Analyzing Longitudinal Data Using Regression Splines

Transcription

1 Analyzing Longitudinal Data Using Regression Splines Zhang Jin-Ting Dept of Stat & Appl Prob National University of Sinagpore August 18, 6 DSAP, NUS p.1/16

2 OUTLINE Motivating Longitudinal Data Parametric Mixed Effects Modeling Nonparametric Mixed-Effects Modeling Regression Splines Regression Spline Mixed-effects Modeling Application DSAP, NUS p.2/16

3 Motivating Longitudinal Data ACTG 388 Data: Collected in an AIDS Clinical Trial Group study 166 HIV-1 infected patients treated with highly active antiretroviral therapy for 12 weeks CD4 cell counts for each patient monitored at baseline and at weeks 4, 8, and every 8 weeks thereafter (up to 12 weeks) Missing data presented due to missing clinical visits or other reasons Remark: CD4 cell count, an important marker for assessing immunologic response of an antiviral regimen. DSAP, NUS p.3/16

4 Motivating Longitudinal Data 1 Raw Curves 1 1 Data quite messy No obvious trend CD4 count DSAP, NUS p.4/16

5 Motivating Longitudinal Data Indiviudal Curves 6 Subj 8 6 Subj Subj Subj Subj Subj 59 6 Measurement errors presented Subject effects presented Missing data presented CD4 count DSAP, NUS p.5/16

6 Parametric Mixed Effects Modeling A Parametric Mixed-Effects Model: Response Parametric Fixed-Effect Parametric Random-Effect Measurement Error Parametric fixed-effects: model overall means Parametric random-effects: model individual (subject) effects Measurement errors: model individual errors Individual responses=overall means+individual effects Advantages: Simple and well studied Fitting methods and software available DSAP, NUS p.6/16

7 Parametric Mixed Effects Modeling Difficulties: Appropriate parametric models, e.g., polynomials, needed for fixed-effects Appropriate parametric models, e.g., polynomials, needed for random-effects But in practice, parametric models may NOT available or appropriate DSAP, NUS p.7/16

8 Nonpar. Mixed Effects Modeling A Nonparametric Mixed-Effects Model: Response curve Nonparametric Fixed-Effect curve Nonparametric Random-Effectcurve Measurement Error Nonparametric fixed-effect curve: model overall mean function Nonparametric random-effect curve: model individual (subject) effect curve Measurement errors: model individual errors Individual curve=fixed-effect curve+ random-effect curve Advantages: No parametric models assumed for both Fixed and Random effects Robust against model misspecification DSAP, NUS p.8/16

9 Nonpar. Mixed Effects Modeling Difficulties: Methods and software rather new, still being developed Need to properly choose one smoothing method Need to properly choose the smoothing parameters Remarks: Popular smoothing methods including Local polynomials, Regression splines, Smoothing splines, and Penalized splines well developed for independent data but rather new for longitudinal data analysis Each smoothing method accompanied with one, two or more smoothing parameters Properly choosing smoothing parameters often challenging DSAP, NUS p.9/16

10 Regression Splines A regression spline is a linear combination of a truncated power basis A truncated power basis is a polynomial basis, plus some truncated power basis functions A -degree polynomial basis consists of Truncated power basis functions with knots can be expressed as The truncated function so called since it equals truncated as negative when if positive and DSAP, NUS p.1/16

11 Regression Splines regression spline linear combination of polynomials linear combination of truncated functions 1 (a) A Regression Spline Basis 4 (b) Three Regression Splines.8 2 y x y x Left: A quadratic truncated power basis with knots, and. Right: Three quadratic regression splines with randomly selected coefficients DSAP, NUS p.11/16

12 RS Mixed Effect Modeling A truncated power basis used for modeling Fixed-Effect curve Another truncated power basis used for modeling Random-Effect curve Numbers of basis functions of the truncated power bases are smoothing parameters Smoothing parameters selected using AIC, BIC and other model selection rules When smoothing parameters fixed, the model becomes a Standard Linear Mixed Effects model A standard LME model can be solved using existing methods and software DSAP, NUS p.12/16

13 Application Methodologies applied to the ACTG 388 data: A quadratic truncated power basis used for Fixed-effect curve, another for Random-effect curve For fixed smoothing parameters, SPLUS function lme can be used to fit the model Smoothing parameters selected using the BIC rule 4 basis functions selected for Fixed-effect curve, and another 4 for Random-effect curve DSAP, NUS p.13/16

14 Application Overall Fits CD4 count (a) Fitted individual functions 5 1 CD4 count (b) Fitted mean function with ± 2 SD x 1 4 (c) Fitted covariance function (d) Fitted correlation function 5 1 Covariance 4 3 Correlation DSAP, NUS p.14/16

15 Application Individual Fits Subj Subj Subj CD4 count Subj Subj Subj 59 raw data population individual 5 1 DSAP, NUS p.15/16

16 End of the Talk Thank You DSAP, NUS p.16/16