Smooth Random Effects Distribution in a Linear Mixed Model

Summary A linear mixed model with a smooth random effects density is proposed. A similar approach to P‐spline smoothing of Eilers and Marx (1996, Statistical Science11, 89–121) is applied to yield a more flexible estimate of the random effects density. Our approach differs from theirs in that the B‐spline basis functions are replaced by approximating Gaussian densities. Fitting the model involves maximizing a penalized marginal likelihood. The best penalty parameters minimize Akaike's Information Criterion employing Gray's (1992, Journal of the American Statistical Association87, 942–951) results. Although our method is applicable to any dimensions of the random effects structure, in this article the two‐dimensional case is explored. Our methodology is conceptually simple, and it is relatively easy to fit in practice and is applied to the cholesterol data first analyzed by Zhang and Davidian (2001, Biometrics57, 795–802). A simulation study shows that our approach yields almost unbiased estimates of the regression and the smoothing parameters in small sample settings. Consistency of the estimates is shown in a particular case.

[1]  A. Wald.  Note on the Consistency of the Maximum Likelihood Estimate , 1949 .

[2]  P. Goel.  A note on the consistency of maximum likelihood estimators , 1974 .

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

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

[5]  H. White.  Maximum Likelihood Estimation of Misspecified Models , 1982 .

[6]  Robert Gray,et al.  Flexible Methods for Analyzing Survival Data Using Splines, with Applications to Breast Cancer Prognosis , 1992 .

[7]  Michael Unser,et al.  On the asymptotic convergence of B-spline wavelets to Gabor functions , 1992, IEEE Trans. Inf. Theory.

[8]  T A Louis,et al.  Random effects models with non-parametric priors. , 1992, Statistics in medicine.

[9]  P. J. Verweij,et al.  Penalized likelihood in Cox regression. , 1994, Statistics in medicine.

[10]  S. Zeger,et al.  A Smooth Nonparametric Estimate of a Mixing Distribution Using Mixtures of Gaussians , 1996 .

[11]  G. Verbeke,et al.  A Linear Mixed-Effects Model with Heterogeneity in the Random-Effects Population , 1996 .

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

[13]  G. Verbeke,et al.  The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data , 1997 .

[14]  M A Newton,et al.  An Estimation Method for the Semiparametric Mixed Effects Model , 1999, Biometrics.

[15]  Wei Shen,et al.  Empirical Bayes Estimation via the Smoothing by Roughening Approach , 1999 .

[16]  John Fox,et al.  Linear Mixed Models , 1999 .

[17]  M Davidian,et al.  Linear Mixed Models with Flexible Distributions of Random Effects for Longitudinal Data , 2001, Biometrics.

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