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Testing against a high-dimensional alternative in the generalized linear model: asymptotic type I error control

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  • Jelle J. Goeman
  • Hans C. van Houwelingen
  • Livio Finos

Abstract

Testing a low-dimensional null hypothesis against a high-dimensional alternative in a generalized linear model may lead to a test statistic that is a quadratic form in the residuals under the null model. Using asymptotic arguments, we show that the distribution of such a test statistic can be approximated by a ratio of quadratic forms in normal variables, for which algorithms are readily available. For generalized linear models, the asymptotic distribution shows good control of type I error for moderate to small samples, even when the number of covariates in the model far exceeds the sample size. Copyright 2011, Oxford University Press.

Suggested Citation

  • Jelle J. Goeman & Hans C. van Houwelingen & Livio Finos, 2011. "Testing against a high-dimensional alternative in the generalized linear model: asymptotic type I error control," Biometrika, Biometrika Trust, vol. 98(2), pages 381-390.
    • Handle: RePEc:oup:biomet:v:98:y:2011:i:2:p:381-390
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    File URL: http://hdl.handle.net/10.1093/biomet/asr016
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    Citations

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    Cited by:

    1. Wang, Siyang & Cui, Hengjian, 2015. "A new test for part of high dimensional regression coefficients," Journal of Multivariate Analysis, Elsevier, vol. 137(C), pages 187-203.
    2. Ma, Yingying & Lan, Wei & Wang, Hansheng, 2015. "Testing predictor significance with ultra high dimensional multivariate responses," Computational Statistics & Data Analysis, Elsevier, vol. 83(C), pages 275-286.
    3. Lan, Wei & Ding, Yue & Fang, Zheng & Fang, Kuangnan, 2016. "Testing covariates in high dimension linear regression with latent factors," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 25-37.
    4. Sardy, Sylvain & Diaz-Rodriguez, Jairo & Giacobino, Caroline, 2022. "Thresholding tests based on affine LASSO to achieve non-asymptotic nominal level and high power under sparse and dense alternatives in high dimension," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    5. Rui Wang & Xingzhong Xu, 2021. "A Bayesian-motivated test for high-dimensional linear regression models with fixed design matrix," Statistical Papers, Springer, vol. 62(4), pages 1821-1852, August.
    6. Jesse Hemerik & Jelle J. Goeman & Livio Finos, 2020. "Robust testing in generalized linear models by sign flipping score contributions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(3), pages 841-864, July.
    7. Bin Guo & Song Xi Chen, 2016. "Tests for high dimensional generalized linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(5), pages 1079-1102, November.
    8. Ningning Xu & Aldo Solari & Jelle J. Goeman, 2023. "Closed testing with Globaltest, with application in metabolomics," Biometrics, The International Biometric Society, vol. 79(2), pages 1103-1113, June.
    9. Ping-Shou Zhong & Tao Hu & Jun Li, 2015. "Tests for Coefficients in High-dimensional Additive Hazard Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(3), pages 649-664, September.
    10. Stefano Bonnini & Michela Borghesi, 2022. "Relationship between Mental Health and Socio-Economic, Demographic and Environmental Factors in the COVID-19 Lockdown Period—A Multivariate Regression Analysis," Mathematics, MDPI, vol. 10(18), pages 1-15, September.
    11. Long Qu & Tobias Guennel & Scott L. Marshall, 2013. "Linear Score Tests for Variance Components in Linear Mixed Models and Applications to Genetic Association Studies," Biometrics, The International Biometric Society, vol. 69(4), pages 883-892, December.
    12. Jiang Dandan & Sun Jianguo, 2017. "Group Tests for High-dimensional Failure Time Data with the Additive Hazards Models," The International Journal of Biostatistics, De Gruyter, vol. 13(1), pages 1-10, May.
    13. Zang, Yangguang & Zhang, Sanguo & Li, Qizhai & Zhang, Qingzhao, 2016. "Jackknife empirical likelihood test for high-dimensional regression coefficients," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 302-316.
    14. Lan, Wei & Zhong, Ping-Shou & Li, Runze & Wang, Hansheng & Tsai, Chih-Ling, 2016. "Testing a single regression coefficient in high dimensional linear models," Journal of Econometrics, Elsevier, vol. 195(1), pages 154-168.
    15. Liu, Yang & Sun, Wei & Hsu, Li & He, Qianchuan, 2022. "Statistical inference for high-dimensional pathway analysis with multiple responses," Computational Statistics & Data Analysis, Elsevier, vol. 169(C).
    16. He, Yi & Jaidee, Sombut & Gao, Jiti, 2023. "Most powerful test against a sequence of high dimensional local alternatives," Journal of Econometrics, Elsevier, vol. 234(1), pages 151-177.
    17. Yi He & Sombut Jaidee & Jiti Gao, 2020. "Most Powerful Test against High Dimensional Free Alternatives," Monash Econometrics and Business Statistics Working Papers 13/20, Monash University, Department of Econometrics and Business Statistics.

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