×

Equivariance and invariance for optimal designs in generalized linear models exemplified by a class of gamma models. (English) Zbl 1478.62217

Summary: The main intention of the present work is to outline the concept of equivariance and invariance in the design of experiments for generalized linear models and to demonstrate its usefulness. In contrast with linear models, pairs of transformations have to be employed for generalized linear models. These transformations act simultaneously on the experimental settings and on the location parameters in the linear component. Then, the concept of equivariance provides a tool to transfer locally optimal designs from one experimental region to another when the nominal values of the parameters are changed accordingly. The stronger concept of invariance requires a whole group of equivariant transformations. It can be used to characterize optimal designs which reflect the symmetries resulting from the group actions. The general concepts are illustrated by models with gamma distributed response and a canonical link. There, for a given transformation of the experimental settings, the transformation of the parameters is not unique and may be chosen to be nonlinear in order to fully exploit the model structure. In this case, we can derive invariant maximin efficient designs for the \(D\)- and the IMSE-criterion.

MSC:

62K05 Optimal statistical designs
62J12 Generalized linear models (logistic models)
62J99 Linear inference, regression

Software:

R
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atkinson, A.; Haines, L.; Ghosh, S.; Rao, C., Designs for nonlinear and generalized linear models, Design and analysis of experiments. Handbook of statistics, 437-475 (1996), Amsterdam: Elsevier, Amsterdam · Zbl 0910.62070
[2] Atkinson, AC; Donev, AN; Tobias, RD, Optimum experimental designs, with SAS (2007), Oxford: Oxford University Press, Oxford · Zbl 1183.62129
[3] Atkinson, AC; Woods, DC; Dean, A.; Morris, M.; Stufken, J.; Bingham, D., Designs for generalized linear models, Handbook of design and analysis of experiments, 471-514 (2015), Boca Raton: Chapman & Hall/CRC Press, Boca Raton · Zbl 1369.62168
[4] Burridge, J.; Sebastiani, P., D-optimal designs for generalised linear models with variance proportional to the square of the mean, Biometrika, 81, 295-304 (1994) · Zbl 0807.62055 · doi:10.1093/biomet/81.2.295
[5] Chernoff, H., Locally optimal designs for estimating parameters, Ann Math Stat, 24, 586-602 (1953) · Zbl 0053.10504 · doi:10.1214/aoms/1177728915
[6] Cox, DR, A note on design when response has an exponential family distribution, Biometrika, 75, 161-164 (1988) · Zbl 0635.62083 · doi:10.1093/biomet/75.1.161
[7] Debusho, LK; Haines, LM, D-optimal population designs for the simple linear random coefficients regression model, Annu Proc S Afr Stat Assoc Conf, 2010, 52-59 (2010)
[8] Dette H (1997) Designing experiments with respect to ‘standardized’ optimality criteria. J R Stat Soc Ser B 59:97-110 · Zbl 0884.62081
[9] Dette, H.; Hoyden, L.; Kuhnt, S.; Schorning, K., Optimal designs for thermal spraying, J R Stat Soc Ser C, 66, 53-72 (2017) · doi:10.1111/rssc.12156
[10] Dette, H.; Martinez Lopez, I.; Ortiz Rodriguez, IM; Pepelyshev, A., Maximin efficient design of experiment for exponential regression models, J Stat Plan Inference, 136, 4397-4418 (2006) · Zbl 1098.62099 · doi:10.1016/j.jspi.2005.06.006
[11] Fahrmeir, L.; Kaufmann, H., Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models, Ann Stat, 13, 342-368 (1985) · Zbl 0594.62058 · doi:10.1214/aos/1176346597
[12] Fedorov, VV; Leonov, SL, Optimal design for nonlinear response models (2013), Boca Raton: CRC Press, Boca Raton · Zbl 1373.62001 · doi:10.1201/b15054
[13] Ford, I.; Torsney, B.; Wu, CFJ, The use of a canonical form in the construction of locally optimal designs for non-linear problems, J R Stat Soc B, 54, 569-583 (1992) · Zbl 0774.62080
[14] Gaffke, N.; Idais, O.; Schwabe, R., Locally optimal designs for gamma models, J Stat Plan Inference, 203, 199-214 (2019) · Zbl 1421.62101 · doi:10.1016/j.jspi.2019.04.002
[15] Gea-Izquierdo, G.; Cañellas, I., Analysis of holm oak intraspecific competition using gamma regression, For Sci, 55, 310-322 (2009)
[16] Graßhoff, U, Holling, H, Schwabe, R (2009). On optimal design for a heteroscedastic model arising from random coeffcients, in: Proceedings of the 6th St. Petersburg Workshop on Simulation, pp. 387-392.
[17] Graßhoff, U.; Schwabe, R., Optimal design for the Bradley-Terry paired comparison model, Stat Methods Appl, 17, 275-289 (2008) · Zbl 1367.62240 · doi:10.1007/s10260-007-0058-4
[18] Grover, G.; Sabharwal, A.; Mittal, J., An application of gamma generalized linear model for estimation of survival function of diabetic nephropathy patients, Int J Stat Med Res, 2, 209-219 (2013)
[19] Heiligers, B.; Schneider, K., Invariant admissible and optimal designs in cubic regression on the v-ball, J Stat Plan Inference, 31, 113-125 (1992) · Zbl 0766.62044 · doi:10.1016/0378-3758(92)90044-S
[20] Idais, O (2021) On local optimality of vertex type designs in generalized linear models. Stat Pap 62: 1871-1898. · Zbl 1477.62200
[21] Idais, O.; Schwabe, R., Analytic solutions for locally optimal designs for gamma models having linear predictors without intercept, Metrika, 84, 1-26 (2021) · Zbl 1457.62233 · doi:10.1007/s00184-019-00760-3
[22] Imhof, L.; Wong, WK, A graphical method for finding maximin efficiency designs, Biometrics, 56, 113-117 (2000) · Zbl 1060.62544 · doi:10.1111/j.0006-341X.2000.00113.x
[23] Kiefer, J., Optimum experimental designs, J R Stat Soc B, 21, 272-304 (1959) · Zbl 0108.15303
[24] Lehmann, E., Testing statistical hypotheses (1959), New York: Wiley, New York · Zbl 0089.14102
[25] Li Y, Deng X (2018) On I-optimal designs for generalized linear models: an efficient algorithm via general equivalence theory. arXiv preprint arXiv:1801.05861
[26] Li, Y. and Deng, X. (2021), An efficient algorithm for Elastic I-optimal design of generalized linear models. Can J Statistics, 49:438-470. · Zbl 07759586
[27] McCullagh, P.; Nelder, JA, Generalized linear models (1989), London: Chapman & Hall, London · Zbl 0588.62104 · doi:10.1007/978-1-4899-3242-6
[28] Nelder, JA; Wedderburn, RWM, Generalized linear models, J R Stat Soc Ser A, 135, 370-384 (1972) · doi:10.2307/2344614
[29] Ng, VK; Cribbie, RA, Using the gamma generalized linear model for modeling continuous, skewed and heteroscedastic outcomes in psychology, Curr Psychol, 36, 225-235 (2017) · doi:10.1007/s12144-015-9404-0
[30] Pitman, E., The estimation of the location and scale parameters of a continuous population of any given form, Biometrika, 30, 3-4, 391-421 (1939) · Zbl 0020.14904 · doi:10.1093/biomet/30.3-4.391
[31] Pronzato, L.; Pázman, A., Design of experiments in nonlinear models: asymptotic normality. Optimality criteria and small-sample properties (2013), New York: Springer, New York · Zbl 1275.62026 · doi:10.1007/978-1-4614-6363-4
[32] Prus, M.; Schwabe, R., Optimal designs for the prediction of individual parameters in hierarchical models, J R Stat Soc B, 78, 175-191 (2016) · Zbl 1411.62235 · doi:10.1111/rssb.12105
[33] Pukelsheim, F., Optimal design of experiments (1993), New York: Wiley, New York · Zbl 0834.62068
[34] Radloff, M.; Schwabe, R.; Kunert, J.; Müller, CH; Atkinson, AC, Invariance and equivariance in experimental design for nonlinear models, mODa 11-advances in model-oriented design and analysis, 217-224 (2016), New York: Springer, New York · doi:10.1007/978-3-319-31266-8_25
[35] R Core Team (2020) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria
[36] Russell, KG; Woods, DC; Lewis, SM; Eccleston, JA, D-optimal designs for Poisson regression models, Stat Sin, 19, 721-730 (2009) · Zbl 1168.62367
[37] Schmidt, D.; Schwabe, R., Optimal design for multiple regression with information driven by the linear predictor, Stat Sin, 27, 1371-1384 (2017) · Zbl 1372.62030
[38] Schwabe, R., Optimum designs for multi-factor models (1996), New York: Springer, New York · Zbl 0858.62058 · doi:10.1007/978-1-4612-4038-9
[39] Silvey, SD, Optimal design (1980), London: Chapman & Hall, London · Zbl 0468.62070 · doi:10.1007/978-94-009-5912-5
[40] Sitter, RR; Torsney, B.; Kitsos, CP; Mülle, WG, D-optimal designs for generalized linear models, MODA4—advances in model-oriented data analysis, 87-102 (1995), Heidelberg: Physica-Verlag, Heidelberg · Zbl 0840.62076 · doi:10.1007/978-3-662-12516-8_9
[41] Tong, L.; Volkmer, HW; Yang, J., Analytic solutions for D-optimal factorial designs under generalized linear models, Electron J Stat, 8, 1322-1344 (2014) · Zbl 1298.62136 · doi:10.1214/14-EJS926
[42] Yang, M.; Stufken, J., Support points of locally optimal designs for nonlinear models with two parameters, Ann Stat, 37, 518-541 (2009) · Zbl 1155.62053 · doi:10.1214/07-AOS560
[43] Yang, M.; Zhang, B.; Huang, S., Optimal designs for generalized linear models with multiple design variables, Stat Sin, 21, 1415-1430 (2011) · Zbl 1223.62136 · doi:10.5705/ss.2009.115
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.