Abstract
The quantile regression has several useful features and therefore is gradually developing into a comprehensive approach to the statistical analysis of linear and nonlinear response models, but it cannot deal effectively with the data with a hierarchical structure. In practice, the existence of such data hierarchies is neither accidental nor ignorable, it is a common phenomenon. To ignore this hierarchical data structure risks overlooking the importance of group effects, and may also render many of the traditional statistical analysis techniques used for studying data relationships invalid. On the other hand, the hierarchical models take a hierarchical data structure into account and have also many applications in statistics, ranging from overdispersion to constructing min-max estimators. However, the hierarchical models are virtually the mean regression, therefore, they cannot be used to characterize the entire conditional distribution of a dependent variable given high-dimensional covariates. Furthermore, the estimated coefficient vector (marginal effects) is sensitive to an outlier observation on the dependent variable. In this article, a new approach, which is based on the Gauss-Seidel iteration and taking a full advantage of the quantile regression and hierarchical models, is developed. On the theoretical front, we also consider the asymptotic properties of the new method, obtaining the simple conditions for an n 1/2-convergence and an asymptotic normality. We also illustrate the use of the technique with the real educational data which is hierarchical and how the results can be explained.
Similar content being viewed by others
References
Lindley D V, Smith A F M. Bayes estimates for the linear model. Journal of the Royal Statistical Society, Series B, 1972, 34: 1–41.
Smith A F M. A general Bayesian linear model. Journal of the Royal Statistical Society, Series B, 1973, 35: 67–75.
Mason W M, Wong G M, Entwistle B. Contextual Analysis Through the Multilevel Linear Model. In: Leinhardt S, ed. Sociological Methodology, San Francisco: Jossey-Bass, 1983, 72–103.
Goldstein H. Multilevel Statistical Models. 2nd ed, New York: John Wiley, 1995
Elston R C, Grizzle J E. Estimation of time response curves and their confidence bands. Biometrics, 1962, 18: 148–159
Laird N M, Ware H. Random-effects models for longitudinal data. Biometrics, 1982, 38: 963–974
Longford N. A fast scoring algorithm for maximum likelihood estimation in unbalanced models with nested random effects. Biometrika, 1987, 74: 817–827
Singer J D. Using SAS PROC MIXED to fit multilevel models, hierarchical models and individual growth models. Journal of Educational and Behavioral Statistics, 1998, 23: 323–355
Rosenberg B. Linear regression with randomly dispersed parameters. Biometrika, 1973, 60: 61–75
Longford N. Random Coefficient Models. Oxford: Clarendon, 1993
Kass R E, Steffey D. Approximate Bayesian inference in conditionally independent hierarchical models (parametric empirical Bayes models). Journal of the American Statistical Association, 1989, 84: 717–726
Dempster A P, Rubin D B, Tsutakawa R K. Estimation in covariance components models. Journal of the American Statistical Association, 1981, 76: 341–353
Hobert J P. Hierarchical models: a current computational perspective. Journal of the American Statistical Association, 2000, 95: 1312–1316
Koenker R, Bassett G. Regression quantiles. Econometrica, 1978, 46: 33–50
Portnoy S, Koenker R. The Gaussian hare and the Laplacian tortoisc: computability of square-error versus absolute-error estimators (with discussion). Statistical Sciences, 1997, 12: 279–300
Koenker R, D’Orey V. A remark on computing regression quantiles. Applied Statistics, 1993, 36: 383–393
Morrison D F. Multivariate Statistical Methods. New York: McGraw-Hill, 1967
Dempster A P, Laird N M, Rubin D B. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 1977, 39: 1–8
Bryk A S, Raudenbush S W, Seltzer M, Congdon R. An Introduction to HLM: Computer Program and User’s Guide. 2nd ed, Chicago: University of Chicago Department of Education, 1988
Billingsley P. Convergence of Probability Measures. New York: John Wiley, 1968
Jurečkovå J, Sen P K. On adaptive scale-equivariant M-estimators in linear models. Statistics & Devisions, 1984, 1(Suppl): 31–46
Ruppert D, Carroll R J. Trimmed least-squares estimation in the linear model. Journal of the American Statistical Association, 1980, 75: 828–838
Jurečkovå J. Asymptotic relations of M-estimates and R-estimates in linear regression model. Annals of Statistics, 1977, 5: 464–472
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tian, M., Chen, G. Hierarchical linear regression models for conditional quantiles. SCI CHINA SER A 49, 1800–1815 (2006). https://doi.org/10.1007/s11425-006-2023-3
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11425-006-2023-3