Experimental Sequential Designs for Logistic Regression Models

Diseños experimentales secuenciales para modelos logísticos de regresión

ARTURO T. DE ZAN1

1Universidad de la Sabana, Facultad de Ingeniería, Departamento de Ingeniería, Chía, Colombia. Assistant Professor. Email: arturo.de.zan@unisabana.edu.co


Abstract

When the usual hypotheses of normality and constant variance do not hold (e.g. in binomial or Bernoulli processes), the problem of choosing appropriate designs creates problems to researches when pursuing a sequential exploration of process. This paper is based on De Zan (2006), where the author proposes two criteria to evaluate design strategies, that take the amount of information as the main evaluation tool. One into account the information of the fitted model, and the other explores the information that is contained on the approximation of a set of the best conditions of factors found on a fitted model. An example of how these strategies work is also given through a simulation using R software.

Key words: Factorial Design, Response Surface Design, Sequential Design of Experiments, Generalized Linear Model, Logistic Regression, Fisher Information Matrix.


Resumen

Cuando los supuestos habituales de normalidad y varianza constante no se cumplen (e.g. en procesos de Bernoulli o binomiales), el problema de la elección de diseños adecuados ocasiona cierta dificultad a los experimentadores, especialmente cuando lo que se persigue es una exploración secuencial del proceso. Este artículo está basado en De Zan (2006), en donde se proponen dos criterios para evaluar estrategias de diseño. Una de ellas toma en cuenta la cantidad de información contenida en el modelo ajustado, mientras que la otra explora la información contenida en las mejores condiciones de experimentación encontradas en el modelo ajustado. Se desarrolla un ejemplo simulado con el paquete R acerca de cómo funcionan estas estrategias.

Palabras clave: diseño factorial, metodología de superficie de respuesta, diseño de experimentos secuenciales, modelo lineal generalizado, regresión logística, matriz de información de Fisher.


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References

1. Atkinson, A. C. (2006), Generalized Linear Models and Response Transformation, Khuri, chapter 8.

2. Box, G. E. P. & Draper, N. R. (1987), Empirical Model Building and Response Surfaces, John Wiley & Sons, New York, United States.

3. Box, G. E. P. & Draper, N. R. (2007), Response Surfaces, Mixtures and Ridge Analysis, second edn, John Wiley & Sons, New York, United States.

4. Box, G. E. P., Hunter, W. G. & Hunter, J. S. (2005), Statistics for Experimenters: Design, Innovation and Analysis, second edn, John Wiley & Sons, New York, United States.

5. Box, G. E. P. & Wilson, K. B. (1951), `On the Experimental Attainment of Optimum Conditions´, Journal of the Royal Statistical Society Series B(13), 1-45.

6. Collett, D. (2002), Modelling Binary Data, second edn, Chapman & Hall/CRC, Boca Raton, United States.

7. Cordeiro, G. M. & De Andrade Lima Neto, E. (2004), Modelos paramétricos, `16o. SINAPE´, Recife, Brasil.

8. Cox, D. R. & Reid, N. (2000), The Theory of the Design of Experiments, Chapman & Hall/CRC, Boca Raton, United States.

9. De Zan, A. T. (2006), Principios de metodología de superficie de respuesta para modelos logísticos, Ph.D. Thesis, Departament d' Estadística i Investigació Operativa, Universitat Politècnica de Catalunya, Barcelona, España.

10. Dobson, A. J. (2002), An introduction to Generalized Linear Models, second edn, Chapman & Hall/CRC, Boca Raton, United States.

11. Firth, D. (1991), Generalized Linear Models, Hinkley, et al., chapter 3.

12. Khuri, A. I. (1993), `Response Surface Methodology within the Framework of GLM´, Journal of Combinatorics Information & System Sciences 18, 193-202.

13. Khuri, A. I. (2001), `An Overview of the Use of Generalized Linear Models in Response Surface Methodology´, Nonlinear Analysis 47, 2023-2034.

14. Khuri, A. I. (2006), Response Surface Methodology and Related Topics, World Scientific Publishing, New Jersey, United States.

15. Khuri, A. I. & Cornell, J. A. (1996), Response Surfaces: Designs and Analyses, second edn, M. Dekker, New York, United States.

16. Khuri, A. I. & Mukhopadhyay, S. (2006), GLM Designs: The Dependence on Unknown Parameters Dilemma, Khuri, chapter 9.

17. Lewis, S. L., Montgomery, D. C. & Myers, R. H. (2001a), `Examples of Designed Experiments with Nonnormal Responses´, Journal of Quality Technology 33, 265-278.

18. Lewis, S. L., Montgomery, D. C. & Myers, R. H. (2001b), `Confidence Interval Coverage for Designed Experiments with GLMs´, Journal of Quality Technology 33, 279-292.

19. Lindsey, J. K. (1997), Applying Generalized Linear Models, Springer, New York, United States.

20. McCullagh, P. M. & Nelder, J. A. (1989), Generalized Linear Models, second edn, Chapman & Hall/CRC, Boca Raton, United States.

21. Montgomery, D. C. (2005), Design and Analysis of Experiments, Sixth edn, John Wiley & Sons, New York, United States.

22. Myers, R. H. (1999), `Response Surface Methodology -Current Status and Future Directions´, Journal of Quality Technology 31, 30-44.

23. Myers, R. H. & Montgomery, D. C. (2002), Response Surface Methodology: Process and Product Optimization Using Designed Experiments, second edn, John Wiley & Sons, New York, United States.

24. Myers, R. H., Montgomery, D. C. & Vining, G. G. (2001a), Generalized Linear Models: With Applications in Engineering and the Sciences, John Wiley & Sons, New York. United States.

25. Myers, R. H., Montgomery, D. C. & Vining, G. G. (2001b), Regression Analysis. Theory, Methods and Applications, Springer, New York, United States.

26. Myers, R. H., Montgomery, D. C., Vining, G. G., Borror, C. M. & Kowalski, S. M. (2004), `Response Surface Methodology: A Retrospective and Literature Survey´, Journal of Quality Technology 36, 53-77.

27. Nelder, J. A. & Wedderburn, R. W. M. (1972), `Generalized Linear Models´, Journal of the Royal Statistical Society Series A(135), 370-384.

28. Pukelsheim, F. (1993), Optimal Design of Experiments, John Wiley & Sons, New York, United States.

29. R Development Core Team, (2006), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. *http://www.R-project.org

30. Robinson, K. S. & Khuri, A. I. (2003), `Quantile Dispersion Graphs for Evaluating and Comparing Designs for Logistic Regression Models´, Computational Statistics & Data Analysis 43, 47-62.


[Recibido en septiembre de 2007. Aceptado en agosto de 2008]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv31n2a09,
    AUTHOR  = {De Zan, Arturo T.},
    TITLE   = {{Experimental Sequential Designs for Logistic Regression Models}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2008},
    volume  = {31},
    number  = {2},
    pages   = {261-291}
}