The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Robust ridge regression for nonlinear mixed effects models with applications to quantitative high throughput screening assay data

비선형 혼합효과모형에서의 로버스트 능형회귀 방법과 정량적 고속 대량 스크리닝 자료에의 응용

  • Yoo, Jiseon (Department of Applied Statistics, Chung-Ang University) ;
  • Lim, Changwon (Department of Applied Statistics, Chung-Ang University)
  • 유지선 (중앙대학교 응용통계학과) ;
  • 임창원 (중앙대학교 응용통계학과)
  • Received : 2017.11.16
  • Accepted : 2017.12.28
  • Published : 2018.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

A nonlinear mixed effects model is mainly used to analyze repeated measurement data in various fields. A nonlinear mixed effects model consists of two stages: the first-stage individual-level model considers intra-individual variation and the second-stage population model considers inter-individual variation. The individual-level model, which is the first stage of the nonlinear mixed effects model, estimates the parameters of the nonlinear regression model. It is the same as the general nonlinear regression model, and usually estimates parameters using the least squares estimation method. However, the least squares estimation method may have a problem that the estimated value of the parameters and standard errors become extremely large if the assumed nonlinear function is not explicitly revealed by the data. In this paper, a new estimation method is proposed to solve this problem by introducing the ridge regression method recently proposed in the nonlinear regression model into the first-stage individual-level model of the nonlinear mixed effects model. The performance of the proposed estimator is compared with the performance with the standard estimator through a simulation study. The proposed methodology is also illustrated using quantitative high throughput screening data obtained from the US National Toxicology Program.

비선형 혼합효과 모형은 다양한 분야에서 반복 측정 자료를 분석할 때 주로 사용된다. 비선형 혼합효과 모형은 개체 내 변동(intra-individual variation)에 대해 고려하는 제 1단계 개별수준모델(individual-level model)과 개체간 변동(inter-individual variation)에 대해 고려하는 제 2단계 개체군모델(population model)의 두 단계로 구성되어 있다. 비선형 혼합효과 모형의 첫 번째 단계인 개별수준모델은 비선형 회귀모형의 모수를 추정하는 것으로 일반적인 비선형 회귀모형과 같고, 주로 보통최소제곱추정 방법을 사용하여 모수를 추정한다. 그러나 최소제곱추정방법은 가정된 비선형 함수가 자료에 의해 명시적으로 드러나지 않는 경우 모수의 추정값과 그 표준오차가 극단적으로 커지는 문제가 발생할 수 있다. 본 논문에서는 최근에 비선형 회귀모형에서 제안된 능형회귀(ridge regression) 방법을 비선형 혼합효과 모형의 제 1단계 개별수준모델에 도입함으로써 이러한 문제를 해결할 수 있는 새로운 추정방법을 제안하였다. 제안된 추정량은 모의실험 연구를 통하여 기존의 표준적인 추정량과 그 성능을 비교하였다. 또한 미국의 National Toxicology Program으로부터 얻어진 정량적 대량고속 스크리닝(quantitative high throughput screening) 실제 자료를 사용하여 추정 방법들을 비교하였다.

Keywords

References

  1. Aggrey, S. E. (2009). Logistic nonlinear mixed effects model for estimating growth parameters, Poultry Science, 88, 276-280.
  2. Arnold, B. C. and Strauss, D. (1991). Pseudolikelihood estimation: some examples, Sankhya, The Indian Journal of Statistics, Series B, 53, 233-243.
  3. Austin, C., Kavlock, R., and Tice, R. (2008). Tox21: Putting a Lens on the Vision of Toxicity Testing in the 21st Century, EPA Science Inventory.
  4. Bishop, C. M. (2007). Pattern Recognition and Machine Learning (5th ed), Springer, New York.
  5. Bogacka, B., Latif, M. A. H. M., Gilmour, S. G., and Youdim, K. (2017). Optimum designs for non-linear mixed effects models in the presence of covariates, Biometrics, 73, 927-937. https://doi.org/10.1111/biom.12660
  6. Calegario, N., Daniels, R. F., Maestri, R., and Neiva, R. (2005). Modeling dominant height growth based on nonlinear mixed-effects model: a clonal Eucalyptus plantation case study, Forest Ecology and Management, 204, 11-21. https://doi.org/10.1016/j.foreco.2004.07.051
  7. Craig, B. A. and Schinckel, A. P. (2001). Nonlinear mixed effects model for swine growth, The Professional Animal Scientist, 17, 256-260. https://doi.org/10.15232/S1080-7446(15)31637-5
  8. Davidian, M. and Gallant, A. R. (1993). The nonlinear mixed effects model with a smooth random effects density, Biometrika, 80, 475-488. https://doi.org/10.1093/biomet/80.3.475
  9. Davidian, M. and Giltinan, D. M. (1993a). Some simple methods for estimating intraindividual variability in nonlinear mixed effects models, Biometrics, 49, 59-73. https://doi.org/10.2307/2532602
  10. Davidian, M. and Giltinan, D. M. (1993b). Some general estimation methods for nonlinear mixed-effects model, Journal of Biopharmaceutical Statistics, 3, 23-55. https://doi.org/10.1080/10543409308835047
  11. Davidian, M. and Giltinan, D. M. (1995). Nonlinear Models for Repeated Measurement Data, Champman & Hall, London.
  12. Davidian, M. and Giltinan, D. M. (2003). Nonlinear models for repeated measurement data: an overview and update. Journal of Agricultural, Biological, and Environmental Statistics, 8, 387-419. https://doi.org/10.1198/1085711032697
  13. Fang, Z. and Bailey, R. L. (2001). Nonlinear mixed effects modeling for slash pine dominant height growth following intensive silvicultural treatments, Forest Science, 47, 287-300.
  14. Garber, S. M. and Maguire, D. A. (2003). Modeling stem taper of three central Oregon species using nonlinear mixed effects models and autoregressive error structures, Forest Ecology and Management, 179, 507-522. https://doi.org/10.1016/S0378-1127(02)00528-5
  15. Gerhard, D., Bremer, M., and Ritz, C. (2014). Estimating marginal properties of quantitative real-time PCR data using nonlinear mixed models, Biometrics, 70, 247-254. https://doi.org/10.1111/biom.12124
  16. Hall, D. B. and Clutter, M. (2004). Multivariate multilevel nonlinear mixed effects models for timber yield predictions, Biometrics, 60, 16-24. https://doi.org/10.1111/j.0006-341X.2004.00163.x
  17. Hill, A. V. (1910). The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves, Journal of Physiology, 40, 4-7.
  18. Huber, P. J. (1981). Robust Statistics, John Wiley & Sons, New York.
  19. Lee, S. Y. and Xu, L. (2004). Influence analyses of nonlinear mixed-effects models, Computational Statistics & Data Analysis, 45, 321-341. https://doi.org/10.1016/S0167-9473(02)00303-1
  20. Lee, Y., Lee, K. Y., and Lee, J. (2006). The estimating optimal number of Gaussian mixtures based on incremental k-means for speaker identification, International Journal of Information Technology, 12, 13-21.
  21. Lim, C. (2015). Robust ridge regression estimators for nonlinear models with applications to high throughput screening assay data, Statistics in Medicine, 34, 1185-1198. https://doi.org/10.1002/sim.6391
  22. Lim, C., Sen, P. K., and Peddada, S. D. (2013a). Robust analysis of high throughput screening (HTS) assay data, Technometrics, 55, 150-160. https://doi.org/10.1080/00401706.2012.749166
  23. Lim, C., Sen, P. K., and Peddada, S. D. (2013b). Robust nonlinear regression in applications, Journal of the Indian Society of Agricultural Statistics, 67, 215-234.
  24. Lindstrom, M. J., and Bates, D. M. (1990). Nonlinear mixed effects models for repeated measures data, Biometrics, 46, 673-687. https://doi.org/10.2307/2532087
  25. Meza, C., Osorio, F., and De la Cruz, R. (2012). Estimation in nonlinear mixed-effects models using heavy-tailed distributions, Statistics and Computing, 22, 121-139. https://doi.org/10.1007/s11222-010-9212-1
  26. Morrell, C. H., Pearson, J. D., Carter, H. B., and Brant, L. J. (1995). Estimating unknown transition times using a piecewise nonlinear mixed-effects model in men with prostate cancer, Journal of the American Statistical Association, 90, 45-53. https://doi.org/10.1080/01621459.1995.10476487
  27. Nguyen, T. T., Bazzoli, C., and Mentre, F. (2012). Design evaluation and optimisation in crossover pharmacokinetic studies analysed by nonlinear mixed effects models, Statistics in Medicine, 31, 1043-1058. https://doi.org/10.1002/sim.4390
  28. Peddada, S. D. (2013). Statistical analysis of data from quantitative high throughput screening (qHTS) assays - Methods and challenges, Journal of the Indian Society of Agricultural Statistics, 67, 141-150.
  29. Rajeswaran, J. and Blackstone, E. H. (2017). A multiphase non-linear mixed effects model: an application to spirometry after lung transplantation, Statistical Methods in Medical Research, 26, 21-42. https://doi.org/10.1177/0962280214537255
  30. Samson, A., Lavielle, M., and Mentre, F. (2006). Extension of the SAEM algorithm to left-censored data in nonlinear mixed-effects model: application to HIV dynamics model, Computational Statistics & Data Analysis, 51, 1562-1574. https://doi.org/10.1016/j.csda.2006.05.007
  31. Smith, C. S., Bucher, J., Dearry, A., Portier, C., Tice, R. R., Witt, K., and Collins, B. (2007). Chemical selection for NTP's high throughput screening initiative (Abstract), Toxicologist, 46, 247.
  32. Stirnemann, J. J., Samson, A., and Thalabard, J. C. (2012). Individual predictions based on nonlinear mixed modeling: application to prenatal twin growth, Statistics in Medicine, 31, 1986-1999. https://doi.org/10.1002/sim.5319
  33. Tice, R. R., Fostel, J., Smith, C. S., Witt, K., Freedman, J. H., Portier, C. J., Dearry, A., and Bucher, J. (2007). The National Toxicology Program high throughput screeing initiative: current status and future directions (Abstract), Toxicologist, 46, 246.
  34. Wang, Y., LeMay, V. M., and Baker, T. G. (2007). Modelling and prediction of dominant height and site index of Eucalyptus globulus plantations using a nonlinear mixed-effects model approach, Canadian Journal of Forest Research, 37, 1390-1403. https://doi.org/10.1139/X06-282
  35. Williams, J. D., Birch, J. B., and Abdel-Salam, A. S. G. (2015). Outlier robust nonlinear mixed model estimation, Statistics in Medicine, 34, 1304-1316. https://doi.org/10.1002/sim.6406
  36. Xia, M., Huang, R., Witt, K. L., Southall, N., Fostel, J., Cho, M. H., Jadhav, A., Smith, C. S., Inglese, J., Portier, C. J., Tice, R. R., and Austin, C. P. (2008). Compound cytotoxicity profiling using quantitative high-throughput screening, Environmental Health Perspectives, 116, 284-291.
  37. Yeap, B. Y., Catalano, P. J., Ryan, L. M., and Davidian, M. (2003). Robust two-stage approach to repeated measurements analysis of chronic ozone exposure in rats, Journal of Agricultural, Biological, and Environmental Statistics, 8, 438-454. https://doi.org/10.1198/1085711032552
  38. Yeap, B. Y. and Davidian, M. (2001). Robust two-stage estimation in hierarchical nonlinear models, Biometrics, 57, 266-272. https://doi.org/10.1111/j.0006-341X.2001.00266.x
  39. Zhao, D., Wilson, M., and Borders, B. E. (2005). Modeling response curves and testing treatment effects in repeated measures experiments: a multilevel nonlinear mixed-effects model approach, Canadian Journal of Forest Research, 35, 122-132. https://doi.org/10.1139/x04-163