Taxonic latent structure
Running head: Taxonic latent structure
The Taxonic Latent Structure and Taxometrics
in Forensic Mental Health
Michael D. Maraun and Stephen D. Hart
Simon Fraser University
Correspondence:
Michael D. Maraun
Department of Psychology
Simon Fraser University
Burnaby, BC
Canada V5A 1S6
Email: maraun@sfu.ca
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Taxonic latent structure
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Abstract
Recently, researchers in the field of forensic mental health have attempted to
address the technical, empirical question of whether important clinical problems, such
as psychopathy or malingering constitute taxa (i.e., discrete conditions). In this paper,
we provide a detailed elucidation of the foundational logic of the quantitative methods
employed to answer this question, focusing on the taxometric procedures developed by
Paul Meehl and colleagues. We attempt to demonstrate that research on taxonicity is
hampered by: (a) researchers’ unfamiliarity with or misunderstanding of the logic
underlying latent variable technologies; and, (b) the fundamental incapacity of
Meehlian procedures to provide a test of taxonicity. We conclude by discussing the
utility of taxometric procedures to research in forensic mental health and, more broadly,
in the field of applied psychological measurement.
KEYWORDS: Taxonic latent structure; taxometrics; dimensional structure;
psychopathy.
Taxonic latent structure
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Taxonic Latent Structure and Taxometrics in Forensic Mental Health
Researchers in the field of forensic mental health have, increasingly, turned their
attention to the question of whether clinically relevant constructs, such as psychopathy
and malingering, constitute latent taxa (i.e., naturally occurring categories); or,
alternatively, whether, in nature, these constructs are distributed dimensionally or
taxonically. We will refer to these two distinct but related questions as Q1 and Q2,
respectively. Published studies addressing Q1 or Q2, as they bear on psychopathy,
include, for example: Edens, Marcus, Lilienfeld, and Poythress (2006); Guay and Knight
(2003); Guay, Ruscio, Knight, and Hare, (2007); Harris, Rice, and Quinsey (1994);
Skilling, Harris, Rice, and Quinsey (2002); Vasey, Kotov, Frick, and Loney (2005); and
Walters, Duncan, and Mitchell-Perez (2007). Studies addressing these questions within
the context of malingering, include: Frazier, Youngstrom, Naugle, Haggerty, and Busch
(2007); Strong, Greene, and Schinka (2000); Walters et al. (2008); and Walters, Berry,
Rogers, Payne, and Granacher (2009).
As Haslam, Holland, and Kuppens (2012) argue, Q1 and Q2 are “crucial scientific
question[s] and not merely a matter of theoretical taste or statistical botanizing” (p. 903).
A construct that is truly taxonic has “category boundaries that exist independent of
social convention or descriptive convenience” (p. 903). This means that measures of
taxonic constructs should permit classification using diagnostic criteria and thresholds
that are not determined on arbitrary or pragmatic grounds (e.g., using rules and cutoffs
of convention or convenience). Also, taxa, as distinct from the additive effects of
multiple causal factors, are more likely to result from “single discrete causal factors” (p.
Taxonic latent structure
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903), something that has implications for both etiological research and clinical
management. The potential theoretical and practical importance of taxonicity is
reflected in the large number of studies- not only within the field of forensic mental
health, but, more generally, within those of the social, medical, and natural, sciences- in
which a primary thrust has been the attempt to detect the presence of taxa. In their
recent review, Haslam et al. (2012) identified 177 such studies.
Let us accept that Q1 and Q2 are scientifically important questions. How do we
go about trying to answer them? Studies to date have relied, most commonly, on a set
of procedures developed by Paul Meehl and colleagues, with names such as MAXCOV,
MAMBAC, MAXSLOPE, and MAXEIG (hereafter, Taxometric Procedures). This is
hardly surprising, for, as suggested by titles such as Detecting Latent Clinical Taxa…
(Meehl, 1965, 1968), Taxometric Analysis: II. Detecting Taxonicity… (Meehl and Yonce,
1996), and The Psychometric Detection of Schizotypy: Do putative Schizotypy Indicators
Identify the same Latent Class (Horan, Blanchard, Gangestad, and Kwapil, 2004), the
taxometric procedures were invented to play the role of, and are widely understood to
be, detectors of latent taxa (taxonic latent structures).
In Meehl’s words: “... a researcher with no theoretical opinion can properly use
taxometrics as a decision procedure [italics added], with one possible outcome being that
the latent structure is nontaxonic” (2004, p.39). In fact, according to Meehl, a chief
motivating force behind his having gone to the trouble of inventing his taxometric
procedures was his belief (incorrect, as we shall see) that linear factor analytic
technology could not, in the service of detecting the taxonic latent structure, be enlisted:
Taxonic latent structure
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“Whenever indicators are valid for a taxon, factor analysis will yield a factor. That
predictable result means factor analysis cannot yield a valid decision as to the taxonic
question” (Meehl, 2004, p.44).
In this paper, we address the issue of Meehlian taxometric procedures, as these
procedures bear on Q1 and Q2. We argue that, in consequence of logical flaws inherent
to their derivation, these procedures are fundamentally unable to provide answers to
Q1 and Q2. We reach the pessimistic conclusion that the taxometric procedures are not,
as Meehl intended them to be, “decision procedures” for employment in the detection
of latent taxa; a conclusion that calls into question the standing of the output of research
programs that feature employment of these procedures.
Turning quickly and unquestioningly, in empirical research, to the employment
of a quantitative procedure, and the computer programs that have, for it, been written,
places the researcher at peril of bypassing the intellectual work that is requisite to the
generation of satisfactory solutions to complicated scientific problems (cf. Freedman,
1985). We, herein, suggest that in their turning to Meehl’s taxometric procedures,
researchers have wholly skirted the consideration of foundational issues, the resolution
of which is prerequisite to a coherent, fruitful, handling of Q1 and Q2.
Most particularly, the quick turn researcher's have made to employing the
taxometric programs written and made available by Meehl and colleagues, is
symptomatic of the deplorably common practice of bypassing, en passant, the task of
mastering the logical principles on which are founded all functioning latent variable
technologies- and on the basis of which technologies such as the taxometric procedures
Taxonic latent structure
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must be adjudged-, in favour of a "getting on with the producing of output" (button
pushing).
Researchers who have not mastered the logical principles that underlie latent
variable technologies are absent the capacity to enter into a meaningful deliberation as
to how these principles must be instantiated in programs of empirical research, in order
that the yield of this research is relevant to the adjudication of, as the case may be, Q1 or
Q2. Thus, although the paper does indeed address the issue of the taxometric
procedures, it does so through a necessary focusing on a much broader issue; to wit, the
logic foundational to the formulation of coherent expressions of Q1 and Q2.
The particulars of the case that we will explain are as follows:
1. To build a tool for employment in the detection of a particular latent structure
requires that this latent structure- the target of detection- be specified1.
2. Neither Meehl and colleagues, nor, to our knowledge, anyone else, has, to
date, specified the latent structure that goes by the name taxonic latent structure (latent
taxon).
3. Consequently, the taxometric procedures are not (and could not be) tools
employable in the service of detecting a latent structure that goes by the name taxonic
latent structure (latent taxon).
1
And, in fact, at the root of each and every extant, properly operating, latent variable technology
(linear factor analysis, quadratic factor analysis, latent profile analysis, item response analysis,
etc.) lies the specification of a particular class of latent structures, the elements of which are, in
respect to the technology, the targets of detection.
Taxonic latent structure
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4. Accordingly, output generated through employment of a taxometric program
can have no implications for decision-making apropos the existence of a latent structure
that goes by the name taxonic latent structure (latent taxon).
5. Because the meanings of Q1 and Q2 depend upon the specification of a latent
structure that goes by the name taxonic latent structure (latent taxon), and this latent
structure remains unspecified, the meanings of Q1 and Q2 are, as it stands,
indeterminate. Hence, Q1 and Q2 are not questions at all; consequently, are not
questions that can be answered through the undertaking of empirical research.
Though we draw the conclusion- patent, given an appreciation of the logical
principles in accordance to which latent variable technologies are created, and
coherently operated- that the taxometric procedures (and associated programs) cannot
be employed in the service of detecting taxonic latent structures- it will not, in fact, be
possible to develop working detectors until there has been specified something to
detect, namely, a latent structure that goes by the name taxonic latent structure-, hence,
cannot be employed to answer either of Q1 or Q2- questions which, in any case, are
currently unanswerable, due to the fact that each lacks a meaning- our aim in so doing is
certainly not to pick a fight with Meehl and colleagues. It is, rather, to suggest to the
forensic researcher the simple truth that, before he or she will be in a position to tackle,
empirically, either of Q1 or Q2, there is some logico-scientific work to be done.
Moreover, to suggest that this work will involve: a) the detailed, technical,
elucidation of the meanings of each of Q1 and Q2, the successful completion of this task,
most fundamentally necessitating that a specification be made of each of taxonic- and
Taxonic latent structure
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dimensional- latent structure; and b) (consequent upon (a) having been adequately
addressed) technical articulation of the manner in which bears on (adequately
elucidated versions of) Q1 and Q2, the output of any computer program put forth as a
candidate for employment in the service of addressing these questions.
A technical, scientific, handling of either of Q1 or Q2- or, for that matter, the issue
of the performance of the taxometric procedures-, requires of the researcher that he or
she possess a sound grasp of the logic of latent variable technologies, for, among other
things, he or she will need to be able to successfully undertake the scientific task of
specifying latent structures. However, a sound grasp of this logic is precisely what the
vast majority of social researchers lack, they depending upon slogans, figurative
language, sundry metaphysical commitments, and primitive heuristics to provide
(very) vague, ambiguous, accounts of their employments of these complicated
technologies. Therefore, in putting forth our case, we will find it necessary to provide
to the reader appropriate technical background; in particular, a surview of the logic on
which is founded all operative latent variable technologies, this logic derived from the
object detection scenarios of the natural sciences.
Accordingly, the organization of the paper is as follows: a) we review the
detection logic on which are founded the object detection protocols of the natural
sciences; b) we elucidate the dependency on this detection logic, of latent variable
technologies, and, importantly, review the manner in which latent structures are
specified; c) we provide technical, scientific considerations of each of Q1 and Q2. As
part of our consideration of Q1, we address the issue of the taxometric procedures.
Taxonic latent structure
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We warn in advance that, for many readers, this paper will be challenging. To
come to an understanding of the logic and mathematics on which rest Q1, Q2, and the
taxometric procedures, is not an easy matter; they are, this logic and mathematics,
complex and unfamiliar. Slow and deliberate reading may be required to avoid
intellectual indigestion. We also warn that our arguments may appear to pin much of
the blame for the problems we identify on the late Paul Meehl. This may seem unfair,
either minimizing Meehl’s contributions to psychology or taking advantage of the fact
that he is now unable to defend himself. Our goal is not to blame anyone, and certainly
not Meehl; if we have built a shoddy structure on the brilliantly creative foundations
provided by Meehl, we have no-one to blame but ourselves.
As a means of illustrating principles, we undertake a small-scale analysis of a set
of PCL-R data. This analysis is intended only to illustrate principles, and should in no
way be understood by the reader as an attempt, by the authors, to weigh in, empirically,
on either of Q1 or Q2. The focus on psychopathy is, of course, in the end, incidental; the
principles discussed, herein, are general, and, accordingly, bear on the problem of
taxonicity as it might arise within any domain of research.
The Object Detection Logic of the Natural Sciences
From the mid-twentieth century onwards, psychologists have followed the lead
of methodological table-setters such as L.L. Thurstone, Lee Cronbach, Paul Meehl, and
Paul Lazarsfeld, in seeing their science through the lense of the empirical realist
philosophy of, among others, Feigl (1950), Hempel (1958), and Sellars (1963). This
Taxonic latent structure
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empirical realist inspired conception of science (hereafter, the ERC2) can be summarized
as follows:
ERC1) psychological phenomena are classifiable as either observable or unobservable3;
ERC2) observables are dependent upon unobservables;
ERC3) the nature of the relational linkage between unobservable and dependent
observable is either that of: a) cause (unobservable) to partially determined consequent
(observable) [e.g., Cattell4 ; Rozeboom5; Meehl6; Mulaik7; McDonald8 ]; or b) conceptual
2
See Maraun, Slaney, and Gabriel (2009) for a review of the role played by the ERC
(called, therein, the Augustinian Conception of Reality) in various of the
methodological orientations indigenous to the social and behavioural sciences.
3
Those within the class of the unobservable- phenomena such as “…a person's anxiety,
extraversion, intelligence, or goal orientation” (Strauss, 1999, p.19), “…general
intelligence…” (Borsboom, 2008, p.27), and “…social class, public opinion or extrovert
personality…” (Everett, 1984, p.2)- referred to, variously, and among others, as latent
constructs (e.g., Lord and Novick, 1968), latent variables (e.g., Green, 1954; Lazarsfeld,
1955), latent concepts (Bollen, 2002), underlying abilities (Lord, 1952), underlying traits
(Lord, 1953), hypothetical variables (e.g., Green, 1954, p. 725), and latent traits (Birnbaum,
1968).
4
"It would seem that in general the variables highly loaded in a factor are likely to be
the causes of those which are less loaded, or, at least that the most highly loaded
measure-the factor itself-is causal to the variables that are loaded in it" (Cattell, 1952,
p.362)
5 "And what we want to learn is not so much F -scores in AM solution-range most
i
closely aligned with scores in P on causal sources of Z as the non-extensional nature of
these causal variables" (Rozeboom, 1988, p.225)
6 "If a causal conjecture substantively entails the existence of a taxon specifying (on
theoretical grounds) its observable indicators, a clear-cut nontaxonic result of
taxometric data analysis discorroborates the causal conjecture"(Meehl, 1992, p. 152)
7 "To use Guttman's measure of indeterminacy in factor analysis, we need not assume
that the factors generating the data [italics added] on the observed variables..." (Mulaik,
1976, p.252). "which set of variables actually are the factors that generated the data [italics
added]..."; "He then establishes a set of empirical operations that will measure that
causal factor [italics added]..." Later in the same paper, in considering the idea of factors
as constructed variates, he states that "such artificial variables in x could not serve as
causal explanations of the variables in n" (Mulaik, 1976, p.254)
Taxonic latent structure
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essence (unobservable) to error-laden exemplar/indicator (observable) [e.g.,
Thurstone9; Guilford10; Green11; McDonald12; McDonald and Mulaik13];
ERC4) fundamental aims of psychological science are to detect and identify
efficacious unobservables; make discoveries as to the natures of the relationships that
"If, in the first place, we are willing to regard the factor model as describing an aspect
of the real world whereby unobserved processes give rise [italics added] to "observable"
random variables, it is then a contradiction to suppose that those processes are not
unique. It is another contradiction to suppose that they are known... In short, most
accounts of the fundamental factor model use the non-mathematical qualifier
"unobservable" to describe the common factor scores. It is not yet proven that it is
philosophically naive to do this." (McDonald, 1972, p.18)
9
"The factorial methods were developed primarily for the purpose of identifying the
principal dimensions or categories of mentality..." (Thurstone, 1947, p.55)
10 "The task of isolating the independent aspects of experience has been a difficult one.
Armchair methods dominated by deductive logic rather than by observation led to the
faculty psychologies, traditionally unacceptable to modern psychology. Direct
observation has likewise failed to arrive at any set of unitary traits which even approach
a universal acceptance. Factor analysis or some similar objective process had to be
brought into the search for the unitary traits of personality [italics added]" (Guilford, 1954,
p.470)
11 "To obtain a more precise definition of attitude, we need a mathematical model that
relates the responses, or observed variables, to the latent variable" (Green, 1954, p. 725)
12 "In using the common factor model, then, with rare exceptions the researcher makes
the simple heuristic assumption that two tests are correlated because they in part
measure the same trait, rather than because they are determined by a common cause, or
linked together in a causal sequence, or related by some other theoretical mechanism. It
is a consequence of this heuristic assumption that we interpret a common factor as a
characteristic of the examinees that the tests measure in common, and, correlatively,
regard the residual (the unique factor) as a characteristic of the examinees that each test
measures uniquely" (McDonald, 1981, p.107); "...I am describing the rule of
correspondence which I both recommend as the normative rule of correspondence, and
conjecture to be the rule as a matter of fact followed in most applications, namely: In an
application, the common factor of a set of tests/items corresponds to their common
property" (McDonald, 1996, p.670)
13 "..the widely accepted aim of factor analysis, namely, the interpretation of a common
factor in terms of the common attribute of the tests that have high loadings on it...";
"what attribute of the individuals the factor variable represents" (McDonald and
Mulaik, 1979, p.298)
8
Taxonic latent structure
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hold among these unobservables; and come to an understanding as to the natures of the
dependencies of observables upon them.
Under the ERC, a need arose for the psychological scientist to possess tools that
could take as input, a) extant knowledge about relationships among observables, and b)
theoretical speculations about the natures of the dependencies of both observables, and
unobservables, on unobservables, and return the desired inferences respecting
scientifically important unobservables. The tools that were seen as satisfying, uniquely
so, this need, were the latent variable technologies; a category which, nowadays, is
populated by such as linear factor analysis, latent profile analysis, quadratic factor
analysis, and item response theory, and, putatively, Meehl’s taxometric procedures.
Lord and Novick put it this way: "...the abilities or traits that psychologists wish
to study are usually not directly measureable; rather they must be studied indirectly,
through measurements of other quantities" (1968, p.13); Latent variable models "...have
in part been offered as a means of linking the more precise but limited mathematical
models of behavior with the more broadly conceived psychological theories" (1968,
p.19); "The factor analytic model is one of a number of models that give concrete form to
the concepts used in theoretical explanations of human behavior in terms of latent traits.
In any theory of latent traits, one supposes that human behavior can be accounted for,
to a substantial degree, by isolating certain consistent and stable human characteristics,
or traits, and by using a person's values on those traits to predict or explain his
performance in relevant situations" (1968, p.537).
Taxonic latent structure
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In Lazarsfeld's words, "Empirical observations locate our objects in a manifest
property space. But this is not what we are really interested in. We want to know their
location in a latent property space. Our problem is to infer this latent space from the manifest
data" (1959, p.490). Latent variable technologies such as Lazarsfeld's latent structure
analysis and Meehl’s taxometric procedures were invented expressly, and for no other
reason than, to address scientific problems implied by the ERC, and these problems are
variations on a theme: The researcher must attempt to "...discover in the factorial
analysis the nature of the underlying order" (Thurstone, 1947, p.56).
The manner in which latent variable technologies address the scientific problems
implied by the ERC is manifest in the logic in accordance to which they are built and
operated. This logic is an object detection logic that was imported from the natural
sciences. It is the very logic that underpins the protocols that have been invented by
natural scientists for use in the detection of perceptually unobservable material entities
such as sub-atomic particles, hidden metal objects, stellar objects, and viruses. Before
turning to an elucidation of the manner in which latent variable technologies are
parasitic on this logic, it will be salutary to provide a brief review the detection
protocols of the natural sciences, a task to which we now turn.
A protocol, D, for employment in the service of detecting the elements u of a
class U of material entities is comprised of four logically distinct components:
1. A specification of the class U, the elements of which are the (unobservable) targets of
detection. A detection protocol is created in response to an identified need to
make decisions in respect whether particular unobservable entities are present.
Taxonic latent structure
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One specifies the class U, the elements u of which are the targets of detection of
D, by laying down a rule that settles the properties that a material entity must
possess in order to qualify as an element of U. If U contains just those material
entities, say, u-things, that are denoted by a particular concept "u", then to specify
U is equivalent to defining the concept "u".
2. A generating proposition that links the presence of an element u of U to an observable
O. Though a u that is an element of U is unobservable, it is, nonetheless, a
material entity. As such, it will have various sorts of impacts upon other
material entities. Some of these impacts may be observable. One must deducethrough a combination of theory, mathematics, and empirical knowledge- a
generating proposition that links the presence of a u at spatio-temporal coordinates
(t,s) to an observable impact, O, of u's being present.
There are three types of generating proposition, each characterized by a
particular type of logical linkage between the presence of a u at (t,s) and an O:
Type I. if u ∈ U is present at (t,s), then O
[i.e., O is a necessary condition of u being present at (t,s)];
Type II. if O, then u ∈ U is present at (t,s)
[i.e., O is a sufficient condition of u being present at (t,s)];
Type III. u ∈ U is present at (t,s) if and only if O
[i.e., O is a necessary and sufficient condition of u being present at (t,s)].
Taxonic latent structure
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3. A tool of detection, T, that yields decisions about whether or not a u is present at a
particular (t,s). A tool of detection T is an implementation of a particular
generating proposition; as such, the type of decision-making it supports is
determined by the logical type of this generating proposition. If T is an
implementation of a Type I generating proposition, then T operates as follows:
when O is not the case, then it can be validly concluded that u is not present at
(t,s). That is to say, in this case, T's employment is as a modus tollens tool of
disconfirmation. If it is an implementation of a generating proposition of Type
II, then T is a confirmatory tool and operates as follows: when O is the case, then
it can be validly concluded that u is present at (t,s). And, finally, if T is an
implementation of a Type III proposition, then it can be employed in both a
confirmatory and disconfirmatory fashion.
4. Side-conditions that must be satisfied in order that tool of detection T functions
properly. A side-condition is a state of affairs (feature of reality) that must hold in
order that T’s generating proposition be true. Importantly, side-conditions are
not, then, properties of the targets of detection u ∈ U, but, rather, have to do with
the correct operation of tool of detection T.
Example: The Detection of Metal Objects
Consider the components of the protocol, D, constructed for employment in the
service of detecting metal objects, and in which is featured the pulse induction metal
detector. Targets of detection. The targets of detection are metal objects; i.e., elements of
the class U of material entities denoted by the concept metal object. Thus, in this case, a
Taxonic latent structure
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specification of the targets of detection is effected by providing a definition of the
concept metal object (to wit, an object made of at least one element, the atoms of which
readily lose electrons). Generating proposition. The generating proposition is, in this
case, of type III: an electro-magnetic impulse of a particular duration was transmitted in
the vicinity of a metal object if and only if (in consequence of the phenomenon of selfinduction) a primary and secondary electro-magnetic impulse occurs in the object
(Kanchev, 2005). Tool of detection. The pulse induction metal detector is an
implementation of this type III generating proposition. It transmits an electro-magnetic
impulse, and if it registers (does not register) a consequent fading impulse, the decision
is made that there exists (does not exist), in the vicinity of the metal detector, a metal
object (Kanchev, 2005). Side conditions. The proper operation of the pulse induction
metal detector is ensured through satisfaction of certain side-conditions; notably, that it
is not operated in the vicinity of televisions, radios, cell phones, and other entities that
produce radio waves (Kanchev, 2005).
Detection Protocols for Latent Structures
Let us now elucidate the dependency of latent variable technologies, upon the
logic of the detection protocols of the natural sciences14 (a technical account of this
dependency is available in Maraun (2016)). To begin, note that: a) the unobservables
mentioned in the ERC are paraphrased technically as latent structures; b) every extant
latent variable technology (linear factor analysis, quadratic factor analysis, latent profile
14
No comment will be made, herein, on the issues of whether either of the ERC, or the
employment of detection-logic based latent variable technologies to address the
problems implied by it, makes sense (is logically coherent). These issues are addressed
elsewhere, e.g., in Maraun (2003) and Maraun, Slaney, and Gabriel (2009).
Taxonic latent structure
17
analysis, the various item response analyses, etc.) arises out of a specification of a
particular class of latent structures, the elements of which are unobservables, the
detection of which has been deemed to be scientifically important.15
Consider, then, a researcher who wishes to build a detection protocol DLS* that
can be employed in the service of making decisions about whether or not an element ls*
of a particular class LS* of latent structures is a latent structure of a set of observable
indicators X. DLS* is, then, comprised of five components, the first four of which are
isomorphic with the components of the object detection protocols of the natural
sciences:
1. A specification of the class LS* of latent structures the elements ls* of which are
the targets of detection.
2. A generating proposition that links the presence of an element ls* of LS* to a
particular observable impact of ls*, these observable impacts, restrictions on the
distribution of X.
3. A tool of detection that is an implementation of the deduced generating
proposition.
4. Side-conditions that must be satisfied in order that the generating proposition
be true.
15
There exists a widespread misconception that latent variable technologies have
something to do with models, and that their employments in research are modeling
exercises (see Maraun, 2003, 2016). Latent variable technologies do not answer to the
logic at root of modelling exercises (the latter of which feature the laying down of
antecedently specified rules of correspondence between constituent parts {t1,t2,...,tp} of
model and constituent parts {a1,a2,...,ap} of that which is modelled), but, rather, to
detection logic (which, conspicuously, features the analytical derivation of generating
propositions).
Taxonic latent structure
18
5. (because the researcher will not possess knowledge of the values assumed by
population parameters) inferential machinery employed to make populationlevel decisions on the basis of sample information.
Let us consider each of these components in turn.
Component a: Specification of Targets of Detection (Class LS* of Latent Structures)
Let
stand for a set of m latent variables, X stand for an arbitrary set of p>m
indicators, f be the marginal distribution of
conditional on
, and f
X|
be the distribution of X
. A latent structure, then, is simply a set of k latent properties ti, a
latent property being a property of either of f
X|
or f . A particular latent structure ls
is then specified by specifying the k latent properties of which it is constituted; i.e., its k
defining properties.
The class of latent structures LS*, the elements ls* of which are the targets
that DLS* was designed to detect, is specified by specifying a set of s latent
properties {t1*,...,ts*} all of which must be possessed by a latent structure in order
that it be an element of LS*. That is to say, one specifies the latent structures ls*
that are the targets of detection by listing the set {t1*,...,ts*} of defining properties
of LS*.
Example (unidimensional, linear factor structures). The class of
unidimensional, linear factor structures LSulf is specified as {t1(ulf), t2(ulf), t3(ulf)} in
which the defining properties are as follows: t1(ulf)
is a single, continuously
distributed, variable with a mean of zero and a variance of unity; t2(ulf)
E(X| =
*)
= + θ*, in which
and
are p × 1 vectors of real numbers; and
Taxonic latent structure
t3(ulf)
C(X| =
*
) = , the p × p conditional covariance matrix of X given
19
, is
diagonal and positive definite.16
Note: Property t2(ulf) states mathematically that the mean of each indicator,
conditional on
on
, is a linear function of
(equivalently, having partialled
; property t3(ulf) states that conditional
from X), the indicators are uncorrelated,
and is a standard paraphrase, within the domain of latent variable technologies,
of the notion that the indicators are causally dependent upon
.
We will encounter t3(ulf), labeled there as LCU- along with a stronger
paraphrase of causal dependency, called latent conditional independence (LCI), that
implies LCU- when we turn, later in the paper, to a consideration of the
taxometric procedures and the specification of the taxonic latent structure.
Example (2-class latent profile structures). The class of 2-class latent profile
structures LS2clp is specified by the set of defining properties {t1(2clp), t2(2clp), t3(2clp)}, in
which: t1(2clp)
is a single, Bernoulli (two-point; values
1
and
2
with probabilities π 1
and π 2=(1- π 1), respectively) distributed variable, with a mean of zero and a variance of
unity; t2(2clp)
E(X| = j) = +
numbers; t3(2clp)
conditional on
C(X| =
j
)=
j,
j={1,2}, in which
and
are p × 1 vectors of real
j,
j={1,2}, the p × p conditional covariance matrix of X
= j, is diagonal and positive definite.
Note: there are only two points of distinction in the specifications of LS2clp and
LSulf. First, in LSulf,
has a continuous distribution, and, in LS2clp, a Bernoulli
distribution. Second, in LSulf , C(X| ) is homoscedastic (constant over the values of
16
Equivalently, the vector of residuals, X-E(X| =θ*), has a p × p covariance matrix
is diagonal and positive definite.
),
that
Taxonic latent structure
20
and in LS2clp, heteroscedastic. From a detection logic perspective, one asks; "what are
the implications of the two targets of detection- the unidimensional, linear, factor
structure, on the one hand, and the two-class latent profile structure, on the otherdiffering in only these ways, for the construction of detection protocols for each?" The
answer will, of course, be found in a mathematical analysis of the restrictions imposed
on the distribution of an arbitrary set of indicators, by each of {t1(2clp), t2(2clp), t3(2clp)} and
{t1(ulf), t2(ulf), t3(ulf)}.
Example (unidimensional, 2-parameter, logistic item response structures). Let
X be a vector of dichotomous [0,1] random variables. The class of unidimensional, 2parameter, logistic item response structures LSu2plir is specified as { t1(u2plir), t2(u2plir),
t3(u2plir)} in which: t1(u2plir)
is a single, continuously distributed, variable with a mean
of zero and a variance of unity; t2(u2plir)
exp( a j
*
(1 + exp( a j
t3(u2plir)
− bj )
*
− b j ))
E(Xj| =
*)
= P(Xj=1| =
*)
=
;
P(X= X*| =
*)
=
p
∏ P(X
j
= 1| =
*
* Xj
) (1 − P(X j = 1| =
*
1− X *j
))
.
j =1
Now, in respect to a set of p continuous indicators, defined in a population P
under study, to state an interest in detecting, say, 2-class latent profile structures, i.e., in
deciding upon whether any of the elements of LS2clp is a latent structure of X, is to state
an interest in detecting just those latent structures, the k defining properties of which
include, among them, the s defining properties {t1(2clp), t2(2clp), t3(2clp)} of LS2clp.
Taxonic latent structure
21
Fact: The detection of the elements of a particular class of latent structures is
inextricably tied to the issue of the dimensionality of the indicators X. In particular: a)
there can be defined a virtually unlimited number of distinct senses of “the indicators X
are r-dimensional” (for each sense, uni-dimensionality as a special case); b) each
particular class of latent structures defines one particular of these senses. Thus, for
example, if an element of LSulf happens to be a latent structure of X, then X is
unidimensional in a linear factor analytic sense. If, on the other hand, an element of the
class of unidimensional, quadratic factor structures happens to be a latent structure of
X, then X is unidimensional in a quadratic factor analytic sense (but two-dimensional in
a linear factor analytic sense (as will be, also, an X that has, as a latent structure, an
element of LS2clp), and p-dimensional in a principal component analytic sense!).
The point, then, is that to decide whether there exists an element ls* of LS* that is a
latent structure to X is, simultaneously, to decide whether X is r-dimensional, wherein
both the sense of dimensionality in play, and the value of r, are determined by LS* .
Component b: Generating Proposition
Let f
X
be the joint distribution of X, and let
be a subset of m parameters of
f (these parameters, standardly, a subset of association parameters such as
X
covariances or correlations). The generating proposition GLS* on which DLS* is founded
is a type I, II, or III proposition that: a) is derived in mathematics; b) links the state of
Taxonic latent structure
affairs that an element ls* of LS* is a latent structure of X, to a restriction on
22
. That is to
say, an unobservable latent structure ls* belonging to LS* manifests itself by restricting
the joint distribution of the indicators in the manner described by the generating
proposition. The generating proposition is the link between the realm of the
unobservable, and that of the observable.
Fact: Each and every latent variable technology is founded on the derivation, in
mathematics, of a generating proposition that links the latent structures that are its
targets of detection, to a restriction on a subset of the parameters of the joint
distribution of an arbitrary set of indicators. For a specified class of latent structures,
there can exist no detectors until one or more generating propositions have been
derived.
Example (generating proposition for LSulf). The generating proposition Gulf that
is foundational of all extant detection protocols Dulf for employment in the service of
detecting unidimensional, linear factor structures, i.e., elements lsulf of LSulf17, was
derived in 1904 by Charles Spearman (cf. Wansbeek and Meijer, 2000, p.150; Mardia,
Kent, and Bibby, 1980, p.257). It is a type III proposition, involves a restriction on the
1
p ( p + 1) unique elements of Σ X 18 (the covariance matrix of the indicators), and can be
2
stated as follows: there exists an element lsulf of LSulf that is a latent structure of X if and
17
And on which is based the linear factor analytic modules of SPSS, SYSTAT, LISREL; all
extant programs invented for the purpose of doing linear factor analysis.
1
18
Accordingly, containing, in this case, just the p variances and p( p − 1) unique
2
covariances, contained within Σ X .
Taxonic latent structure
only if there exists a p × 1 vector of real numbers
definite matrix
, such that Σ X =
23
and a p × p diagonal, positive
'+ .
Example (generating proposition for LSu2plir). In this case,
contains the
2 p probabilities P(X= X*) that constitute the joint distribution of the p dichotomous [0,1]
variables contained in X. The type I generating proposition is as follows: if an element
lsu2plir of LSu2plir is a latent structure of X, then there exists 2p parameters {a1, a2,…, ap, b1,
b2,…, bp} such that the numerical value of each element of
, i.e., each P(X= X*), is given
as
P( X = X * ) =
X *j
⎡ exp( a j − b j ) ⎤ ⎡
exp( a j − b j ) ⎤
1
−
⎢
⎥
⎢
∫ ⎢ (1 + exp( a j − b j )) ⎥ ⎢ (1 + exp( a j − b j )) ⎥⎥
−∞ ⎣
⎦ ⎣
⎦
∞
1 − X *j
dF(θ ) .
Holland (e.g., 1990) has shown that this condition is equivalent to the condition
of
lying on a very particular nonlinear manifold (i.e., tracing out a very particular
shape in its ( 2 p -1)-dimensional embedding space). Accordingly, an observable
manifestation of the presence of an element lsu2plir of LSu2plir (in fact, the observable
manifestation that is the basis of most of the extant detection protocols for LSu2plir) is just
that
happens to lie on this nonlinear manifold.
Component c: Tool of Detection
Each and every latent variable technology features a tool of detection that is an
implementation of a mathematically deduced generating proposition. Let TLS * be a tool
of detection for LS* constructed on the basis of a deduced generating proposition GLS*.
The type of decision-making possible with TLS * is determined by the type of proposition
that GLS* is.
Taxonic latent structure
24
Gulf, for example, is the basis for the type III tool Tulf that is featured in all
detection protocols Dulf employed in the service of detecting elements lsulf of LSulf19. The
manner in which Tulf operates is as follows: if there exists a p × 1 vector of real numbers
and a p × p diagonal, positive definite matrix
, such that Σ X =
'+ , then it is
concluded that there exists an element lsulf of LSulf that is a latent structure of X; else, the
conclusion is made that there does not exist an element lsulf of LSulf that is a latent
structure of X.
Component d: Side Conditions
Side conditions are potential properties of f that must obtain empirically in
X
order that the generating proposition on which a tool of detection is based is true;
equivalently, in order that the detection protocol in which is featured the tool of
detection, be properly operating. A detection protocol need not feature any side
conditions. For example, because no side conditions need hold in order that Gulf be true
(Gulf is unconditionally true), there are no side conditions associated with a detection
protocol Dulf, the tool of detection of which is an implementation of Gulf .
Component e: Inferential Machinery
Finally, DLS* will, of necessity, have an inferential component. This is because,
when it comes time to employ tool of detection TLS * to make a decision about whether or
not an element ls* of LS* is a latent structure of some particular set of indicators X*, they
distributed in a particular population P, the researcher will not know the numerical
value of
19
(it being a population parameter). Consequently, he or she will not know
And, once again, is implemented in all extant linear factor analytic programs.
Taxonic latent structure
whether
25
satisfies the restriction mentioned in GLS*. He or she will, instead, have to
make, on the basis of a sample drawn from P, an inferential decision as to whether the
restriction is satisfied in P. The necessity of making an inferential decision as to
whether the restriction is satisfied in P, introduces into the process of detecting
elements of LS*, sampling error.
Example (a detection protocol Dulf for LSulf). Let the vector X* of indicators contain the
twenty items of the PCL-R (Hare, 1991), and let this vector be distributed in a
population P of offenders newly admitted to correctional services in British Columbia
and the Yukon Territory.
Component a. The targets of detection will be the elements lsulf of LSulf, LSulf
specified by the set of defining characteristics {t1(ulf), t2(ulf), t3(ulf)}.20 Thus, the detection
problem is stated as follows: we wish to decide whether or not, in population P, there
exists an element lsulf of LSulf that is a latent structure of X*; i.e., whether X* has, as a
latent structure, the unidimensional, linear, factor structure.
Components b, c, and d. Our detection protocol Dulf will feature the type III tool
of detection Tulf that is an implementation of the type III generating proposition Gulf.
Thus, the decision-making will be as follows: Evidence that there exists a p × 1 vector of
real numbers
and a p × p diagonal, positive definite matrix
, such that Σ X * =
'+
decision: there exists an element lsulf of LSulf that is a latent structure of X*; else
decision: there does not exist an element lsulf of LSulf that is a latent structure of X*.
20
Given that the items of the PCL-R have three-point response scales, it might seem
unlikely that the item-latent variable regressions will be linear. Be that as it may, there
exists no logical prohibition against testing whether there exists an element of LSulf that
is a latent structure of X*.
Taxonic latent structure
26
As noted previously, there are no side-conditions that must be satisfied in order that
this detection tool functions properly (i.e., no conditions that must be satisfied in order
that Gulf be true).
Component e. We will need, then, to make an inferential decision as to whether
holds in the population under investigation, the restriction named by Gulf; i.e., that there
exists a 20 by 1 vector of real numbers
matrix
, such that Σ X* =
and a 20 by 20 diagonal, positive definite
'+ . In other words, we will require an inferential test of
the hypothesis pair [H0: Σ X* =
'+ , H1: Σ X* ≠
'+ ]21. To this end, we will employ
the program LISREL (Joreskog and Sorbom, 1993). With, as input, a sample covariance
matrix S, calculated on the basis of a sample of size n drawn from P, LISREL employs a
numerical algorithm to search for a 20 × 1 vector of real numbers a and a 20 × 20
diagonal matrix D, such that aa’+D is as close as possible to S (the sense of closeness in
play, determined by the loss function selected, but always a function of the matrix of
residuals, S-(aa’+D); herein, we make the selection of a maximum likelihood loss
function).
Within the context of structural equation modeling, a great deal has been written
on the topic of "model fit" and "fit indices." Presuming, of course, that LISREL succeeds
in producing an admissible pair [a,D]22, what is relevant to the making of an inferential
decision as to whether, in population P, there exists an element lsulf of LSulf that is a
21
Alternatively, in the event that rmsea-style thinking is adopted, a test of the pair [H0:
D( , Ω ulf )≤c, H1: D( , Ω ulf )>c], wherein D is a distance measure, and c is a positive
number, could be undertaken.
22
Admissibility, in this case, equivalent to D containing only positive elements; i.e.,
containing no Heywood cases.
Taxonic latent structure
27
latent structure of X are: a) the matrix of standardized residuals; b) useful scalar
functions of the raw residuals, such as the root mean-square error of approximation
(RMSEA) (see, e.g., Steiger and Lind, 1980). Herein, the decision will be made that
there exists an element lsulf of LSulf that is a latent structure of X* if: a) there are, at most,
a very small number of standardized residuals, the numerical values of which are
greater, in absolute value, than two; b) a (sufficiently narrow) 90% confidence interval
for RMSEA contains the value .05.23 Else, it will be decided that there does not exist an
element lsulf of LSulf that is a latent structure of X*.
A sample of size 176 drawn from P, yielded the sample covariance matrix S
shown in Table 1(a). With this matrix as input, and the choice of a maximum likelihood
loss function, LISREL produced an admissible pair [a,D] (Table 1(b)) for which: a) 72 of
190 standardized residuals lie outside of the [-2,2] interval; b) the point estimate of
RMSEA is equal to .141 (90% CI: [.131,.151]). The decision, then, is made that the
restriction Σ X* =
'+ does not hold in population P; accordingly, that the
unidimensional, linear, factor analytic structure is not, within this particular population,
a latent structure of the indicators.
Psychopathy and the Taxonic Latent Structure
The Issue of Q1
With, now, set in plain view, the detection logic on which is founded all latent
variable technologies, let us turn to Q1, the question of whether it is the case that
This, of course, is a heuristic to guide subjective judgment, the general principle being
that the closer to zero is the estimate of the population RMSEA, the stronger is the
evidence that there exists an element lsulf of LSulf that is a latent structure of X. On the
other hand, the narrower the interval, the better is the quality of the estimation.
23
Taxonic latent structure
28
psychopathy is taxonic. Stated technically: Is it the case that a particular set of
psychopathy indicators X*, say the items of the PCL-R24, has as a latent structure, in a
particular population P under study, something called a taxonic latent structure
(hereafter, TLS)?25
A researcher wishing to answer this question would have to:
1. Give the question a sense (meaning) by specifying the class LStax of latent structures
the elements of which- those latent structures that are the unobservable targets of
detection- go by name of TLS. To specify LStax, the researcher would have to specify its
s defining characteristics, say, {tTLS(1),...,tTLS(s)}. Now, specification is not an empirical
issue, but, rather, a conceptual issue (just as is a conceptual issue, the issue of what it is
that is to be detected in a detection of metal objects). And just as it was the job of the
scientific community of astronomers to come to a resolution on the conceptual issue of
whether or not Pluto should remain a member of the category of planets, it is the job of
the scientific community of forensic researchers- at least as regards Q1- to come to a
resolution on the issue of which set of latent properties are the defining properties of
LStax (i..e, which latent structures should receive the label TLS).
24
In this paper, we do not address the issue of which variables constitute the optimal
set of psychopathy indicators, but only the detection problems Q1 and Q2 consequent
upon the choice of such a set. Thus, the reader is free to substitute, in place of the PCLR, any set of variables of his or her choosing.
25
Of course, because it is a particular crossing of set of indicators and population P that
determines the joint distribution of X, hence, determines which latent structures X does
and does not have, there is no reason to expect that distinct crossings of psychopathy
indicators and populations P will have the same latent structures.
Taxonic latent structure
29
2. (Given a specification of LStax) Deduce in mathematics- or, alternatively, find in the
psychometric literature- a generating proposition that links the specified defining
characteristics {tTLS(1),...,tTLS(s)} of LStax to a restriction on the joint distribution of X*.
3. (Given deduction of a generating proposition) Test, using a sample on X* drawn from
the population P under study, whether the named restriction holds in P.
In the next sections, we will: a) bring to the researcher’s attention the fact that
LStax has never been specified, the consequence being that Q1 does not have a meaning
and so is not, as matters stand, an empirical question at all (i.e., it cannot be answered);
b) explain why it is that the taxometric procedures cannot work (a general corollary
being that, because LStax has not been specified, there cannot have been derived for LStax
any generating propositions, hence, any tools that could be employed in the service of
detecting a latent structure that goes by the name TLS); c) offer some speculations as to
why researchers have failed to specify LStax; and d) consider some candidate
specifications of LStax, and, for these candidates, discuss extant theory apropos
generating propositions and tools of detection.
The failure to specify LStax. In the absence of a specification of the defining
properties of LStax, Q1 is not an empirical question at all, but simply a meaningless
assemblage of words. And, of course, there can be no answer given to an assemblage of
words that is absent a meaning. It is significant, then, that nowhere within the forensic
publications in which the attempt has been made to address Q1, is there to be found a
specification of the defining properties of LStax. These publications, instead, hold to an
approach in which is provided a very loose, non-technical, recounting of the received
Taxonic latent structure
30
dogma pertaining to the general issue of taxonicity, followed by a sweeping referencing
of the work of Paul Meehl and colleagues.
Thus, for example, prior to asserting, without justification, that "...taxometric
procedures are needed to reveal the underlying taxonic structure of a construct" (2005,
p.413), Vasey et. al offer the following by way of an explanation of taxonic latent
structure: "The term taxon is synonymous with such terms as 'category' and 'syndrome'"
(2005, p.413). According to Walters et. al. (2007, p.271), "The goal of the categorical
perspective is to identify a taxon, which Meehl and Golden (1982) define as 'an entity,
type, syndrome, species, disease, or, more generally, a nonarbitrary class'... The goal of
the dimensional perspective, on the other hand, is to identify and map out the
psychopathic dimensions of human behaviour." These authors assert that the aim of
their study is to "...explore the taxometric structure of the PCL-R..." (2007, p.271). In
lieu of a specification of LStax, Guay et. al. (2007, p.701) state that "...taxonicity implies
[italics added] both a nonarbitrary latent category and a particular causal structure..."
and speak vaguely about psychopathy being "...distributed as a taxon..."
Given the quick referencing of Meehl and colleagues by forensic, and other,
researchers, it might be thought that the former, they being the inventors of the
taxometric procedures, had already settled the matter of the specification of LStax, and
that the latter were simply deferring to this specification. But this is not so; regardless of
how matters might, to the applied researcher, appear, Meehl and colleagues have never specified
LStax. In the foundational papers on the topic of the taxometric procedures (e.g., Meehl
(1973, 1995); Meehl and Golden (1982); Meehl and Yonce (1994, 1996)), the very papers
Taxonic latent structure
31
that are, in fact, cited by forensic researchers attempting to grapple with Q1, a total of
ten different latent properties are put forth as having something to do with LStax. But
these properties: a) vary paper to paper; b) are, within individual papers, addressed in
obscure and contradictory fashions; and c) are, when taken as a set, mutually
incompatible26.
That is to say, nowhere within these papers is it settled which set of latent
properties constitutes the defining set of LStax. Nor is the issue settled in the otherwise
comprehensive 1998 book by Waller and Meehl, Multivariate Taxometric Procedures:
Distinguishing Types from Continuua. One can, in fact, search as long and hard as one
pleases, but one will not find in the literature devoted to the taxometric procedures, a
specification of LStax.
The taxometric programs. Prior to the development of computer programs that
implement a latent variable technology, there must be a latent variable technology. This
means that there must have been deduced in mathematics, a logically true generating
proposition that links a particular class of latent structures to a restriction on the joint
distribution of a set of indicators. Most essentially, what computer programs such as
LISREL, Latent Gold, EQS, and Multilog offer to the researcher is a means of conducting
26
For a comprehensive, step-by-step, mathematical sustaining of these assertions (due
to its technical nature, unsuitable for inclusion in the current article), see Maraun,
Halpin, and Gabriel (2007).
Taxonic latent structure
32
inferential tests of whether the restriction stated in a particular generating proposition
holds in particular populations under investigation.27
What have appeared in the work of Meehl and colleagues on the taxometric
procedures, are putative generating propositions. Here are the putative propositions at
root of MAXCOV, MAMBAC, and MAXSLOPE:
GMAXCOV: If an element of LStax is a latent structure of X, then the covariance of any pair
of indicators Xi and Xj, conditional on a third Xk, i.e., C(Xi,Xj|Xk=x), is a single-peaked
function of x;
GMAMBAC: If an element of LStax is a latent structure of X, then the mean of any indicator
Xi, conditional on a second indicator Xj being greater than a value x, minus the mean of
Xi, conditional on Xj being less than x, i.e., d(x)=E(Xi| Xj≥x)- E(Xi|Xj≤x), is a singlepeaked function of x;
GMAXSLOPE: If an element of LStax is a latent structure of X, then
d
E(Xi|Xj=x), i.e., the
dx
derivative (slope) of the conditional mean function of one indicator Xi given a second Xj,
is a single-peaked function of x.
In association with these putative generating propositions, there have been
written computer programs- the taxometric programs- that take data as input and
return decisions as to whether hold, the restrictions mentioned in these propositions;
i.e., that C(Xi,Xj|Xk=x) is a single-peaked function of x, that d(x)=E(Xi| Xj≥x)-
27
Programs implementing linear factor analysis, for example, carry out tests of hypothesis pairs
of the form [H0: Σ X =
'+ , H1: Σ X is an arbitrary gramian matrix] just because H0 is the
restriction stated in the generating proposition Gulf.
Taxonic latent structure
E(Xi|Xj≤x) is a single-peaked function of x, and that
33
d
E(Xi|Xj=x) is a single peaked
dx
function of x. If the putative generating propositions- GMAXCOV, GMAMBAC, GMAXSLOPE,
etc.- were true, the decisions returned by the programs would be, in respect LStax, modus
tollens disconfirmatory in nature. However, because, in consequence of the failure to
specify LStax, there does not, for these propositions, exist an antecedent, each
proposition can neither be true, nor false. Each proposition is a meaningless
assemblage of words.
Consequently, a test of whether holds empirically, the consequent- i.e., the
named restriction-, of any of these putative generating propositions, has no implications.
Such a test- consequently, the output of the taxometric programs- could not possibly
say anything about the presence or absence of an element of LStax, because it has not
been settled what these elements are (and it not possible to judge the presence or
absence of an entity that cannot be identified).
The Monte Carlo simulations. The Monte Carlo simulations conducted by
Meehl and Yonce, as part of the development of MAXCOV and MAMBAC, are
frequently cited as establishing that the taxometric procedures "work." Indeed, the
single-peaked graphs that are presented in the 1994 and 1996 papers give the strong
impression that the generating propositions of MAXCOV and MAMBAC are true.
These simulations, however, are a paradigm case of how Monte Carlo studies can
mislead.
So, then, how is one to reconcile the Monte Carlo results with the state of affairs
that LStax has not been specified; consequently, that the putative generating propositions
Taxonic latent structure
34
of the taxometric procedures have no meaning? Two distinct issues must be considered.
On the one hand, there is the issue of the defining characteristics of LStax. On the other
hand, there is the issue of the properties on the basis of which were- in fact- generated
the data in Meehl’s and Yonce’s Monte Carlo simulations.
As we have noted, the defining characteristics of LStax have never been specified.
What, then, were the properties that Meehl and Yonce assigned to the data generated in
their Monte Carlo simulations? The properties were, as a matter of fact: B
Bernoulli (two-point) distribution28; VAL
E(Xr| =T) - E(Xr| =T')>0, r=1..p29; LCU
C(X| ) is diagonal and positive definite, for
fXr| is, for
has a
={T',T}; CN (conditional normality)
={T',T}, r=1..p, a normal distribution; HV (homogeneity of variance)
σ X2 |T = σ X2 |T ' , r=1..p.
r
r
Now, although Meehl and Yonce expressly deny that CN and HV have anything
to do with LStax30, Maraun and Slaney (2005) prove that the addition of CN and HV to
the set {B ,VAL,LCU} is sufficient to yield a C(Xi,Xj|Xk=x) that is a single-peaked
function of x. Thus, the very properties that Meehl and Yonce insist are not defining
characteristics of LStax, and portray, rather, as incidental data features of their Monte
28
The two values of which can, for convenience, be labeled T (denoting individuals in
the taxonic class) and T’ (denoting individuals in the complement class)
29
Called, sometimes, the validity condition (Meehl and Yonce, 1994, 1996).
30
"Although our Monte Carlo data were generated by a Gaussian algorithm assigning
equal variances SDt2, SDc2 to taxon and complement classes, none of the core
derivations underlying MAXCOV are thus restrictive. The conjectured structure...is
highly general, that of two overlapping unimodal frequency distributions. The
mathematics speaks for itself, and it was developed by Meehl with psychopathology in
mind, where skewness and heterogeneity of variance are common" (Meehl and Yonce,
1996, p.1097)
Taxonic latent structure
35
Carlo simulations, are, in fact, essential to making these simulations yield graphs that
give the impression that the taxometric procedures work.
The work of Ruscio and colleagues. It is impossible to discuss published
evidence apropos the performance of the taxometric procedures without making
comment on the recent work of Ruscio and colleagues (e.g., Ruscio, Ruscio, and Keane,
2004; Ruscio, 2007; Ruscio and Marcus, 2007; Ruscio and Kaczetow, 2009). And
although the output of these authors on the topic of the taxometric procedures is
prolific, we address it with feelings of vexation. The approach taken by Ruscio and
colleagues in their assessment of the performance of the taxometric procedures, is to
conduct Monte Carlo simulations. And though it can, potentially, upon complicated
quantitative issues, shed light, this potential is realized only if the Monte Carlo study is
grounded in a bed of technical detail and theory, sufficient to allow for meaningful
interpretation of the results produced.
Unfortunately, technical detail and theory is all but absent from the work of
Ruscio and colleagues. In particular, the precision inherent to the specification of latent
structures is even lower- the uncertainty over the latent properties in play, even greaterthan that encountered in the work of Meehl and colleagues. In our view, this lack of
precision renders the Monte Carlo simulations presented in the papers of Ruscio and
colleagues virtually meaningless; there is simply no way to ascertain what the results
say about the performance of the taxometric procedures in respect the detection of the
elements of particular classes of latent structures.
Taxonic latent structure
36
Perhaps this is not surprising, for it is badly distorted, the scientific metatheory
in which is couched the consideration offered by Ruscio and colleagues of the
taxometric procedures; consonant neither with Meehl, nor the ERC, nor the forensic
researcher's understanding of the taxonic detection problem- at least, say, as manifest in
the paper by Hare and Neuman (2008); nor, finally, with the very logic on which are
founded, all operational latent variable technologies.
Throughout the work of Ruscio and colleagues, the application of the taxometric
procedures is portrayed as a model fitting exercise, the aim being to proclaim as winner,
from amongst a slate of candidates, a most adequate model31. However, the application
of a latent variable technology is not a modelling exercise. A model represents, to some
degree of adequacy, an extant something that can be independently identified (for which
there is present in the language, an independent criterion of identity). A latent
structure, then, is not a model, for it doesn’t represent anything. It is, instead, a possibly
extant unobservable.
The fact of a latent structure's unobservability implies that decision-making as to
whether it does or does not exist (whether it is or is not present) is a detection problem.
Latent variable technologies do not yield models; they yield decisions as to whether are
present- underlying, in a particular population, a particular set of variables- certain,
particular, latent structures; consequently, are detection protocols. The possible state of
nature Σ X =
'+ that is focal in linear factor analytic technology, for example, is not a
statement of representation; it is neither a model for data, nor for covariance matrix. It
31
See Maraun (2003) for a comprehensive sustaining of this verdict.
Taxonic latent structure
37
is, rather, a testable (observable) consequence (deduced in mathematics) of a set of
indicators having, as an unobservable latent structure, the linear factor structure. One
constructs a representation or model for an extant something; but one does not
construct, for a set of indicators, a particular latent structure. Instead, a set of indicators
either does, or does not, have, as a latent structure, some particular, nominated, latent
structure. Whether it does or does not, must be revealed through the employment of
appropriate detection protocols (latent variable technologies).
Ruscio and Kaczetow (2009, p.263) state, correctly, that the aim inherent to model
construction is “..to represent something real and simplify it in a useful way.” But the
taxonic detection problem is not about representing something "real", or known to exist;
but, rather, about deciding whether there is extant, and underlying a set of variables, a
particular unobservable that goes by the name, taxonic latent structure. The complaint of
Ruscio and Kaczetow that “...if tests of a model fail to meet criteria to support an
inference of taxonic latent structure, there seems to be no way to determine whether a
model of dimensional structure provides better fit...” (2009, p. 262), is very much telling
of a failure to grasp the detection logic on which latent variable technologies are
founded32; roughly akin to complaining that, if a metal detector fails to turn up a metal
object, then, at the least, it should provide the seeker of metal with something else, a
wooden object, perhaps. That is simply not how detectors work! Every detection
32
And is, moreover, very much in keeping with the scientific zeitgeist, it amounting to
an insistence that there should be no possibility of failure; that, in the application of a
latent variable technology, the researcher should be guaranteed of some sort of success,
something with which to go forth.
Taxonic latent structure
38
protocol, including each and every latent variable technology, is founded on a true
generating proposition which links a specific class of unobservables to an observable
consequence. There would be absolutely no need to go about deducing, in
mathematics, the generating propositions (observable consequences) that are at root of
all functioning latent variable technologies, if the aim were simply to represent or
model data.
Pace Ruscio and Kaczetow (2009), Meehl's entire conception of science- his
empirical realism and the fundamental concern with unobservable causal structures,
that, from it, arises- is a rebuke to the weak, model fitting, interpretation they assign to
the employment of taxometric procedures. In marked contradistinction to the portrayal
authored by Ruscio and Kaczetow (2009), Meehl: harped on the theme that a taxonic
latent structure, should it exist, is a real entity, a "natural kind" (e.g., Meehl, 1992, p.122),
to be contrasted with an "arbitrary class" (1992, p. 122), is "...in some sense really out
there, whether human scientists identify it or not" (1992, p. 122); concerned himself with
the problem of pseudo-taxa, in which "..data sets may behave taxonically when
examined by whatever taxometric method, but that are in some sense spurious,
artifactual, not "real entities"" (e.g., 1992, p.142); explicitly assigned to the taxonic latent
structure the property of causal efficacy (e.g., 1992, p. 126); spoke of "Conjecturing the
existence of a latent taxon" (e.g., 2004, p. 42); acknowledged the oft-times usefulness of
categorization schemes arising out of pragmatic aims, say, as a product of numerical
taxonomy, but distinguished the categories thus derived from categories that are latent
taxa on the grounds that the latter "...would exist in the world as a real category if
Taxonic latent structure
39
human psychologists had not had a purpose in sorting for it" (e.g., 1992, p.123);
described the problem that the taxometric procedures were invented to address as a
detection problem (e.g., 1992, p. 129, p. 131); likened this problem to problems extant
within the natural sciences in which "...events or processes observable in principle
[cannot be] observed because of spatial or temporal remoteness from the scientist..."
(1992, p. 133); justified the search for latent taxa on the grounds that ".. if there are real
taxa in a domain, theoretical science should come to know them" (e.g., 1992, p. 161); and
dismissed cluster analysis- along with other putative competitors of the taxometric
procedures- as inappropriate for employment in the taxonic detection problem, just
because it merely represents data (e.g., 1992, p. 130).
Speculations on the issue of why LStax has not been specified. In our view,
there are two chief explanations for the failure of researchers interested in the “taxonic
detection problem”, to provide a specification of LStax. The first reason is simply that
they, and psychologists as a group, have shown little interest in mastering the logic on
which are founded the latent variable technologies that they regularly employ;
consequently, are in no position to notice, let alone do justice to, the individual
components of this logic as they come into play in specific scientific contexts. The
second reason has to do with the unique scientific metatheory in which Paul Meehl
embedded his treatment of the taxonic detection problem, and, in particular, the
comments he made on the topic of open concepts.
In the next sections we will: a) provide examples of the illogicalities and
confusions that have been engendered by the endemic failure of researchers to master
Taxonic latent structure
40
the detection logic on which are erected latent variable technologies; and b) address
Meehl's conception of open concepts.
Confusions and illogicalities.
Failure to grasp relationship between latent variable technology and latent structure.
The failure to grasp the foundational logic of latent variable technologies inculcates
profound misunderstandings about what exactly a latent variable technology can
contribute to the scientific enterprise. Perhaps most tellingly, researchers continue to
believe that latent variable technologies are general, open-ended tools of observation,
that can be employed in the service of “…studying the underlying structure…” (Vasey,
et.al., 205, p.413) of sets of variables.
Nothing could be further from the truth. Each and every latent variable
technology is a detection protocol, and, as such, rests on a generating proposition,
deduced in mathematics, that expresses a linkage between a particular class of latent
structures (the elements of which are the targets of detection) and a particular restriction
on the distribution of a set of indicators. It follows, then, that a particular latent variable
technology can support decision-making in respect the presence of only those latent
structures that are elements of the particular class named by its generating proposition.
This principle might usefully be called, the specificity principle of latent structure
detection.
Failure to grasp what latent structures are. The researcher’s failure to grasp the
foundational logic of latent variable technologies consigns him or her to a practice of
speaking in a fashion, ambiguous and nontechnical, about the focal concept of latent
Taxonic latent structure
41
structure; as when he or she takes it to mean construct, underlying variable, and the like
(e.g., “..a non-taxonic (i.e., purely dimensional) construct produces a relatively flat
line…” (Vasey et. al., 2005, p.413)). This, in the first place, means that his or her
employments of latent variable technologies- and associated programs- will be absent
the requisite specification of latent structures. But this is equivalent to employing tools
of detection in absence of an awareness of what it is that they are capable of detecting,
and renders, virtually meaningless, scientific work featuring latent variable
technologies.
In the second place, an ambiguous and nontechnical handling of the concept of
latent structure tends to generate a great deal of science-undermining confusion, three
examples of which we will now provide.
Example 1: As will be recalled, a latent structure is the intersection of a set of
latent properties, these properties, its defining characteristics. It has become endemic to
research in which latent variable technologies are employed to misportray defining
characteristics (characteristics that must be given in order that a latent structure be
specified) as assumptions (cf. Meehl, 1992, pp. 135-136), a practice the ubiquity of which
may, in fact, explain why researchers have failed to even notice that LStax remains
unspecified.
The property of linearity of conditional mean function, i.e., E(X| =θ*) = + θ*,
for example, is no more an assumption attendant to the employment of linear factor
analytic technology, than is the disposition of "electron loss" an assumption attendant to
the employment of the pulse induction metal detector. Both linearity of E(X| =θ*) and
Taxonic latent structure
42
the disposition of electron loss are defining characteristics of particular, unobservable,
targets of detection; the former of the unidimensional linear factor structure, the latter of
the metal object. One does not assume linearity of E(X| =θ*) when one employs linear
factor analytic technology. Rather, linearity of E(X| =θ*) is a constituent part of the
unidimensional linear factor structure, that particular latent structure the detection of
which is the raison d'être of linear factor analytic technology.
When the aim is to detect, in nature, the presence of metal objects, the property
of electron loss is not an assumption; it is, instead, the very basis for identifying metallic
objects, and distinguishing them from non-metallic objects. One must be is possession
of the grammatical fact that a metallic object is an object that loses electrons, in order to
deduce observable consequences of the presence of such objects.
Example 2: Many researchers who concern themselves with the taxonic
detection issue seem to believe that there exists, in respect this issue, a “problem of
nuisance covariance” (see, e.g., Guay et. al., 2007, p. 703). But this belief is as confused
as, say, the belief that the points of a triangle are nuisance features inherent to attempts to
perceive a circle. There does not exist a problem of nuisance covariance. Let us see why.
To begin, let it be the case that it has been decided that LCU- the property of
C(X| ) being diagonal and positive definite, for
={T',T}- is a defining characteristic of
some particular specification of LStax. By “nuisance covariance”, researchers mean,
then, the property of ~LCU, i.e., the property of C(X| ) being non-diagonal. Now,
what it means for ~LCU to be present in an investigation of a particular set of indicators,
is that the latent structures of X are to be found among those latent structures, the
Taxonic latent structure
43
defining characteristics of which include ~LCU. Thus, it is rank confusion to portray
~LCU as interfering with the detection of LStax. The fact of ~LCU means that LStax is not a
latent structure of X. "Nuisance covariance" is, in reality, the property ~LCU, which is a
defining characteristic of particular latent structures that do not belong to LStax.
Consequently, when present, ~LCU rules out (in nature) the presence of LStax.
Example 3: Attempts to express, in a nontechnical fashion, technical, scientific
ideas, generate incoherence with a regularity, truly astonishing. As an example,
consider the comment made by Guay et. al. (2003, p.702), to the effect that “…either or
both factors may be distributed as taxa.” This statement is incoherent. Let us see why.
Taxon is a (nontechnical) name for LStax, and LStax is a (currently unspecified)
latent structure, one of the defining characteristics of which is B (that the latent
variable
is Bernoulli distributed). Factor is, on the other hand, a name for the latent
variable that appears in the specification of LSulf. Clearly, then, it incoherent to speak of
factors as being distributed as taxa, for taxon is not a kind of distribution. But even if the
statement were altered so as to read, "...either or both factors are Bernoulli
distributed...", it would still make no sense. Factors are simply the latent variables that
appear in the specifications of linear factor structures, and, by specification, the latent
variables of linear factor structures are continuously distributed. Accordingly, they cannot
be Bernoulli distributed.
Failure to appreciate modus tollens logic. Let us pretend that, for some particular
specification of LStax, the generating propositions GMAXCOV, GMAMBAC, etc., of the
taxometric procedures happened to be true. Then, on the basis of each generating
Taxonic latent structure
44
proposition could be constructed, for LStax, a modus tollens tool of disconfirmation.
The decision-making would be as follows:
Evidence that C(Xi,Xj|Xk=x) [d(x)] {
d
E(Xi|Xj=x)} is not a single-peaked function of x
dx
decision: there does not exist an element of LStax that is a latent structure of X.
Evidence that C(Xi,Xj|Xk=x) [d(x)] {
d
E(Xi|Xj=x)} is a single-peaked function of x
dx
decision: no decision made.33
It is, however, endemic to researchers' employments of the taxometric
procedures (see, e.g., Vasey et. al., 205, pp.413-417; Guay et. al., 2007, p.703) that
evidence that C(Xi,Xj|Xk=x) [d(x)] {
d
E(Xi|Xj=x)} is a single-peaked function of x, is
dx
taken to be supportive of, or even confirmatory for, the hypothesis that there exists an
element of LStax that is a latent structure of X.
Even if it were the case that the putative generating propositions were true,
hence, that the taxometric procedures were, in fact, tools employable in the service of
detecting the elements of a specified class LStax, researchers who employed these
procedures in this confirmatory fashion would be guilty of the logical fallacy of
affirming the consequent.
The "open concept" rationalization. Throughout his published work, Meehl has
argued the case that there cannot be given a definition of taxon, for the reason that it is
33
That is to say, the fact that there exists evidence that the restriction holds in the
population under investigation does not count as evidence either for or against the
proposition “there exists an element of LStax that is a latent structure of X”.
Taxonic latent structure
45
an open concept: “Taxonicity is a nice example of the logician’s open concepts…” (Meehl,
1992, p. 128); “We accept taxonicity as an open concept” (Meehl, 1972). In our
estimation, it is a category error, Meehl’s belief that the putative status of the concept
taxon as an open concept bears on the issue of the detection of the taxonic latent
structure. Though it might, coherently, be argued that the concept of taxon is
undefinable by virtue of its openness, no such argument can, logically, be set forth as
bearing on a latent structure. For a latent structure is not a concept, but, rather, an
intersection of latent properties; moreover, a possibly extant constituent of natural
reality. A latent structure, however it be labeled, is specified through a listing of its
defining characteristics. A latent structure that is not specified, is not a latent structure;
accordingly, is not something the existence of which, in nature, could be in question;
obviously, then, is not something that could play the role of target of detection.
Candidates, generating propositions, and tools of detection.
Candidates and their generating propositions. Let {tTLS(1),...,tTLS(s)} be the s
defining properties of a particular candidate specification of LStax. We take it for
granted that our readers will agree to taking tTLS(1) to be B . Whatever other properties
be assigned to LStax, B - the property that, within population P under study, there exist
two sub-classes of individuals- is at the very heart of the notion of taxonicity. As we
have noted, the problem is that, as yet, it has not been resolved what the remaining
properties of LStax should be. The failure to specify these properties has meant that LStax
has not, as of yet, been specified; consequently, that the taxonic detection problem is
absent a target of detection. In this section, we will put forth several candidate
Taxonic latent structure
46
specifications of LStax and, for each, review what is known about generating
propositions and consequent tools of detection.
Of course, the matter of the specification of LStax, at least as it is relates to either
of Q1 or Q2, is a conceptuo-scientific issue; accordingly, must be settled by the
community of forensic researchers, rather than by the psychometrician or statistician.
The quantitative theory, herein, reviewed will have little if any relevance until have
settled the issue of the specification of LStax, forensic researchers interested in answering
either of Q1 or Q2. Our hope is that what we, herein, offer, will be suggestive of the
work that, once such a specification has been achieved, these researchers will have to
undertake.
{B ,VAL,LCU}. It has already been noted that Meehl and colleagues mention, in
their work on the taxometric procedures, a great many latent properties, and that these
properties vary paper to paper, are often mutually contradictory, and are related to the
accompanying mathematics in non-transparent ways. Though it is true that latent
structures are simply sets of latent properties, the latent structures that are, to the social
scientist, of importance, have been specified in accordance with a recipe that involves:
a) the specification of the distribution of
; b) a quantitative paraphrase of the notion
that “the indicators are causally dependent upon
dependency of the indicators on
34
”; and c) a statement of the stochastic
.34
Even under this recipe, in consequence of the latitude that exists under each of (a), (b),
and (c), a remarkable diversity of latent structures are specifiable.
Taxonic latent structure
47
The influence of this recipe is evident in the papers of Meehl and colleagues,
wherein (b) is frequently taken to be LCU, and (c), VAL35. In conjunction with B , these
choices yield candidate specification {B ,VAL, LCU}.
Now, for latent structure {B ,VAL,LCU}: i) putative generating propositions
GMAXCOV, GMAMBAC, and GMAXSLOPE are false (Maraun, Halpin, and Gabriel, 2007); ii) the
following generating proposition is true (Maraun, Slaney,and Goddyn, 2003): if there
exists an element of {B ,VAL, LCU} that is a latent structure of X, then there exists a p ×
1 vector of real numbers
ΣX =
and a p × p diagonal, positive definite matrix
, such that
'+ . 36
{B ,VAL,LCU,CN,HV}. As will be recalled, this is the latent structure under
which the data were generated in the Monte Carlo simulations of Meehl and Yonce
(1994, 1996). It is also a latent structure that Meehl rejected as a specification of LStax, he
being disinclined, reasonably enough, to bind the notion of taxonic latent structure to
distributional particulars such as conditional normality.
For {B ,VAL,LCU,CN,HV}: i) GMAXCOV is true37 (Maraun and Slaney, 2005); ii)
Because {B ,VAL,LCU,CN,HV} is a subclass of {B ,VAL,LCU}, the following
35
See, e.g., Meehl and Yonce (1994, 1996): "The core idea motivating the procedure is that,
if two observable variables ("indicators") tend to discriminate, i.e., are valid for, a latent
category ("taxon") and they do not covary otherwise (no "nuisance covariance" within
the latent classes), then any observed correlation is due solely to category mixture."
(Meehl and Yonce, 1996, p.1096).
36
It will be recalled that this restriction is also placed on the distribution of X by LSulf.
37
This is chiefly because, under CN, LCU becomes a much stronger paraphrase of the
notion of “the indicators are causally dependent upon ”, namely, the property of
p
latent conditional independence (LCI): fX = ∏ fX (read as follows: the joint distribution
j =1
j
Taxonic latent structure
48
generating proposition is true: if there exists an element of {B ,VAL,LCU,CN,HV} that
is a latent structure of X, then there exists a p × 1 vector of real numbers
diagonal, positive definite matrix
, such that Σ X =
and a p × p
'+ .
{B ,VAL,LCU,CN}. Save for the absence of the property HV, this class of latent
structures is identical to the previous one. If there exists an element of
{B ,VAL,LCU,CN} that is a latent structure of X, then: i) C(Xi,Xj|Xk=x) is either a one- or
two-peaked function of x (see Maraun and Slaney, 2005); ii) there exists a p × 1 vector of
real numbers
and a p × p diagonal, positive definite matrix
, such that Σ X =
'+ .
{B ,VAL,LCI}. In our opinion, this latent structure is far and away the most
reasonable candidate specification of LStax. In addition to the uncontroversial B , it has,
as defining properties, Meehl’s VAL property (the conditional mean of each indicator
given
is not flat-line), and the widely accepted paraphrase of causal dependency that
is LCI. Moreover, it: a) is not tied to any, always questionable, claims about the
conditional distribution of X given
; and b) contains, as special cases, all of the sub-
classes of latent structures thus far considered, the implication being that any tools of
disconfirmation derived for {B ,VAL,LCI} are also tools of disconfirmation for
{B ,VAL,LCU,CN}, {B ,VAL, LCU}, etc.
For {B ,VAL,LCI}: i) Putative generating propositions GMAXCOV, GMAMBAC, and
GMAXSLOPE are false (Maraun and Slaney, 2005; Maraun, Halpin, and Gabriel, 2007); ii)
the following generating proposition is true: if there exists an element of {B ,VAL,
of the indicators is the product of, or factors into, the marginal (individual)
X
distributions of the indicators. That is to say, holding constant renders statistically
independent the indicators).
f
Taxonic latent structure
49
LCU} that is a latent structure of X, then there exists a p × 1 vector of real numbers
and a p × p diagonal, positive definite matrix
, such that Σ X =
'+ .
Now, in the course of an intensive mathematical study of the taxometric
procedures, the first author deduced, for {B ,VAL,LCI}- on the basis of theory drawn
from the area of the Mantel-Haenszel based detection of differential item functioning
(see, e.g., Holland and Thayer, 1988)-, a rather interesting generating proposition, the
focal quantity of which, as with GMAXCOV, is a conditional covariance function.
Here is a description of what was discovered:
i.
Let there be a set of (p+2) indicators;
ii.
Partition the set of indicators into two subsets, set A consisting of two indicators
(labeled Xi and Xj) and set B consisting of the remaining p indicators;
iii.
p
Form the sum t = ∑ X k of the indicators contained in set B;
k =1
iv.
Now, if there exists an element of {B ,VAL,LCI} that is a latent structure of X,
then C(Xi,Xj|t=x) = D Tx (1- D Tx )
i
v.
=[E(Xi| =T) - E(Xi| =T')], and
i
j
j
, in which D Tx =P( =T|t=x),
=[E(Xj| =T) - E(Xj| =T')];
It can be proven (Maraun and Slaney, 2005) that, if there exists an element of
{B ,VAL,LCI} that is a latent structure of X, then, as p becomes large,
D Tx converges to
-1
⎛ (1 - Π T )
⎞
exp(ax2 + bx + c) ⎟ ,
⎜ 1+
ΠT
⎝
⎠
Taxonic latent structure
50
a function that is either one- or two-peaked38;
vi.
Thus, we have the following generating proposition: As long as the single sidecondition of large p is satisfied39, if there exists an element of
{B ,VAL,LCI} that is a latent structure of X, then C(Xi,Xj|t=x) is either one- or
two-peaked.
Tools of detection. The generating propositions we have just reviewed can be
implemented in a range of tools of detection, several of which we, now, note.
Tool 1: a modus tollens tool of disconfirmation for {B ,VAL,LCI}, {B ,VAL,LCU,CN},
{B ,VAL,LCU,CN,HV}, and {B ,VAL,LCU}. Evidence that there does not exist a p × 1
vector of real numbers
and a p × p diagonal, positive definite matrix
, such that Σ X =
'+
decision: there does not exist an element of either {B ,VAL,LCI}, {B ,VAL,LCU,CN},
{B ,VAL,LCU,CN,HV}, or {B ,VAL,LCU} that is a latent structure of X;
Evidence that there exists a p × 1 vector of real numbers
positive definite matrix
, such that Σ X =
and a p × p diagonal,
'+
decision: no decision made.
38
In which a =
2
c=
39
2
t|T
2
t|T
-
t|T'
2
(ó 2 t|T' - ó 2 t|T )
2ó 2 t|T ó 2 t|T'
2
2
t|T
2
t|T'
t|T'
, b=
(ì
t|T'
ó 2 t|T - ì
t|T
ó 2 t|T ó 2 t|T'
ó 2 t|T' )
, and
2
+ ln
t|T
2
.
t|T'
Monte Carlo work within the area of Bayesian statistics (e.g., Basawa and Rao, 1980)
indicates that the convergence will occur at approximately p=6 indicators. That set B
contains 6 or more indicators is the single assumption of our procedure.
Taxonic latent structure
51
Note: As will be remembered, linear factor analytic technology features as- for LSulf- its
primary tool of detection, the testing of the hypothesis H0: Σ X =
'+ ,
diagonal and
positive definite, a fact that contradicts Meehl’s (e.g., 2004) belief in the irrelevance of
linear factor analytic technology to the taxonic detection problem.
Tool 2: a modus tollens tool of disconfirmation for {B ,VAL,LCU,CN}. Evidence that
C(Xi,Xj|Xk=x) is neither a one- or two-peaked function of x
decision: there does not exist an element of {B ,VAL,LCU,CN} that is a latent structure
of X;
Evidence that C(Xi,Xj|Xk=x) is neither a one- nor a two-peaked function of x
decision: no decision made.
Tool 3: a modus tollens tool of disconfirmation for {B ,VAL,LCU,CN,HV}. Evidence
that C(Xi,Xj|Xk=x) is not a single-peaked function of x
decision: there does not exist an element of {B ,VAL,LCU,CN,HV} that is a latent
structure of X;
Evidence that C(Xi,Xj|Xk=x) is a single-peaked function of x
decision: no decision made.
Tool 4: a modus tollens tool of disconfirmation for {B ,VAL,LCI}. [Given satisfaction
of the side-condition of large p] Evidence that C(Xi,Xj|t=x) is neither a one- nor a twopeaked function of x
decision: there does not exist an element of {B ,VAL,LCI} that is a latent structure of X;
Evidence that C(Xi,Xj|t=x) is a single-peaked function of x
decision: no decision made.
Taxonic latent structure
52
Example (a detection protocol Dtax for LStax ≡ {B ,VAL,LCI}). Once again, let the
vector X* of indicators contain the twenty items of the PCL-R and be distributed in a
population P of offenders newly admitted to correctional services in British Columbia
and the Yukon Territory. Let us pretend that LStax has been specified as {B ,VAL,LCI},
so that, by agreement, taxonic latent structures- the targets of detection- are the
elements of this class of latent structures.
We seek, then, a detection protocol Dtax that can be employed for the purpose of
making valid decisions as to whether X* has, within P, a latent structure that is an
element of LStax, i.e., whether it has a taxonic latent structure. We could, of course,
employ Tool 4 as, in respect to {B ,VAL,LCI}, a basis for disconfirmatory decisionmaking. However, in this particular instance, there is no need for additional testing.
The reader will, no doubt, recall, from earlier in the manuscript, the test, carried out
using LISREL, of the hypothesis pair [H0: Σ X =
'+ , H1: Σ X is an arbitrary gramian
matrix]. This test can, now, be seen as an employment of Tool 1; for {B ,VAL,LCI}, a
modus tollens tool of disconfirmation. Because the decision made was to reject
H0 : Σ X =
'+ , the decision can, now, be made that there does not exist an element of
{B ,VAL,LCI} (hence, nor either of {B ,VAL,LCU,CN}, {B ,VAL,LCU,CN,HV}, or
{B ,VAL,LCU}) that is a latent structure of X*.
Fact: Because, under employment of a modus tollens tool of disconfirmation, the
decisions possible are "disconfirm" and "no decision made", the elements of a set of such
tools can never yield contradictory results. On the other hand, because the input into
each is a sample drawn from a population, there is the possibility, with respect each, of
Taxonic latent structure
53
an incorrect disconfirmation (an inferential error). This implies the truth of what Paul
Meehl has argued; to wit, that the greater the number of disconfirmations yielded by
multiple distinct Type I tools, the stronger is the disconfirmatory case made.
The Issue of Q2: “Dimensional” Latent Structures
Question Q2, the question of whether psychopathy is taxonic or dimensional,
can be stated technically as follows: In a particular population P under study, is it the
case that a particular set of psychopathy indicators X*, say the items of the PCL-R, has,
as a latent structure, something called a taxonic latent structure (an element of LStax) or,
alternatively, something called a dimensional latent structure (hereafter, an element of
LSdim)? We have already noted that researchers, to date, have not specified LStax, and
this state of affairs is sufficient to render indeterminate, Q2. However, the situation
with Q2 is even murkier, for, additionally, researchers interested in this question have
failed to specify LSdim. In the previous sections, we considered a number of candidate
specifications of LStax. We, now, do the same for LSdim; equivalently, consider the issue
of a sensible paraphrase of “dimensional latent structure.”
To begin, note that, because
contains m latent variables, it is distributed in an
m-dimensional embedding space, the reference axes of which are called dimensions.
Within its m-dimensional embedding space,
can be distributed either discretely or
continuously. In their employment of the ambiguous term “dimensional latent
structure”, forensic researchers seem to be making reference, uncertain though it is, to
the class LScon of latent structures, the set of defining characteristics of which includes
the property that
is continuously distributed. However, LScon is a vast and
Taxonic latent structure
54
enormously diverse class of latent structures. Which subclass of LScon does the forensic
researcher wish to be taken to be LSdim? Because this issue has not been settled, LSdim, at
least as it features in Q2, is unspecified.
We might speculate that the forensic researcher would desire attention to be
restricted to subclasses of LScon for which
contains but one latent variable, i.e., the
subclass LSucon of unidimensional latent structures for which
has a continuous
distribution. But even LSucon is a vast and immensely varied class of latent structures.
In addition to the subclasses LSulf, LS2clp, and LSu2plir that we have, heretofore,
encountered- each of these subclasses containing countless special cases-, there is
contained within LSucon, for example, the mth degree polynomial factor structures,
which are specified as follows: t1(mpf)
zero and variance of unity; t2(mpf)
of real numbers,
has a continuous distribution with a mean of
E(X| =θ*) =
+ f(θ*), in which
is a p × 1 vector
is a p by m matrix of real numbers, and f(θ*) contains a set of m
functions (linear, quadratic, ..., mth degree polynomial) of
, the p × p conditional covariance matrix of X given
*;
and t3(mpf)
C(X| =θ* ) =
=θ*, is diagonal and positive
definite.
The behaviours of these manifold classes of latent structures- consequently, the
detection theory required to reveal their presences in the joint distributions of
indicators- is, to say the least, widely varying. The point, then, is that in order to give
Q2 a meaning, thereby rendering it amenable to being answered as a matter of scientific
inquiry, the egregiously loose notion of “dimensional latent structure” will have to be
dispensed with in favour of a specification of LSdim; equivalently, the nomination of one
Taxonic latent structure
55
of the subclasses of LScon as the subclass to which the label “dimensional latent
structure” will be applied. It appears that a chief reason for the abject failure of
researchers to take this step has been their, and Paul Meehl’s, wildly misguided
proclivity to tacitly treat LScon as if it were pretty much the same thing as LSulf.
But, now, let us imagine that researchers take the step of providing a
specification of each of LStax and LSdim . Q2 would, then, no longer be indeterminate,
but, rather, ill-formed. The cause of this ill-formation would be that, no matter how
specified, the union of the classes LStax and LSdim would not be exhaustive of latent
structures; equivalently, it would not be the case that LSdim was equivalent to ~LStax (the
complement of the class LStax). Countless sub-classes of latent structures- e.g., the m>2
class latent profile structures-, would lie outside of the union of LStax and LSdim.
The point is that, even if the community of forensic researchers were to arrive at
a specification of both LStax and LSdim, Q2 would turn out to be a true false dichotomy; in
consequence, unanswerable. What this illustrates is that if the forensic researcher
wishes to construct a coherent version of Q2, one that is amenable to being answered
through the employment of the tools of science, logical work remains to be done.
Psychometric confusions attendant to Q2. To finish off, we would like to point
out two fundamental misconceptions of a psychometric nature that are consistently
observed in extant literature bearing on the issue of Q2, and that are likely to hinder
progress even if, in future, forensic researchers do take the step of specifying each of
LStax and LSdim.
Taxonic latent structure
56
The belief that nontaxonic latent structures yield flat conditional covariance
functions. The belief that nontaxonic latent structures yield C(Xi,Xj|Xk=x) that are flat is
patently false. Let us pretend, for a moment, that LStax has been specified, one of its
defining characteristics being the indisputable B . Then the claim "nontaxonic latent
structures yield flat C(Xi,Xj|Xk=x)" is equivalent to the claim, "all latent structures
belonging to ~LStax yield flat C(Xi,Xj|Xk=x)", a claim that is easily disproven by
counterexample (see, e.g., Maraun and Slaney, 2005; Miller, 1996).
Consider, for example, the- commonplace- unidimensional, nonlinear factor
structure specified as {t1, t2, t3}, in which: t1
contains a single, uniformly
(continuously) distributed (on [-3,3]), variable; t2
LCI; t3
the distributions of Xj,
−1
⎡
⎛ x − α j (θ ) ⎞ ⎤
j=1..3, conditional on , are logistic, i.e., fX |θ (x ) = 1 − ⎢1 + exp ⎜
⎟ ⎥ , with
j
⎜
⎟⎥
β
⎢⎣
j
⎝
⎠⎦
parameters (the α j (θ ) , conditional means, and the β j ,
3
π
times the conditional
⎡
⎤
exp(.7θ + 5)
standard deviations) as follows: ⎢α1(θ ) =
, β1 = .25⎥ ,
1 + exp(.7θ + 5)
⎣
⎦
⎡α 2(θ ) = .7θ 2 + .5θ + 1, β2 = .75⎤ , and ⎡α (θ ) = .8θ + .5, β = 1.75⎤ .
3
⎣ 3
⎦
⎣
⎦
This latent structure yields the decidedly non-flat, quadratically shaped, C(Xi,Xj|Xk=x)
shown in Figure 1.
To appreciate just how outlandish a claim it is, the claim that "nontaxonic latent
structures yield flat C(Xi,Xj|Xk=x)", and how intellectually careless are those who have
made it, the reader might simply list off a number of the subclasses of ~LStax- e.g., LSulf,
Taxonic latent structure
57
LS2clp, LSu2plir, the entire subclass of multidimensional latent structures- and note that in
order for the claim to be true, each and every latent structure contained within these
(and countless other) subclasses would have to yield a flat C(Xi,Xj|Xk=x). It is doubtful
that those who have claimed that "nontaxonic latent structures yield flat C(Xi,Xj|Xk=x)"
have investigated the behaviour of C(Xi,Xj|Xk=x) under any of these subclasses, let
alone all of them.
Now, even if the claim were narrowed so that it referred only to the subclass
LSucon ⊂ ~LStax , it would still be patently false (see, e.g., Maraun and Slaney, 2005; Miller
1996). And so, too, for that matter, if it were narrowed, even further, to refer only to
LSulf ⊂ ~LStax . For it is not the case that every latent structure that is an element of the
class of unidimensional, linear, factor structures yields a flat C(Xi,Xj|Xk=x). It is
provably the case that, for the very narrow subclass of LSulf, for which the indicators
have a multivariate normal distribution, C(Xi,Xj|Xk=x) is flat.
The belief that the taxometric procedures can be employed to distinguish
between taxonic and dimensional cases. This belief, most recently expressed in a paper
by McGrath and Walters (2012; see also Ruscio and Kaczetow, 2009) is patently false. In
the first place, as we have seen, the taxometric procedures can have implications for
neither something called a taxonic latent structure, nor a dimensional latent structure,
for the simple reason that these structures have not been specified (this lack of a
specification rendering indeterminate the putative generating propositions on which
the procedures are based).
Taxonic latent structure
58
But let us imagine, for a moment, that: a) specifications had been given for each
of LStax and LSdim; b) for these particular specifications, GMAXCOV, GMAMBAC, and
GMAXSLOPE were true; and c) the following proposition were true: If an element of LSdim is
a latent structure of X, then C(Xi,Xj|Xk=x) [d(x)] {
d
E(Xi|Xj=x)} is flat. Now, the belief
dx
that "the taxometric procedures can be employed to distinguish between taxonic and
dimensional cases" is equivalent to the belief that these generating propositions support
the following decision-making:
Evidence that C(Xi,Xj|Xk=x) [d(x)] {
d
E(Xi|Xj=x)} is a single-peaked function of x
dx
decision: there exists an element of LStax that is a latent structure of X;
Evidence that C(Xi,Xj|Xk=x) [d(x)] {
d
E(Xi|Xj=x)} is a flat
dx
decision: there exists an element of LSdim that is a latent structure of X.
And this belief is false, it containing two instances of the logical fallacy of
affirming the consequent. In fact, the generating propositions described would yield
two distinct sets of modus tollens tools of disconfirmation, one set for each of LStax and
LSdim. The decision-making that would be supported is describable as follows:
Evidence that C(Xi,Xj|Xk=x) [d(x)] {
d
E(Xi|Xj=x)} is not a single-peaked function of x
dx
decision: there does not exist an element of LStax that is a latent structure of X;
Evidence that C(Xi,Xj|Xk=x) [d(x)]{
d
E(Xi|Xj=x)} is a single-peaked function of x
dx
decision: no decision made about LStax.
Taxonic latent structure
Evidence that C(Xi,Xj|Xk=x) [d(x)] {
59
d
E(Xi|Xj=x)} is not flat
dx
decision: there does not exist an element of LSdim that is a latent structure of X;
Evidence that C(Xi,Xj|Xk=x) [d(x)]{
d
E(Xi|Xj=x)} is flat
dx
decision: no decision made about LSdim.
Thus, far from providing a basis for distinguishing between LStax and LSdim, the
decision reached in the employment of these tools might well be- as, for example, when
there exists evidence that C(Xi,Xj|Xk=x) [d(x)] {
d
E(Xi|Xj=x)} is neither single-peaked,
dx
nor flat- that there does not exist an element from either of LStax or LSdim that is a latent
structure of X.
Conclusion
It is commonly understood that the taxometric procedures provide the researcher
with a means of doing many different things, including estimating the base rate of
taxonic classes, and assigning individuals to taxonic and complement classes.
However, it makes little sense to, say, estimate the base rate of a latent taxon in the
absence of evidence that such a thing exists; i.e., that a taxonic latent structure does, in
fact, underlie a set of indicators. And just as the sine qua non of linear factor analytic
technology is the test- carried out by testing whether the observable restriction
ΣX =
'+ , , diagonal and positive definite, holds empirically- of whether a linear
factor structure underlies a set of indicators, the sine qua non of each taxometric
Taxonic latent structure
60
procedure is the test of whether something called a taxonic latent structure underlies a
set of indicators.
That is to say, the essence of each taxometric procedure is a test of whether an
observed restriction- in the case of MAXCOV, MAMBAC, and MAXSLOPE, whether
C(Xi,Xj|Xk=x), d(x), and
d
E(Xi|Xj=x), respectively, is single peaked- holds
dx
empirically. But each of these tests is a modus tollens test of disconfirmation of the
presence of something called a taxonic latent structure only if the corresponding
observable restriction is the consequent of a true material implication, the antecedent of
which is a specified set of defining latent properties of a class LStax, the elements of
which are, definitionally, taxonic latent structures. And no such a specification of LStax
has ever been made.
Consequently, the tests carried out in applications of the taxometric procedures
are not disconfirmatory tests of the presence of something called the taxonic latent
structure. As it stands, it is not possible to test for the presence of a taxonic latent
structure, because there is nothing called a taxonic latent structure (but only a label for
such a thing). The taxometric procedures, therefore, cannot play the role, in science, for
which they were invented to play; they can have no implications for the adjudication of
either of Q1 or Q2. More generally, as a consequence of the failure to specify LStax (and,
for Q2, additionally, LSdim), each of Q1 and Q2 is indeterminate; though each has the
ring of an empirical question, neither is so.
Given acceptance of the ERC conception of science, to wit, that there are in play
in nature unobservable latent structures possessing of causal potentiality, the basic
Taxonic latent structure
61
motivation that brought into existence the taxometric procedures- the development of
tools of detection that can be employed to reveal the presence of an unobservable latent
structure called the taxonic latent structure- is sound. But prior to the development of
such tools- and a requirement for each of Q1 and Q2 to have a meaning-, there must be
such a latent structure, the presence of which is at issue. Thus, for those researchers
interested in the taxonic detection problem, everything begins with a specification of
LStax.
Once researchers interested in Q1 and Q2 have agreed upon a specification of
LStax, the ultimate role that Meehl's taxometric procedures can play will be settled. If it
happens that the specification agreed upon implies the observed restriction tested by a
particular taxometric procedure, then that procedure can be employed in the service of
detecting taxonic latent structures, specified as such. Alternatively, if the agreed upon
specification does not imply the observed restriction, then the procedure can play no
role in the detection of taxonic latent structures; in this case, non-Meehlian detection
protocols (perhaps of the sort described, herein, under the heading of Tool 4) will have
to be invented.
We urge researchers interested in Q1 and Q2 to remember that the taxonic
detection problem, though pioneeringly addressed by Paul Meehl, is a free-standing
problem in latent structure detection; consequently, is in no way bound to the
employment of the taxometric procedures invented by Meehl and colleagues.
Moreover, it is the responsibility of these researchers, and not Paul Meehl's, to assign a
Taxonic latent structure
62
meaning to each of Q1 and Q2 through a specification of LStax and LSdim. Once LStax and
LSdim have been specified, the search for employable detection protocols can begin.
Taxonic latent structure
63
References
Basawa, I., & Rao, B. (1980). Statistical inference for stochastic processes. New
York:Academic Press.
Birnbaum, A. (1968). Some Latent Trait Models. In F.M. Lord & M.R. Novick (Eds.)
Statistical Theories of Mental Test Scores. Reading, Massachusetts: AddisonWesley. pp. 397-424.
Bollen, K. (2002). Latent variables in psychology and the social sciences. Annual Review
of Psychology, 53, 605-634.
Borsboom, D. (2008). Latent Variable Theory. Measurement, 6, 25-53.
Edens, J., Marcus, D., Lilienfeld, S., & Poythress, N. (2006). Psychopathic, not
psychopathic: Taxometric evidence for the dimensional structure of
psychopathy. Journal of Abnormal Psychology, 115(1), 131–144.
Everitt, B. S. (1984). An Introduction to Latent Variable Models. New York: Chapman
and Hall.
Feigl, H. (1953). Existential hypotheses: realistic versus phenomenalistic interpretations.
Philosophy of Science, 17, 36-62.
Frazier, T., Youngstrom, E., Naugle, R., Haggerty, K., & Busch, R. (2007). The latent
structure of cognitive symptom exaggeration on the Victoria Symptom Validity
Test. Archives of Clinical Neuropsychology, 22(2), 197-211.
Taxonic latent structure
64
Freedman, D. A. (1985). Statistics and the Scientific Method. In W. Mason & S. Fienberg
(Eds.) Cohort Analysis in Social Research. New York: Springer-Verlag New York
Inc.
Green, B. F. (1954). Attitude measurement. In G. Lindzey (Ed.), Handbook of social
psychology, Vol. 1. Reading, MA: Addison-Wesley. pp. 335 – 369.
Guay, J. P., & Knight, R. A. (2003, July). Assessing the Underlying Structure of
Psychopathy Factors using Taxometrics. Poster presented at the Developmental
and Neuroscience Perspectives on Psychopathy conference, Madison, Wisconsin.
Guay, J. P., Ruscio, J., Knight, R. A., & Hare, R. D. (2007). A taxometric analysis of the
latent structure of psychopathy: Evidence for dimensionality. Journal of
Abnormal Psychology, 116(4), 701-716.
Hare, R. D. (1991). The Psychopathy Checklist-Revised manual. Toronto, Canada: MultiHealth Systems.
Hare, R. D., & Neumann, C. S. (2008). Psychopathy as a clinical and empirical construct.
Annual Review of Clinical Psychology, 4, 217-246.
Hempel, K. (1958). The theoretician's dilemma: A study in the logic of theory
construction, in C. G. Hempel, 1965, Aspects of Scientific Explanation and Other
Essays. New York: The Free Press.
Holland, P. W., & Thayer, D. T. (1988). Differential item functioning and the MantelHaenszel procedure. In H. Wainer & H. I. Braun (Eds.), Test validity (pp. 129-145).
Hillsdale, NJ: Erlbaum.
Taxonic latent structure
65
Holland, P. (1990). On the sampling theory foundations of item response theory models.
Psychometrika, 55, 577-601.
Horan, W. P., Blanchard, J. J., Gangestad, S. W., & Kwapil, T. R. (2004). The
psychometric detection of schizotypy: Do putative schizotypy indicators identify
the same latent class? Journal of Abnormal Psychology, 113, 339-357.
Joreskog, K., & Sorbom, D. (1993). LISREL 8: User’s guide. Chicago: Scientific
Software.
Kanchev, I. (2005).
http://www.igkelectronics.com/theory_Pulse_induction_metal_detectors
_by_iliya_kanchev.html
Lazarsfeld, P.F. (1950). The logical and mathematical structure of latent structure
analysis. In S.A. Stouffer, L. Guttman, E.A. Suchman, P.F. Lazarsfeld,
S.A.Star, and J.A. Clausen (Eds.), Measurement and Prediction. NewYork:
John Wiley & Sons, Inc.
Lazarsfeld, P.F. (1959). Latent structure analysis. In S. Koch (Ed.), Psychology: A
study of a science, Vol.3. Toronto: McGraw-Hill Book Co. pp. 476-543.
Lord, F. (1952). A Theory of Test Scores (Psychometric Monograph No. 7). Richmond,
VA: Psychometric Corporation.
Lord, F. (1953). The relation of test score to the trait underlying the test.
Educational and Psychological Measurement, 13, 517-548.
Lord, F., & Novick, M. (1968). Statistical Theories of Mental Test Scores. London:
Addison-Wesley Publishing Company.
Taxonic latent structure
66
Mardia, K., Kent, J., and Bibby J. (1980). Multivariate Analysis. Academic Press.
Maraun, M. (2003). Myths and confusions: Psychometrics and the latent variable model.
Unpublished Manuscript.
Maraun, M., Halpin, P., & Gabriel, S. (2007). On the Detection of Latent Structures:An
Analytic consideration of Meehl’s MAXCOV, MAMBAC, and MAXSLOPE
procedures. Unpublished manuscript.
http://www.sfu.ca/~maraun/Microsoft%20Word%20-%20taxlogicrev.pdf
Maraun, M., & Slaney, K. (2005). An analysis of Meehl’s MAXCOV-HITMAX procedure
for the case of continuous indicators. Multivariate Behavioral Research, 40(4),
489-518.
Maraun, M., Slaney, K., & Gabriel, S. (2009). The Augustinian methodological family of
psychology. New Ideas in Psychology, 27, 148-162.
Maraun, M., Slaney, K., & Goddyn, L. (2003). An analysis of Meehl's MAXCOVHITMAX procedure for the case of dichotomous items. Multivariate Behavioral
Research, 38(1), 81-112.
Maraun, M. (2016). The object detection logic of latent variable technologies. Quality
and Quantity.
McGrath, R. E., & Walters, G. D. (2012). Taxometric analysis as a general strategy for
distinguishing categorical from dimensional latent structure. Psychological
Methods, 17(2), 284-293.
Meehl, P. E. (1965). Seer over sign: The first good example. Journal of Experimental
Research in Personality, 1, 27-32.
Taxonic latent structure
67
Meehl, P. E. (1968). Detecting latent clinical taxa, II: A simplified procedure, some
additional hitmax cut locators, a single-indicator method, and miscellaneous
theorems (Report No. PR-68-4). Minneapolis: University of Minnesota, Research
Laboratories of the Department of Psychiatry.
Meehl, P. E. (1972). Specific genetic etiology, psychodynamics and therapeutic nihilism.
International Journal of Mental Health, 1, 10-27.
Meehl, P. E. (1973). Psychodiagnosis: Selected papers. Minneapolis, MN: University of
Minnesota Press.
Meehl, P. E. (1992). Factors and taxa, traits and types, differences of degree and
differences in kind. Journal of Personality, 60, 117-174.
Meehl, P. E. (1995). Bootstraps taxometrics: Solving the classification problem in
psychopathology. American psychologist, 50(4), 266-275.
Meehl, P. E. (2004). What's in a taxon? Journal of Abnormal Psychology, 113, pp. 39-43.
Meehl, P. E., and Golden, R. (1982). Taxometric methods. In P. Kendall & J. Butcher
(Eds.), Handbook of research methods in clinical psychology (pp. 127-181). New
York: Wiley.
Meehl, P. E., & Yonce, L. J. (1994). Taxometric analysis: I. Detecting taxonicity with two
quantitative indicators using means above and below a sliding cut (MAMBAC
procedure). Psychological Reports, 74(3, Pt 2), 1059-1274.
Meehl, P. E., & Yonce, L. J. (1996). Taxometric analysis: II. Detecting taxonicity using
covariance of two quantitative indicators in successive intervals of a third
Taxonic latent structure
68
indicator (MAXCOV PROCEDURE). Psychological reports, 78(3, pt 2), 1091-1227,
Monograph Supplement.
Miller, M. (1996). Limitations of Meehl’s MAXCOV-HITMAX procedure. American
Psychologist, 51, 554-556.
Ruscio, J., & Kaczetow, W. (2009). Differentiating categories and dimensions: Evaluating
the robustness of taxometric analyses. Multivariate Behavioral Research, 44(2),
259-280.
Sellars, W. (1963). Empiricism and the philosophy of mind, in Wilfrid Sellars, Science,
Perception and Reality. London: Routledge & Kegan Paul.
Skilling, T. A., Harris, G. T., Rice, M. E., & Quinsey, V. L. (2002). Identifying persistently
antisocial offenders using the Hare Psychopathy Checklist and the DSM
antisocial personality disorder criteria. Psychological Assessment, 14, 27–38.
Steiger, J. H., & Lind, J. C. (1980). Statistically-based tests for the number of common
factors. Paper presented at the annual Spring Meeting of the Psychometric
Society in Iowa City. May 30, 1980.
Strauss, B. (1999). Latent trait and latent class models. International Journal of Sport
Psychology, 30(1), 17-40.
Strong, D. R., Greene, R. L., & Schinka, J. A. (2000). A taxometric analysis of MMPI-2
infrequency scales [F and F(p)] in clinical settings. Psychological Assessment, 12,
166-173.
Thurstone, L. L. (1947). Multiple Factor Analysis. Chicago: The University of Chicago
Press.
Taxonic latent structure
69
Vasey, M., Kotov, R., Frick, P., & Loney, B. (2005). The latent structure of psychopathy
in youth: A taxometric investigation. Journal of Abnormal Child Psychology,
33(4), 411-429.
Waller, N. G., & Meehl, P. E. (1998). Multivariate taxometric procedures: Distinguishing
types from continuua. Thousand Oaks California: SAGE publications.
Walters, G.D., Berry, D.T.R., Rogers, R., Payne, J.W. & Granacher, R.P. (2009). Feigned
neurocognitive deficit: Taxon or dimension? Journal of Clinical and Experimental
Neuropsychology, 31, 584-593.
Walters, G. D., Rogers, R., Berry, D. T. R., Miller, H. A., Duncan, S. A., McCusker, P. J.,
et al. (2008). Malingering as a categorical or dimensional construct: The latent
structure of feigned psychopathology as measured by the SIRS and MMPI-2.
Psychological Assessment, 20, 238–247.
Walters, G., Duncan, S., & Mitchell-Perez, K. (2007). The latent structure of
psychopathy: A taxometric investigation of the Psychopathy Checklist-Revised
in a heterogeneous sample of male prison inmates. Assessment, 14, 270-278.
Wansbeek, T., & Meijer, E. (2000). Measurement error and and latent variables in
econometrics. Amsterdam: Elsevier.
Taxonic latent structure
70
Table 1
1a) Sample Covariance Matrix of 20 PCL-R Items
-------------------------------------------------------------------------------------------------------------------------------------------Item
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
-------------------------------------------------------------------------------------------------------------------------------------------1
.52
2
.31 .62
3
.21 .20 .61
4
.21 .24 .12 .51
5
.21 .20 .14 .28 .64
6
.17 .23 .09 .23 .25 .51
7
.20 .29 .20 .23 .19 .26 .62
8
.14 .19 .09 .22 .28 .36 .26 .56
9
.11 .08 .25 .14 .19 .15 .21 .15 .50
10
.14 .17 .24 .05 .06 .10 .23 .11 .19 .67
11
.08 .11 .14 .13 .20 .14 .13 .21 .09 .06 .74
12
.17 .18 .27 .20 .17 .16 .17 .13 .21 .27 .17 .68
13
.16 .20 .29 .14 .16 .18 .25 .14 .25 .28 .10 .26 .65
14
.15 .06 .33 .11 .09 .08 .17 .07 .19 .28 .11 .23 .28 .59
15
.13 .11 .20 .12 .22 .15 .24 .18 .21 .16 .15 .11 .22 .22 .47
16
.10 .18 .05 .13 .13 .24 .18 .18 .07 .08 .10 .09 .11 .06 .07 .41
17
.16 .06 .09 .11 .09 .09 .03 .07 .03 .07 .19 .11 .00 .10 .11-.01 .72
18
.17 .19 .32 .23 .14 .18 .16 .13 .25 .22 .13 .36 .25 .24 .12 .10 .10 .73
19
.13 .06 .32 .09 .07 .11 .06 .07 .22 .26 .08 .25 .21 .28 .18 .05 .17 .45 .81
20
.10 .07 .28 .12 .14 .12 .09 .14 .20 .27 .10 .27 .22 .25 .17 .09 .21 .35 .47 .77
1b) Admissible Pair [a,D] from LISREL
-------------------------------------------------------------------------------------------------------------------------------------------Item
a
D (diagonal elements)
-------------------------------------------------------------------------------------------------------------------------------------------1
.39
.37
2
.42
.45
3
.50
.36
4
.40
.35
5
.42
.47
6
.42
.34
7
.48
.40
8
.40
.40
9
.43
.33
10
.43
.50
11
.30
.65
12
.49
.45
13
.50
.40
14
.43
.40
15
.40
.32
16
.26
.34
17
.21
.68
18
.53
.45
19
.46
.61
20
.46
.56
Taxonic latent structure
Figure 1. Conditional Covariance Function of a Nonlinear Factor Structure
71