Function-on-Scalar Regression with the refund Package

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1 University of Haifa From the SelectedWorks of Philip T. Reiss July 30, 2012 Function-on-Scalar Regression with the refund Package Philip T. Reiss, New York University Available at:

2 Function-on-scalar regression with the refund package Philip T. Reiss New York University and Nathan Kline Institute Joint Statistical Meetings San Diego, July 30, 2012 Joint work with Lei Huang (NYU / Johns Hopkins), Lan Huo (NYU), Fabian Scheipl (LMU Munich) Research supported in part by grant DMS , U.S. National Science Foundation 1 / 22

3 2 / 22 Outline The refund package Function-on-scalar regression: raw vs. preprocessed responses Within-function dependence Hypothesis testing

4 3 / 22 Regression functions in the refund package (Crainiceanu et al., 2012) Predictors Scalar Functional Responses Scalar fpcr, (l)pfr, (l)peer, wcr, wnet Functional fosr, pffr pffr

5 4 / 22 A function-on-scalar regression example: Midsagittal corpus callosum width

6 5 / 22 We ll consider corpus callosum width functions derived from 98 controls 70 subjects with very mild dementia, and 28 with mild dementia, based on MRI scans from the Open Access Series of Imaging Studies (OASIS) database (Marcus et al., 2007). Image processing at Nathan Kline Institute: each individual s corpus callosum was extracted by an automated procedure developed by Babak Ardekani; width at each of 99 points, from anterior to posterior, along the midsagittal plane was computed by Alvin Bachman.

7 We thereby obtain 196 corpus callosum width curves (functional responses)... Effect age Effect Control Very mild Mild Position Position... which we are primarily interested in regressing on diagnostic group. 6 / 22

8 7 / 22 Function-on-scalar regression with raw-data responses Given response functions y 1,..., y n each observed at s 1,..., s T S, and scalar predictor matrix X = (x im ) 1 i n,1 m p (usually x i1 1), we consider the repeated-measures varying-coefficient model y i (s t) = x im β m(s t) + ε i (s t), (1) where the error functions ε i arise from a mean-zero process on S. Representing the coefficient functions as β m(s) = θ T mb(s) for some θ m R K (m = 1,..., p), where b(s) = [b 1 (s),..., b K (s)] T denotes basis functions (often B-splines), we reduce (1) to the n T matrix equation Y = XΘB T + E where Y = [y i (s t)] 1 i n,1 t T, Θ (p K ) is the coefficient matrix, B (T K ) is the basis function evaluation matrix, and E = [ε i (s t)] 1 i n,1 t T. Our estimate of Θ is the minimizer of a penalized double SSE, say n T [ ] 2 [y i (s t) (XΘB T ) it ] 2 + λ m θ T mb (s) ds. i=1 t=1 }{{} β m (s)2

9 8 / 22 Criterion n i=1 T [y i (s t) (XΘB T ) it ] 2 + t=1 [ 2 λ m θ T mb (s)] ds can be rewritten in the generalized ridge regression form where vec(y T ) (X B)vec(Θ T ) 2 + vec(θ T ) T (Λ P)vec(Θ T ) = y X B θ 2 + θ T P λ θ, (2) y = vec(y T ), θ = vec(θ T ), X B = X B, [ ] P λ = Λ P = diag(λ 1,..., λ p) b k (s)b l (s)ds For given λ 1,..., λ p, (2) is minimized by ˆθ = (X T BX B + P Λ ) 1 X T By. 1 k,l K.

10 9 / 22 From raw to preprocessed responses Sometimes (e.g., large data sets) one works with preprocessed response functions given by an n K matrix C of basis coefficients such that y i (s) = c T i b(s) for i = 1,..., n. fda package (Ramsay and Silverman, 2005; Ramsay et al., 2009) chooses Θ to minimize the penalized integrated sum of squared errors n [ 2 [c T i b(s) x T i Θb(s)] 2 ds + λ m θ T mb (s)] ds. i=1 Solution can be derived (Reiss et al., 2010) as limit of raw-response fit θ = (X T BX B + P Λ ) 1 X T By for uniform grid s 1,..., s T as T : 1 T X T BX B = 1 T (X 1 B)T (X B) = T (X T X) (B T B) [ ] = (X T 1 T X) b k (s t)b l (s t) T t=1 [ ] (X T X) b k (s)b l (s)ds fosr works with either raw or preprocessed responses. 1 k,l K 1 k,l K.

11 We simulated 3 groups 50 functional responses from y i (s) = µ(s) + β gp(i) (s) + η i (s) }{{} GP(0,σ 2 η ρ s t ) + ξ i (s) }{{} WN(σ 2 ξ =0.22 ) σ η = 0.2 σ η = and estimated µ and β 1, β 2, β 3 using raw vs. preprocessed responses. 10 / 22

12 1000 mean integrated squared error: raw vs. preprocessed σ η = 0.2 σ η = 0.5 T = 30 T = 120 µ β 1 β 2 β µ β 1 β 2 β µ β 1 β 2 β µ β 1 β 2 β / 22

13 12 / 22 Extension to function-on-concurrent-function regression? The raw-response design matrix X (n p) B (T K ) has (i, m) block x im B = [x im b k (s t )] 1 t T,1 k K (1 i n, 1 m p). For the concurrent function-on-function model y i (s) = x im (s)β m (s) + ε i (s) (i.e., time-varying predictors), the (i, m) block becomes [x im (s t )b k (s t )] 1 t T,1 k K. For preprocessed responses, the extension is more complex!

14 13 / 22 Callosum thickness example Intercept Very mild Mild Coefficient function Coefficient function Coefficient function Position Position Position Brain size Female Male Age Coefficient function Coefficient function Coefficient function Position Position Position

15 14 / 22 Residual dependence When minimizing the penalized OLS criterion n y i i=1 2 x im β m + λ m β m(s) 2 ds, where y i = [y i (s 1 ),..., y i (s T )] T, β m = [β m(s 1 ),..., β m(s T )] T, selection of λ 1,..., λ p by REML or GCV is not straightforward since in general Σ = Cov(y i x i1,..., x ip ) I. Three ways to take this within-function dependence into account in smoothness selection: 1. leave-one-function-out cross-validation 2. penalized generalized least squares 3. functional mixed-effects models extensions of repeated-measures methodology to varying-coefficient models.

16 15 / 22 Method 1: Leave-one-function-out cross-validation This is the classical method (Rice and Silverman, 1991; Ramsay and Silverman, 2005): minimize n (y i ŷ ( i) i ) 2. (3) i=1 For given λ, fosr uses the leave-one-function-out lemma y i ŷ ( i) i = (I H ii ) 1 (y i ŷ i ) to compute (3) without refitting. But, there s no standard algorithm for minimizing (3) wrt multiple λ s.

17 16 / 22 Method 2: Penalized generalized least squares Reiss et al. (2010) consider replacing the penalized OLS criterion n y i i=1 2 x im β m + λ m β m(s) 2 ds with the penalized GLS criterion [ n y i i=1 ] T [ x im β ˆΣ 1 m y i ] x im β m + λ m β m(s) 2 ds, where ˆΣ is an estimate of Cov(y i x i1,..., x ip ) (cf. Krafty et al., 2008). Intuition: Equivalent to penalized OLS for prewhitened responses ˆΣ 1/2 y 1,..., ˆΣ 1/2 y n. Viewing these as approximately IID lets us use mgcv (Wood, 2006, 2011) for multiple smoothing parameter selection. Can iterate between updated estimates of Σ and of β 1,..., β p.

18 17 / 22 Method 3: Functional mixed effects Idea: y i (s) = x im β m (s)+η i (s) + ξ i (s), i.e., decompose residual function ε i (s) into random effect function η i (s) + noise function ξ i (s). Express η i (s) as a linear combination of the leading functional principal components. This is implemented for pffr, but not yet in fosr (for version 0.1-6).

19 Details for functional mixed effects The penalized OLS criterion can be written as n y i (x i1 B... x ip B) i=1 θ 1. θ p Consider instead the doubly penalized criterion 2 + λ mθ T mpθ m. β u i {}}{{}}{ X i n θ {}}{ 1 Z i {}}{ y i (x i1 B... x ip B). ( ρ 1 φ 1... u i1 ρ C φ C ). θ p u ic i=1 2 n + λ mθ T mpθ m+λ u u i 2, i=1 where φ j = [φ j (s 1 ),..., φ j (s T )] T is the jth eigenfunction of the residual covariance operator, and ρ j is the jth eigenvector ( u ij s iid). 18 / 22

20 Hypothesis testing Test M 0 vs. M A via pointwise F-statistics (Ramsay and Silverman, 2005) F(s) = [SSEM (s) 0 SSEM (s)]/(p A A p 0 ). SSE MA /(n p A ) Declare significance at t such that F(t) > 100(1 α) percentile, }{{} estimated by permutation under M 0, of max F(s). s Diagnostic group effect F statistics Location 19 / 22

21 20 / 22 Great Books

22 21 / 22 And many thanks to Thank you! Eva Petkova, Lassell Lu and Aaron Chen (NYU), Todd Ogden (Columbia), Ciprian Crainiceanu (Johns Hopkins) and Giles Hooker (Cornell), for valuable discussions; the NIH, for supporting refund via grant 5R01EB (PI: Ogden) and for funding the OASIS project via grants P50 AG05681, P01 AG03991, R01 AG021910, P50 MH071616, U24 RR021382, R01 MH56584; Babak Ardekani, Al Bachman and Sang Han Lee (Nathan Kline Institute), for providing the corpus callosum data derived from the OASIS database.

23 22 / 22 References Crainiceanu, C. M., P. T. Reiss, J. Goldsmith, L. Huang, L. Huo, and F. Scheipl (2012). refund: Regression with functional data. R package version Krafty, R. T., P. A. Gimotty, D. Holtz, G. Coukos, and W. Guo (2008). Varying coefficient model with unknown within-subject covariance for analysis of tumor growth curves. Biometrics 64, Marcus, D., T. Wang, J. Parker, J. Csernansky, J. Morris, and R. Buckner (2007). Open Access Series of Imaging Studies (OASIS): cross-sectional MRI data in young, middle aged, nondemented, and demented older adults. Journal of Cognitive Neuroscience 19(9), Ramsay, J. O., G. Hooker, and S. Graves (2009). Functional Data Analysis with R and MATLAB. New York: Springer. Ramsay, J. O. and B. W. Silverman (2005). Functional Data Analysis (2nd ed.). New York: Springer. Reiss, P. T., L. Huang, and M. Mennes (2010). Fast function-on-scalar regression with penalized basis expansions. International Journal of Biostatistics 6(1), article 28. Rice, J. and B. Silverman (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. Journal of the Royal Statistical Society, Series B 53(1), Wood, S. N. (2006). Generalized Additive Models: An Introduction with R. Boca Raton, FL: Chapman & Hall. Wood, S. N. (2011). Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. Journal of the Royal Statistical Society: Series B 73(1), 3 36.