×

zbMATH — the first resource for mathematics

Double hierarchical generalized linear models (with discussion). (English) Zbl 05188732
Summary: We propose a class of double hierarchical generalized linear models in which random effects can be specified for both the mean and dispersion. Heteroscedasticity between clusters can be modelled by introducing random effects in the dispersion model, as is heterogeneity between clusters in the mean model. This class will, among other things, enable models with heavy-tailed distributions to be explored, providing robust estimation against outliers. The h-likelihood provides a unified framework for this new class of models and gives a single algorithm for fitting all members of the class. This algorithm does not require quadrature or prior probabilities.

MSC:
62-XX Statistics
Software:
GAMLSS; MLwiN
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1023/A:1008847820371 · doi:10.1023/A:1008847820371
[2] DOI: 10.1111/1467-9868.00201 · Zbl 0951.62091 · doi:10.1111/1467-9868.00201
[3] Bjornstad J. F., J. Am. Statist. Ass. 91 pp 791– (1996)
[4] DOI: 10.2307/2290687 · Zbl 0775.62195 · doi:10.2307/2290687
[5] DOI: 10.2307/2337629 · doi:10.2307/2337629
[6] DOI: 10.2307/2669837 · Zbl 1064.62515 · doi:10.2307/2669837
[7] Chernoff H., Ann. Math. Statist. 25 pp 573– (1954)
[8] Cox D. R., J. R. Statist. Soc. 49 pp 1– (1987)
[9] Davidian M., J. R. Statist. Soc. 50 pp 74– (1988)
[10] Diggle P. J., Analysis of Longitudinal Data (1994) · Zbl 0825.62010
[11] DOI: 10.1111/1467-9868.00218 · Zbl 0945.62084 · doi:10.1111/1467-9868.00218
[12] Engel R. E., ARCH (1995)
[13] Goldstein H., Multilevel Statistical Models (1995) · Zbl 1014.62126
[14] Green P. J., Nonparametric Regression and Generalized Linear Models: a Roughness Penalty Approach (1994) · Zbl 0832.62032 · doi:10.1007/978-1-4899-4473-3
[15] DOI: 10.1093/biomet/92.3.717 · Zbl 1152.62379 · doi:10.1093/biomet/92.3.717
[16] DOI: 10.1093/biomet/88.1.233 · Zbl 1033.62096 · doi:10.1093/biomet/88.1.233
[17] DOI: 10.1023/A:1014839723865 · Zbl 1037.62100 · doi:10.1023/A:1014839723865
[18] DOI: 10.1002/bimj.200390039 · doi:10.1002/bimj.200390039
[19] Harvey A. C., Forecasting, Structural Time Series Models and the Kalman Filter (1989)
[20] DOI: 10.2307/2297980 · Zbl 0805.90026 · doi:10.2307/2297980
[21] DOI: 10.2307/2286796 · Zbl 0373.62040 · doi:10.2307/2286796
[22] DOI: 10.2307/1911491 · Zbl 0547.62077 · doi:10.2307/1911491
[23] Henderson C. R., Biometrics 31 pp 423– (1975)
[24] Hougaard P., Analysis of Multivariate Survival Data (2000) · Zbl 0962.62096
[25] S. J. Hudak, A. Saxena, R. J. Bucci, and R. C. Malcom (1978 ) Development of standard methods of testing and analyzing fatigue crack growth rate data .Technical Report AFML-TR-78-40. Westinghouse R&D Center, Westinghouse Electric Corporation, Pittsburgh.
[26] DOI: 10.1111/1467-937X.00050 · Zbl 0910.90067 · doi:10.1111/1467-937X.00050
[27] DOI: 10.2307/2286284 · Zbl 0391.62029 · doi:10.2307/2286284
[28] Lange K. L., J. Am. Statist. Ass. 84 pp 881– (1989)
[29] DOI: 10.1023/B:LIDA.0000036389.14073.dd · Zbl 1054.62122 · doi:10.1023/B:LIDA.0000036389.14073.dd
[30] DOI: 10.1080/02664760020016136 · Zbl 0991.62013 · doi:10.1080/02664760020016136
[31] DOI: 10.1191/1471082X04st070oa · Zbl 1111.62054 · doi:10.1191/1471082X04st070oa
[32] Lee Y., J. R. Statist. Soc. 58 pp 619– (1996)
[33] Lee Y., Can. J. Statist. 26 pp 95– (1998)
[34] Lee Y., Appl. Statist. 49 pp 591– (2000)
[35] DOI: 10.1093/biomet/88.4.987 · Zbl 0995.62066 · doi:10.1093/biomet/88.4.987
[36] DOI: 10.1191/147108201128050 · Zbl 1004.62080 · doi:10.1191/147108201128050
[37] Lee Y., Statist. Comput. (2005)
[38] Lee Y., Statist. Oper. Res. Trans. 29 pp 141– (2005)
[39] Lee Y., Comput. Statist. 20 pp 295– (2005)
[40] Lin X., J. Am. Statist. Ass. 91 pp 1007– (1996)
[41] DOI: 10.1016/j.csda.2004.09.011 · Zbl 1431.62219 · doi:10.1016/j.csda.2004.09.011
[42] DOI: 10.2307/2532087 · doi:10.2307/2532087
[43] Longford N., Random Coefficient Models (1993) · Zbl 0859.62064
[44] DOI: 10.2307/1269661 · Zbl 0775.62271 · doi:10.2307/1269661
[45] DOI: 10.1093/biomet/90.1.157 · Zbl 1035.62114 · doi:10.1093/biomet/90.1.157
[46] McLachlan G. J., Appl. Statist. 36 pp 318– (1987)
[47] McLachlan G. J., The EM Algorithm and Extensions (1997) · Zbl 0882.62012
[48] A. Medina, H. Lakhina, I. Matta, and J. Byers (2001 ) BRITE: universal topology generation from a user’s perspective .Technical Report. Boston University, Boston.
[49] Nelder J. A., Appl. Stochast. Mod. Data Anal. 7 pp 107– (1991)
[50] Nelder J. A., J. R. Statist. Soc. 54 pp 273– (1992)
[51] DOI: 10.2307/2336136 · doi:10.2307/2336136
[52] Nelder J. A., J. R. Statist. Soc. 135 pp 370– (1972)
[53] M. Noh, and Y. Lee (2004 ) REML estimation for binary data in generalized linear mixed models . To be pub-lished.
[54] DOI: 10.1002/gepi.20078 · doi:10.1002/gepi.20078
[55] Noh M., Genet. Epidem. (2004)
[56] DOI: 10.1093/biomet/90.1.239 · Zbl 1039.62068 · doi:10.1093/biomet/90.1.239
[57] DOI: 10.2307/2333763 · doi:10.2307/2333763
[58] Pawitan Y., In All Likelihood: Statistical Modelling and Inference using Likelihood (2001) · Zbl 1013.62001
[59] DOI: 10.1093/biomet/87.2.425 · Zbl 0954.62091 · doi:10.1093/biomet/87.2.425
[60] Rigby R. A., GLIM4 Macro Library Manual, Release 2.0 pp 48– (1996)
[61] Robinson G. K., Statist. Sci. 6 pp 15– (1991)
[62] DOI: 10.1023/A:1026509432144 · Zbl 0971.62061 · doi:10.1023/A:1026509432144
[63] DOI: 10.1111/j.0006-341X.2005.030833.x · Zbl 1077.62083 · doi:10.1111/j.0006-341X.2005.030833.x
[64] Schall R., Biometrika 40 pp 917– (1991)
[65] Schumacher M., Statist. Med. 6 pp 773– (1987)
[66] Shephard N., Time Series Models in Econometrics, Finance and Other Fields (1996)
[67] DOI: 10.1093/biomet/84.3.653 · Zbl 0888.62095 · doi:10.1093/biomet/84.3.653
[68] DOI: 10.1111/1467-9868.00353 · Zbl 1067.62010 · doi:10.1111/1467-9868.00353
[69] DOI: 10.2307/2532086 · Zbl 0712.62048 · doi:10.2307/2532086
[70] Verbeke G., Linear Mixed Models for Longitudinal Data (2000) · Zbl 0956.62055
[71] DOI: 10.1111/1467-9876.00154 · Zbl 0956.62062 · doi:10.1111/1467-9876.00154
[72] DOI: 10.1093/biomet/83.2.447 · Zbl 0878.62019 · doi:10.1093/biomet/83.2.447
[73] DOI: 10.1198/016214502753479400 · Zbl 1073.62591 · doi:10.1198/016214502753479400
[74] Wahba G., Spline Models for Observational Data (1990) · Zbl 0813.62001 · doi:10.1137/1.9781611970128
[75] Wakefield J. C., Appl. Statist. 43 pp 201– (1994)
[76] DOI: 10.2307/2334725 · doi:10.2307/2334725
[77] Welham S., Statist. Med. (2004)
[78] Wolfinger R. D., Communs Statist. Simuln Computn 22 pp 1079– (1993)
[79] DOI: 10.1016/S0167-9473(03)00033-1 · Zbl 1429.62331 · doi:10.1016/S0167-9473(03)00033-1
[80] Yun S., Statist. Med. (2004)
[81] Yun S., J. Appl. Statist. (2004)
[82] Zeger S. L., Biometrics 50 pp 689– (1994)
[83] Zimmerman D., Test 10 pp 1– (2001)
[84] Aalen O. O., Statist.Med. 7 pp 1121– (1988)
[85] DOI: 10.2307/2335549 · doi:10.2307/2335549
[86] DOI: 10.1111/1467-9868.00282 · Zbl 0983.60028 · doi:10.1111/1467-9868.00282
[87] Bayarri M. J., Statistical Decision Theory and Related Topics IV (1988)
[88] DOI: 10.1191/1471082X05st095oa · Zbl 1111.62058 · doi:10.1191/1471082X05st095oa
[89] Birnbaum A., J. Am. Statist. Ass. 57 pp 269– (1962)
[90] Bjornstad J. F., J. Am. Statist. Ass. 91 pp 791– (1996)
[91] DOI: 10.2307/2290687 · Zbl 0775.62195 · doi:10.2307/2290687
[92] DOI: 10.2307/2337629 · doi:10.2307/2337629
[93] W. J. Browne (1998 ) Applying MCMC methods to multi-level models.PhD Thesis. University of Bath, Bath.
[94] DOI: 10.1016/S0167-9473(01)00058-5 · Zbl 1132.62312 · doi:10.1016/S0167-9473(01)00058-5
[95] Carlin B. P., Bayesian and Empirical Bayesian Methods for Data Analysis (2000) · doi:10.1201/9781420057669
[96] DOI: 10.1214/aos/1018031098 · Zbl 04556018 · doi:10.1214/aos/1018031098
[97] Cox D. R., J. R. Statist. Soc. 49 pp 1– (1987)
[98] DOI: 10.1093/biomet/91.3.729 · Zbl 1162.62365 · doi:10.1093/biomet/91.3.729
[99] DOI: 10.2307/2951768 · Zbl 0783.62099 · doi:10.2307/2951768
[100] DOI: 10.2307/3318481 · Zbl 0836.62107 · doi:10.2307/3318481
[101] Efron B., J. Am. Statist. Ass. 81 pp 709– (1986)
[102] Gallant A., Econometr. Theory 12 pp 657– (1996)
[103] DOI: 10.2307/2336270 · doi:10.2307/2336270
[104] Goldstein H., J. R. Statist. Soc. 159 pp 505– (1996)
[105] Gourieroux C., J. Appl. Econometr. 8 pp S85– (1993)
[106] DOI: 10.1093/biomet/92.3.717 · Zbl 1152.62379 · doi:10.1093/biomet/92.3.717
[107] Hougaard P., Analysis of Multivariate Survival Data (2000) · Zbl 0962.62096 · doi:10.1007/978-1-4612-1304-8
[108] DOI: 10.2307/2530043 · doi:10.2307/2530043
[109] DOI: 10.1191/1471082X04st070oa · Zbl 1111.62054 · doi:10.1191/1471082X04st070oa
[110] Lee Y., J. R. Statist. Soc. 58 pp 619– (1996)
[111] Lee Y., Appl. Statist. 49 pp 413– (2000)
[112] DOI: 10.1093/biomet/88.4.987 · Zbl 0995.62066 · doi:10.1093/biomet/88.4.987
[113] DOI: 10.1191/147108201128050 · Zbl 1004.62080 · doi:10.1191/147108201128050
[114] Lee Y., Statist. Oper. Res. Trans. 29 pp 141– (2005)
[115] Lee Y., Statist. Comput. (2005)
[116] Leppik I. E., Neurology 37 pp 963– (1987) · doi:10.1212/WNL.37.6.963
[117] DOI: 10.2307/2336267 · doi:10.2307/2336267
[118] Lin X., J. Am. Statist. Ass. 91 pp 1007– (1996)
[119] Lindsay B. G., Contemp. Math. 80 pp 221– (1988) · doi:10.1090/conm/080/999014
[120] R. Ma (1999 ) An orthodox BLUP approach to generalized linear mixed models.PhD Thesis. University of British Columbia, Vancouver.
[121] MacKenzie G., Biostatistics (2006)
[122] MacKenzie G., Select. Proc. 2nd Int. Wrkshp Correlated Data Modelling (2006)
[123] DOI: 10.1086/296519 · doi:10.1086/296519
[124] Molenberghs G., Models for Discrete Longitudinal Data (2005) · Zbl 1093.62002
[125] Nelder J. A., Proc. R. Soc. Lond. 283 pp 147– (1965)
[126] Nelder J. A., Appl. Stochast. Mod. Data Anal. 7 pp 107– (1991)
[127] Nelder J. A., J. R. Statist. Soc. 135 pp 370– (1972) · doi:10.2307/2344614
[128] Noh M., Statist. Med. (2006)
[129] Noh M., Manuscript (2005)
[130] DOI: 10.1002/gepi.20078 · doi:10.1002/gepi.20078
[131] DOI: 10.1093/biomet/90.1.239 · Zbl 1039.62068 · doi:10.1093/biomet/90.1.239
[132] Payne R. W., COMPSTAT 2004: Proc. Computational Statistics pp 1629– (2004)
[133] Payne R. W., Scand. J. Statist. 19 pp 3– (1992)
[134] DOI: 10.1093/biomet/87.2.425 · Zbl 0954.62091 · doi:10.1093/biomet/87.2.425
[135] DOI: 10.2307/2333689 · doi:10.2307/2333689
[136] Rasbash J., MLwiN (Version 2.0) (2004)
[137] DOI: 10.1111/j.0006-341X.1999.00137.x · Zbl 1059.62601 · doi:10.1111/j.0006-341X.1999.00137.x
[138] DOI: 10.1111/j.1467-9876.2005.00510.x · Zbl 05188697 · doi:10.1111/j.1467-9876.2005.00510.x
[139] Rodriguez G., J. R. Statist. Soc. 158 pp 73– (1995)
[140] DOI: 10.1016/S0304-4076(98)00016-5 · Zbl 0937.62110 · doi:10.1016/S0304-4076(98)00016-5
[141] DOI: 10.1111/1467-9884.00227 · doi:10.1111/1467-9884.00227
[142] DOI: 10.1002/(SICI)1099-095X(199911/12)10:6<695::AID-ENV385>3.0.CO;2-M · doi:10.1002/(SICI)1099-095X(199911/12)10:6<695::AID-ENV385>3.0.CO;2-M
[143] DOI: 10.2307/2532086 · Zbl 0712.62048 · doi:10.2307/2532086
[144] DOI: 10.1093/biomet/92.3.519 · Zbl 1183.62037 · doi:10.1093/biomet/92.3.519
[145] DOI: 10.2307/2334725 · doi:10.2307/2334725
[146] Wolfinger R., J. Statist. Computn Simuln 48 pp 233– (1993)
[147] Xia N., Acta Math. Appl. Sin. (2005)
[148] Xia N., Working Paper (2005)
[149] DOI: 10.1002/sim.964 · doi:10.1002/sim.964
[150] Yates F., J. Agric. Sci. 28 pp 556– (1938)
[151] DOI: 10.1111/1467-9868.00327 · Zbl 1015.62073 · doi:10.1111/1467-9868.00327
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.