Abstract

Considered here is a hypothesis test for coefficients in change-plane

models to detect the existence of a change plane, which in practice can guide

personalized treatment recommendations. The considered test is from a class of

problems where some parameters are not identifiable under the null hypothesis.

Classic exponential average tests do not work well in practice. To overcome this, a

novel test statistic is proposed by taking the weighted average of the squared score

test statistic (WAST) over the grouping parameter’s space, which has a closed

form with an appropriate weight. The WAST significantly improves power in

practice. Asymptotic distributions of the WAST are derived under the null and

alternative hypotheses. A bootstrap method for approximating critical values

is investigated and theoretically guaranteed. Moreover, the method is extended

to the generalized estimating equation (GEE) framework and multiple change

planes. The WAST performs well in simulations, and its performance is shown

further by applying it to three medical datasets.

Information

Preprint No.SS-2025-0155
Manuscript IDSS-2025-0155
Complete AuthorsXu Liu, Jian Huang, Yong Zhou, Feipeng Zhang, Panpan Ren
Corresponding AuthorsPanpan Ren
Emailspanpanren@stu.sufe.edu.cn

References

  1. Andrews, D. W. K. (1993). Tests for parameter instability and structural change with unknown change point. Econometrica 61(4), 821–856.
  2. Andrews, D. W. K. and W. Ploberger (1994). Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62(6), 1383–1414.
  3. Andrews, D. W. K. and W. Ploberger (1995). Admissibility of the Likelihood Ratio Test When a Nuisance Parameter is Present Only Under the Alternative. The Annals of Statistics 23(5), 1609 – 1629.
  4. Antoniou, A. C., S. Casadei, and et al. (2014). Breast-cancer risk in families with mutations in palb2. New England Journal of Medicine 371(6), 497–506.
  5. Assmann, S. F., S. J. Pocock, L. E. Enos, and L. E. Kasten (2000). Subgroup analysis and other (mis)uses of baseline data in clinical trials. The Lancet 355(9209), 1064–1069.
  6. Cheng, C., X. Feng, J. Huang, and X. Liu (2022). Regularized projection score estimation of treatment effects in high-dimensional quantile regression. Statistica Sinica 32(1), 23–41.
  7. Davies, R. B. (1977). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64(2), 247–254.
  8. Davies, R. B. (2002). Hypothesis testing when a nuisance parameter is present only under the alternative: linear model case. Biometrika 89(2), 484–489.
  9. Fan, A., S. Rui, and W. Lu (2017). Change-plane analysis for subgroup detection and sample size calculation. Journal of the American Statistical Association 112(518), 769–778.
  10. Fan, C., W. Lu, R. Song, and Y. Zhou (2017). Concordance-assisted learning for estimating optimal individualized treatment regimes. Journal of the Royal Statistical Society Series B: Statistical Methodology 79(5), 1565–1582.
  11. Feng, X., X. He, and J. Hu (2011). Wild bootstrap for quantile regression. Biometrika 98(4), 995–999.
  12. Fu, Z. and Y. Hong (2019). A model-free consistent test for structural change in regression possibly with endogeneity. Journal of Econometrics 211(1), 206–242.
  13. Fu, Z., Y. Hong, and X. Wang (2023). Testing for structural changes in large dimensional factor models via discrete fourier transform. Journal of Econometrics 233(1), 302–331.
  14. Hu, B., Q. Wei, C. Zhou, M. Ju, L. Wang, L. Chen, Z. Li, M. Wei, M. He, and L. Zhao (2020). Analysis of immune subtypes based on immunogenomic profiling identifies prognostic signature for cutaneous melanoma. International Immunopharmacology 89, 107162.
  15. Hu, X., J. Huang, L. Liu, D. Sun, and X. Zhao (2021). Subgroup analysis in the heterogeneous cox model. Statistics in Medicine 40(3), 739–757.
  16. Huang, Y., J. Cho, and Y. Fong (2021). Threshold-based subgroup testing in logistic regression models in two phase sampling designs. Journal of the Royal Statistical Society Series C: Applied Statistics 70(2), 291–311.
  17. Kang, S., W. Lu, and R. Song (2017). Subgroup detection and sample size calculation with proportional hazards regression for survival data. Statistics in Medicine 36(29), 4646–4659.
  18. Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer.
  19. Kosorok, M. R. and R. Song (2007). Inference under right censoring for transformation models with a change-point based on a covariate threshold. The Annals of Statistics 35(3), 957 – 989.
  20. Lee, S., Y. Liao, M. H. Seo, and Y. Shin (2018). Oracle estimation of a change point in highdimensional quantile regression. Journal of the American Statistical Association 113(523), 1184–1194.
  21. Lee, S., M. H. Seo, and Y. Shin (2011). Testing for threshold effects in regression models. Journal of the American Statistical Association 106(493), 220–231.
  22. Li, J., Y. Li, B. Jin, and M. R. Kosorok (2021). Multithreshold change plane model: Estimation theory and applications in subgroup identification. Statistics in Medicine 40(15), 3440– 3459.
  23. Liao, M., F. Zeng, Y. Li, Q. Gao, M. Yin, G. Deng, and X. Chen (2020, July). A novel predictive model incorporating immune-related gene signatures for overall survival in melanoma patients. Scientific reports 10(1), 12462.
  24. Liu, P., Y. Li, and J. Li (2025). Change surface regression for nonlinear subgroup identification with application to warfarin pharmacogenomics data. Biometrics 81(1), ujae169.
  25. Liu, R. Y. (1988). Bootstrap procedures under some non-I.I.D. models. The Annals of Statistics 16(4), 1696 – 1708.
  26. Mammen, E. (1993). Bootstrap and wild bootstrap for high dimensional linear models. The Annals of Statistics 21(1), 255 – 285.
  27. Shen, J. and A. Qu (2020). Subgroup analysis based on structured mixedeffects models for longitudinal data. Journal of Biopharmaceutical Statistics 30(4), 607–622.
  28. Siegel, R. L., K. D. Miller, H. E. Fuchs, and A. Jemal (2021). Cancer statistics, 2021. CA: A Cancer Journal for Clinicians 71(1), 7–33.
  29. Song, R., M. R. Kosorok, and J. P. Fine (2009). On asymptotically optimal tests under loss of identifiability in semiparametric models. The Annals of Statistics 37(5A), 2409 – 2444.
  30. Su, L., W. Lu, R. Song, and D. Huang (2020). Testing and estimation of social network dependence with time to event data. Journal of the American Statistical Association 115(530), 570–582.
  31. Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large. Transactions of the American Mathematical Society 54(3), 426– 482.
  32. Wan, Q., J. Tang, H. Lu, L. Jin, Y. Su, S. Wang, Y. Cheng, Y. Liu, C. Li, and Z. Wang (2020). Six-gene-based prognostic model predicts overall survival in patients with uveal melanoma. Cancer Biomark 27(3), 343–356.
  33. Wang, J., J. Li, Y. Li, and W. K. Wong (2019). A model-based multithreshold method for subgroup identification. Statistics in medicine 38(14), 2605–2631.
  34. Wei, S. and M. R. Kosorok (2018, 10). The change-plane Cox model. Biometrika 105(4), 891–903.
  35. Wu, C. F. J. (1986). Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis. The Annals of Statistics 14(4), 1261 – 1295.
  36. Wu, R.-f., M. Zheng, and W. Yu (2016). Subgroup analysis with time-to-event data under a logistic-cox mixture model. Scandinavian Journal of Statistics 43(3), 863–878.
  37. Xiong, J., Z. Bing, and S. Guo (2019). Observed survival interval: A supplement to tcga pan-cancer clinical data resource. Cancers 11(3), 280.
  38. Yu, P. and X. Fan (2020). Threshold regression with a threshold boundary. Journal of Business and Economic Statistics.
  39. Zhang, E., Y. Chen, S. Bao, X. Hou, J. Hu, O. Y. N. Mu, Y. Song, and L. Shan (2021). Identification of subgroups along the glycolysis-cholesterol synthesis axis and the development of an associated prognostic risk model. Human genomics 15(1), 1–20.
  40. Zhang, Y., H. J. Wang, and Z. Zhu (2022). Single-index thresholding in quantile regression. Journal of the American Statistical Association 117(540), 2222–2237.
  41. Zhao, L., L. Tian, T. Cai, B. Claggett, and L.-J. Wei (2013). Effectively selecting a target population for a future comparative study. Journal of the American Statistical Association 108(502), 527–539.

Acknowledgments

The authors would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the qual-

ity of this paper. Liu’s work is supported by the National Natural Science

Foundation of China (12271329, 72331005), Guangxi Natural Science Foundation under Grant No. 2025GXNSFDA04240010, Program for Innovative

Research Team of SUFE, and the Shanghai Research Center for Data Science and Decision Technology. Zhou’s work is supported by the National

Key R&D Program of China (2021YFA1000100, 2021YFA1000101), State

Key Program of National Natural Science Foundation of China (72531003),

Natural Science Foundation of Shanghai (23JS1400500), and Shanghai Municipal Education Commission (2024AI01002). Huang’s work is supported

by the National Natural Science Foundation of China (72331005) and the

research grants from The Hong Kong Polytechnic University (P0046811,