Analyzing Longitudinal Data Using Regression Splines

Analyzing Longitudinal Data Using
Regression Splines
Zhang Jin-Ting
Dept of Stat & Appl Prob
National University of Sinagpore
August 18, 2006
DSAP, NUS – p.1/16

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

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 120 weeks
CD4 cell counts for each patient monitored at baseline and at weeks 4, 8,
and every 8 weeks thereafter (up to 120 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

Motivating Longitudinal Data
1400
Raw Curves
1200
1000
Data quite messy
No obvious trend
CD4 count
800
600
400
200
0
0 20 40 60 80 100 120
Week
DSAP, NUS – p.4/16

Motivating Longitudinal Data
Indiviudal Curves
600
Subj 8
600
Subj 11
400
400
200
200
0
0 20 40 60 80 100 120
Subj 21
600
0
0 20 40 60 80 100 120
Subj 36
600
400
400
200
200
0
0 20 40 60 80 100 120
Subj 39
600
0
0 20 40 60 80 100 120
Subj 59
600
400
200
Measurement errors presented
Subject effects presented
Missing data presented
0
0 20 40 60 80 100 120
CD4 count
400
200
0
0 20 40 60 80 100 120
Week
DSAP, NUS – p.5/16

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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

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

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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

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

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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
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expressed as
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The truncated function so called since it equals
truncated as negative
when
if
positive and
DSAP, NUS – p.10/16

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Regression Splines
regression spline
linear combination of polynomials
linear combination of truncated functions
1
(a) A Regression Spline Basis
4
(b) Three Regression Splines
0.8
2
y
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
x
y
0
−2
−4
0 0.2 0.4 0.6 0.8 1
x
Left: A quadratic truncated power basis with knots
, and
.
Right: Three quadratic regression splines with randomly selected
coefficients
DSAP, NUS – p.11/16

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

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

Application
Overall Fits
CD4 count
1000
800
600
400
200
(a) Fitted individual functions
0
0 50 100
Week
CD4 count
(b) Fitted mean function with ± 2 SD
450
400
350
300
250
200
150
0 50 100
Week
x 10 4
(c) Fitted covariance function
(d) Fitted correlation function
5
1
Covariance
4
200 2 3
100
Week
0
0
100
50
Week
150
Correlation
0.9
0.8
0.7
200
100
Week
0
0
100
50
Week
150
DSAP, NUS – p.14/16

Application
Individual Fits
Subj 8
600
500
400
300
200
0 50 100
Subj 21
400
300
200
0 50 100
Subj 39
400
300
200
100
0 50 100
CD4 count
600
500
400
300
200
400
300
200
100
400
300
200
100
Subj 11
0 50 100
Subj 36
0 50 100
Subj 59
raw data
population
individual
0
0 50 100
Week
DSAP, NUS – p.15/16

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