Motivational Example

Transcription

1 Motivational Example Data: Observational longitudinal study of obesity from birth to adulthood. Overall Goal: Build age-, gender-, height-specific growth charts (under 3 year) to diagnose growth abnomalities. Specific Aim: Estimate the reference range for age-, gender-, height-specific weight. A simple version: Estimate the reference range for age-, gender-specific BMI. Statistical Problem: Estimate covariate specific quantiles in a reference group.

2 New method: Semiparametric Models Allow the shape depend on X and do not specify specific distribution for Y. Model µ(x) and σ(x) parametricly to gain efficiency. General model Y i = µ(x i ; θ) + σ(x i ; θ)ε(x i ), µ(x i ; θ) is location parameter, σ(x i ; θ) is scale parametre, i.e. (Var(Y i X i )) and ε(x i ) is from baseline distribution with mean zero, unit varaince. Denote baseline distribution function by F 0 (z, X) = P(ε(X) z X), we have Y α (X; θ, F 0 ) = µ(x i ; θ) + σ(x i ; θ)z α (X), Z α (X) is the αth quantile of ε(x), i.e. F 0 (Z α (X), X) = α.

3 Model details We can use splines to model µ(x) and σ(x). Let θ = {β 1,, β p, γ 1,, γ q }. µ(x) = log{σ(x)} = p β k R k (X) k=1 q γ k S k (X) k=1 R(X) = {R 1 (X),, R p (X)} and S(X) = {S 1 (X),, S q (X)} are pre-specified regression spline basis functions.

4 Quasi-likelihood and GEE 2 Consider general moment restricted models as below: E[f (Y, X, θ) X] = 0. All RAL estimator for θ must be solution from E[A(X; θ)f (Y, X, θ)] = 0. The most efficient selection will be A(X; θ) = E[ f (Y, X, θ) X]Var[f (Y, X, θ) X] 1. θ

5 Quasi-likelihood and GEE 2 A special case, location and scale model, θ = (β, γ): f 1 (Y, X, θ) = Y µ(x; β) f 2 (Y, X, θ) = (Y µ(x; β)) 2 σ 2 (X; γ). We have E[ f 1(Y,X,θ) γ X] = E[ f 2(Y,X,θ) X] = 0, so estimating β functions should be µ(x; β) Var(Y X) 1 [Y µ(x; β)] β σ 2 (X; γ) Var[(Y µ(x; β)) 2 X] 1 [(Y µ(x; β)) 2 σ 2 (X; γ)] γ

6 Quasi-likelihood and GEE 2 Using independent working correlation and use normal distribution as working model for Y X, we obtain quasi-likelihood score equation. Var[(Y µ(x; β)) 2 X] = 2Var(Y X) 2 0 = i 0 = i µ(x i ; β) Y i µ(x i ; β) β σ 2 (X i ; γ) σ 2 (X i ; γ) γ (Y i µ(x i ; β)) 2 σ 2 (X i ; γ) 2σ 4 (X i ; γ)

7 Quasi-likelihood and GEE 2 For our model specification, we have µ(x i ; β) = R(X i ) T β σ 2 (X i ; β) = 2S(X i ) T σ 2 (X i ; β) β 0 = i 0 = i R(X i ) T Y i µ(x i ; β) σ 2 (X i ; γ) S(X i ) T (Y i µ(x i ; β)) 2 σ 2 (X i ; γ) σ 2 (X i ; γ)

8 Estimate Baseline Function Obtain consistent estimates of µ(x) and σ(x) as above Obtain the estimated residual ê i (X i ) = Y i µ(x i ; ˆθ) σ(x i ; ˆθ), then estimate the baseline function F 0 (z, X) from ê i (X i ).

9 Estimate Baseline Function Speical case: F 0 does not depend on X. ˆF 0 (z, X) = n 1 n I(ê i z). i=1 Step function continuous function. ˆF 0 (z, X) = n 1 n K λ2 (z, ê i ), i=1 where K λ2 (z, ê i ) = K {(z ê i )/λ 2 } and K ( ) is any continuous distribution function. lim K {(z ê i)/λ 2 } = I(ê i < z) + K (0)I(ê i = z). λ 2 0 ˆF 0 (z, X) is monotonicly increasing in z.

10 Estimate Baseline Function Speical case: F 0 does not depend on X. ˆF 0 (z, X) = n i=1 w λ 1 (X, X i )I(ê i z) n i=1 w, λ 1 (X, X i ) where w λ1 (X, X i ) = W ((X X i )/λ 1 ) and W ( ) can be any kernel function satisfy Continuous version: W (x) 0 W (x)dx = 1 σ 2 W = x 2 W (x)dx < ˆF 0 (z, X; λ 1, λ 2 ) = n i=1 w λ 1 (X, X i )K λ2 (z, ê i ) n i=1 w, λ 1 (X, X i )

11 Bandwith and Kernel Function We will use following kernels (for simplicity, not most efficient): w λ1 (X, X i ) = φ((x X i )/λ 1 ) K λ2 (z, ê i ) = Φ((z ê i )/λ 2 ) For λ 1 and λ 2, we can use either fixed value or allow they depend on X.

12 Comparing to kernel estimator from Y i A nonparametric way will be use ˆF(z, X) = n i=1 w λ 1 (X, X i )I(Y i z) n i=1 w, λ 1 (X, X i ) For certain z, X and λ 1, assuming we have a uniform distribution for X, the bias will be W (u/λ1 )[F(z, X + u) F(z, X)]du W (u/λ1 )du W (u/λ1 )[Ḟ(z, X)u F(z, X)u 2 ]du W (u/λ1 )du = 1 4 F(z, X)λ 2 1 σ2 W So whether semiparametric method gain depend on comparison of F(z, X) and F 0 (z, X) or F(z, X) 2 df X and F0 (z, X) 2 df X.

13 Optimal band width For certain z, X and λ 1, the variance is approximate (nλ 1 ) 1 F (z, X) W (u) 2 du. The mean square error (MSE) will be 1 4 F(z, X) 2 λ 4 1 σ4 W + (nλ 1) 1 F(z, X) W (u) 2 du λ 1 = ( n 1 F(z, X) ) 1/5 W (u) 2 du F(z, X) 2 σw 4 The estimator is n 2/5 consistent for both ˆF and ˆF 0. Semiparametric method and nonparametric one has same convergence rate!

14 Multiple covariates Estimating equation part: Use two set of spline basis R(X 1 ), R(X 2 ) and S(X 1 ), S(X 2 ). Use their tensor product to generate the new basis R(X) = R(X 1 ) R(X 2 ), S(X) = [S(X 1 ), S(X 2 )]. Baseline estimator: Not approximate indicator function by continuous one. For kernel W (X), can use any multivariate kernel, for example the tensor product of two univariate kernel W 1 (X 1 ) W 2 (X 2 ). In application, they use W (X) = W 1 (X 1 ).

15 Example BMI Age BMI Age Figure: Fitted mean (top) and standard deviation (bottom) function

16 Example Residual Age Figure: Residual quantile regression for fitted 1st, 5th, 10th, 25th, 50th, 75th, 90th, 95th and 99th quantiles

17 Example Normal Q-Q Plot Sample Quantiles Theoretical Quantiles Figure: Residual Q-Q plot

18 Example Baseline not depend on X BMI Age Baseline depends on X BMI Age Figure: Fitted 1st, 5th, 10th, 25th, 50th, 75th, 90th, 95th and 99th quantiles with (top) or without (bottom) allow baseline depend on X

19 Simulation: Methods Comparing We will compare following methods in univariate covariate setting: 1. Bin and smooth quantile 2. Nonparametric quantile regression 3. Locally weighted method (empirical one) 4. Parametric method (LMS model) 5. Semiparametric method assuming same baseline (empirical one) 6. Semiparametric method allowing baseline depend on X (empirical one) For methods using bandwidth, we will choose several bandwidths (0.05, 0.1, 0.2).

20 Simulation: Methods Comparing For simplicity, we use a linear term for all parametric parts. True model is generated with semiparametric model with n = 5000, where µ(x) = X log(σ(x)) = 3 + 2X, X follow standard uniform distribution and the e(x) from following distributions 1. Standard normal N(0, 1). 2. Standardized log normal distribution. 3. Mixture normal N( 0.5, 1), N(0.5, 1).

21 Simulation: Methods Comparing We are interested in estimating following things from M = 1000 simulations 1. Bias, variance and MSE of the estimator for specific quantiles (90%, 95%, 99%) and specific covariate values (0.05, 0.25, 0.5, 0.75, 0.95) 2. Integrated mean square error of the estimator for specific quantiles