IDEAS home Printed from https://ideas.repec.org/a/gam/jstats/v5y2022i3p53-933d917108.html
   My bibliography  Save this article

Robust Permutation Tests for Penalized Splines

Author

Listed:
  • Nathaniel E. Helwig

    (Department of Psychology, University of Minnesota, Minneapolis, MN 55455, USA
    School of Statistics, University of Minnesota, Minneapolis, MN 55455, USA)

Abstract

Penalized splines are frequently used in applied research for understanding functional relationships between variables. In most applications, statistical inference for penalized splines is conducted using the random effects or Bayesian interpretation of a smoothing spline. These interpretations can be used to assess the uncertainty of the fitted values and the estimated component functions. However, statistical tests about the nature of the function are more difficult, because such tests often involve testing a null hypothesis that a variance component is equal to zero. Furthermore, valid statistical inference using the random effects or Bayesian interpretation depends on the validity of the utilized parametric assumptions. To overcome these limitations, I propose a flexible and robust permutation testing framework for inference with penalized splines. The proposed approach can be used to test omnibus hypotheses about functional relationships, as well as more flexible hypotheses about conditional relationships. I establish the conditions under which the methods will produce exact results, as well as the asymptotic behavior of the various permutation tests. Additionally, I present extensive simulation results to demonstrate the robustness and superiority of the proposed approach compared to commonly used methods.

Suggested Citation

  • Nathaniel E. Helwig, 2022. "Robust Permutation Tests for Penalized Splines," Stats, MDPI, vol. 5(3), pages 1-18, September.
  • Handle: RePEc:gam:jstats:v:5:y:2022:i:3:p:53-933:d:917108
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2571-905X/5/3/53/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2571-905X/5/3/53/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Lauren N. Berry & Nathaniel E. Helwig, 2021. "Cross-Validation, Information Theory, or Maximum Likelihood? A Comparison of Tuning Methods for Penalized Splines," Stats, MDPI, vol. 4(3), pages 1-24, September.
    2. Freedman, David & Lane, David, 1983. "A Nonstochastic Interpretation of Reported Significance Levels," Journal of Business & Economic Statistics, American Statistical Association, vol. 1(4), pages 292-298, October.
    3. White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-838, May.
    4. Simon N. Wood, 2013. "On p-values for smooth components of an extended generalized additive model," Biometrika, Biometrika Trust, vol. 100(1), pages 221-228.
    5. Tapio Nummi & Jianxin Pan & Tarja Siren & Kun Liu, 2011. "Testing for Cubic Smoothing Splines under Dependent Data," Biometrics, The International Biometric Society, vol. 67(3), pages 871-875, September.
    6. Simon N. Wood, 2013. "A simple test for random effects in regression models," Biometrika, Biometrika Trust, vol. 100(4), pages 1005-1010.
    7. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506, October.
    8. Giampiero Marra & Simon N. Wood, 2012. "Coverage Properties of Confidence Intervals for Generalized Additive Model Components," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 39(1), pages 53-74, March.
    9. Ping Ma & Jianhua Z. Huang & Nan Zhang, 2015. "Efficient computation of smoothing splines via adaptive basis sampling," Biometrika, Biometrika Trust, vol. 102(3), pages 631-645.
    10. Cyrus J. DiCiccio & Joseph P. Romano, 2017. "Robust Permutation Tests For Correlation And Regression Coefficients," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1211-1220, July.
    11. Young‐Ju Kim & Chong Gu, 2004. "Smoothing spline Gaussian regression: more scalable computation via efficient approximation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(2), pages 337-356, May.
    12. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167, October.
    13. Ciprian Crainiceanu & David Ruppert & Gerda Claeskens & M. P. Wand, 2005. "Exact likelihood ratio tests for penalised splines," Biometrika, Biometrika Trust, vol. 92(1), pages 91-103, March.
    14. Zack W. Almquist & Nathaniel E. Helwig & Yun You, 2020. "Connecting Continuum of Care point-in-time homeless counts to United States Census areal units," Mathematical Population Studies, Taylor & Francis Journals, vol. 27(1), pages 46-58, January.
    15. Hui Zou & Trevor Hastie, 2005. "Addendum: Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(5), pages 768-768, November.
    16. Hui Zou & Trevor Hastie, 2005. "Regularization and variable selection via the elastic net," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(2), pages 301-320, April.
    17. Inyoung Kim & Noah D. Cohen & Raymond J. Carroll, 2003. "Semiparametric Regression Splines in Matched Case-Control Studies," Biometrics, The International Biometric Society, vol. 59(4), pages 1158-1169, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Marra, Giampiero & Radice, Rosalba, 2017. "Bivariate copula additive models for location, scale and shape," Computational Statistics & Data Analysis, Elsevier, vol. 112(C), pages 99-113.
    2. Giampiero Marra & Rosalba Radice & Till Bärnighausen & Simon N. Wood & Mark E. McGovern, 2017. "A Simultaneous Equation Approach to Estimating HIV Prevalence With Nonignorable Missing Responses," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 484-496, April.
    3. Lauren N. Berry & Nathaniel E. Helwig, 2021. "Cross-Validation, Information Theory, or Maximum Likelihood? A Comparison of Tuning Methods for Penalized Splines," Stats, MDPI, vol. 4(3), pages 1-24, September.
    4. Øystein Sørensen & Anders M. Fjell & Kristine B. Walhovd, 2023. "Longitudinal Modeling of Age-Dependent Latent Traits with Generalized Additive Latent and Mixed Models," Psychometrika, Springer;The Psychometric Society, vol. 88(2), pages 456-486, June.
    5. Marra Giampiero & Radice Rosalba, 2017. "A joint regression modeling framework for analyzing bivariate binary data in R," Dependence Modeling, De Gruyter, vol. 5(1), pages 268-294, December.
    6. Marra, Giampiero & Wyszynski, Karol, 2016. "Semi-parametric copula sample selection models for count responses," Computational Statistics & Data Analysis, Elsevier, vol. 104(C), pages 110-129.
    7. Wojtyś, Magorzata & Marra, Giampiero & Radice, Rosalba, 2016. "Copula Regression Spline Sample Selection Models: The R Package SemiParSampleSel," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 71(i06).
    8. Xiao Ni & Daowen Zhang & Hao Helen Zhang, 2010. "Variable Selection for Semiparametric Mixed Models in Longitudinal Studies," Biometrics, The International Biometric Society, vol. 66(1), pages 79-88, March.
    9. Longhi, Christian & Musolesi, Antonio & Baumont, Catherine, 2014. "Modeling structural change in the European metropolitan areas during the process of economic integration," Economic Modelling, Elsevier, vol. 37(C), pages 395-407.
    10. Chen, Haiqiang & Fang, Ying & Li, Yingxing, 2015. "Estimation And Inference For Varying-Coefficient Models With Nonstationary Regressors Using Penalized Splines," Econometric Theory, Cambridge University Press, vol. 31(4), pages 753-777, August.
    11. Nadja Klein & Thomas Kneib & Giampiero Marra & Rosalba Radice & Slawa Rokicki & Mark E. McGovern, 2018. "Mixed Binary-Continuous Copula Regression Models with Application to Adverse Birth Outcomes," CHaRMS Working Papers 18-06, Centre for HeAlth Research at the Management School (CHaRMS).
    12. Nott, David J., 2008. "Predictive performance of Dirichlet process shrinkage methods in linear regression," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3658-3669, March.
    13. Maike Hohberg & Francesco Donat & Giampiero Marra & Thomas Kneib, 2021. "Beyond unidimensional poverty analysis using distributional copula models for mixed ordered‐continuous outcomes," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 70(5), pages 1365-1390, November.
    14. Roel Verbelen & Katrien Antonio & Gerda Claeskens, 2018. "Unravelling the predictive power of telematics data in car insurance pricing," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 67(5), pages 1275-1304, November.
    15. Sylvie Charlot & Riccardo Crescenzi & Antonio Musolesi, 2014. "Augmented and Unconstrained: revisiting the Regional Knowledge Production Function," SEEDS Working Papers 2414, SEEDS, Sustainability Environmental Economics and Dynamics Studies, revised Aug 2014.
    16. Thomas Kneib & Nadja Klein & Stefan Lang & Nikolaus Umlauf, 2019. "Modular regression - a Lego system for building structured additive distributional regression models with tensor product interactions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(1), pages 1-39, March.
    17. Marra, Giampiero & Radice, Rosalba, 2013. "Estimation of a regression spline sample selection model," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 158-173.
    18. Ghosal, Rahul & Maity, Arnab, 2022. "A Score Based Test for Functional Linear Concurrent Regression," Econometrics and Statistics, Elsevier, vol. 21(C), pages 114-130.
    19. Feng Li & Lu Lin & Yuxia Su, 2013. "Variable selection and parameter estimation for partially linear models via Dantzig selector," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(2), pages 225-238, February.
    20. Karol Wyszynski & Giampiero Marra, 2018. "Sample selection models for count data in R," Computational Statistics, Springer, vol. 33(3), pages 1385-1412, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jstats:v:5:y:2022:i:3:p:53-933:d:917108. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.