Taxometric
presented by
Panya Sirichote
Ph.D.
(Educational Research Measurement and Statistic )
BUU. Thailand
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Types/Kinds
Human
Behavior
Unobserved
[HB]
Latent
Traits
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Construct
HB
Latent Variables
X = T + e
Indentify
Classify
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Taxonomy
• Distinguish
[Degree]
•
•
•
•
•
Identification
Nomenclature
Classification
Descriptive
Relationships
Quicke, D. L. J. (1993)
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What is a taxometric?
•Methodology
•Analytical Techniques
[Procedures]
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Numerical Taxonomy = Taxometrics
Taxon + Metric = Taxometric
Distinguish degree
Sneath, P. H. A.(1973)
• Taxa [Taxon]
• Taxonic [categories]
• Dimension [continuum]
Meehl, P. E. (1992)
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Taxometric
Procedures
Paul Meehl (1992)
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The Efficiency Indentifying
of Basic Education School
Quality : Data Envelopment,
Cluster Analysis, and
Taxometric Analysis.
Panya Sirichote
(2011)
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We will begin by considering
A Difficult Task
Waller, N.G. (2008)
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Carving Nature at Its Joints
Applied scientists are often faced with the
difficult task of determining whether data have
been sampled from a single population or from
a finite set of homogenous populations;
whether observations differ in kind or differ in
degree; or whether individual differences arise
from types or traits.
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carving nature at the joints [I]
• Natural categories or fundamental dimensions: On
carving nature at the joints and the rearticulating
of psychopathology.
• The question of whether to view psychopathology as
categorical or dimensional continues to provoke
debate. We review the many facets of this
argument. These include the pragmatics of
measurement; the needs of clinical practice; our
ability to distinguish categories from dimensions
empirically; methods of analysis appropriate to
each and how they relate; and the potential
theoretical biases associated with each approach.
We conclude that much of the debate is
misconceived in that we do not observe pathology
directly; rather, we observe its properties.
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carving nature at the joints [II]
• The same pathology can have some
properties that are most easily understood
using a dimensional conceptualization while
at the same time having other properties
that are best understood categorically. We
suggest replacing Meehl's analogy involving
qualitatively distinct species with an
alternative analogy with the "duality" of
light, a phenomenon with both wave- and
particle-like properties.
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What are traits?
Traits: factors, dimensions, continua
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Many of you are familiar with
Common methods for identifying traits
• Factor Analysis (Spearman, 1904, Thurstone, 1937).
• Item Response Theory (Lord, 1955, 1980).
• Multidimensional Scaling (Roger Sheppard, 1962).
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What are types?
Types: natural classes, natural kinds,
nonarbitrary clusters, taxa, species,
latent classes.
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Common Methods
Identifying
carvingfor
Nature
cont.‘Types’
• Parametric Mixture Models (Aitken & Rubin, 1985; McLachlan &
Basford, 1988; Pearson, 1895; Roeder, 1994).
• Cluster Analysis (e.g., Blashfield & Aldenderfer, 1988, p. 460,
*note there are over 300 varieties of cluster analysis).
• Neural Network Models for unsupervised learning (Waller, et al,
1998).
• Latent Class Analysis (Lazarsfeld & Henry, 1968).
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Taxometrics
Psychiatric nosologists are increasingly using taxometric
procedures for distinguishing
♦ types: taxa, species, entities, latent classes, natural kinds,
from.
♦ continua: dimensions, latent traits, factors.
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Taxometric Search Procedures
-Procedure that tests the hypothesis that
a set of data came from two dichotomous
groups (called taxa)
-Seeking mathematical relationships that
will exist only if the taxa exist
-These procedures have shown great
promise in the study of psychopathology
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Taxometric Procedures
•
MAXCOV-HITMAX (Meehl, 1973)
•
MAXEIG-HITMAX (Waller & Meehl, 1998)
•
L-Mode (Waller & Meehl, 1998)
•
MAMBAC (Meehl & Yonce, 1994)
•
MAXSLOPE (Grove & Meehl, 1993)
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distinguishing feature
A distinguishing feature of Taxometric
methods is that they are designed to
identify nonarbitrary types (in contrast
to many cluster analysis algorithms
which always find partitions in data).
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nonarbitrary
• adj.
: not subject to individual
determination
[syn: {nonarbitrary}, {unarbitrary}]
[ant: {arbitrary}]
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What is a nonarbitrary Type?
“An … obvious reason for the collapse of the classical
typologies in human personology is the ambiguity of the
concept of a type” (Grant Dahlstrom, 1972, p. 4).
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Definition:
Taxon
The word ‘taxon,’ like many scientific terms,
cannot be precisely defined. It has been used
contextually to imply a “type,” a “natural
category,” or a “nonarbitrary class” (Meehl,
1992, p. 120).
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Three Meanings of the term TAXA
(1) Common Sense
(2) Causal Origin
(3) Formal-Numerical
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(1) Common Sense Meaning of Taxa
The common sense or standard science usage does not claim semantic
rigor, but is satisfactory for many purposes. The familiar words are
“species,” “syndrome,” “disease entity,” “type,” “category,” “latent
class.” What they have in common across various disciplines of the life
sciences is that they intend to designate a nonarbitrary class, a natural
kind, and the user of such a taxonic expression intends to “carve nature at
its joints” (Plato). Among accepted taxa that are not in controversy, we
have such categories as chipmunks and gophers, igneous and sedimentary
rocks, cesium and rubidium.
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(2) Causal Origin Meaning of Taxa
A second meaning, refers to the causal origin of a natural kind
or nonarbitrary class. Thus, some disputes in the taxonomy of
plants and animals are approached from the standpoint of their
phyletic origin on the evolutionary tree. Disease entities in
organic medicine are specified jointly by their pathology and
etiology when both are known, and by pathology when the
etiology is as yet undiscovered (Meehl, 1973, pp. 285-287).
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(3) Formal-Numerical Meaning of Taxa
A third meaning, clearer than the first, but more outcomeneutral than the second, is a purely formal-numerical definition.
Like multiple factor analysis, multidimensional scaling, cluster
analysis, latent class analysis, and other well-known statistical
procedures, it sits loose with respect to causality and focuses on
the numerical relations among the various candidate indicators of
a conjectured taxon.
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Misconceptions & Confusions
about Taxons
Before getting into specifics, let us clarify some
Misconceptions and Confusions about
Taxonic Constructs
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Bimodality Not Required!
Bimodality is neither a necessary nor a sufficient condition for latent
taxonicity (see, e.g., Grayson, 1987; Murphy, 1964), a point that is easily
illustrated. Consider the following situation. The next figure portrays a
composite and component frequency distributions for a mixture (base
rate equals one-half) of two latent (normal) distributions of equal
variance and a mean difference of two standard deviations. Please notice
in this figure that the smoothed composite distribution is unimodal; that
is, there are no obvious discontinuities in the observed scores.
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Bimodality Not Required!
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Mix of 2 Distributions
Bimodality not sufficient
Bimodality Not Sufficient
Golden
Grayson
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Latent Distribution
B
0.8
0.6
0.4
0.0
0.2
P (keyed response)
0.3
0.2
0.0
0.1
^
fx
10 IRFs
1.0
0.4
A
-3
-1
0
1
2
3
-3
Test Characteristic Curve
D
-2
-1
0
1
2
3
Observed Distribution
0.30
Density
0.10
0.20
8
6
4
0.00
2
0
Expected Oberved Score
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C
-2
-3
-2
-1
0
1
2
3
0
2
4
6
8
10
Total Score
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Bimodality/IRT
1: Mathematical Foundations
An Intuitive look at the
Mathematical Foundations of
Multivariate Taxometrics
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A Scatterplot of Data
1 group?
2 groups?
Scatterplot of mixed data
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scatter--nontaxon
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scatter--taxon
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scatter --combined
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regression, composite
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regressions in tax/ntax
The General Covariance Mixture Theorem
The covariance of a pair of taxon indicators, say, x and y, can be expressed as
an estimable linear function of the following terms:
cov(xy) = P covt(xy) + Q covc(xy) + PQ( x t – xc )( y t – y c )
where:
cov(xy) is the covariance of x and y in the total (i.e., mixed) sample;
P is the base rate of taxon members in the total sample;
Q = 1 – P equals the base rate of complement (nontaxon) members in the total sample;
P covt(xy) is the weighted indicator covariance in the taxon (t) class;
Q covc(xy) is the weighted indicator covariance in the complement (c) class; and
PQ( x t – xc )( y t – y c ) is the weighted cross-product of the latent class mean differences.
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1a General Cov Mix Theorem
Meehl Gen Cov Mix Theorem
Meehl’s General Covariance Mixture Theorem
covxy = P covtaxon + Q covnontaxon
+ PQ( xtaxon –
xnontaxon )( y
–
taxon
y
)
nontaxon
where: P = taxon base rate, Q = 1 – P
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Reduced Form of the General Covariance Mixture Theorem
cov(xy) = PQ( xt – xc )( y t –
yc )
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Reduced form
Variances in Mixed Samples
Because the variance (var) of a variable is the covariance of the
variable with itself, thus the theorem also implies that:
var(x) = P var(xt) + Q var(xc) + PQ( xt – xc )
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or in standard score form:
( x t  xc )
var(xt )
var(
x
)
c
var(xz )  1.00  P
Q
 PQ
var(x)
var(x)
var( x)
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1b Vars in mixed samples
Correlations in Mixed Samples
(xt  xc)( y  y )
covt (xy)
cov
(
xy
)
c
c
t
r(xy)  P
Q
 PQ
sd(x)sd( y)
sd( x)sd( y)
sd( x)sd( y)
1c Corrs in mixed samples
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Q&A
NEXT TIME: will be back
Thank you.
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