

of existential hypotheses (later expounded inFeigl (1950)). Their very form of words

(hypothetical construct) bespoke a marriage of Carnaps construct concept with Feigls. In

extending the construct concept to test theory,Cronbach and Meehl (1955)aimed to rescue

psychometrics from operationalism and to establish MacCorquodales and Meehls hypothetical

construct concept within that discipline. Hypothetical constructs were proposed as an

alternative to operationally defined concepts: Construct validationis involved whenever a test is

to be interpreted as a measure of some attribute or quality which is not operationally defined

(Cronbach & Meehl, 1955, p. 282). Their intended alternative was unworkable, however,

because, like his mentor, Meehl never overcame the presupposition that constructs absorb at

least some meaning from that of observable concepts, which is incompatible with his view that

hypothetical constructs are proposed as real entities or processes linked causally to those same

observable concepts. Despite Meehls attitude, the wider psychological community did not see

any need to abandon its operationalism. It appropriated Meehl s term, construct validity, but at

the same time required that constructs be operationally defined via mental tests.

This approach also found an unintended justification in Cronbach s and Meehls insistence that

the meaning of theoretical constructs is set forth by stating the laws in which they occur (1955,

p. 294). Of course, psychometrics had no laws, only postulated connections between

constructs, and correlation coefficients between relevant tests. Cronbach and Meehl allowed

that a set of intercorrelations, expressed as factor loadings, substituted for networks of laws and

allowed that in psychology, our incomplete knowledge of the laws of nature produces a

vagueness in our constructs (1955, p. 294). If incomplete knowledge produces vagueness,

then ignorance produces emptiness. They believed that psychologists would eventually be able
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to define constructs, such as anxiety when all of the laws involving it (Cronbach & Meehl,

1955, p. 294) were in, but psychologists took their words as justification for treating constructs

as empty promissory notes(Messick, 1989, p. 23) for theoretical concepts, the understanding

of which might be indefinitely deferred, while in the interim, operational definitions sufficed for

the pragmatic purpose of anchoring them to observable criteria.

However, ignorance regarding the meaning of constructs is a significant obstacle if one wants to

claim that tests measurethem. As I have argued elsewhere ([Michell, 1990],[Michell,

1997]and[Michell, 1999]&2007a), in claiming to measure constructs, psychologists3typically

display a debilitating false consciousness regarding the concept of measurement, on the one

hand explicitly defining it trivially, followingStevens (1946), as the assignment of numerals to

objects or events according to rules, while on the other hand implicitly presuming the standard

scientific definition because test theories, such as factor analytic and item response theories

require that the relevant theoretical attributes possess continuous quantitative structure. In

quantitative sciences, such as physics ([Maxwell, 1891],[Terrien, 1980]and[Wildhack, 2005]),

it is understood that the measurementof a magnitude of a quantitative attribute is an estimate

of the ratio between that magnitude and whichever magnitude of the same attribute is taken as

the unit of measurement ([Michell, 2007a]and[Michell, 2007b], p. 74), that the concepts of

measurement and continuous quantitative structure are logically interconnected (Hlder, 1901),

and that we cannot measure what we are ignorant of because measurement presumes

knowledge, viz., that the relevant concept is a continuous quantitative attribute.

However, if it is ruled that the meaning of constructs is given by relationships between test

scores, as Cronbach and Meehl suggest, the fact that test scores are discrete quantities could

easily lull adherents ofStevens (1946)definition into thinking that measurement, far from being

prevented by ignorance, is already a fait accompli. Psychologists named constructs using

familiar mental terms, such as anxiety, without dwelling upon the construct s meaning, thinking

it sufficient that constructs were operationalized via test scores. A constructs nomological

network became a tissue of operational definitions, often involving scores on a range of tests

as, for example, general intelligence is thought of as operationally defined via factor analysis

of scores on several tests (Flynn, 2007). Thought of this way, constructs are really dispositional

concepts, that is, defined in terms of their putative effects, but, as has already been stressed,

effects cannot satisfactorily define constructs. For example, general intelligence is sometimes

defined as abstract problem-solving ability, which says nothing about the character of

intelligence per se, but merely specifies it as the otherwise unknown cause of successful

abstract problem-solving behaviour. Furthermore, the constructs dispositional label (e.g.,

abstract reasoning ability), because it seems to admit of degrees of more and less, is taken,

fallaciously (Michell, 2009b), to vouchsafe the constructs presumed quantitative structure.

Psychological constructs, shorn of their operational and dispositional accretions, are merely

conceptual simulacra, lacking intrinsic meaning except for their presumed continuous

quantitative structure.Borsboom, Cramer, Kievit, Zand Scholten, and Franic (2009)note that

we have rarely come across a clear description of what something should be like in order to
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deserve the label construct (p. 150), but it is exactly this characteristic, viz., the construct

concepts lack of definite characteristics, beyond that of purported measurability, that is really

its raison dtrewithin psychology. It gives the appearance of scientific theorising without any

substance while at the same time appearing to support the claim that psychological tests

measure something.

While the construct concept was not universally endorsed and realist-oriented psychologists

like Loevinger saw that a constructconnotes construction and artifice, whereas, traits and

abilities, if real are exactly what the psychologist does not construct (1957, p. 642), the

mainstream view is that literally a construct is something that the scientist puts together from

his own imagination, justified by the misconception that all theories in science mainly concern

statements about constructs (Nunnally, 1967, p. 85). It is supposed that constructs are

deliberately and consciously invented or adopted for a special scientific purpose, namely, so

that they can be observed and measured via psychological tests (Kerlinger & Lee, 2000, p.

40). However, what is not recognised is that constructs, in being operationally observed and

measured via tests, are deliberately left empty, empty that is, except for the requirement that

they be continuous quantities.

It is worth noting that while psychologists have established methods ofconstruct validation, so

called (e.g.,Jonson & Plake, 1998), it is also not recognised that these are insensitive to the

existence or non-existence of measurability in whatever theoretical attributes actually produce

test performance (Michell, 2009a). Attributes are only measurable if they are continuous

quantities, a feature involving specific empirical conditions. Before anyone can safely conclude

that hypothesised attributes possess continuous quantitative structure, it is necessary to

consider evidence diagnostic of quantitative structure and presently, such evidence is lacking

([Michell, 2000]and[Michell, 2008]). While anyone is free to consider the hypothesis that some

particular attribute possesses this sort of structure, there is no scientific advantage to be gained

by postulating its truth a priori.

3. Theorising, construction and inference

Psychologists embraced the construct concept because they accepted the logical empiricists

view that theorising about hidden causes required some kind of special methodological device.

However, theorising requires no special device. Considered at its most general, science has but

two methodological devices,observationand reason, and as anyone familiar with its history

knows, these are sufficient to deal with hidden causes (Crombie, 1994). By observationis

meant the process whereby knowledge of things is achieved via the senses, augmented,

perhaps, by scientific instruments, such as telescopes, etc., and by reason, the process whereby

knowledge advances through deriving implications of propositions already accepted as true,

using patterns of deductive and probable inference.

Given that we can only consider things in terms of what can be said about them, i.e., in

propositions (Anderson, 1927, p. 242), there are only a limited number of ways to consider
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anything: (a) observation might show it to be true or false; (b) it might be inferred from

propositions already accepted; and (c) it might form one amongst a set of propositions tested by

the truth or falsity of implications. It follows then that regarding things that we cannot directly

observe, we must come to know them via reason. We have nothing else to reason from other

than what we already consider true. That is, given observations and general views about the

logic of things (i.e., the logic of causality, of quantity, etc.), scientists must reason their way to

conclusions about things not directly observed. That is, scientists can only get to know about

hidden causes via (b) and (c) above.

The logical empiricists did not see things exactly this way. They emphasised (c) at the expense

of (b) because, asBechtel (1988, p. 66)notes, they made a sharp distinction between the

contexts of discovery and justification. Discovery was assumed to be a nonrational process and

so philosophical attention was directed at the question of how theories could be justified, not

how they were initially developed. Somewhat romantically, theories were understood as

imaginatively, and even irrationally, constructed. AsPopper (1959, p. 31)echoed this point, the

act of conceiving or inventing a theory, seems to me to neither call for logical analysis nor to be

susceptible to it. The logical empiricists did not think that theories were in general discovered

via a rational, inferential process and, so, ruled out (b) above. That is, they rejected the idea that

theories could be inferred from propositions already accepted as true. On the other hand, the

appraisal of theories was seen as a rational process, involving only (c) above. This is

the hypothetico-deductivemodel of justification. Despite its considerable virtues, focussing on it

alone is needlessly fighting a battle one-handed.

A successful theory can only be reached by first considering phenomena to be explained.

Because explanation involves a deductive relationship between theory and phenomena,

theories must be tailored logically to yield fitting implications regarding phenomena.

Furthermore, it is unnecessary for any theory to contain more than required to explain relevant

phenomena and be consistent with other, already accepted, conclusions. Hence, the process of

producing viable theories is first and foremost a process of reasoning from phenomena, that is,

ofinference([Catton, 2004]and[Haig, 2009]; &Musgrave, 2009) and only secondarily, in the

absence of sufficient information and only ever in part if at all, one of imaginativeconstruction.

An inference from the phenomenarequires an adequate conception of the character of

phenomena. Phenomena are not the same as data.Woodward (1989)has clarified this

distinction. First, data are particular, while phenomena may be general. The scientist does not

want a theory that only explains a specific body of data. The scientist wants a theory capable of

explaining general facts about the things investigated and, so, what are to be explained include

general propositions, not only particular ones. One important role of data in science is to

indicate the character of the phenomena to be explained. Second, for any body of data, it is

always likely that it contains errors of observation and, so, if a theory were constructed to

explain all features of data, it would likely be false because it would entail errors. It is a

misconception that theories are falsified by failing to account for all features of data. A

conception of phenomena is needed that distinguishes structure from error in data and this
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inevitably involves a degree of idealisation on the scientist s part. What features of data are

erroneous and what reflect genuine phenomena may not always be clear and that, itself, is

something that the scientist needs to discover. The scientist seeks structure in data as a clue to

genuine phenomena and eschews as error that which obscures structure.

A complete analysis of the phenomena involved in psychological testing and the kinds of

theories that might be inferred would be too large an undertaking to pursue here. Instead, I will

consider just one, central issue, viz., in general, is it reasonable to infer from the phenomena of

testing that the theoretical concept (or attribute) assessed by a psychological test possesses

continuous quantitative structure?

Note first that, a priori, there is no logical necessity that attributes underlying test performance

must be continuous quantities. Despite this, from its commencement, the testing movement,

somewhat grandiloquently, referred to scores as measures (e.g.,[Binet,

1905]and[Spearman, 1904]), convinced that test scores measure something(Kelley, 1929, p.

86), even though, as Kelley added, it was never clear what. Furthermore, all testing concepts

were brought into line with this conceit. For example, the problem of validity became that of

the determination of what a test measures(Courtis, 1921, p. 80; my italics), rather than, more

neutrally, determination of what is assessed. These were not harmless terminological excesses.

In those days, beforeStevens (1946)proposed the vacuous definition of measurement that

most psychologists later accepted ([Michell, 1997],[Michell, 2007a]and[Michell, 2007b]), in

scientific contexts, the term measurementmeant only what today is called ratio

scalemeasurement (i.e., discovery or estimation of ratios of magnitudes of continuous

quantitative attributes) and, so, the effect of cementing this terminology in place was to

foreclose consideration of the issue of whether the attributes that tests assess really are

measurable quantities (Michell, 2009a).

Typically, test scores are frequencies (e.g., number of correct responses). Under the same

conditions, on the same test, different people may get different scores. It is reasonable to

conclude that such differences result from psychological differences between those involved.

Now, a difference between test scores is a discrete quantity. From this fact, does it follow that

the underlying, causally relevant psychological difference must be a difference between

magnitudes of some continuous quantitative attribute? Considered as a purely logical question,the answer is nobecause a discrete quantitative difference need not be caused by a

quantitative factor at all, let alone one that is a continuous quantity. For example, it is possible to

imagine two families alike in most respects except for the fact that (a) the first set of parents

believes that contraception is immoral while the second does not and (b) the first set of parents

has more children than the second. In such circumstances, it is possible that the difference in

(a), a non-quantitative factor, is the cause of the difference in (b), a discrete quantitative

outcome. That is, differences in belief may cause differences in numbers of children.

Of course, causality is a complex relation, involving three terms, the cause, the fieldwithin

which the cause operates, and the effect(Anderson, 1938). If the effect is quantitative, there
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must be something quantitative in either the cause or the field to produce that quantitative

effect. If the causes are non-quantitative, as in our example, they can only produce a

quantitative difference by acting differentially upon quantitative conditions within the causal

fields. However, as in our example, these quantitative conditions (say, the number of

opportunities for conception) may be the same for both couples and what makes the difference

is different beliefs about the morality of contraception, which in the case of one couple but not

the other, allows birth control methods to be applied.

Where an effect is a difference in frequency, the cause may be (a) a non-quantitative difference

(as shown); or (b) a difference between discrete quantities (as, for example, if our couples

differed in the number of opportunities for conception) or (c) a difference with respect to some

relevant continuous quantity (say, consistent differences in body temperature might affect

fertility). It is only in the latter kind of case that issues to do with measurement, in the scientific

sense, arise. So what follows regarding the causal factors involved in testing?

An answer to this question can be illustrated by the case of cognitive tests. Consider the

phenomenon of a person, a, correctly answering a test item, k, and suppose that as response

is not an error.4In such a case, the cause is the presentation ofkto a, the causal field consists

ofas cognitive resources plus other conditions (physical, social, and psychological) needed

forato respond to k, and the effect is as response. Because as response is both correct and

not an error, amust possess cognitive resources (e.g., knowledge, strategies, and skills) that at

least match ks demands. Suppose, now, that on a test of titems, aanswers just nitems

correctly and another person, b, only mitems (where nand mare any two whole numbers in the

range 0to t). If nothing more is known about the structure of the items and if it is assumed

(1)

that apart from their cognitive resources, the causal fields involving aand bare similar

in all relevant respects;

(2)

that relative to each person, the causal fields remain unchanged throughout the testing

session;

(3)

that no errors have occurred (i.e., that the answers given by aand baccurately reflect

their cognitive resources); and

(4)

that the items are such that for any pair of people, getting the same item correct is only

possible through possessing the same cognitive state;

all that can be inferred validly is that apossesses nand b, mof the tcognitive states required to

get all items correct. That is, we are able to infer validly no more than that the difference in

number of correct responses on this test is caused by the difference in number of relevant

cognitive states possessed. The situation looks like a case of discrete quantity caused by

discrete quantity and, so, no issue of scientific measurement is involved.
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The best we can do in such a case is to classify people according to the combination of

cognitive states possessed and, if the items are binary (i.e., scored correct or incorrect),

there are 2t such combinations possible. That is, the relevant attribute is a partial order

consisting of2t cognitive state combinations (a partial order because if one person gets correct

all items another gets correct, then the first person s combination of cognitive states includes

the second persons). The usual expedient of classifying people according to total score is sub-

optimal because it may class together people with different combinations of cognitive states,

that is, people who differ cognitively relative to this test but, nevertheless, get the same total

score.

However, with certain tests, viz., unidimensional tests, things may seem more promising for

those aspiring to measurement. A unidimensional test is one in which all items assess the same

ability, although items may differ in degree 5of difficulty. Consider the performance ofaand bon

such a test under the same assumptions as above. It not only follows

that apossesses nand b, mof the relevant cognitive states, but also, where n> m,

that apossesses the same mstates that bpossesses plus nmmore (i.e., the test is

aGuttman (1944)scale). In such cases, the relevant set of cognitive states is structured, such

that if a person possesses the cognitive state needed to pass any item, then that person

possesses all of the states needed to pass all less difficult items and, so, instead of the relevant

attribute yielding 2tpartially ordered degrees, it delivers t+ 1 strictly ordered degrees. Now

while such an attribute is not necessarily quantitative (quantity also including additive structure

as well as a strict order), it is half way to that goal (Michell, 2009b). But is the goal necessarily

attainable in this sort of case?

Progressing from any specific degree of such an attribute to higher degrees usually requires

coming to possess whatever new knowledge, strategies, and skills are needed to pass more

difficult items. This may be a problem for the hypothesis that the relevant ability is quantitative

because an increase in knowledge, in available strategies, or in skill is not necessarily an

increase in some homogeneous quality, as is an increase in, say, length, where an increase of

any magnitude is in principle possible from any initial length. Cognitive abilities may not be

homogeneous in this way. In at least some unidimensional tests, say, a test of mathematical

ability, as item difficulty increases, corresponding increases in cognitive resources required for

correct responses are mutually qualitatively heterogeneous. For any pair of items of distinct

difficulties, the difference between the cognitive resources required by the more versus the less

difficult one is not just more of exactly the sameknowledge, samestrategies, orsameskills

already required for the less difficult item, for if it were, it would be no cognitive increase at all

and, so, could not explain the difference in difficulty between the items. Thus, for tests of this

kind, the cognitive attribute assessed is one in which differences between degrees are mutually

heterogeneous.

Now, if the differences between the degrees of some attribute are heterogeneous, then such

differences, being of different kinds, are not intrinsically less than, equal to or greater than one

another; but such intrinsic order relations between differences are exactly what quantitative
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structure requires (Michell, 2009b). Therefore, in what is thought of as a test suitable for

psychometric measurement (viz., a unidimensional cognitive test of this kind), quantitative

structure is ruled out and, so, measurement is not possible.

Of course, with some unidimensional tests, increasingly difficult items may differ from one

another quantitatively (e.g., items in the digit span subtest of the Wechsler Adult Intelligence

Scale). However, the point at issue does not concern item differences as such; it concerns

differences between the cognitive states necessary and sufficient for correct responses. Even

where item differences are quantitative, if it is the case that, for example, qualitatively different

strategies lead to different levels of performance (as different memory strategies may for digit

span tests [e.g.,Engle & Marshall, 1982]), then the hierarchy of cognitive states constituting the

relevant ability may involve heterogeneous differences between degrees and, if so, cannot

constitute degrees of a quantitative attribute.

None of this is to deny that responses to a unidimensional test might sometimes display

statistical fit to a quantitative, probabilistic measurement model. Statistical fit to such a model

does not by itself entail that the relevant attribute is quantitative because it is entirely possible

that such fit could occur when the structure of the attribute is no more than ordinal

(e.g.,Kyngdon & Richards, 2007). Quantitative structure within the underlying attribute can only

be safely inferred from the phenomena when,

(1)

differences between the content of increasingly difficult items require homogeneous

quantitative differences between the cognitive resources necessary for correct

responses; and

(2)

there is evidence for additive structure within the attribute, such as might be obtained

indirectly by testing the cancellation conditions of additive conjoint measurement theory

(Krantz, Luce, Suppes, & Tversky, 1971; &Michell, 1990) in relation to items specifically

tailored to reveal additive relations (Michell, 2009b).

Kant (Sutherland, 2004), in his writings on measurement,von Kries (1882), in his critique of

psychophysical measurement, andKeynes (1921), in his dissertation on probability, all noted

that ordered attributes possessing heterogeneous differences between degrees are notquantitative and, so, not measurable; andCollingwood (1933)explored this matter in more

depth. However, the possibility of ordinal, non-quantitative attributes is not often acknowledged

in psychometrics. While some have recognised, there may be distinguishable qualitative levels

between the extremes (Wilson, 2005, p. 6) of any psychological attribute, and others have

asked, How can such qualitative changes be hypothesized to fall so many units apart on one

particular trait?(Yen, 1986, p. 312), implications regarding the very possibility of measurement

are ignored. While there are exceptions (e.g.,Cliff & Keats, 2003), psychometricians often write

as if, in assessment, just two kinds of structure are possible, viz., classifications (sometimes

called categories) and continuous quantities (sometimes called dimensions or continua) (see,
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for example,[Cronbach and Meehl, 1955]and[Loevinger, 1957], andDe Boeck, Wilson, and

Acton (2005)). Since the time ofStevens (1946), ordinal scales have been widely recognised,

but these are often seen merely as portents of interval scales (e.g.,Harwell & Gatti, 2001;

andNunnally, 1967).

Interestingly, however, the sorts of ordinal attributes mentioned above mesh more smoothly with

current cognitive theory than do continuous quantitative attributes. An ordered attribute, the

degrees of which are composed of, say, knowledge, strategies, and skills, is defined in terms

similar to those already commonly invoked in cognitive psychology. On the other hand,

continuous quantities, while not incommensurate with the concepts of cognitive science, raise

the issue of how the two sorts of concepts are connected. More than twenty years ago,Snow

and Lohman (1989, p. 317)wrote,

The evidence from cognitive psychology suggests that test performances are comprised of

complex assemblies of component information-processing actions that are adapted to task

requirements during performance. The implication is that sign-trait interpretations of test scores

and their intercorrelations are superficial summaries at best. At worst, they have misled

scientists, and the public, into thinking of fundamental, fixed entities, measured in amounts.

That is, within cognitive psychology it has been found that performance differences are most

usefully accounted for in terms of qualitative differences between relevant cognitive resources

and not in terms of differences between magnitudes of the sorts of quantitative constructs

invoked in psychometrics. I suggest that psychological constructs remain empty concepts

precisely because of the difficulties in specifying, first, what the homogeneous stuff is that

people differing in some continuously quantitative construct have different amounts of and,

second, how quantitative differences in this stuff translate into differences in the cognitive

resources that test performances require.

Therefore, my conclusion is that there seems little basis within the phenomena of testing from

which to infer that the theoretical concepts assessed by tests are continuous quantities. Indeed,

the fact that for at least some unidimensional cognitive tests, differences between degrees of

the relevant attribute are heterogeneous entails that the relevant attributes, although ordered,

must actually be non-quantitative. It remains to be seen how far this conclusion generalizes.

Some signs suggest that it generalizes beyond abilities. Consider social attitudes: in cases

where items are so constructed as to ensure unidimensionality, positive increases in attitude

toward some social issue are generally accompanied by heterogeneous differences between

attitude content (e.g.,Kyngdon & Richards, 2007). This suggests that a series of attitudes

towards some social issue ordered from favourable to unfavourable will display heterogeneous

differences between degrees and, so, be non-quantitative as well.

Thus, the hypothesis that the attributes underlying test performance are quantitative cannot be

taken as thea priorior default position for psychometrics. Instead, the quantitative hypotheses

characterising psychometrics require evidence specifically diagnostic of quantitative structureover and above that of mere order. In so far as quantitative structure is taken to be part and
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Constructs Inferences and Mental Measurement Critica a La Medida en Psicologia

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