Charting the future of Configural Frequency Analysis: The development of a statistical method

Psychology Science, Volume 45, 2003 (2), p. 217-222 Charting the future of Configural Frequency Analysis: The development of a statistical method ALEXANDER VON EYE 1, ERWIN LAUTSCH 2 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 Event 1904 first discussion of contingence tables (Pearson) 1922 combination of symptoms beyond expectation (Pfaundler & von Sehr) 1950 first discussion of the concept of configurations (Meehl) 1968 CFA proposed (Lienert) 1969 first discussion of log-linear models (Bishop & Fienberg) 1971 X 2 -test for CFA proposed (Lienert) 1971 first CFA model for two groups of variables (2-sample CFA; Lienert) 1973 Binomial test proposed for use in CFA (Krauth & Lienert) 1973 α-adjustment proposed (Krauth) 1975 first dedicated CFA software (Roeder) 1988 hierarchical log-linear models proposed as base models for CFA (von Eye) 1989 log-linear quasi-independence models proposed as a new approach to CFA (Victor) 1994 Bayesian CFA proposed (Wood, Sher, & von Eye) 1995 alternative concepts of deviation from independence discussed (von Eye, Spiel, & Rovine) 1998 discussion of the relationship between sampling schemes and the selection of CFA base models (von Eye & Schuster) 2000 use of β-error for evaluation of test performance in CFA (von Weber) 2000 Covariates introduced in CFA (Glück & von Eye) 1 Prof. Dr. Alexander von Eye, Michigan State University, Department of Psychology, 119 Snyder Hall, East Lansing, MI 48824-1117; E-mail: voneye@msu.edu 2 Prof. Dr. Dr. Erwin Lautsch, Universität Kassel, FB 5: Gesellschaftswissenschaften, Nora-Platiel-Str. 1, D-34127 Kassel; E-mail: erla@uni-kassel.de

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; Gutiérrez- Peña, & 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

220 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 χ 2 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. References 1. Bishop, Y.M.M., & Fienberg, S.E. (1969. Incomplete two-dimensional contingency tables. Biometrics, 25, 119-128. 2. Bortz, J. (2002). Interaktionsstrukturanalyse (ISA) bei nicht orthogonalen Kontingenztafeln. Vortrag auf dem G.A. Lienert Gedächtnissymposium: die Konfigurationsfrequenzanalyse in Theorie und Anwendung. Wien: Universität. 3. DuMouchel, W. (1999). Bayesian data mining in large frequency tables, with an application to the FDA spontaneous reporting system. The American Statistician, 53, 177-190. 4. Glück, J., & von Eye, A. (2000). Including covariates in Configural Frequency Analysis. Psychologische Beiträge, 42, 405-417. 5. Gutiérrez-Peña, E., & von Eye, A. (2000). A Bayesian approach to Configural Frequency Analysis. Journal of Mathematical Sociology, 24, 151-174. 6. Kieser, M., & Victor, N. (1999). Configural Frequency Analysis (CFA) revisited - A new look at an old approach. Biometrical Journal, 41, 967-983. 7. Krauth, J. (1973). Inferenzstatistischer Nachweis von Typen und Syndromen. In J. Krauth, & G.A. Lienert. KFA. Die Konfigurationsfrequenzanalyse und ihre Anwendung in Psychologie und Medizin (pp. 39-51). Freiburg: Alber. 8. Krauth, J., & Lienert, G.A. (1973). Nichtparametrischer Nachweis von Syndromen durch simultane Binomialtests. Biometrische Zeitschrift, 15, 13-20. 9. Lienert, G.A. (1969). Die Konfigurationsfrequenzanalyse als Klassifikationsmethode in der klinischen Psychologie. In M. Irle (Ed.), Bericht über den 16. Kongress der Deutschen Gesellschaft für Psychologie in Tübingen 1968 (pp. 244-255). Göttingen: Hogrefe. 10. Lienert, G.A. (1971). Die Konfigurationsfrequenzanalyse I. Ein neuer Weg zu Typen und Syndromen. Zeitschrift für Klinische Psychologie und Psychotherapie, 19, 99-115. 11. Lienert, G.A., & Krauth, J. (1973). Die Konfigurationsfrequenzanalyse V. Kontingenz- und Interaktionsstrukturanalyse multinär skalierter Merkmale. Zeitschrift für Klinische Psychologie und Psychotherapie, 21, 26-39. 12. Meehl, P.E. (1950). Configural scoring. Journal of Consulting Psychology, 14, 165-171. 13. Pearson, K. (1904). On the theory of contingency and its relation to association and normal correlation. Draper s Company Research Memoris, Biometric Series I. 14. Roeder, B. (1975). KFA-Programm. Dortmund, Pädagogische Hochschule. Unpublished software. 15. Victor, N. (1989). An alternative approach to configural frequency analysis. Methodika, 3, 61-73. 16. von Eye, A. (1988). The General Linear Model as a framework for models in Configural Frequency Analysis. Biometrical Journal, 30, 59-67.

222 A. von Eye, E. Lautsch 17. von Eye, A. (2002). The odds favor antitypes - A comparison of tests for the identification of configural types and antitypes. Methods of Psychological Research - online, 7, 1-29. 18. von Eye, A. (in press). A comparison of tests used in 2 x 2 tables and in two-sample CFA. Psychologische Beiträge. 19. von Eye, A., & Gardiner, J.C. (in preparation). Locating deviations from multivariate normality. 20. von Eye, A., & Lautsch, E. (2000). A brief history of Configural Frequency Analysis. Psychologische Beiträge, 42, 241-249. 21. von Eye, A., & Schuster, C. (1998). On the specification of models for Configural Frequency Analysis - Sampling schemes in Prediction CFA. Methods of Psychological Research - online, 3, 55-73. 22. von Eye, A., Spiel, C., & Rovine, M. J. (1995). Concepts of nonindependence in Configural Frequency Analysis. Journal of Mathematical Sociology, 20, 41-54. 23. von Weber, S. (2000). Ein Vergleich der in der KFA verwendeten Tests mittels Simulationsrechnungen. Psychologische Beiträge, 42, 260-272. 24. Wood, P. K., Sher, K., & von Eye, A. (1994). Conjugate methods in Configural Frequency Analysis. Biometrical Journal, 36, 387-410. 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)