Restricted Likelihood Ratio Tests for Functional Effects in the - PowerPoint PPT Presentation

Restricted Likelihood Ratio Tests for Functional Effects in the Functional Linear Model by Swihart, Goldsmith, and Crainiceanu (2014) September 1, 2017 1 / 16

Modeling Framework Consider { Y i , X i , W i } modeled with a Functional Linear Model (FLM) � E [ Y i ] = α + X i β + W i ( s ) γ ( s ) ds (1) Y i : Continuous, scalar response for the i -th subject X i : Non-functional covariates with coefficients β W i ( s ): Functional predictor for s ∈ [0 , 1] α : Mean parameter γ ( s ): Coefficient function of interest Objective: Determine if a functional predictor should be included in FLM. 2 / 16

Hypothesis Test 1. Test of functional form H 0 : E ( Y i ) = α + X i β + ¯ W i β W (2) � H A : E ( Y i ) = α + X i β + W i ( s ) γ ( s ) ds Does γ ( s ) need to be written as a function, or is the mean sufficient? Equivalently, H 0 : γ ( s ) = c vs H A : γ ( s ) � = c for all c ∈ R . 3 / 16

Hypothesis Test 2. Test of inclusion � H 0 : E ( Y i ) = α + X i β + W i 1 ( s ) γ 1 ( s ) ds (3) � � H A : E ( Y i ) = α + X i β + W i 1 ( s ) γ 1 ( s ) ds + W i 2 ( s ) γ 2 ( s ) ds Is the second functional predictor necessary? Equivalently, H 0 : γ 2 ( s ) = 0 vs H A : γ s � = 0. 4 / 16

Outline of Testing Procedure The authors propose 2 testing procedures. The basic procedure is: Step 1. Decompose the functional predictor W i ( s ) using functional principal components (FPC). Step 2. Express the coefficient function, γ ( s ), using basis functions. Step 3. Rewrite the FLM as a linear model or linear mixed model. Step 4. Test H 0 under the linear (mixed) model using a (restricted)-likelihood ratio test. 5 / 16

Method 1: Functional Principal Components Reg. (FPCR) Step 1. Use FPC to decompose the functional predictor W i ( s ). ∞ � cov [ W i ( s ) , W i ( s ′ )] = λ k ψ k ( s ) ψ k ( s ′ ) k =1 By a truncated Karhunen-Lo´ eve approximation: K w � W i ( s ) = µ ( s ) + c ik ψ k ( s ) k =1 λ k : (Non-decreasing) k -th eigenvalue ψ k : k -th eigenfunction µ ( s ): Mean function � c ik = { W i ( s ) − µ ( s ) } ψ k ( s ) ds ; k -th score for i -th subject. K w : Truncation parameter 6 / 16

Method 1: Functional Principal Components Reg. (FPCR) Deviating from standard FPCR, subtract the subject-specific predictor means. K g W i ( s ) − ¯ � c ∗ W i = ik ψ k ( s ) k =1 W i : Predictor mean for i -th subject, defined as ¯ ¯ � W i = W i ( s ) ds { W i ( s ) − ¯ � c ∗ ik = W i } ψ k ( s ) ds This is important for re-formulating the linear model to test H 0 . 7 / 16

Method 1: Functional Principal Components Reg. (FPCR) Step 2. Express γ ( s ) using basis set φ ( s ). Define the PC basis φ ( s ) = { φ 1 ( s ) , . . . , φ K g ( s ) } , and let K g � γ ( s ) = φ ( s ) γ = γ k φ k ( s ) k =1 Where γ = { γ 1 , . . . , γ K g } are fixed basis coefficients. 8 / 16

Method 1: Functional Principal Components Reg. (FPCR) Step 3. Rewrite the FLM as a linear model. � E [ Y i ] = α + X i β + W i ( s ) γ ( s ) ds � = α + X i β + ¯ c ∗ T ψ T ( s ) φ ( s ) γ ds W i γ 0 + i W i γ 0 + c ∗ T = α + X i β + ¯ γ i Let β T = [ α, β, γ 0 , γ ] and X [ i , ] = � � c ∗ T Then, 1 X i i E ( Y ) = X β 9 / 16

Method 1: Functional Principal Components Reg. (FPCR) Step 4. Test under the linear model E [ Y i ] = α + X i β + ¯ W i γ 0 + c ∗ T γ . i 1. Test of Functional form H 0 : E ( Y i ) = α + X i β + ¯ W i β W ⇔ γ = 0 � H A : E ( Y i ) = α + X i β + W i ( s ) γ ( s ) ds ⇔ γ � = 0 2.Test of Inclusion � H 0 : E ( Y i ) = α + X i β + W i 1 ( s ) γ 1 ( s ) ds ⇔ γ 0 = 0 and γ = 0 � � H A : E ( Y i ) = α + X i β + W i 1 ( s ) γ 1 ( s ) ds + W i 2 ( s ) γ 2 ( s ) ds ⇔ γ 0 � = 0 or γ � = 0 10 / 16

Method 2: Penalized Functional Regression (PFR) Step 1. Use FPC to decompose the functional predictor W i ( s ). ∞ � cov [ W i ( s ) , W i ( s ′ )] = λ k ψ k ( s ) ψ k ( s ′ ) k =1 By a truncated Karhunen-Lo´ eve approximation: K w � W i ( s ) = µ ( s ) + c ik ψ k ( s ) k =1 λ k : (Non-decreasing) k -th eigenvalue ψ k : k -th eigenfunction µ ( s ): Smooth mean function � c ik = { W i ( s ) − µ ( s ) } ψ k ( s ) ds ; k -th score for i -th subject. K w : Truncation parameter 11 / 16

Method 2: Penalized Functional Regression (PFR) Step 2. Express γ ( s ) using basis φ ( s ). Define the B-spline basis φ ( s ) = { φ 1 ( s ) = 1 , φ 2 ( s ) , . . . , φ K g ( s ) } , and let K g � γ ( s ) = φ ( s ) g = γ 0 + g k φ k ( s ) k =1 Where g = { γ 0 , g 1 , . . . , g K g } are basis coefficients, γ 0 is fixed, and g k are random (from mixed models). Use a modified first-order random walk prior for g k , where g 1 ∼ N (0 , σ 2 g ) and g l ∼ N ( g l − 1 , σ 2 g ). 12 / 16

Method 2: Penalized Functional Regression (PFR) Step 3. Rewrite the FLM as a linear mixed model. � E [ Y i ] = α + X i β + W i ( s ) γ ( s ) ds � c ∗ T ψ T ( s ) φ ( s ) γ ds = α + X i β + a + i = α + X i β + a + c ∗ T Mg i � � Where M [ m , n ] = ψ m ( s ) φ n ( s ) ds and a = µ ( s ) γ ( s ) ds . 13 / 16

Method 2: Penalized Functional Regression (PFR) In matrix notation, let β T = [ α + a , β, γ 0 ] � � ( c ∗ T X [ i , ] = 1 X i M ) [1] i Z [ i , ] = ( c ∗ T M ) [2: K g ] i u T = { g k } K g k =1 Then, E [ Y | X , u ] = X β + Zu u ∼ N ( 0 , σ 2 g D ) Where D is the penalty matrix induced by the random walk prior. 14 / 16

Method 2: Penalized Functional Regression (PFR) Step 4. Test under the linear mixed model g D ) , β T = [ α, β, γ 0 , γ ] E [ Y | X , u ] = X β + Zu , u ∼ N ( 0 , σ 2 1. Test of Functional form H 0 : E ( Y i ) = α + X i β + ¯ ⇔ σ 2 g = 0 W i β W � ⇔ σ 2 H A : E ( Y i ) = α + X i β + W i ( s ) γ ( s ) ds g � = 0 2.Test of Inclusion � H 0 : E ( Y i ) = α + X i β + W i 1 ( s ) γ 1 ( s ) ds ⇔ γ 0 = 0 and σ 2 g = 0 � � H A : E ( Y i ) = α + X i β + W i 1 ( s ) γ 1 ( s ) ds + W i 2 ( s ) γ 2 ( s ) ds ⇔ γ 0 � = 0 or σ 2 g � = 0 15 / 16

Comparison of Approaches Method 1: Functional Principal Components Regression (FPCR) Testing is done with standard likelihood-ratio tests of fixed effects. Selection of K g controls the smoothness of γ ( s ) and is very important. The authors suggested using cross-validation. Overall, simpler and more straightforward than PFR to implement. Method 2: Penalized Functional Regression (PFR) Testing is done with non-standard likelihood-ratio test for random and fixed effects (Crainiceanu and Ruppert (2004), Greven et al. (2008). Smoothness of γ ( s ) is induced through the mixed model framework for g , for sufficiently large # of PCs. Overall, more flexible than FPCR but more complex to implement. 16 / 16