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Self-Modeling Regression for
Longitudinal Data with Time-
Invariant Covariates
Naomi S. Altman
Penn State University
naomi_altman@stat.psu.edu
Julio Villarreal
EdVision
Experiments in Which the
Response is a Curve
• growth curves
• longitudinal
data
• blood
concentration
curves
Features
• multiple curves with similar shape
• covariates or treatments (“time
invariant”)
Objectives
• flexible nonparametric model for shape
• interpretable parameters for treatment
effects
• test statistics for treatment effects
• test statistics for effect of treatment on
shape
Shape-Invariant Regression
yi (t) = ααααi0 + Ai1 µµµµ0(bi0 +Bi1t) + εεεεit
(Lawton et al 1972)
• Common shape: µµµµ0 (t)
• Parameters: θθθθi’ = (ααααi0, ααααi1, ββββi0, ββββi1)
• Ai1= exp(ααααi1)
• Bi1= exp(ββββi1)
Why Use Shape-Invariant
Regression?
• If shape is not of interest:
– treatment effects can be summarized by a model
for the parameters
– common summaries (e.g. height of maximum,
time to maximum) depend on shape only through
functionals which are constant over “i”
• Shape can be estimated nonparametrically
– this allows a test of treatment effect on shape
– a functional form for shape may be suggested
Outline
• Nestling Growth Experiments
• Fitting a SIM model (algorithm)
• Testing effects (simulation)
• Results for Tree Swallow Growth
• Does the model fit?
Nestling Growth Experiments
• Several data sets and experimental conditions and covariates. Many response variables.
• Questions of interest:
a) Are there treatment effects?
b) Does the shape of the growth curve vary
with response variable?
c) Do treatment effects for the response variables differ? - e.g. If a treatment delays growth of tarsus length, does it delay growth of head circumference.
Experiment Design
For Tree Swallow Study
• 2 to 8 times per curve (mean 6)
• 297 nestlings
• A split plot design with whole plot
(nest) factors:
– covariate HatchDate
– dietary supplement/none
(courtesy of Matt Wasson, Cornell)
Growth Curves
Age
Tarsus Length
0 2 4 6 8 10 12 14
46
810
12
Age
Wing Length
0 2 4 6 8 10 12 14
10
20
30
40
50
60
Age
Mass
0 2 4 6 8 10 12 14
510
15
20
25
Age
Head Diameter
0 2 4 6 8 10 12 14
15
20
25
tarsus
transformed time
Growth Curve
0 2 4 6
45
67
89
wing
transformed time
Growth Curve
0.0 0.05 0.10 0.15 0.20 0.25 0.30
-5*10^13
05*10^13
10^14
mass
transformed time
Growth Curve
0 5*10 -̂6 10 -̂5 1.5*10 -̂5
0.2
0.4
0.6
0.8
1.0
head
transformed time
Growth Curve
2 4 6 8 10
12
14
16
18
tarsus lengthwing lengthtarsus length wing length
mass masshead head
size
age transformed time
Raw Data Fitted Growth Curve
Back to
Shape-Invariant Regression
yi (t) = ααααi0 + Ai1 µµµµ0(ββββιιιι0 +Bi1t) + εεεεit
(Lawton et al 1972)
• Common shape: µµµµ0 (t)
• Parameters: θθθθi’ = (ααααi0, ααααi1, ββββi0, ββββi1)
• Ai1= exp(ααααi1)
• Bi1= exp(ββββi1)
Fitting the SIM Model
Starting with θθθθij = (0,0,0,0)
1. Let Y*ij(t) = (Yij(t) – aaaaij0)/exp(aij1) “aligned
response”
t* = bij0 + exp(bij1 )t “aligned time”
2. Use a nonparametric smoother to regress Y* on t*.
Call this m(t*).
3. Use nonlinear mixed models to fit the model
yij (t) = ααααij0 + Aij1 m(ββββijijijij0 +Bij1t*) + εεεεijt
4. Check for convergence. If not converged, go to 1.
Notes
• In fitting a complicated SIM model such as the Bird Growth Data, it is convenient to fit first without the linear model for the parameters. In the last step, the linear model for the parameters can be fitted.
• The convergence criterion used was the change in predicted values.
• It is convenient to smooth using a penalized cubic spline. This can be done using a linear mixed models routine.
Penalized Cubic Spline
• Pick a dense set of equally spaced time points – in a typical study with 4-12 time points per curve, 20 points will do.
• Fit a linear mixed model:
cubic polynomial in time is the fixed effect
are the random effects
• The result is similar to a smoothing spline, but computationally simpler.
(Carroll and Ruppert, 1997; Eilers and Marx, 1997)
tj
K
j j
i
i i ttt ητγδµ +−+= +== ∑∑ 3
1
3
10 )()(
3)( +− it τ
subject to γ’γ ≤ C
It turns out that this is readily fitted by considering the
δδδδ’s to be fixed effects and the γγγγ’s to be random effects with common variance .
Why? Computationally very simple and fast compared
to other smoothing techniques.
This has two nice consequences:
The shape is a polynomial if =0.
The treatment effects on the curves can be readily
modeled by using the same linear model that we used
for the parameters.
2
γσ
2
γσ
Fitting the Penalized Spline
Mixed Models for the Parameters
• Suppose that we now add another level
to the model:
• yi (t) = ααααi0 + Ai1 µµµµ0(ββββi0 +Bi1t) + εεεεit
θθθθi’ = (ααααi0, ααααi1, ββββi0, ββββi1)
• Ai1= exp(ααααi1) Bi1= exp(ββββi1)
θθθθij = Xjdj + Zj Dj• where Xj and Zj are observed
predictors; dj and Dj are fixed and
random effects
Estimation of the Mixed Model
Parameters
is readily incorporated into the algorithm by either:
• adding the mixed model to the NLME step during iteration
(unconditional method)
• iterating the basic SIM model until convergence and then fitting the mixed model in the final NLME step
(conditional method)
Testing the Mixed Model
Parameters
• Conditional on the fitted shape, we
have a NLME. So, the LRT from the
conditional method should be
asymptotically chi-squared.
• How asymptotic is this?
• What about the unconditional method?
• Why not fit the entire model as one
huge NLME?
Distribution of the Conditional
LRT
Time
mu(t)
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
Curve 1
Curve 2
Curve 3
Simulation Study
0.100.05σσσσεεεε
0.100.05σσσσb1
0.300.10σσσσb0
0.100.05σσσσa1
0.300.10σσσσa0
Large
Error
Small
Error
How Good is the Fit?
The ASE is an order of magnitude smaller than the fit
obtained by fitting each curve individually.
23
45
67
20 30 50
Number of Curves
AS
E*1
0000
23
45
67
23
45
67
20 30 50
Number of CurvesA
SE
*100
00
23
45
67
23
45
67
20 30 50
Number of Curves
AS
E*1
0000
23
45
67
510
1520
2530
20 30
Number of Curves
AS
E*1
0000
510
1520
2530
510
1520
2530
20 30
Number of Curves
AS
E*1
0000
510
1520
2530
510
1520
2530
20 30
Number of Curves
AS
E*1
0000
510
1520
2530
20 points/curve
ASE*10000
5 10 15 20 25 30
ASE*10000
2 3 4 5 6 7
ASE*10000
2 3 4 5 6 7
ASE*10000
5 10 15 20 25 30
ASE*10000
5 10 15 20 25 30
20 30 50
Number of CurvesASE*10000
2 3 4 5 6 7
20 30 50
Number of Curves
20 30 50
Number of Curves
30 points/curve
20 30 20 30 20 30
How Good are the Parameter
Estimates?
0.2
0.4
0.6
0.8
1.0
20 30 50
Number of Curves
Cor
rela
tion
0.2
0.4
0.6
0.8
1.0
Curve 1- Small Variance A0
0.2
0.4
0.6
0.8
1.0
20 30 50
A1
Number of Curves
Cor
rela
tion
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
20 30 50
B0
Number of Curves
Cor
rela
tion
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
20 30 50
B1
Number of Curves
Cor
rela
tion
0.2
0.4
0.6
0.8
1.0correlation
0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
20 30
B1
Number of Curves
Cor
rela
tion
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
20 30
B1
Number of Curves
Cor
rela
tion
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
20 30
Number of Curves
Cor
rela
tion
0.2
0.4
0.6
0.8
1.0
Curve 1- Large Variance A0
0.2
0.4
0.6
0.8
1.0
20 30
A1
Number of Curves
Cor
rela
tion
0.2
0.4
0.6
0.8
1.0
a1 b1 a1 b1
Small Variance Large Variance
a0 b0 a0 b0
Correlation Among Estimates:
-0.5
0.0
0.5
r(A0,A1) r(A0,B0) r(A0,B1) r(A1,B0) r(A1,B1) r(B0,B1)
Cor
rela
tion
Curve 1 - Small Variance
-0.5
0.0
0.5
r(A0,A1) r(A0,B0) r(A0,B1) r(A1,B0) r(A1,B1) r(B0,B1)
Cor
rela
tion
Curve 1 - Large Variance
correlation
-0.5 0.0 0.5
correlation
-0.5 0.0 0.5
r(a0,a1) r(a0,b0) r(a0,b1) r(a1, b0) r(a1, b1) r(b0,b1)
r(a0,a1) r(a0,b0) r(a0,b1) r(a1, b0) r(a1, b1) r(b0,b1)
Distribution of the LRT
m=n=20 m=30 n=50
The top curves are the observed
percentiles of the CLRT versus
chi-square
The lower curves are the
observed percentiles of the
CLRT versus the LRT with the
correct parametric form.
chi-square percentile
CLRT percentile
CLRT percentile
CLRT percentile
CLRT percentile
parametric LRT percentile
Power of the CLRT
(level=.05)
89%87%1.0 σ
36%33%0.5 σ
unconditionalconditionalshift
99%91%1.0 σ
70%67%0.5 σ
unconditionalconditionalshift
20 curves, 20 time points
50 curves, 30 time points
Why not fit the whole thing as
one big mixed model?
SIM Model for the Nestling
Growth Study
yijk (t) = ααααij0 +exp( ααααij1)µµµµ0 (exp( ββββij1 )t) + εεεεijt
θθθθij =γγγγ0 + φφφφi(HatchDateij) + Treatmentij
HatchDate and Treatment are time invariant
and are applied to every bird in the nest
Note: ααααijk ↑ larger birds
ββββij1 ↑ faster growth
Tarsus Length versus Age
Tarsus
Time
Tarsus
0 2 4 6 8 10 12 14
46
810
12
Tarsus
transformed time
Transform
ed Tarsus
0 1 2 3 4
45
67
8
transformed time
Growth Curve
0 2 4 6
45
67
89
time
size
2 4 6 8
46
810
12
26 knots
Random effects only
Some individual
growth curves
Aggregate Aligned
Parameters versus Hatch Datea0
Hatch Date
a0
30 40 50 60
-1.5
-0.5
0.5
a1
Hatch Date
a1
30 40 50 60
-10^-12
-4*10^-13
04*10^-13
b1
Hatch Date
b1
30 40 50 60
-0.2
-0.1
0.0
0.1
Hatch Date
Hatch Date
Hatch Date
Parameters versus Treatment
-1.5
-0.5
0.5
1.0
Control Calcium
a0
a0
-3*10^-16
02*10^-16
Control Calcium
a1
a1
-0.2
-0.1
0.0
0.1
Control Calcium
b1
b1
cor(a0,a1)=.88
cor(a0, b1)=.60
control calcium control calcium
control calcium
ao
-1.5 -0.5 0.5 1.0
a1*1016
-3 -1 1
b1
-0.2 -0.1 0.0 0.1
SIM Model for the Nestling
Growth Study
yijk (t) = ααααij0 +exp( ααααij1)µµµµ0 (exp( ββββij1 )t) + εεεεijt
θθθθij =γγγγ0 + ρρρρ1 HatchDateij + ρρρρ2222 HatchDateij2
+ Treatmentij + ζζζζij
Conditional Likelihoods
• Conclusion: Both hatch date (quadratic) and
treatment have an effect on nestling growth.
• Similarly, despite the small variance
component for αααα1, the fit is significantly worse without it.
0.04
0.07
0.02
p-value
19-1832full
13-1839treatment
16-1836hatch date
10-1843null
d.f.likelihoodmodel
Does the Model Fit?
• Does the Treatment Affect Shape?
• A simple idea: Fit a linear mixed model
to the LME for shape.
Does the Model Fit?
• Does the Treatment Affect Shape?
• A simple idea: Fit a linear mixed model
to the LME for shape.
Crainiceanu and Ruppert (2003) show that the LRT cannot be
used to test for polynomial versus p-spline, unless the design
matrices are orthogonalized.
Xu (2003) found that for test equality of curves, P-spline fit of
full model under the null hypothesis is WORSE than the fit of
the null model (although the models are nested) unless the
design matrices are orthogonalized.
Does the Model Fit?
• Does the Treatment Affect Shape?
• A simple idea: Fit a linear mixed model
to the LME for shape.
Crianiceanu and Ruppert (2003)
Xu (2003)
Good News: There is never a shortage of research problems.
Does the Model Fit?
∆LRT=56
P<0.05 (for d.f. < 40)
estimated d.f. @ 5
Time
Why consider the SIM Model?
• can be used in a variety of problems:– growth
– sera concentration (hormones, drugs)
– bio-equivalence
– materials deformation
• more flexible than polynomial or other parametric fits
• just as easy to use and interpret as parametric nonlinear mixed model
• can be used to test goodness-of-fit of parametric models (particularly easy for polynomials)
• can be used to suggest parametric shapes
• can be used to compare across curves with different shapes but similar treatment effects
Main References
• Crainiceanu et al (2005) Biometrika
• Ke andWang, (2001) JASA
• Kneip and Gasser (1998)
• Lawton, Sylvestre, and Maggio (1972)
Technometrics
• Lindstrom (1995) Statistics in Medicine
• Murphy and van der Vaart (2000) JASA
And many thanks to ...
Chuck McCulloch (penalized splines)
Matt Wasson (data)
Doug Bates and Jose Pinheiro (lme)
JCSS editors, associate editors and
reviewers
The awards committee
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