Longitudinal Penalized Functional Regression

Longitudinal Penalized Functional Regression Jeff Goldsmith Johns Hopkins Bloomberg School of Public Health

Joint work with Ciprian Crainiceanu, Brian Caffo, and Daniel Reich

Outline

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Longitudinal Functional Data

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Real Data Example

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Longitudinal Penalized Functional Regression

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Application Results

microns2 ms 1.0 1.5 2.0 2.5 3.0

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Cognitive Function Score 15 0.5

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Data Structure

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Data Structure

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Subject 1; Visit 1 Subject 1; Visit 2 Subject 1; Visit 3 Subject 1; Visit 4 Subject 1; Visit 5

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Neuroimaging Application I I I

White vs. Grey Matter Multiple Sclerosis Diffusion Tensor Imaging [4]

Outcome and Predictors I

Corpus callosum and corticospinal tract profiles

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PASAT-3

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FA

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FA Profile

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Distance Along Tract

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Study Population

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100 subjects (66 women; 34 men)

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Age between 21 and 70 years at first visit

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Between 2 and 8 visits each (median = 3)

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340 total visits


Observe data [Yij , Wij1 (s), . . . , WijL (s), Xij ]. We propose the longitudinal functional regression model [2] Yij

∼ EF(µij , η)

g(µij ) = Xij β + Zij bi +

L Z X l=1

1

Wijl (s)γl (s)ds

0

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Adds random effects to functional regression

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Adds functional effects to GLMMs

(1)

Functional Contribution

For the functional contributions

R1 0

Wijl (s)γl (s)ds

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Express (or estimate) the functional predictors using a principal components decomposition.

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Express the functional coefficient in terms of a spline basis and induce smoothness via a mixed-effects model [6].

Functional Contribution: Principal Components Decomposition Let Wl

Σ (s, t) = Cov[Wijl (s), Wijl (s)] =

∞ X

λkl ψkl (s)ψkl (s),

k =1

with λ1l ≥ λ2l ≥ . . . non-increasing eigenvalues and ψ l (·) = {ψkl (·) : k ∈ Z+ } corresponding eigenfunctions. Approximate the functional predictors using Wijl (s) = µl (s) +

Kw X k =1

cijkl ψkl (s).

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Cognitive Function Score

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Functional Contribution: Predictor Estimation

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Functional Contribution: γl (s)

Let φ(s) = {φ1 (s), φ2 (s), . . . , φKg (s)} be a truncated power series spline basis, so that γl (s) = φ(s)g l = g1l + g2l t +

Kg X

gkl (t − κk )+

k =3 K

g where g l = {g1l , . . . , gKg l }T and {κk }k =3 are knots.

Functional Contribution

The integral Z

R1 0

Wijl (s)γl (s)ds can now be written

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Z Wijl (s)γl (s)ds = al +

0

0

Z 0

1

c 0ijl ψ T (s)φ(s)g l ds = al + c 0ijl M l g l

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W l (s)γl (s)ds = al + C l M l g l


Let I

X = [1 X (C 1 M 1 )[,1:2] . . . (C L M L )[,1:2] ];

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Z = [Z1 (C 1 M 1 )[,3:Kg ] . . . (C L M L )[,3:Kg ] ];

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β = [α, β, g11 , g21 , . . . , g1L , g2L ];

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g g u = [{bi }Ii=1 , {gk 1 }k =3 , . . . , {gkL }k =3 ]

K

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Longitudinal Penalized Functional Regression Using all the above, the longitudinal generalized functional regression model can be posed: Y

∼ EF (µ, γ)

g(µ) = X β + Z b +

L Z X l=1

1

Wijl (s)γl (s)ds

0

= Xβ + Zu    σb2 I I 0     0   0    u ∼ N  ..  ,  ..  .   . 0 0

0 ... 0 2 σg 1 I Kg −2 0 .. .. . . 0 . . . σg2 L I Kg −2

    .  

Joint Modeling

! !2g

g C ! !2X !2Y

Y

!2b b

W

Joint Modeling

h i ∼ N X ij β + Z ij b + c 0ij Mg, σY2 ; σY2 ∼ IG [AY , BY ] h i Wij (t) ∼ N µ(t) + c 0ij ψ(t)T , σX2 I ; σX2 ∼ IG [AX , BX ]

Yij c 0ij g b β

∼ N [0, Λ] ; λk ∼ IG [Aλ , Bλ ] for 1 ≤ k ≤ Kw ∼ N 0, σg2 D ; σg2 ∼ IG Ag , Bg ∼ N h0, σb2 I i; σb2 ∼ IG [Ab , Bb ] ∼ N 0, σβ2 I

Application

Recall our application: I

100 MS patients with 340 total visits

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Regress PASAT-3 on corpus callosum and corticospinal tract profiles

Application Results

Multivariate Analysis

Univariate Analyses

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γ(t)

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Likelihood-Based 95% Confidence Interval Posterior Mean 95% Credible Interval

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γ(t)

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PD of Right Corticospinal

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MD of Corpus Callosum

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PD of Right Corticospinal

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MD of Corpus Callosum

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Software

model = lpfr(Y, subject, covariates, funcs, ...) I

Available in refund package on CRAN [5]

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Returns γˆ (s), confidence intervals, random effects, spline and PC bases.

Future Work

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Tailored PC decompositions [1, 3]

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Images as predictors

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Expanded software capabilities

Acknowledgements and Contact

This work was supported in part by Award Number R01NS060910 from the National Institute Of Neurological Disorders And Stroke and by the Intramural Research Program of the National Institute of Neurological Disorders and Stroke.

Contact : [email protected] Website : www.biostat.jhsph.edu/~jgoldsmi

References C.Z. Di, C.M. Crainiceanu, B.S. Caffo, and N.M. Punjabi. Multilevel functional principal component analysis. Annals of Applied Statistics, 4:458–288, 2009. J. Goldsmith, C. Crainiceanu, B. Caffo, and D. Reich. A case study of longitudinal association between disability and neuronal tract measurements. Under Review, 2011. S. Greven, C. Crainiceanu, B. Caffo, and D. Reich. Longitudinal functional principal component analysis. Electronic Journal of Statistics, 2010. S. Mori and P. Barker. Diffusion magnetic resonance imaging: its principle and applications. The Anatomical Record, 257:102–109, 1999. P. Reiss, L. Huang, and J. Goldsmith. refund: Regression with Functional Data, 2010. R package version 0.1-2. D. Ruppert, M.P. Wand, and R.J. Carroll. Semiparametric Regression, volume 66. Cambridge: Cambridge University Press, 2003.

Thank You