Double hierarchical generalized linear models (with discussion)

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.

[1]  William G. Cochran,et al.  The analysis of groups of experiments , 1938, The Journal of Agricultural Science.

[2]  H. Chernoff.  On the Distribution of the Likelihood Ratio , 1954 .

[3]  Leonard J. Savage,et al.  The foundations of statistical inference : a discussion , 1962 .

[4]  A. Birnbaum.  On the Foundations of Statistical Inference , 1962 .

[5]  G. Patil.  A characterization of the exponential - type distribution , 1963 .

[6]  Calyampudi R. Rao,et al.  The theory of least squares when the parameters are stochastic and its application to the analysis of growth curves. , 1965, Biometrika.

[7]  John A. Nelder,et al.  The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance , 1965, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[8]  George E. P. Box,et al.  The Bayesian estimation of common parameters from several responses , 1965 .

[9]  P. McCullagh,et al.  Generalized Linear Models , 1972, Predictive Analytics.

[10]  R. W. Wedderburn.  Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method , 1974 .

[11]  C. R. Henderson,et al.  Best linear unbiased estimation and prediction under a selection model. , 1975, Biometrics.

[12]  D. Harville.  Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems , 1977 .

[13]  Ashok Saxena,et al.  Development of Standard Methods of Testing and Analyzing Fatigue Crack Growth Rate Data , 1978 .

[14]  N. Laird.  Nonparametric Maximum Likelihood Estimation of a Mixing Distribution , 1978 .

[15]  P. W. Lane,et al.  Analysis of covariance and standardization as instances of prediction. , 1982, Biometrics.

[16]  O. Barndorff-Nielsen.  On a formula for the distribution of the maximum likelihood estimator , 1983 .

[17]  J. Heckman,et al.  A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration Data , 1984 .

[18]  H. Goldstein.  Multilevel mixed linear model analysis using iterative generalized least squares , 1986 .

[19]  S. Zeger,et al.  Longitudinal data analysis using generalized linear models , 1986 .

[20]  B. Efron.  Double Exponential Families and Their Use in Generalized Linear Regression , 1986 .

[21]  D. Cox,et al.  Parameter Orthogonality and Approximate Conditional Inference , 1987 .

[22]  G. McLachlan.  On Bootstrapping the Likelihood Ratio Test Statistic for the Number of Components in a Normal Mixture , 1987 .

[23]  M. Jacobs,et al.  A controlled study of progabide in partial seizures , 1987, Neurology.

[24]  J. Nelder,et al.  An extended quasi-likelihood function , 1987 .

[25]  M. Schumacher,et al.  The impact of heterogeneity on the comparison of survival times. , 1987, Statistics in medicine.

[26]  Marie Davidian,et al.  A Note on Extended Quasi-Likelihood , 1988 .

[27]  O. Aalen,et al.  Heterogeneity in survival analysis. , 1988, Statistics in medicine.

[28]  Jeremy MG Taylor,et al.  Robust Statistical Modeling Using the t Distribution , 1989 .

[29]  P. Thall,et al.  Some covariance models for longitudinal count data with overdispersion. , 1990, Biometrics.

[30]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[31]  G. Wahba.  Spline models for observational data , 1990 .

[32]  E. Seneta,et al.  The Variance Gamma (V.G.) Model for Share Market Returns , 1990 .

[33]  D. Bates,et al.  Nonlinear mixed effects models for repeated measures data. , 1990, Biometrics.

[34]  D. Duffie,et al.  Simulated Moments Estimation of Markov Models of Asset Prices , 1990 .

[35]  John A. Nelder,et al.  Generalized linear models for the analysis of Taguchi-type experiments , 1991 .

[36]  G. Robinson.  That BLUP is a Good Thing: The Estimation of Random Effects , 1991 .

[37]  R. Schall.  Estimation in generalized linear models with random effects , 1991 .

[38]  R. Payne,et al.  General balance, combination of information and the analysis of covariance , 1992 .

[39]  John A. Nelder,et al.  Likelihood, Quasi-likelihood and Pseudolikelihood: Some Comparisons , 1992 .

[40]  N. Breslow,et al.  Approximate inference in generalized linear mixed models , 1993 .

[41]  R. Wolfinger.  Covariance structure selection in general mixed models , 1993 .

[42]  C. Joseph Lu,et al.  Using Degradation Measures to Estimate a Time-to-Failure Distribution , 1993 .

[43]  B. Silverman,et al.  Nonparametric Regression and Generalized Linear Models: A roughness penalty approach , 1993 .

[44]  R. Wolfinger,et al.  Generalized linear mixed models a pseudo-likelihood approach , 1993 .

[45]  B. Silverman,et al.  Nonparametric regression and generalized linear models , 1994 .

[46]  P. Diggle,et al.  Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters. , 1994, Biometrics.

[47]  B. Silverman,et al.  Nonparametric Regression and Generalized Linear Models: A roughness penalty approach , 1993 .

[48]  Adrian F. M. Smith,et al.  Bayesian Analysis of Linear and Non‐Linear Population Models by Using the Gibbs Sampler , 1994 .

[49]  N. Shephard,et al.  Multivariate stochastic variance models , 1994 .

[50]  N. Shephard,et al.  Stochastic Volatility: Likelihood Inference And Comparison With Arch Models , 1996 .

[51]  Noreen Goldman,et al.  An assessment of estimation procedures for multilevel models with binary responses , 1995 .

[52]  A. Gallant,et al.  Which Moments to Match? , 1995, Econometric Theory.

[53]  N. Breslow,et al.  Bias correction in generalised linear mixed models with a single component of dispersion , 1995 .

[54]  E. Eberlein,et al.  Hyperbolic distributions in finance , 1995 .

[55]  Harvey Goldstein,et al.  Improved Approximations for Multilevel Models with Binary Responses , 1996 .

[56]  E. Vonesh,et al.  A note on the use of Laplace's approximation for nonlinear mixed-effects models , 1996 .

[57]  J. F. Bjørnstad.  On the Generalization of the Likelihood Function and the Likelihood Principle , 1996 .

[58]  N. Shephard.  Statistical aspects of ARCH and stochastic volatility , 1996 .

[59]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[60]  J. Nelder,et al.  Hierarchical Generalized Linear Models , 1996 .

[61]  N. Breslow,et al.  Bias Correction in Generalized Linear Mixed Models with Multiple Components of Dispersion , 1996 .

[62]  M. Pitt,et al.  Likelihood analysis of non-Gaussian measurement time series , 1997 .

[63]  Jianqing Fan,et al.  Smoothing spline models for the analysis of nested and crossed samples of curves. Commentaries. Authors' reply , 1998 .

[64]  J. Rice,et al.  Smoothing spline models for the analysis of nested and crossed samples of curves , 1998 .

[65]  Siem Jan Koopman,et al.  Time Series Analysis of Non-Gaussian Observations Based on State Space Models from Both Classical and Bayesian Perspectives , 1999 .

[66]  Marco Alfò,et al.  Regression models for binary longitudinal responses , 1998, Stat. Comput..

[67]  Siem Jan Koopman,et al.  Estimation of stochastic volatility models via Monte Carlo maximum likelihood , 1998 .

[68]  John A. Nelder,et al.  Generalized linear models for the analysis of quality‐improvement experiments , 1998 .

[69]  B. Everitt,et al.  Analysis of longitudinal data , 1998, British Journal of Psychiatry.

[70]  Steven G. Gilmour,et al.  The analysis of designed experiments and longitudinal data by using smoothing splines - Discussion , 1999 .

[71]  D. Firth,et al.  Estimating Intraclass Correlation for Binary Data , 1999, Biometrics.

[72]  M. Kenward,et al.  The Analysis of Designed Experiments and Longitudinal Data by Using Smoothing Splines , 1999 .

[73]  Jane-Ling Wang,et al.  Dimension reduction for censored regression data , 1999 .

[74]  Kani Chen,et al.  Strong consistency of maximum quasi-likelihood estimators in generalized linear models with fixed and adaptive designs , 1999 .

[75]  Rasmus Waagepetersen,et al.  Contribution to the discussion of Besag, J. and Higdon, D. (1999) , 1999 .

[76]  Gordon K. Smyth,et al.  Adjusted likelihood methods for modelling dispersion in generalized linear models , 1999 .

[77]  Renjun Ma.  An orthodox blup approach to generalized linear mixed models , 1999 .

[78]  J. Besag,et al.  Bayesian analysis of agricultural field experiments , 1999 .

[79]  M. Pourahmadi.  Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix , 2000 .

[80]  John A. Nelder,et al.  Two ways of modelling overdispersion in non‐normal data , 2000 .

[81]  David J. Spiegelhalter,et al.  Consensus and controversy in pharmaceutical statistics - Discussion , 2000 .

[82]  Stephen Senn.  Consensus and Controversy in Pharmaceutical Statistics , 2000 .

[83]  John A. Nelder,et al.  The relationship between double‐exponential families and extended quasi‐likelihood families, with application to modelling Geissler's human sex ratio data , 2000 .

[84]  M E Robinson,et al.  Bayesian Methods for a Growth-Curve Degradation Model with Repeated Measures , 2000, Lifetime data analysis.

[85]  Youngjo Lee,et al.  Hierarchical likelihood approach for frailty models , 2001 .

[86]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[87]  J. Nelder,et al.  Hierarchical generalised linear models: A synthesis of generalised linear models, random-effect models and structured dispersions , 2001 .

[88]  Ana Ivelisse Avilés,et al.  Linear Mixed Models for Longitudinal Data , 2001, Technometrics.

[89]  M. Kenward,et al.  Parametric modelling of growth curve data: An overview , 2001 .

[90]  X Xue.  Analysis of childhood brain tumour data in New York City using frailty models. , 2001, Statistics in medicine.

[91]  Youngjo Lee.  Can we recover information from concordant pairs in binary matched pairs? , 2001 .

[92]  Anukool Lakhina,et al.  BRITE: Universal Topology Generation from a User''s Perspective , 2001 .

[93]  Youngjo Lee,et al.  Modelling and analysing correlated non-normal data , 2001 .

[94]  A. Kuk,et al.  Robust estimation in generalized linear mixed models , 2002 .

[95]  Harvey Goldstein,et al.  Likelihood methods for fitting multilevel models with complex level-1 variation , 2002 .

[96]  Eric R. Ziegel,et al.  Generalized Linear Models , 2002, Technometrics.

[97]  Youngjo Lee,et al.  Hierarchical-Likelihood Approach for Mixed Linear Models with Censored Data , 2002, Lifetime data analysis.

[98]  James R. Kenyon,et al.  Analysis of Multivariate Survival Data , 2002, Technometrics.

[99]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[100]  Y. Pawitan.  In all likelihood : statistical modelling and inference using likelihood , 2002 .

[101]  Dibyen Majumdar,et al.  Conditional Second-Order Generalized Estimating Equations for Generalized Linear and Nonlinear Mixed-Effects Models , 2002 .

[102]  Taesung Park,et al.  Joint Modelling of Repeated Measures and Survival Time Data , 2003 .

[103]  Gilbert MacKenzie,et al.  On modelling mean‐covariance structures in longitudinal studies , 2003 .

[104]  Daniel Krewski,et al.  Random effects Cox models: A Poisson modelling approach , 2003 .

[105]  M. Crowder,et al.  Covariates and Random Effects in a Gamma Process Model with Application to Degradation and Failure , 2004, Lifetime data analysis.

[106]  Estimating intraclass correlation for binary data using extended quasi-likelihood , 2004 .

[107]  Youngjo Lee,et al.  Comparison of hierarchical and marginal likelihood estimators for binary outcomes , 2004, Comput. Stat. Data Anal..

[108]  D. Cox,et al.  A note on pseudolikelihood constructed from marginal densities , 2004 .

[109]  R. Rigby,et al.  Generalized additive models for location, scale and shape , 2005 .

[110]  Adrian Bowman,et al.  Generalized additive models for location, scale and shape - Discussion , 2005 .

[111]  Risto Lehtonen,et al.  Multilevel Statistical Models , 2005 .

[112]  C. Varin,et al.  A note on composite likelihood inference and model selection , 2005 .

[113]  Youngjo Lee,et al.  Comparison of hierarchical likelihood versus orthodox best linear unbiased predictor approaches for frailty models , 2005 .

[114]  Ruggero Bellio,et al.  A pairwise likelihood approach to generalized linear models with crossed random effects , 2005 .

[115]  G. Molenberghs,et al.  Models for Discrete Longitudinal Data , 2005 .

[116]  Youngjo Lee,et al.  Robust ascertainment‐adjusted parameter estimation , 2005, Genetic epidemiology.

[117]  John A. Nelder,et al.  Likelihood for random-effect models , 2005 .

[118]  Maengseok Noh,et al.  HGLM modelling of dropout process using a frailty model , 2005, Comput. Stat..

[119]  Sudhir Paul,et al.  Bias-corrected maximum likelihood estimator of the negative binomial dispersion parameter. , 2005, Biometrics.

[120]  P. J. Lindsey,et al.  Multivariate distributions with correlation matrices for nonlinear repeated measurements , 2006, Comput. Stat. Data Anal..

[121]  Robust estimation in mixed linear models with non‐monotone missingness , 2006, Statistics in medicine.

[122]  John A. Nelder,et al.  Fitting via alternative random-effect models , 2006, Stat. Comput..

[123]  M. Kenward,et al.  The Analysis of Longitudinal Data Using Mixed Model L‐Splines , 2006, Biometrics.

[124]  Y Pawitan,et al.  Multicomponent variance estimation for binary traits in family‐based studies , 2006, Genetic epidemiology.

[125]  Jianxin Pan,et al.  Quasi-Monte Carlo estimation in generalized linear mixed models , 2007, Comput. Stat. Data Anal..

[126]  Optimal Joint Mean-Covariance Modelling , 2007 .

[127]  J. F. Bjørnstad.  ----On the Generalization of the Likelihood Function and the Likelihood Principle , 2008 .