×

zbMATH — the first resource for mathematics

Fitting psychometric models with methods based on automatic differentiation. (English) Zbl 1306.62401
Summary: 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.

MSC:
62P15 Applications of statistics to psychology
Software:
revolve
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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. · Zbl 1274.62486
[2] Birkes, D., & Dodge, Y. (1993). Alternative methods of regression. New York: Wiley. · Zbl 0850.62528
[3] Burden, R.L., & Faires, J.D. (2005). Numerical analysis (8th ed.). Belmont, CA: Thompson Brooks/Cole. · Zbl 0671.65001
[4] Chinchalkar, S. (1994). The application of automatic differentiation to problems in engineering analysis. Computer Methods in Applied Mechanics and Engineering, 118, 197–207. · Zbl 0842.73079
[5] Dayton, C.M., & Macready, G.B. (1988). Concomitant-variable latent-class models. Journal of the American Statistical Association, 83, 173–178.
[6] Donaldson, J.R., & Schnabel, R.B. (1987). Computational experience with confidence regions and confidence intervals for nonlinear least squares. Technometrics, 29, 67–82. · Zbl 0611.62034
[7] Dunnill, M. (2000). The Plato of Praed Street: The life and times of Almroth Wright. London: Royal Society of Medicine Press.
[8] Fischer, H. (1993). Automatic differentiation and applications. In E. Adams, & U. Kurlisch, (Eds.), Scientific computing with automatic result verification (pp. 105–142). San Diego, CA: Academic Press. · Zbl 0789.65011
[9] Froemel, E.C. (1971). A comparison of computer routines for the calculation of the tetrachoric correlation coefficient. Psychometrika, 36, 165–173.
[10] 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. · Zbl 1121.62613
[11] Griewank, A. (2000). Evaluating derivatives: Principles and techniques of algorithmic differentiation. Philadelphia: Society for Industrial and Applied Mathematics. · Zbl 0958.65028
[12] 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. · Zbl 1137.65330
[13] Guilford, J.P., & Fruchter, B. (1973). Fundamental statistics in psychology and education (5th ed.). New York: McGraw-Hill. · Zbl 0266.62061
[14] Hadgu, A., & Qu, Y. (1998). A biomedical application of latent class models with random effects. Applied Statistics, 47, 603–616. · Zbl 0913.62105
[15] Hamdan, M.A. (1970). The equivalence of tetrachoric and maximum likelihood estimates of {\(\rho\)} in 2 {\(\times\)} 2 tables. Biometrika, 57, 212–215. · Zbl 0193.16704
[16] Hammer, R., Hocks, M., Kulisch, U., & Ratz, D. (1991). Numerical toolbox for verified computing I: Basic numerical problems. New York: Springer-Verlag. · Zbl 0796.65001
[17] Hansen, J.W., Caviness, J.S., & Joseph, C. (1962). Analytic differentiation by computer. Communications of the Association for Computing Machinery, 5, 349–355. · Zbl 0113.32901
[18] 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. · Zbl 0891.65016
[19] 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. · Zbl 1072.62659
[20] Hui, S.L., & Zhou, X.H. (1998). Evaluation of diagnostic tests without gold standards. Statistical Methods in Medical Research, 7, 354–370.
[21] Jerrell, M.E. (1997). Automatic differentiation and interval arithmetic for estimation of disequilibrium models. Computational Economics, 10, 295–316. · Zbl 0892.90042
[22] Juedes, D. (1991). A taxonomy of automatic differentiation tools. In A. Griewank & G.F. Corliss (Eds.), Automatic differentiation of algorithms: Theory, implementation, and application (pp. 315–329). Philadelphia: SIAM. · Zbl 0782.65029
[23] Kalaba, R., & Tishler, A. (1984). Automatic derivative evaluation in the optimization of nonlinear models. Review of Economics and Statistics, 66, 653–660. · Zbl 0518.49021
[24] Kendall, M. (1980). Multivariate analysis (2nd ed.). London: Charles Griffin. · Zbl 0519.62040
[25] Lau, T.-S. (1997). The latent class model for multiple binary screening tests. Statistics in Medicine, 16, 2283–2295.
[26] Lewis, D. (1960). Quantitative methods in psychology. New York: McGraw-Hill.
[27] 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. · JFM 32.0238.01
[28] Pearson, K. (1904, November 19). Antityphoid inoculation. British Medical Journal, 1432.
[29] 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. · Zbl 0845.65001
[30] 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. · Zbl 0875.62551
[31] Seber, G.A.F., & Wild, C.J. (1989). Nonlinear regression. New York: Wiley. · Zbl 0721.62062
[32] Schittkowski, K. (2002). Numerical data fitting in dynamical systems. Dordrecht: Kluwer Academic. · Zbl 1018.65077
[33] Simpson, R.J.S., & Pearson, K. (1904, November 5). Report on certain enteric fever inoculation statistics. British Medical Journal, 1243–1246.
[34] 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. · Zbl 04576089
[35] 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/
[36] 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.
[37] Vermunt, J.K. (2003). Multilevel latent class models. Sociological Methodology, 33, 213–239.
[38] Wengert, R.E. (1964). A simple automatic derivative evaluation program. Communications of the Association for computing Machinery,7 463–464. · Zbl 0131.34602
[39] Wilkins, R.D. (1964). Investigation of a new analytical method for numerical derivative evaluation. Communications of the Association for Computing Machinery, 7, 465–471. · Zbl 0131.34603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.