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Psychology Science, Volume 45, 2003 (2), p. 217-222
Charting the future of Configural Frequency Analysis:
The development of a statistical method
ALEXANDER VON EYE1, ERWIN LAUTSCH2
1. Configural Frequency Analysis - the past
The first 35 years of the development of Configural Frequency Analysis (CFA; Lienert,
1969) were characterized by a rapid expansion of possibilities. Full of enthusiasm, researchers
developed new designs that allow one to answer increasingly specific questions. The areas of
categorical variable analysis, parametric and non-parametric statistics, significance testing,
modeling, sampling, α-protection, frequentist and Bayesian statistics, and many other
domains were combed with the goal of identifying methods, models, and techniques that could
be adopted for use in CFA. In addition, the advent of CFA triggered the development of new
methods, in particular in the areas of significance testing and α protection. Table 1 presents a
non-exhaustive time table of CFA-related innovations.
Table 1:
CFA-related innovations in the second millennium
Year
1904
1922
1950
1968
1969
1971
1971
1973
1973
1975
1988
1989
1994
1995
1998
2000
2000
1
2
Event
first discussion of contingence tables (Pearson)
combination of symptoms beyond expectation (Pfaundler & von Sehr)
first discussion of the concept of configurations (Meehl)
CFA proposed (Lienert)
first discussion of log-linear models (Bishop & Fienberg)
X2-test for CFA proposed (Lienert)
first CFA model for two groups of variables (2-sample CFA; Lienert)
Binomial test proposed for use in CFA (Krauth & Lienert)
α-adjustment proposed (Krauth)
first dedicated CFA software (Roeder)
hierarchical log-linear models proposed as base models for CFA (von Eye)
log-linear quasi-independence models proposed as a new approach to CFA (Victor)
Bayesian CFA proposed (Wood, Sher, & von Eye)
alternative concepts of deviation from independence discussed (von Eye, Spiel, & Rovine)
discussion of the relationship between sampling schemes and the selection of CFA base
models (von Eye & Schuster)
use of β-error for evaluation of test performance in CFA (von Weber)
Covariates introduced in CFA (Glück & von Eye)
Prof. Dr. Alexander von Eye, Michigan State University, Department of Psychology, 119 Snyder Hall, East
Lansing, MI 48824-1117; E-mail: [email protected]
Prof. Dr. Dr. Erwin Lautsch, Universität Kassel, FB 5: Gesellschaftswissenschaften, Nora-Platiel-Str. 1,
D-34127 Kassel; E-mail: [email protected]
218
A. von Eye, E. Lautsch
These efforts paid off greatly. CFA now belongs to the arsenal of generally accepted
methods of analysis. The method finds applications in all areas of the empirical sciences.
Empirical articles in which data are analyzed using CFA appear in the best journals. Textbooks
on CFA have been published by reputed publishers, computer programs have been published,
and CFA as a method is covered by entries in recent and upcoming encyclopedias. In other
words, CFA as a method for the exploration of cross-classifications is known to be a useful
method that is employed widely (for a brief history of CFA see von Eye & Lautsch, 2000).
2. Configural Frequency Analysis - the future
At least as important as the recognition and the use of a statistical method is its continuous
development. In the history of most methods of statistics, the presentation of a new method is
followed by a period of euphoria. During this period, the basics of the method are established,
and researchers explore fields of application. The possibilities provided by the new method
are charted. Soon, limits become apparent and misuses become known. Researchers learn that
there are optimal data characteristics for the application of a method, but that there are also
conditions under which an application is less promising. For example, data bodies may be too
small or too large, distributional characteristics may not meet requirements, or the questions
asked by researchers cannot be answered using a particular method.
In the case of CFA, the bases have been established, as can be seen from the brief time line
in Table 1. The method finds widespread application. In addition, methodologists are now in a
phase in which the characteristics of elements of CFA are examined under various conditions.
Six fields of research on the method of CFA can currently be distinguished:
1. Simulation studies that center on the behavior of statistical tests that are used to make type/
antitype decisions (von Eye, 2002; in press; von Weber, 2000); more studies are under way
(see below).
2. Studies concerning the dependency structure of tests performed in CFA. First studies
exist (Victor, 1989), in which the authors propose that there be at least 3 or 4 degrees of
freedom for each type/antitype in a cross-classification. More studies on this topic are
being undertaken (see below).
3. Studies concerning the size of tables that can be meaningfully explored using CFA.
Stimulated by a paper by duMouchel (1999), studies are being undertaken with the goal
to determine the maximum and the minimum size of tables for which CFA is a suitable
method of analysis (see below).
4. The statistical bases of CFA are being expanded. The original approach to CFA is based
on methods for the estimation of expected cell frequencies that reside in what is known
as χ2-analysis. These methods have been put in the context of hierarchical log-linear
modeling (von Eye, 1988), and in the context of the more general log-linear models of
quasi-independence (Victor, 1989; Kieser & Victor, 1999). In addition, CFA has been
reformulated as a method of Bayesian statistics (Wood, Sher, & von Eye, 1994; GutiérrezPeña, & von Eye, 2000). The earlier approaches used noninformative priors. We are
waiting for these researchers to present Bayesian CFA methods that employ different
concepts of priors.
5. First attempts exists at formulating a new version of Interaction Structure Analysis (ISA;
Lienert & Krauth, 1973) that is based on the General Linear Model instead of the General
The future of CFA
219
Log-linear Model (Bortz, 2002). These attempts are underway, and we look forward to
seeing first written reports.
6. Existing computer programs are continuously being improved. Current foci include
improved procedures of α-protection, the incorporation of estimates of β-errors, and the
automatized determination of continuity corrections (see below). First attempts have also
been made to base the estimation of expected cell frequencies on multivariate distributional
assumptions (von Eye & Gardiner, in preparation).
These six topics of further development of CFA indicate that this method not only found
broad fields of application, but it possesses great potential for further development and for
users such that an even wider range of questions can be answered, and tailored solutions are
provided for even more problems.
3. The topics of the current issue
The current Special Issue reflects the trends described for the development and application
of CFA. The contributions present (1) interesting and innovative applications of CFA, (2)
new developments of the method of CFA, and (3) discuss CFA in comparison with existing
other methods of data analysis. The contributions are grouped in two sections. The first is
applications of CFA. This section contains seven articles in which existing methods are
employed. The second section proposes developments of the method of CFA. This domain
contains eleven articles that reflect the lines of development highlighted in Section 2.
3.1 Applications of CFA
The first article in this section presents a re-analysis of data that Janke analyzed using
multiple regression methods in the years 1963 - 1966. These were the years immediately
before the first version of CFA was proposed by Lienert (1968). The authors, Janke and Ising,
show CFA-specific results and compare CFA with regression analysis.
The second article, contributed by Ising, employs CFA as a method for the detection of
genetic associations for complex diseases. This article illustrates the usefulness of CFA as an
exploratory method in case-control studies and in family-based association studies.
The third article is authored by Wagner-Menghin. This contribution centers around the
possibility of using CFA for the identification of achievement motivation types from data
collected with the Work Style test battery, a short, computer-assisted test battery.
Bäumler and Stemmler study an interesting socio-genetic hypothesis in the fourth article.
The authors ask whether mate selection in Germany 200 years ago can be retraced from
physical characteristics of athletes in the 20st century. CFA methods are used to confirm this
hypothesis.
In the fifth article, Lautsch and Thöle use data from the Shell Youth Study, 2000, to
classify and explain life concepts in adolescents. CFA is used for both goals of analysis.
On the interface of application and development of a method is the comparison of
statistical methods using empirical data. Two articles are included that address this topic. In
the first of these two, Reuter, Hüppe, Netter, and Hennig compare the methods of CFA and
of Structural Equations Modeling in the sixth article of this section. The authors conclude
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A. von Eye, E. Lautsch
that both methods, while providing congruent findings, yield non-redundant results. In the
second, Lautsch and Plichta compare CFA, correspondence analysis, and latent class analysis.
The authors conclude that these three methods complement each other in the analysis of the
structure of types.
3.2 New Developments of CFA
This section presents new and classical methodological and conceptual developments
of CFA. In the first contribution, Krauth asks whether dichotomization, a popular method
of categorizing continuous scales, is a suitable procedure that can lead to appropriate CFA
applications. Artifacts are pointed out and illustrated.
A topic that is central to the interpretability of CFA results is the dependency of CFA tests.
Krauth shows in the second article, using the base model of first order CFA, that tests in small
tables are dependent. Bounds for the percentage of possible type structures are provided.
Related to this topic is the third article which was contributed by von Weber, Lautsch, and
von Eye. The authors present conceptual and simulation results on the question of whether the
application of the first order or the zero order CFA base models is meaningful in 2 x 2 tables.
Another two simulation studies follow. The first of these articles, also authored by von
Weber, Lautsch, and von Eye, focuses on the performance of CFA tests in tables of varying
sizes. In addition, this study presents a new method for the determination of continuity
corrections that help researchers keep the α-level constant, and the study shows the magnitude
of the β-errors one faces when performing CFA. The last simulation study in this group,
presented by von Eye in the fifth article in this section, focuses on the performance of tests
used for the 2 x 2 tables of interest in 2-sample CFA. This work focuses on relative power and
on the distributional characteristics of the test statistics.
In the sixth article of this section, Lautsch and von Weber propose a new procedure for use
in CFA. This procedure uses Victor’s and Bayesian concepts of CFA. Numerical simulations
show that the procedure performs well in comparison with established procedures.
Critical notes about the coefficient of determination as applied in CFA are presented in the
seventh article, by Betzin and Bollmann-Sdorra.
Stemmler and Bingham take up the topic of how to analyze improvement scores in prepost designs. The authors propose CFA methods for analysis in the eighth article of this
section, specifically, CFA methods of group comparisons.
New methods for the analysis of change using CFA are proposed by Stemmler and von
Eye in article nine. The authors propose using marginal homogeneity models and compare the
new approach with methods of Directed CFA and Prediction CFA.
In article ten, Lautsch, von Eye, and von Weber present a comparison of currently actively
developed software programs for CFA.
This section concludes with three articles from the fundus of unpublished CFA papers. It is
well known that a large number of articles on CFA exists in draft form, but was never pursued
until publication. Three of these articles are presented here, authored by Krauth. These
articles provide the mathematical foundation of CFA. The first of the three articles deals with
Lancaster’s χ2 decomposition model as the basis for Lienert’s Association Structure Analysis.
The second article discusses the bases of methods for α protection. The third paper provides
an inferential basis for two- and multisample CFA. These three articles are of dual importance.
First, they show the mathematical bases of a method that has been discussed largely from an
The future of CFA
221
applied perspective. Second, these articles are of historical value. They show that from the
beginning of the development of CFA, the mathematical foundation of CFA as a statistical
method was discussed. Current efforts to describe the characteristics of the methods of CFA,
exemplified, for instance by Krauth’s paper on type structures or by the simulation studies in
this Special Issue, can be viewed as a continuation of the attempts to develop CFA as a method
of defensible mathematical and statistical characteristics.
Thus, this Special Issue reflects the two streams of work that characterize current work in
the domain of CFA. On the one hand, there is a large field of application. On the other hand,
there is continuous development of CFA as a method.
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Acknowledgements. About half of the articles that are included in this Special Issue are
based on the presentations that the authors made at the conference that Lautsch, Lantermann,
and von Eye had organized to commemorate the first anniversary of G.A. Lienerts death, in
Kassel, May 2002. The other articles are contributions written for this Special Issue. The editors
of this Special Issue are indebted to the authors. Their efforts result in this most attractive
Special Issue which demonstrates clearly that research with and on the method of CFA is most
active and most promising. The editors are also indebted to the G.A. Lienert Foundation for
financial support of the conference in Kassel. Finally, we would like to thank the publisher and
the editor of this journal, W. Pabst and K. Kubinger, respectively, for providing us with the
opportunity to present this exciting issue to the readership of the journal.
Alexander von Eye (East Lansing) and Erwin Lautsch (Kassel)