Selection of Designs in Ordinal Regression Models under Linear Predictor Misspecification
The purpose of this article is to find a method
of comparing designs for ordinal regression models using
quantile dispersion graphs in the presence of linear predictor
misspecification. The true relationship between response variable
and the corresponding control variables are usually unknown.
Experimenter assumes certain form of the linear predictor of the
ordinal regression models. The assumed form of the linear predictor
may not be correct always. Thus, the maximum likelihood estimates
(MLE) of the unknown parameters of the model may be biased due to
misspecification of the linear predictor. In this article, the uncertainty
in the linear predictor is represented by an unknown function. An
algorithm is provided to estimate the unknown function at the
design points where observations are available. The unknown function
is estimated at all points in the design region using multivariate
parametric kriging. The comparison of the designs are based on
a scalar valued function of the mean squared error of prediction
(MSEP) matrix, which incorporates both variance and bias of the
prediction caused by the misspecification in the linear predictor. The
designs are compared using quantile dispersion graphs approach.
The graphs also visually depict the robustness of the designs on the
changes in the parameter values. Numerical examples are presented
to illustrate the proposed methodology.
[1] M. A. Heise and R. H. Myers, “Optimal designs for bivariate logistic
regression,” Biometrics, vol. 52, pp. 613–624, 1996.
[2] S. S. Zocchi and A. C. Atkinson, “Optimum experimental designs for
multinomial logistics models,” Biometrics, vol. 55, pp. 437–443, 1999.
[3] S. Mukhopadhyay and A. I. Khuri, “Comparison of designs for
multivariate generalized linear models,” Journal of Statistical Planning
and Inference, vol. 138, pp. 169–183, 2008.
[4] L. Fahrmeir and G. Tutz, Multivariate Statistical Modelling Based on
Generalized Linear Models, 2nd ed. New York: Springer, 2001.
[5] G. Tutz, Regression for categorical data. Cambridge University Press,
2011, vol. 34.
[6] A. J. Adewale and X. Xu, “Robust designs for generalized linear
models with possible overdispersion and misspecified link functions,”
Computational Statistics and Data Analysis, vol. 54, pp. 875–890, 2010.
[7] A. J. Adewale and D. P. Wiens, “Robust designs for misspecified logistic
models,” Journal of Statistical Planning and Inference, vol. 139, pp.
3–15, 2009.
[8] S. N. Lophaven, H. B. Nielsen, and J. Sondergaard, “A matlab kriging
toolbox,” Technical University of Denmark, DK-2800 Kgs. Lyngby,
Denmark, Tech. Rep., 2002.
[9] T. J. Santner, B. J. Williams, and W. Notz, The Design And Analysis of
Computer Experiments. Springer-Verlag, 2003.
[10] A. Wald, “Tests of statistical hypotheses concerning several parameters
when the number of observations is large,” Trans. Amer. Math. Soc.,
vol. 54, pp. 426–482, 1943.
[1] M. A. Heise and R. H. Myers, “Optimal designs for bivariate logistic
regression,” Biometrics, vol. 52, pp. 613–624, 1996.
[2] S. S. Zocchi and A. C. Atkinson, “Optimum experimental designs for
multinomial logistics models,” Biometrics, vol. 55, pp. 437–443, 1999.
[3] S. Mukhopadhyay and A. I. Khuri, “Comparison of designs for
multivariate generalized linear models,” Journal of Statistical Planning
and Inference, vol. 138, pp. 169–183, 2008.
[4] L. Fahrmeir and G. Tutz, Multivariate Statistical Modelling Based on
Generalized Linear Models, 2nd ed. New York: Springer, 2001.
[5] G. Tutz, Regression for categorical data. Cambridge University Press,
2011, vol. 34.
[6] A. J. Adewale and X. Xu, “Robust designs for generalized linear
models with possible overdispersion and misspecified link functions,”
Computational Statistics and Data Analysis, vol. 54, pp. 875–890, 2010.
[7] A. J. Adewale and D. P. Wiens, “Robust designs for misspecified logistic
models,” Journal of Statistical Planning and Inference, vol. 139, pp.
3–15, 2009.
[8] S. N. Lophaven, H. B. Nielsen, and J. Sondergaard, “A matlab kriging
toolbox,” Technical University of Denmark, DK-2800 Kgs. Lyngby,
Denmark, Tech. Rep., 2002.
[9] T. J. Santner, B. J. Williams, and W. Notz, The Design And Analysis of
Computer Experiments. Springer-Verlag, 2003.
[10] A. Wald, “Tests of statistical hypotheses concerning several parameters
when the number of observations is large,” Trans. Amer. Math. Soc.,
vol. 54, pp. 426–482, 1943.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:73056", author = "Ishapathik Das", title = "Selection of Designs in Ordinal Regression Models under Linear Predictor Misspecification", abstract = "The purpose of this article is to find a method
of comparing designs for ordinal regression models using
quantile dispersion graphs in the presence of linear predictor
misspecification. The true relationship between response variable
and the corresponding control variables are usually unknown.
Experimenter assumes certain form of the linear predictor of the
ordinal regression models. The assumed form of the linear predictor
may not be correct always. Thus, the maximum likelihood estimates
(MLE) of the unknown parameters of the model may be biased due to
misspecification of the linear predictor. In this article, the uncertainty
in the linear predictor is represented by an unknown function. An
algorithm is provided to estimate the unknown function at the
design points where observations are available. The unknown function
is estimated at all points in the design region using multivariate
parametric kriging. The comparison of the designs are based on
a scalar valued function of the mean squared error of prediction
(MSEP) matrix, which incorporates both variance and bias of the
prediction caused by the misspecification in the linear predictor. The
designs are compared using quantile dispersion graphs approach.
The graphs also visually depict the robustness of the designs on the
changes in the parameter values. Numerical examples are presented
to illustrate the proposed methodology.", keywords = "Model misspecification, multivariate kriging,
multivariate logistic link, ordinal response models, quantile
dispersion graphs.", volume = "9", number = "12", pages = "759-8", }