Hostname: page-component-89b8bd64d-7zcd7 Total loading time: 0 Render date: 2026-05-05T20:07:11.820Z Has data issue: false hasContentIssue false
  • English
  • Français

Fitting Psychometric Models with Methods Based on Automatic Differentiation

Published online by Cambridge University Press:  01 January 2025

Robert Cudeck
Robert Cudeck*
Affiliation:
Ohio State University
*
Correspondence should be addressed to R. Cudeck, Psychology Department, Ohio State University, 240K Lazenby Hall, Columbus, OH 43210, USA. E-mail: cudeck.1@osu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Quantitative psychology is concerned with the development and application of mathematical models in the behavioral sciences. Over time, models have become more complex, a consequence of the increasing complexity of research designs and experimental data, which is also a consequence of the utility of mathematical models in the science. As models have become more elaborate, the problems of estimating them have become increasingly challenging. This paper gives an introduction to a computing tool called automatic differentiation that is useful in calculating derivatives needed to estimate a model. As its name implies, automatic differentiation works in a routine way to produce derivatives accurately and quickly. Because so many features of model development require derivatives, the method has considerable potential in psychometric work. This paper reviews several examples to demonstrate how the methodology can be applied.

Keywords

Estimationmodel fittingdifferentiation

Information

Type
2005 Presidential Address
Information
Psychometrika , Volume 70 , Issue 4 , December 2005 , pp. 599 - 617
Copyright
Copyright © 2005 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

From the Presidential Address delivered at the 70th Annual Meeting of the Psychometric Society, Tilburg University, The Netherlands, July 5–8, 2005.

References

Albert, P.S., Dodd, L.E. (2004). A cautionary note on the robustness of latent class models for estimating diagnostic error without a gold standard. Biometrics, 60, 427–435.CrossRefGoogle ScholarPubMed
Birkes, D., Dodge, Y. (1993). Alternative methods of regression. New York: Wiley.CrossRefGoogle Scholar
Burden, R.L., Faires, J.D. (2005). Numerical analysis (8th ed.). Belmont, CA: Thompson Brooks/Cole.Google Scholar
Chinchalkar, S. (1994). The application of automatic differentiation to problems in engineering analysis. Computer Methods in Applied Mechanics and Engineering, 118, 197–207.CrossRefGoogle Scholar
Dayton, C.M., Macready, G.B. (1988). Concomitant-variable latent-class models. Journal of the American Statistical Association, 83, 173–178.CrossRefGoogle Scholar
Donaldson, J.R., Schnabel, R.B. (1987). Computational experience with confidence regions and confidence intervals for nonlinear least squares. Technometrics, 29, 67–82.CrossRefGoogle Scholar
Dunnill, M. (2000). The Plato of Praed Street: The life and times of Almroth Wright. London: Royal Society of Medicine Press.Google Scholar
Fischer, H. (1993). Automatic differentiation and applications. In Adams, E., Kurlisch, U. (Eds.), Scientific computing with automatic result verification (pp. 105–142). San Diego, CA: Academic Press.CrossRefGoogle Scholar
Froemel, E.C. (1971). A comparison of computer routines for the calculation of the tetrachoric correlation coefficient. Psychometrika, 36, 165–173.CrossRefGoogle Scholar
Goetghebeur, E., Liinev, J., Boelaert, M., Van der Stuyft, P. (2000). Diagnostic test analyses in search of their gold standard: Latent class analyses with random effects. Statistical Methods in Medical Research, 9, 231–248.CrossRefGoogle ScholarPubMed
Griewank, A. (2000). Evaluating derivatives: Principles and techniques of algorithmic differentiation. Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar
Griewank, A., Walther, A. (2000). Algorithm 799: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM Transactions on Mathematical Software, 26, 19–45.CrossRefGoogle Scholar
Guilford, J.P., Fruchter, B. (1973). Fundamental statistics in psychology and education (5th ed.). New York: McGraw-Hill.Google Scholar
Hadgu, A., Qu, Y. (1998). A biomedical application of latent class models with random effects. Applied Statistics, 47, 603–616.Google Scholar
Hamdan, M.A. (1970). The equivalence of tetrachoric and maximum likelihood estimates of ρ in 2 × 2 tables. Biometrika, 57, 212–215.Google Scholar
Hammer, R., Hocks, M., Kulisch, U., Ratz, D. (1991). Numerical toolbox for verified computing I: Basic numerical problems. New York: Springer-Verlag.Google Scholar
Hansen, J.W., Caviness, J.S., Joseph, C. (1962). Analytic differentiation by computer. Communications of the Association for Computing Machinery, 5, 349–355.CrossRefGoogle Scholar
Hovland, P., Bischof, C., Spiegelman, D., Casella, M. (1997). Efficient derivative codes through automatic differentiation and interface contraction: An application in biostatistics. SIAM Journal on Scientific Computing, 18, 1056–1066.CrossRefGoogle Scholar
Huang, W., Zeger, S.L., Anthony, J.C., Garrett, E. (2001). Latent variable model for joint analysis of multiple repeated measures and bivariate event times. Journal of the American Statistical Association, 96, 906–914.CrossRefGoogle Scholar
Hui, S.L., Zhou, X.H. (1998). Evaluation of diagnostic tests without gold standards. Statistical Methods in Medical Research, 7, 354–370.CrossRefGoogle ScholarPubMed
Jerrell, M.E. (1997). Automatic differentiation and interval arithmetic for estimation of disequilibrium models. Computational Economics, 10, 295–316.CrossRefGoogle Scholar
Juedes, D. (1991). A taxonomy of automatic differentiation tools. In Griewank, A., Corliss, G.F. (Eds.), Automatic differentiation of algorithms: Theory, implementation, and application (pp. 315–329). Philadelphia: SIAM.Google Scholar
Kalaba, R., Tishler, A. (1984). Automatic derivative evaluation in the optimization of nonlinear models. Review of Economics and Statistics, 66, 653–660.CrossRefGoogle Scholar
Kendall, M. (1980). Multivariate analysis (2nd ed.). London: Charles Griffin.Google Scholar
Lau, T.-S. (1997). The latent class model for multiple binary screening tests. Statistics in Medicine, 16, 2283–2295.3.0.CO;2-T>CrossRefGoogle ScholarPubMed
Lewis, D. (1960). Quantitative methods in psychology. New York: McGraw-Hill.CrossRefGoogle Scholar
Pearson, K. (1900). Mathematical contributions to the theory of evolution. VII. On the correlation of characters not quantitatively measurable. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 195, 1–47.Google Scholar
Pearson, K. (1904, November 19). Antityphoid inoculation. British Medical Journal, 1432.CrossRefGoogle Scholar
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P. (1992). Numerical recipes in C: The art of scientific computing (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
Qu, Y., Tan, M., Kutner, M.H. (1996). Random effects models in latent class analysis for evaluating accuracy of diagnostic tests. Biometrics, 52, 797–810.CrossRefGoogle ScholarPubMed
Seber, G.A.F., Wild, C.J. (1989). Nonlinear regression. New York: Wiley.CrossRefGoogle Scholar
Schittkowski, K. (2002). Numerical data fitting in dynamical systems. Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
Simpson, R.J.S., & Pearson, K. (1904, November 5). Report on certain enteric fever inoculation statistics. British Medical Journal, 1243–1246.Google Scholar
Skaug, H.J. (2002). Automatic differentiation to facilitate maximum likelihood estimation in nonlinear random effects models. Journal of Computational and Graphical Statistics, 11, 458–470.CrossRefGoogle Scholar
Skaug, H.J., & Fournier, D. (2004, October). Automatic evaluation of the marginal likelihood in nonlinear hierarchical models. Unpublished research report. Bergen, Norway: Institute of Marine Research. Available at http://bemata.imr.no/.Google Scholar
Tateneni, K. (1998). Use of automatic and numerical differentiation in the estimation of asymptotic standard errors in exploratory factor analysis. Unpublished doctoral dissertation, Columbus, OH: Psychology Department, Ohio State University.Google Scholar
Vermunt, J.K. (2003). Multilevel latent class models. Sociological Methodology, 33, 213–239.CrossRefGoogle Scholar
Wengert, R.E. (1964). A simple automatic derivative evaluation program. Communications of the Association for computing Machinery, 7, 463–464.CrossRefGoogle Scholar
Wilkins, R.D. (1964). Investigation of a new analytical method for numerical derivative evaluation. Communications of the Association for Computing Machinery, 7, 465–471.CrossRefGoogle Scholar