Skip to main content
SpringerLink
Log in

R-optimal designs for multi-factor models with heteroscedastic errors

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

In this paper, we consider the R-optimal design problem for multi-factor regression models with heteroscedastic errors. It is shown that a R-optimal design for the heteroscedastic Kronecker product model is given by the product of the R-optimal designs for the marginal one-factor models. However, R-optimal designs for the additive models can be constructed from R-optimal designs for the one-factor models only if sufficient conditions are satisfied. Several examples are presented to illustrate and check optimal designs based on R-optimality criterion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Berger MPF, Wong WK (2009) An introduction to optimal designs for social and biomedical research. Wiley, New York

    Book  MATH  Google Scholar 

  • Biedermann S, Dette H, Woods DC (2011) Optimal design for additive partially nonlinear models. Biometrika 98:449–458

    Article  MATH  MathSciNet  Google Scholar 

  • Dette H (1997) Designing experiments with respect to ‘standardized’ optimality criteria. J R Stat Soc Ser B 59:97–110

    Article  MATH  MathSciNet  Google Scholar 

  • Dette H, Röder I (1996) Optimal product designs for multivariate regression with missing terms. Scand J Stat 23:195–208

    MATH  MathSciNet  Google Scholar 

  • Dette H, Röder I (1997) Optimal discrimination designs for multifactor experiments. Ann Stat 25:1161–1175

    Article  MATH  MathSciNet  Google Scholar 

  • Dette H, Röder I (2000) Online calculation of efficient designs for multi-factor models. Biometrical J 42:349–362

    Article  MATH  MathSciNet  Google Scholar 

  • Dette H, Trampisch M (2010) A general approach to \(D\)-optimal designs for weighted univariate polynomial regression models. J Korean Stat Soc 39:1–26

    Article  MATH  MathSciNet  Google Scholar 

  • Dette H, Wong WK (1996) Optimal Bayesian designs for models with partially specified heteroscedastic structure. Ann Stat 24:2108–2127

    Article  MATH  MathSciNet  Google Scholar 

  • Fedorov VV (1972) Theory of Optimal Experiments. Academic, New York

    Google Scholar 

  • Graßhoff U, Großmann H, Holling H, Schwabe R (2007) Design optimality in multi-factor generalized linear models in the presence of an unrestricted quantitative factor. J Stat Plan Inference 137:3882–3893

    Article  MATH  MathSciNet  Google Scholar 

  • Hoel PG (1965) Minimax designs in two dimensional regression. Ann Math Stat 36:1097–1106

    Article  MATH  MathSciNet  Google Scholar 

  • Kleinbaum DG, Kupper LL, Muller KE, Nizam A (1998) Applied regression analysis and multivariable methods. Duxbury Press, Belmont

    Google Scholar 

  • Kutner M, Nachtsheim C, Neter J, Li W (2005) Applied linear statistical models, 5th edn. McGraw-Hill/Irwin, New York

    Google Scholar 

  • Liu X, Yue R-X (2013) A note on \(R\)-optimal designs for multiresponse models. Metrika 76:483–493

    Article  MATH  MathSciNet  Google Scholar 

  • Liu X, Yue R-X, Chatterjee K (2014) A note on \(R\)-optimal designs for multi-factor models. J Stat Plan Inference 146:139–144

  • Liu X, Yue R-X, Chatterjee K (2016) Algorithmic construction of \(R\)-optimal designs for second-order response surface models. J Stat Plan Inference 178:61–69

  • Montepiedra G, Wong WK (2001) A new design criterion when heteroscedasticity is ignored. Ann Inst Stat Math 53:418–426

    Article  MATH  MathSciNet  Google Scholar 

  • Rodríguez C, Ortiz I (2005) \(D\)-optimum designs in multi-factor models with heteroscedastic errors. J Stat Plan Inference 128:623–631

    Article  MATH  MathSciNet  Google Scholar 

  • Rodríguez C, Ortiz I, Martínez I (2015) Locally and maximin optimal designs for multi-factor nonlinear models. Statistics 49:1157–1168

    Article  MATH  MathSciNet  Google Scholar 

  • Rodríguez C, Ortiz I, Martínez I (2016) \(A\)-optimal designs for heteroscedastic multifactor regression models. Commun Stat: Theory Methods 45:757–771

    Article  MATH  MathSciNet  Google Scholar 

  • Schwabe R (1996) Optimum designs for multi-factor models, lecture notes in statistics, vol 113. Springer, New York

    Book  MATH  Google Scholar 

  • Schwabe R (2000) Optimal designs for a one-way layout with covariates. J Stat Plan Inference 85:49–59

    Article  MATH  MathSciNet  Google Scholar 

  • Wong WK (1994) \(G\)-optimal designs for multifactor experiments with heteroscedastic errors. J Stat Plan Inference 40:127–133

    Article  MATH  Google Scholar 

  • Zhang C-H, Zhang SS (2014) Confidence intervals for low dimensional parameters in high dimensional linear models. J R Stat Soc Ser B 76:217–242

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Lei He & Rong-Xian Yue

  2. Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, 200234, China

    Rong-Xian Yue

Authors
  1. Lei He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rong-Xian Yue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rong-Xian Yue.

Additional information

This work was supported by NSFC Grant 11471216.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, L., Yue, RX. R-optimal designs for multi-factor models with heteroscedastic errors. Metrika 80, 717–732 (2017). https://doi.org/10.1007/s00184-017-0624-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00184-017-0624-1

Keywords

Search

Navigation