Smoothing and mixed models

SummarySmoothing methods that use basis functions with penalisation can be formulated as maximum likelihood estimators and best predictors in a mixed model framework. Such connections are at least a quarter of a century old but, perhaps with the advent of mixed model software, have led to a paradigm shift in the field of smoothing. The reason is that most, perhaps all, models involving smoothing can be expressed as a mixed model and hence enjoy the benefit of the growing body of methodology and software for general mixed model analysis. The handling of other complications such as clustering, missing data and measurement error is generally quite straightforward with mixed model representations of smoothing.

[1]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[2]  David Ruppert,et al.  Variable Selection and Function Estimation in Additive Nonparametric Regression Using a Data-Based Prior: Comment , 1999 .

[3]  Yuedong Wang,et al.  Semiparametric Nonlinear Mixed-Effects Models and Their Applications , 2001 .

[4]  R. Wolfinger,et al.  Spatial Regression Models, Response Surfaces, and Process Optimization , 1997 .

[5]  N. Draper,et al.  Applied Regression Analysis: Draper/Applied Regression Analysis , 1998 .

[6]  Peter J. Diggle,et al.  Spatial and Longitudinal Data Analysis: Two Histories with a Common Future? , 1997 .

[7]  Terry Speed,et al.  [That BLUP is a Good Thing: The Estimation of Random Effects]: Comment , 1991 .

[8]  J. Ibrahim.  Incomplete Data in Generalized Linear Models , 1990 .

[9]  R. Tibshirani,et al.  Varying‐Coefficient Models , 1993 .

[10]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[11]  David Ruppert,et al.  Theory & Methods: Spatially‐adaptive Penalties for Spline Fitting , 2000 .

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

[13]  Zehua Chen.  Fitting Multivariate Regression Functions by Interaction Spline Models , 1993 .

[14]  P. Green.  Penalized Likelihood for General Semi-Parametric Regression Models. , 1987 .

[15]  Peter Green.  Linear models for field trials, smoothing and cross-validation , 1985 .

[16]  J. Marron,et al.  SiZer for Exploration of Structures in Curves , 1999 .

[17]  M. Wand,et al.  ADDITIVE MODELS WITH PREDICTORS SUBJECT TO MEASUREMENT ERROR , 2005 .

[18]  M. Wand,et al.  Generalized additive models for cancer mapping with incomplete covariates. , 2004, Biostatistics.

[19]  R. Tibshirani,et al.  Generalized additive models for medical research , 1986, Statistical methods in medical research.

[20]  F. O’Sullivan.  Fast Computation of Fully Automated Log-Density and Log-Hazard Estimators , 1988 .

[21]  Noel A. C. Cressie,et al.  Statistics for Spatial Data: Cressie/Statistics , 1993 .

[22]  Roger Woodard,et al.  Interpolation of Spatial Data: Some Theory for Kriging , 1999, Technometrics.

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

[24]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

[25]  R. Tibshirani,et al.  Bayesian backfitting (with comments and a rejoinder by the authors , 2000 .

[26]  Joseph L Schafer,et al.  Analysis of Incomplete Multivariate Data , 1997 .

[27]  Thomas S. Shively,et al.  Variable Selection and Function Estimation in Additive Nonparametric Regression Using a Data-Based Prior , 1999 .

[28]  H. D. Patterson,et al.  Recovery of inter-block information when block sizes are unequal , 1971 .

[29]  P. Diggle,et al.  Analysis of Longitudinal Data , 2003 .

[30]  S. Lipsitz,et al.  Missing responses in generalised linear mixed models when the missing data mechanism is nonignorable , 2001 .

[31]  D. Bates,et al.  Mixed-Effects Models in S and S-PLUS , 2001 .

[32]  D. Ruppert,et al.  Measurement Error in Nonlinear Models , 1995 .

[33]  P. J. Huber.  Robust Estimation of a Location Parameter , 1964 .

[34]  C. Kelly,et al.  Monotone smoothing with application to dose-response curves and the assessment of synergism. , 1990, Biometrics.

[35]  S. R. Searle,et al.  Generalized, Linear, and Mixed Models , 2005 .

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

[37]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[38]  G. Wahba.  Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression , 1978 .

[39]  Rob J Hyndman,et al.  Mixed Model-Based Hazard Estimation , 2002 .

[40]  C. Ansley,et al.  The Signal Extraction Approach to Nonlinear Regression and Spline Smoothing , 1983 .

[41]  M. E. Johnson,et al.  Minimax and maximin distance designs , 1990 .

[42]  M. Wand,et al.  Smoothing with Mixed Model Software , 2004 .

[43]  N. Draper,et al.  Applied Regression Analysis , 1966 .

[44]  Yuedong Wang.  Mixed effects smoothing spline analysis of variance , 1998 .

[45]  Douglas W. Nychka,et al.  Design of Air-Quality Monitoring Networks , 1998 .

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

[47]  Gareth M. James,et al.  Functional linear discriminant analysis for irregularly sampled curves , 2001 .

[48]  X. Lin,et al.  Inference in generalized additive mixed modelsby using smoothing splines , 1999 .

[49]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[50]  N. Draper,et al.  Applied Regression Analysis , 1967 .

[51]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[52]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[53]  Catherine A. Sugar,et al.  Principal component models for sparse functional data , 1999 .

[54]  D. Ruppert.  Selecting the Number of Knots for Penalized Splines , 2002 .

[55]  D. Rubin,et al.  Statistical Analysis with Missing Data. , 1989 .

[56]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[57]  Paul H. C. Eilers,et al.  Flexible smoothing with B-splines and penalties , 1996 .

[58]  Wing K. Fung,et al.  Influence diagnostics and outlier tests for semiparametric mixed models , 2002 .

[59]  Jianqing Fan,et al.  Regularization of Wavelet Approximations , 2001 .

[60]  Yuedong Wang.  Smoothing Spline Models with Correlated Random Errors , 1998 .

[61]  G. Wahba.  Partial and interaction spline models for the semiparametric estimation of functions of several variables , 1986 .

[62]  R. Gray,et al.  Spline-based tests in survival analysis. , 1994, Biometrics.

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

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

[65]  M. Wand,et al.  Simple Incorporation of Interactions into Additive Models , 2001, Biometrics.