Semiparametric Modeling, Penalized Splines, and Mixed Models
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1 Semi 1 Semiparametric Modeling, Penalized Splines, and Mixed Models David Ruppert Cornell University January 24 Joint work with Babette Brumback, Ray Carroll, Brent Coull, Ciprian Crainiceanu, Matt Wand, Yan Yu, and others
2 Semi 2 Example (data from Hastie and James, this analysis in RWC) spinal bone mineral density age (years)
3 Semi 3 Possible Model SBMD i,j is spinal bone mineral density on ith subject at age equal to age i,j SBMD i,j = U i + m(age i,j ) + ɛ i,j, i = 1,, m = 23, j = i,, n i U i is the random intercept for subject i {U i } are assumed iid N(, σu 2 )
4 Semi 4 Underlying philosophy 1 minimalist statistics keep it as simple as possible 2 build on classical parametric statistics 3 modular methodology
5 Semi 5 Reference Semiparametric Regression by Ruppert, Wand, and Carroll (23) Lots of examples from biostatistics
6 Semi 6 Recent Example April 17, 23 Canfield et al (23) Intellectual impairment and blood lead longitudinal (mixed model) nine covariates (modelled linearly) effect of lead modelled as a spline (semiparametric model) disturbing conclusion
7 Semi Quadratic IQ Spline lead (microgram/deciliter) Thanks to Rich Canfield for data and estimates
8 Semi 8 Semiparametric regression Partial linear or partial spline model: Y i = W T i β W + m(x i ) + ɛ i m(x) = X T i β X + B T (x)b B T (x) = ( B 1 (x) B K (x) ) Eg, X T i = ( X i X p i ) B T (x) = { (x κ 1 ) p + (x κ K ) p + }
9 Semi 9 Example m(x) = β + β 1 x + b 1 (x κ 1 ) b K (x κ K ) + slope jumps by b k at κ k
10 Semi 1 Linear plus function 2 plus fn 18 derivative
11 Semi 11 Fitting LIDAR data with plus functions log ratio range
12 Semi 12 Generalization m(x) = β +β 1 x+ +β p x p +b 1 (x κ 1 ) p ++ +b K (x κ K ) p + pth derivative jumps by p! b k at κ k first p 1 derivatives are continuous
13 Semi Quadratic plus function plus fn derivative 2nd derivative
14 Semi 14 Raw Data Ordinary Least Squares 2 knots 3 knots 5 knots knots 2 knots 5 knots 1 knots
15 Semi 15 Penalized least-squares Minimize n { Y (W T i β W + X T i β X + B T (X i )b) } 2 + λ b T Db i=1 Eg, D = I
16 Semi 16 Raw Data Penalized Least Squares 2 knots 3 knots 5 knots knots 2 knots 5 knots 1 knots
17 Semi 17 Ridge Regression From previous slide: n { Y (W T i β W + X T i β X + B T (X i )b) } 2 + λ b T Db i=1 Let X have row ( Wi T X T i B T (X i ) ) Then β W β = { X X b T X + λ blockdiag(,, D) } 1 X T Y Also, a BLUP in a mixed model and an empirical Bayes estimator
18 Semi 18 where b is N(, σ 2 b Σ b ) Linear Mixed Models Y = Xβ + Zb + ε Xβ are the fixed effects and Zb are the random effects Henderson s equations ( ) β b = ( X T X X T Z Z T X Z T Z + λσ 1 b λ = σ2 ɛ σ 2 b ) 1 ( X T Y Z T Y )
19 Semi 19 From previous slides: Let X have row ( Wi T X T i B T (X i ) ) Then β W β = { X X b T X + λ blockdiag(,, D) } 1 X T Y Linear mixed model: ( ) ( β X T X X T Z = b Z T X = Z T Z + λσ 1 b ) 1 ( X T Y Z T Y { ( X Z ) T ( X Z ) + λ blockdiag(, Σ 1 b )} 1 ( X Z ) T Y )
20 Semi 2 Selecting λ 1 cross-validation (CV) 2 generalized cross-validation (GCV) 3 ML or REML in mixed model framework
21 Semi 21 Selecting the Number of Knots (a) SpaHet, j = 3, typical data set (b) MASE comparisons y True full search relative MASE K fixed nknots myopic full search frequency 1 5 ASE K= number of knots (coded) n = ASE K=5
22 Semi 22 5 (a) SpaHetLS, j = 3, n = 2, 115 (b) MASE comparisons y 5 True full search relative MASE K fixed nknots myopic full search x 1 3 frequency ASE K= number of knots (coded) n = 2, ASE K=5 x 1 3
23 Semi 23 x MSE MSE 1 Variance Bias df fit (λ) Optimal n = 1,, 2 knots, quadratic spline
24 Semi 24 Return to spinal bone mineral density study spinal bone mineral density age (years) SBMD i,j = U i + m(age i,j ) + ɛ i,j, i = 1,, m = 23, j = i,, n i
25 Semi 25 X = 1 age 11 1 age 1n1 1 age m1 1 age mnm
26 Semi 26 Z = 1 (age 11 κ 1 ) + (age 11 κ K ) + 1 (age 1n1 κ 1 ) + (age 1n1 κ K ) + 1 (age m1 κ 1 ) + (age m1 κ K ) + 1 (age mnm κ 1 ) + (age mnm κ K ) +
27 Semi 27 u = U 1 U m b 1 b K
28 Semi 28 spinal bone mineral density age (years) Variability bars on m and estimated density of U i
29 Semi 29 Broken down by ethnicity Hispanic White spinal bone mineral density Asian Black age (years)
30 Semi 3 Model with ethnicity effects SBMD ij = U i + m(age ij ) + β 1 black i + β 2 hispanic i Asian is the reference group +β 3 white i + ε ij, 1 j n i, 1 i m
31 Semi 31 Only requires an expansion of the fixed effects by adding the columns black 1 hispanic 1 white 1 black 1 hispanic 1 white 1 black m hispanic m white m black m hispanic m white m
32 Semi 32 contrast with Asian subjects Black Hispanic White
33 Semi 33 In this model, the age effects curve for the four ethnic groups are parallel Could we model them as non-parallel? Might be problematic in this example because of the small values of the n i But the methodology should be useful in other contexts
34 Semi 34 Add interactions between age and black, hispanic, and white These are fixed effects Then add interactions between black, hispanic, white, and asian and the linear plus functions in age These are mean-zero random effects with their own variance component This variance component control the amount of shrinkage of the enthicity-specific curves to the overall effect
35 Semi 35 Penalized Splines and Additive Models Additive model: Y i = m 1 (X 1,i ) + + m P (X P,i ) + ɛ i
36 Semi 36 Bivariate additive spline model Y i = β +β x,1 X i + b x,1 (X i κ x,1 ) b x,k (X i κ x,kx ) + + β z,1 Z i + b z,1 (Z i κ z,1 ) b z,k (Z i κ z,kz ) + + ɛ i no need for backfitting computation very rapid no identifiability issues inference is simple
37 Semi 37 Bayesian methods The linear mixed model is half-bayesian The random effects have a prior The parameters without a prior are: fixed effects give them diffuse normal priors variance components give them diffuse inverse gamma priors
38 Semi 38 Bayesian methods Can be easily implemented in WinBUGS or programmed in, say, MATLAB Allows Bayes rather than empirical Bayes inference Uncertainty due to smoothing parameter selection is taken into account
39 Semi 39 The Bias-Variance Trade-off and Confidence Bands lambda= lambda=1 log ratio -8-4 log ratio range range lambda=3 lambda=1 log ratio -8-4 log ratio range range
40 Semi 4 How does one adjust confidence intervals for bias? undersmooth so variance dominates and bias can be safetly ignored
41 Semi 41 x n=1, 2 knots σ=3 MSE MSE 1 5 Variance Bias log(λ) optimal
42 Semi 42 Adjustment for bias continued estimate bias by a higher order method and subtract off bias (essentially the same as above) Wahba/Nychka Bayesian intervals bias is random so adds to posterior variance interval is widened but there is no offset
43 Semi 43 Wahba/Nychka Bayesian Intervals y = Xβ + Zu + ε, Cov [ ] u ε [ σ 2 = u I σεi 2 ], C = ( X Z ) β and ũ are BLUPs
44 Semi 44 ([ β ] u) Cov ũ = σε(c 2 T C+ σ2 ε D) 1 C T C(C T C+ σ2 σu 2 ε D) 1 σu 2 (Frequentist variance Ignores bias) ([ Cov ]) β ũ u = σ 2 ε(c T C + σ2 ε σ 2 u D) 1 (Bayesian posterior variance Takes bias into account)
45 Semi 45 strontium ratio age (million years)
46 Semi Effect of measurement error y x plus error W = X + error and Var(X) = Var(error)
47 Semi 47 Correction for measurement error Relatively little research in this area Fan and Truong (1993): deconvolution kernels first work inefficient in finite-sample studies no inference strictly for 1-dimensional smoothing Carroll, Maca, Ruppert functional SIMEX methods and structural spline methods more efficient than Fan and Truong
48 Semi 48 Berry, Carroll, and Ruppert (JASA, 22) fully Bayesian smoothing or penalized splines rather efficient in finite-sample studies inference available scales up semiparametric inference is easy structural
49 Semi 49 Berry, Carroll, and Ruppert starts with mixed-model spline formulation but fully Bayesian conjugate priors true covariates are iid normal but surprisingly robust normal measurement error in Gibbs, only sampling of true (unknown) covariates requires a Hastings-Metropolis step
50 Semi Effect of measurement error y x plus error W = X + error and Var(X) = Var(error)
51 Semi 51 Correction for measurement error Solid: true Dotted: uncorrected Dashed: corrected
52 Semi 52 Measurement Error, continued Ganguli, Staudenmayer, Wand: EM maximum likelihood estimation in BCR model Works about as well as the fully Bayesian approach Extension to additive models
53 Semi 53 Generalized Regression Extension to non-gaussian responses is conceptually easy Get a GLLM However, GLIM s are not trivial Can use: Monte Carlo EM Or MCMC
54 Semi 54 Single-Index Models Y i = g(x T i θ) + Z T i β + ɛ i Yu and Ruppert (22, JASA) Let g(x) = γ + γ 1 x + + γ p x p +c 1 (x κ 1 ) p c K (x κ K ) p + Becomes a nonlinear regression model Y i = m(x i, Z i, θ, β, γ, c) + ɛ i
Semiparametric Modeling, Penalized Splines, and Mixed Models David Ruppert Cornell University
Semiparametric Modeling, Penalized Splines, and Mixed Models David Ruppert Cornell University Possible Model SBMD i,j is spinal bone mineral density on ith subject at age equal to age i,j lide http://wwworiecornelledu/~davidr
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