Nonparametric Mixed-Effects Models for Longitudinal Data Zhang Jin-Ting Dept of Stat & Appl Prob National University of Sinagpore University of Seoul, South Korea, 7 p.1/26
OUTLINE The Motivating Data Various Parametric/Nonparametric ME Models Various Fitting Approaches Smoothing Parameter Selection Real Data Application Other ME Models University of Seoul, South Korea, 7 p.2/26
The Motivating Data The ACTG 388 Data (Park and Wu 4) 1 Raw Curves 1 1 8 CD4 count 6 Response CD4 cell counts Covariate Time 2 4 6 8 1 12 Week patients. Total Number of Observations University of Seoul, South Korea, 7 p.3/26
The Motivating Data Six Selected Subjects 6 Subj 8 6 Subj 11 2 4 6 8 1 12 Subj 21 6 2 4 6 8 1 12 Subj 36 6 2 4 6 8 1 12 Subj 39 6 2 4 6 8 1 12 Subj 59 6 2 4 6 8 1 12 CD4 count 2 4 6 8 1 12 Week University of Seoul, South Korea, 7 p.4/26
The Motivating Data Pointwise Means 1 Pointwise Raw Means 1 1 8 6 CD4 count 6 2 4 6 8 1 12 Week University of Seoul, South Korea, 7 p.5/26
The Motivating Data For the ACTG 388 data, the following Population-Mean ME Model is proper: GP where : measurement errors for -th measurement of -th subject : smooth fixed-effects function : smooth random-effects function of -th subject : individual function of -th subject Aim: Estimate, and ; Predict and University of Seoul, South Korea, 7 p.6/26
Various Parametric/Nonparametric ME Models Parametric Mixed-Effects Models In classical longitudinal data analysis, parametric mixed-effects models are often used. Linear Mixed-effects Models: i.i.d : Parametric fixed-effects : Parametric random-effects : Measurement errors University of Seoul, South Korea, 7 p.7/26
Various Parametric/Nonparametric ME Models Nonlinear Mixed-effects Models: i.i.d where is some known nonlinear function. See Davidian and Giltinan (1995), Vonesh and Chinchilli (1996), among others. University of Seoul, South Korea, 7 p.8/26
Various Parametric/Nonparametric ME Models Advantages: May take many covariates into account Easy to fit and analyze via EM algorithm Methodologies well developed Disadvantages: Need valid parametric assumptions May lead to misleading conclusions Not robust against model misspecification University of Seoul, South Korea, 7 p.9/26
Various Parametric/Nonparametric ME Models Nonparametric Mixed-Effects Models Recently, various nonparametric mixed-effects models are proposed. Population-Mean ME Model GP GP : nonparametric fixed-effects function : nonparametric random-effects function : Measurement error process See Zhang et al. (1998), Rice and Wu (1), Wu and Zhang (2, 6) among others University of Seoul, South Korea, 7 p.1/26
Various Parametric/Nonparametric ME Models Varying Coefficient ME Model GP GP where is the vector of some unknown smooth coefficient functions. See Wu and Zhang (6) University of Seoul, South Korea, 7 p.11/26
Various Parametric/Nonparametric ME Models Random Coefficient ME Model GP GP -th random coefficient function. is the where the University of Seoul, South Korea, 7 p.12/26
Various Parametric/Nonparametric ME Models Advantages: Flexible to fit longitudinal data Robust against model misspecification Disadvantages: Only involve a few covariates May computationally intensive Methodologies under developing See Wu and Zhang (6) and the references therein. University of Seoul, South Korea, 7 p.13/26
Various Fitting Approaches In the above nonparametric ME models, the nonparametric components such as should be fitted using some smoothing technique. The major smoothing techniques include Regression Spline Method (Eubank, 1999) and Smoothing Spline Method (Wahba, 199, Green and Silverman 1994) Penalized Spline Method (Ruppert, Wand and Carroll, 3) Local Polynomial Method (Fan and Gijbels, 1996) To adopt the above smoothing methods, we shall use the Population Mean ME model as an example. University of Seoul, South Korea, 7 p.14/26
Various Fitting Approaches respectively. the The Regression Spline Method Key Ideas: Approximate the nonparametric FE component by a regression spline, and approximate the nonaparametric RE components by regression splines and are two regression spline bases of dimensions and Then Population-Mean ME model can be approximated by where where and. This is a Standard Linear Mixed-effects (LME) Model. See Rice and Wu (1), Wu and Zhang (6, Chapter 5) for more details. University of Seoul, South Korea, 7 p.15/26
Various Fitting Approaches The Smoothing Spline Method Key Ideas: Take into account the roughness of and simultaneously via introducing roughness penalty. For example, for the cubic smoothing spline method, it is to find and to minimize the following criterion: Loglik where Loglik is the log-likelihood function evaluated at the design time points, and are the associated smoothing parameters. See Brumback and Rice (1998), Wu and Zhang (6, Chapter 6) for more details. University of Seoul, South Korea, 7 p.16/26
Various Fitting Approaches The Penalized Spline Method Key Ideas: Take into account the roughness of approximating and by regression splines penalizing the associated coefficients. That is to find following criterion: and and simultaneously via and, and to minimize the Loglik where Loglik is the log-likelihood function evaluated at the design time points and based on the regression spline approximations, and are the associated smoothing parameters. See Wu and Zhang (6, Chapter 7) for more details. University of Seoul, South Korea, 7 p.17/26
Various Fitting Approaches The Local Polynomial Method Key Ideas: At any fixed time point, the Population-Mean model can be approx- imated by a standard LME model via approximating and by polynomials of some order. The resulting LME can be fitted by the existing approaches for LME models. See Wu and Zhang (2, 6, Chapter 4) for more details. University of Seoul, South Korea, 7 p.18/26
Smoothing Parameter Selection Mixed Effects Fits Let the above methods and for the Population Mean Model, the Fixed-Effects Fits at can be expressed as be all the design time points. For where is the smoother matrix for, and the Random Effects Fits at is where the smoother matrix for all the random-effects evaluated at the design time points. University of Seoul, South Korea, 7 p.19/26
Smoothing Parameter Selection Goodness of Fit and Model Complexity Smoothing Parameter Selection attempts to trade off Goodness of Fit and Model Complexity. Goodness of Fit can be measured by the Log-likelihood: Loglik Const Cov (1) The larger the Loglik, the better Goodness of Fit of the modeling, indicating that the data are fitted very closely by the model. Model Complexity can be measured by the Trace of Smoother Matrix Model Complexity for Fixed-Effects df tr, indicating how complicate of the model is for fitting the fixed effects. Model Comlexity for Random-Effects df tr, indicating how complicate of the model is for fitting the random effects. University of Seoul, South Korea, 7 p.2/26
Smoothing Parameter Selection AIC and BIC A Criterion should trade off Goodness of Fit and Model Complexity for Fixed-effects and Random-effects. For example, for the Population Mean Model and for the regression spline method, we shall define AIC Loglik df df, BIC Loglik df df, where and are the number of knots for and. For the smoothing spline and penalized spline methods, and should be replaced by and respectively. University of Seoul, South Korea, 7 p.21/26
Real Data Application AIC and BIC for ACTG 388 Data 4.465 x (a) AIC 14 4.46 4.455 4.45 4.445 4.44 4.435 Value 4.72 x (b) BIC 14 4.7 4.68 4.66 4.64 K v =1 K v =2 K v =3 4.43 2 4 6 8 1 4.62 2 4 6 8 1 K seems a good choice University of Seoul, South Korea, 7 p.22/26
Real Data Application Overall Fits: CD4 count 1 8 6 (a) Fitted individual functions 5 1 Week CD4 count (b) Fitted mean function with ± 2 SD 45 35 3 25 15 5 1 Week x 1 4 (c) Fitted covariance function (d) Fitted correlation function 5 1 Covariance 4 3 Correlation.9.8 2 1 Week 1 5 Week 15.7 1 Week 1 5 Week 15 University of Seoul, South Korea, 7 p.23/26
Real Data Application Individual Fits: Subj 8 6 5 3 5 1 Subj 21 3 5 1 Subj 39 3 1 5 1 CD4 count 6 5 3 3 1 3 1 Subj 11 5 1 Subj 36 5 1 Subj 59 raw data population individual 5 1 Week University of Seoul, South Korea, 7 p.24/26
Other ME Models The Methodologies proposed above can be applied to other ME Models, e.g.: Semiparametric ME Models: GP Generalized Nonparametric ME Model: is known. where Other ME models.. University of Seoul, South Korea, 7 p.25/26
End of the Talk Thank You University of Seoul, South Korea, 7 p.26/26