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The history of the past four decades of the theory and application of additive conjoint
measurement (ACM) is characterized by vivid developments of its theoretical foundation
(cf. Luce and Tukey, 1964; Krantz et al., 1971, 2006; Narens, 1974), industrious developments
of statistical and computational implementations (cf. Karabatsos and Ullrich, 2002;
Karabatsos and Sheu, 2004; Karabatsos, 2005; Myung et al., 2005) and heated debates
about its applicability and significance in psychology (cf. Michell, 1997, 2009; Borsboom
and Mellenbergh, 2004; Barrett, 2008; Borsboom and Scholten, 2008; Kyngdon, 2008a;
Trendler, 2009). What started as a promising foundation to solve the everlasting debate
about the quantitative nature of psychological attributes (Ferguson et al., 1939)
ended in perseverative debates with very little transfer to mainstream psychological
science still being dominated by structural equation modeling (SEM) and item response
theory (IRT). After reading the aforementioned articles, and comparing their implications
with the day-to-day business of mainstream psychological science, even an unbiased
reader would certainly agree with Cliff (1992) that ACM was a “… revolution that never
happened” (p. 186).
It is not the aim of this article, to discredit the efforts of mathematical psychology
and proponents of ACM in particular. I just want to address the naïve but relevant
question why ACM as a stringent way to formalize and to test the requirements of quantitative
measurement in psychology has not been embraced by mainstream psychology as a means
to an end to test what they always claim: that most of the attributes (e.g., intelligence
and personality factors) are quantitative.
An attribute possessing a quantitative structure is required to satisfy the three
conditions of ordinality (transitivity, antisymmetry, and strong connexity) and the
six conditions of additivity (associativity, commutativity, monotonicity, solvability,
positivity, and the Archimedean condition; cf. Michell, 1990, p. 52f.). Most of these
conditions are testable hypotheses but I have never seen any empirical test in psychological
articles before data were analyzed with SEM or IRT models, which already assume the
quantitative structure of the attributes under consideration as argued below. Somewhere
during my psychology studies at the university I learned that psychology is an empirical
science and that there is therefore no room for claims that should just be believed.
However, given the assumed but almost never tested quantitative nature of most of
the psychological attributes as reflected in factor analysis, SEM and IRT models,
I must have missed or misunderstood something.
Resistance toward inconvenient truth
The question arises why debates about testing the assumption of quantitative measurement
more rigorously emerge from time to time without any broader impact on psychological
measurement with a few exceptions (Luce, 2000; Kyngdon, 2011). Any attempt to answer
this question will, of course, be incomplete, so that I will suggest a factor that
might be of special importance: psychologist's avoidance toward falsifiability and
hence, toward inconvenient truth.
A number of authors state (cf. Borsboom and Mellenbergh, 2004; Borsboom and Scholten,
2008; Fisher, 2011) that the axiomatic structure of ACM is too restrictive with respect
to the regularities in the order relations of the items, the examinees, and an ordinal
index of the probability of a correct response. ACM relates to situations in which
one attribute (P; e.g., the probability of getting an item correct) is related additively
to two others (A the ability and B the item difficulty) such that P = f(A + B) (where
f is any positive monotonic function). In fact, the requirements of ACM are rarely
fulfilled in applied psychological data (Cliff, 1992; Michell, 2009) because the data
must satisfy the highly restrictive conditions of double cancelation, solvability,
and the Archimedian axiom (cf. Michell, 1990). Satisfaction of these requirements
implies that A and B are additive and are therefore quantitative (cf. Krantz et al.,
1971).
I therefore agree with the argument that it is more than questionable why such rigorous
measurement structures could be found in psychological data. As illustrated elsewhere
(cf. Schönemann, 1994; Heene, 2011) psychology seemed to be overwhelmed by the successful
application of mathematics in classical physics and invented “… models with close
reference to those of classical physics, which were then applied to psychological
observations” (Heene, 2011, p. 53; italics in the original). This approach ignores
that the development of mathematical models has been closely interwoven with the empirical
observation of invariant phenomena in physics implying that the mathematical models
have often been derived from those phenomena (see also Sherry, 2011).
On the other hand, the tools of mainstream psychology such as SEM and IRT make exactly
these strong assumptions about the quantitative structure of psychological attributes.
But avoiding any tests of quantitative measurement but applying methods making the
assumption of quantity appears to be nothing more than a self-delusion that one bears
something valuable instead of being in fact empty-handed. This all too strong tendency
to avoid falsification is probably deeply rooted in the scientifically unhealthy political/economical
aspiration of psychology (Vautier et al., 2012) which keeps the machine for paper-producing
and grant-funding well-oiled but also leading to a severe publication bias. Consider
Levine et al. (2009) who showed that effect size and sample size are negatively correlated
in 80% of meta-analyses. Consider Fanelli (2010, p. 4) who found that “… the odds
of reporting a positive result were around five times higher for papers published
in Psychology and Psychiatry and Economics and Business than in Space Science” (see
also Fanelli, 2009, 2012; Bones, 2012). Despite these numbers, the possibly best evidence
of my claims comes from a logical argument: has anyone ever seen articles using SEM,
IRT, or Rasch models in which the author admitted the falsification of his/her hypotheses?
On the contrary, it appears that stringent model tests are mostly carefully avoided
in favor of insensitive “goodness-of-fit indices” (cf. Karabatsos, 2001; Heene et
al., 2011).
Given that the empirical foundation for ACM might seldom be given it is then reasonable
to apply more flexible measurement models such as the Rasch model (Rasch, 1981) which
some authors regard as a probabilistic formulation of ACM (Perline et al., 1979) and
also leading to interval-level measurement. Kyngdon (2008b), however, argues that
there is no basis for this claim by showing that parameters of IRT and Rasch models
are only invariant against positive monotone transformations. Thus, if both the Rasch
model and the more general three-parameter logistic model fit a data set, only the
order upon the person ability estimates produced by these models remains invariant.
Hence, as only order is preserved under positive monotone transformation (Narens,
1981), the fit of an IRT or a Rasch model, respectively, may in fact not be indicative
of quantity, but of order.
Moreover, justification for using the Rasch model relates frequently to the argument
that random error forms a fundamental that is, non-ignorable feature of every psychological
response process and must therefore be included in any model formulation (cf. Borsboom
and Scholten, 2008; Fisher, 2011). Since the Rasch model as a probabilistic model
accounts for random error it seems to be the panacea of the measurement problems in
psychology. However, the magic of obtaining an interval-scale for items and examinees
comes with a price because the Rasch model's status as a quantitative theory is derived
exclusively through the error term as Michell (2008) pointed out. With the Rasch model,
if the error was eliminated, the slope of the item response curves would become infinite,
resulting in step-functions of the Guttman model and the “measurements” of the Rasch
model reduce only to mere order. But eliminating error must by definition lead to
better measurement, not the impossibility of measurement. Nevertheless, Sijtsma (2012)
has recently argued that this reasoning is incorrect:
The Guttman model divides the latent variable scale into disjoint and exhaustive intervals
in which differences Θ − δ
j
do not affect response probabilities. The Rasch model assumes these differences to
have a monotone relationship to response probabilities. From the viewpoint of IRT,
the Guttman model ignores the information contained in the intervals, thus paying
the price of a lower measurement level. (p. 14)
I do not see why this line of argumentation refutes Michell's (2008) “Rasch paradox”.
Sijtsma's reasoning presupposes that the latent trait is continuous. Furthermore,
we can only ignore information “… contained in the intervals” when there already is
interval-level information, but this is not at all self-evident but simply an assumption
of IRT.
This uncomfortable situation that psychometric models cannot work without “error,”
has lead in my opinion, to great statistical hand wringing and argumentative acrobatics
to avoid falsification of the quantitaty assumption. This line of argumentation is
often linked to the demonstration of correspondences between psychology and physics.
For instance, Fisher (2011) claims that the probabilistic nature of the Rasch model
reflects the physical phenomenon of stochastic resonance (SR) within a biological
system. Simply put, SR states that an output signal-to-noise ratio of a nonlinear
threshold system is improved by moderate values of input noise intensity (cf. McNamara
and Wiesenfeld, 1989). The weak and normally undetectable signal becomes then detectable
due to resonance between the signal and the added stochastic noise because the added
noise will occasionally lead to an exceeding of a threshold value of the periodic
force (see Gammaitoni et al., 1998, for illustrative examples). A plethora of physical,
biological and neurophysiological systems, as well as some phenomena from linguistics
and visual perception can be described by SR which has been indirectly shown by applying
both the signal and the noise externally to receptors and neurons or by data simulations
(cf. Simonotto et al., 1997; Gammaitoni et al., 1998; Moskowitz and Dickinson, 2002).
Although it is intriguing to regard SR as a valid justification for probabilistic
item response models in order to capture randomness, such an extrapolation is far-fetched
because it is not at all self-evident why and how such micro-level phenomena can be
extrapolated to the macro-level of item responses. Moreover, because present results
on SR in biological systems bear on indirect evidence, the general applicability of
SR to such systems is far from being clear as noted by McDonnell and Abbott (2009):
Adding noise to external stimuli cannot prove that neurons or brain function depend
on consistently available internal sources of randomness, i.e., on endogenous neural
noise. The challenge is to devise an experiment that can remove naturally occurring
healthy variability and demonstrate that function is impaired solely due to that removal.
(p. 6)
It appears that borrowing examples from the natural sciences and relating them to
the (error) structure of probabilistic item response models might be a persuading
analogy but is not a convincing justification for the probabilistic nature of item
response models. Explicit cognitive theories of the test item response process are
needed, but psychometrics is profoundly lacking in such theories (Kyngdon, 2011).
Furthermore, no experimental evidence currently exists which shows why and how such
system-inherent error might occur in the item response process.
Finally, I just wonder why psychometricians have yet ignored the success ACM has within
theories of utility and decision making in psychology (“prospect theory”; Kahneman
and Tversky, 1979) in which ACM served as a formal proof. While it is true that human
choice behavior did not strictly follow the requirements of ACM and research has discovered
paradoxes of human choice behavior (Birnbaum, 2008), it is also clear that these observations
have led to falsifications of old theories of choice behavior and the development
of new ones that account for persistent violations of coalescing and first order stochastic
dominance (e.g., Birnbaum, 2008; Luce et al., 2008). Frankly speaking, I have very
rarely seen such an attitude within mainstream psychometrics be it IRT/Rasch or SEM
where items are omitted from tests, powerless but flattering item-fit statistics are
commonly used (Karabatsos, 2001), and correlated error terms are specified (Cole et
al., 2007) to get a reasonable model-fit and to construct support for one's own the
theory despite doubtful consequences (cf. Bones, 2012; Ferguson and Heene, 2012).
Conclusion
Altogether, it is possible that human cognitive abilities and personality traits simply
are not quantitative. ACM might be in fact too severe for practical testing purposes.
However, psychometricians continue to argue that cognitive abilities are quantitative
and measurable “latent traits” (Markus and Borsboom, 2012). If this argument is correct,
then once item response error is controlled, test score response data should be consistent
with the cancellation axioms of ACM. Thus, more direct experimentation is needed instead
of more sophisticated IRT models.
It is still unclear and an unsolved problem what SEM and IRT models, notably the Rasch
model, add to the clarification of the quantity problem in psychology. It is furthermore
unclear what insights into empirical phenomena it provides as even attempts to explain
the error structure seem to be premature. It is mostly forgotten that Rasch himself
did not derive his model from empirical observations but “… within [Rasch's] own mathematical
playground—with no relation to any actual item analysis problem!” (Rasch, 1979). It
is not necessarily wrong to develop mathematical models independently from empirical
observations. But, it is also not at all self-evident that empirical insights will
result from such models, be it an IRT, SEM, or ACM. However, by avoiding tests of
the assumption of a quantitative structure of psychological attributes, psychologists
have yet failed to make progress on the basis of the fundamental scientific principle
of falsification and in regard to their most fundamental assumptions of quantitative
psychological attributes.

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Mark McDonnell, Derek Abbott (2009)

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Daniele Fanelli (2010)

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H Steiger, Michael Cole, A Cieśla (2007)

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