A Semi-Parametric Block Bootstrap Approach for Clustered Data Ray - PowerPoint PPT Presentation

A Semi-Parametric Block Bootstrap Approach for Clustered Data Ray Chambers & Hukum Chandra University of Wollongong SAE2011, Trier, Germany 12 August 2011

Overview of Presentation Background and Motivation  Bootstrap Methods  Empirical Evaluations  Application to a Real Dataset  Concluding Remarks  2

Background • Bootstrap technique: a computer intensive and general way of measuring the accuracy of estimators (Efron, 1979; Efron & Tibshirani, 1993) • Originally developed for parameter estimation given data values that are independent and identically distributed ( iid ) • Random effects models for hierarchically dependent data , e.g. clustered data, are now widely used (e.g. in SAE) • With such data, it is important to use bootstrap techniques that allow for the hierarchical dependence structure • Parametric bootstrap based on the assumed hierarchical random effects model is widely used o very effective if model is correctly specified 3

Background • If the variability assumptions of the model, e.g. the assumption that the random effects are iid Normal random variables, are violated, then it is hard to justify use of the parametric bootstrap (Rasbash et al., 2000; Carpenter et al., 2003) • A semi-parametric bootstrap for multilevel modelling is described in Carpenter et al. (2003) • We describe an alternative semi-parametric block bootstrap for clustered data o 'semi-parametric' because although the marginal model is bootstrapped parametrically, the dependence structure in the model residuals is non-parametrically block bootstrapped o 'block' because we are interested in a bootstrap procedure that is robust to within cluster heterogeneity Focus on bootstrap confidence interval performance 4

Bootstrap Percentile Confidence Interval • α 2 values of the bootstrap distribution are used to Upper and lower 100(1 − α )% confidence interval construct a ˆ • θ L , α 2 denotes the value such that a fraction α 2 of all bootstrap Let ˆ θ U , α 2 be the value such that a estimates are smaller, and likewise let α 2 of all bootstrap estimates are larger; then an approximate fraction θ L , α 2 , ˆ ˆ ⎡ ⎤ confidence interval for θ is θ U , α 2 ⎦ ⎣ ( ) θ θ U , α 2 − ˆ ˆ θ U , α 2 − ˆ ˆ • θ L , α 2 and standardised width = θ L , α 2 Width = 5

Random Effects Model for Clustered Data Two-level model y ij = x ij T β + u i + e ij , j = 1,..., n i ; i = 1,.., D IID N (0, σ u 2 ) u i  Group-specific random effects (level 2) IID N (0, σ e u i ⊥ e ij | x ij 2 ) , e ij  Individual level errors (level 1) Focus on bootstrap distributions for estimates ˆ β , ˆ σ u σ e 2 and ˆ 2 6

Parametric 2-Level Bootstrap (Para) ˆ • β , ˆ σ u σ e 2 and ˆ 2 ML/REML estimates IID *  σ u i = 1,.., D N (0, ˆ 2 ) , • u i Simulate level 2 errors IID * ⊥ e ij *  σ e 2 ) , j = 1,..., n i ; i = 1,.., D , N (0, ˆ * • e ij u i Simulate level 1 errors * = x ij * + e ij T ˆ β + u i * • y ij Bootstrap Data Refit model, obtain bootstrap parameter estimates ˆ β * , ˆ σ u σ e 2* and ˆ 2* • Repeat B times ⇒ B sets of bootstrap estimates • Generate bootstrap distributions of ˆ β , ˆ σ u 2 and ˆ σ e 2 • • Bootstrap CIs 'read off' from bootstrap distributions 7

Semi-Parametric 2-Level Bootstrap (CGR) (Carpenter, Goldstein and Rasbash, 2003) ˆ • β , ˆ σ u σ e 2 and ˆ 2 ML/REML estimates • i = 1,.., D Level 2 residuals (EBLUPs) ˆ u i , T ˆ • e ij = y ij − x ij β − ˆ u i , j = 1,..., n i ; i = 1,.., D ˆ Level 1 residuals • Centre and then rescale residuals ˆ u i and ˆ e ij Rescaling • The empirical variance-covariance matrices of the level 1 and level 2 residuals can be different from their corresponding ML/REML estimates • Rescale residuals to make these the same before bootstrapping 8

Rescaling (Cholesky decomposition) Estimate the variance-covariance matrix ˆ Σ of model errors ˆ • U (either level 2 or level 1) via ML/REML Σ = AA T calculated, where Cholesky decomposition ˆ • A is a lower triangular matrix ˆ V = empirical variance/covariance matrix of ˆ • U V = BB T calculated, where Cholesky decomposition ˆ • B is a lower triangular matrix ( ) U * = ˆ T Rescaled residuals ˆ C = AB − 1 • UC where 9

• Sample independently with replacement from these two new sets of centred and rescaled residuals ( ) { } , m = 1 * = srswr u h ; h = 1,.... D • ˆ u i { } * = srswr ˆ • e hj , j = 1,..., n i ; h = 1,.., D e ij * = x ij * + e ij T ˆ β + u i * • y ij Bootstrap Data Refit model, obtain bootstrap parameter estimates ˆ β * , ˆ σ u σ e 2* and ˆ 2* • • Repeat this process B times Generate bootstrap distributions for ˆ β , ˆ σ u σ e 2 and ˆ 2 • • Bootstrap CIs 'read off' from bootstrap distributions 10

Simple Semi-Parametric Block Bootstrap (SBB) T ˆ Estimate ˆ • β with residuals : r ij = y ij − x ij β , j = 1,..., n i ; i = 1,.., D ∑ n h • − 1 r h = n h h = 1,.., D r hj Level 2 residuals : Group averages , j = 1 (1) = r hj − r h • ⇒ r h (1) vector of size r hj n i Level 1 residuals : • Sample independently with replacement from these two sets of residuals ( ) { } , m = 1 * = srswr r h ; h = 1,.... D r i o ( ) { } , m = 1 h ( i ) = srswr 1,......, D o ( ) { } , m = n i (1)* = srswr (1) r i r h ( i ) o Block/group structure : * = x i T ˆ • β + r * 1 n i + r i (1)* y i Bootstrap Data : i 11

SBB + Post Bootstrap Adjustment (SBB. Post ) • Multivariate bootstrap distribution of the variance component estimates is first transformed ('tilted') in order to ensure that the bootstrap estimates of these components are uncorrelated • All bootstrap distributions of model parameter estimates are then 'tethered' to the original estimate values, using either a mean correction (for estimates, e.g. regression coefficients, defined on the entire real line) or a ratio correction (for estimates, e.g. variance components, that are strictly positive) 12

Tilting and Tethering (post-bootstrapping) ( ) B = ( ) B − av B ˆ ( ) ⎡ ⎤ ˆ β k 1 B + ˆ ˆ β k β k β k ** * * • ⎦ ⎣ { } ( ) B = ( ) B × ˆ ( ) − 1 2 av B ˆ σ u σ u σ u σ u ˆ ˆ 2** *mod2 *mod2 • { } ( ) B = ( ) B × ˆ ( ) − 1 2 av B ˆ σ e σ e σ e σ e ˆ 2** ˆ *mod2 *mod2 • where { } { } × D B ( ) B ˆ ( ) B ( ) C B ( ) * + * − M B − 1/2 ⎡ ⎤ σ u σ e ⎦ = exp M B ˆ *mod2 *mod2 * * * S B ⎣ ( ) 1 B av B log ˆ ( ) 1 B ∗ = av B log ˆ ⎡ ⎤ ∗ 2 ∗ 2 σ u σ e ⎦ M B ⎣ ( ) B ( ) B ∗ = ⎡ ⎤ ∗ 2 ∗ 2 σ u σ e log ˆ log ˆ ⎦ S B ⎣ ( ) ∗ = cov B S B ∗ C B ( ) 1 B sd B log ˆ ( ) 1 B ∗ = sd B log ˆ ⎡ ⎤ ∗ 2 ∗ 2 σ u σ e ⎦ D B ⎣ 13

Semi-Parametric Block Bootstrap with Centred and Rescaled Residuals (SBB.Prior) T ˆ Estimate ˆ • β with residuals r ij = y ij − x ij β , j = 1,..., n i ; i = 1,.., D ∑ n h − 1 • r h = n h h = 1,.., D r hj Level 2 residuals: Group-wise averages , j = 1 (1) = r hj − r h • ⇒ r h (1) vector of size r hj n i Level 1 residuals: • (1) r h and r hj Centre and rescale Level 1 and Level 2 residuals: • Apply SBB to these centred and rescaled residuals • No post-bootstrap adjustment 14

SBB + External Calibration to Covariance Matrix of Variance Components (SBB.Prior.Adj) • Calibrate covariance matrix of bootstrap estimates of variance components to ML/REML estimate of the covariance matrix of variance components obtained from the model - Cholesky decomposition • Post-bootstrap adjustment where bootstrap distribution of variance components is tilted to recover ML/REML estimate of variance/ covariance matrix of estimated variance components 15

Bootstrap Methods Used in Simulations Type Description Para Parametric 2-level bootstrap CGR Semi-Parametric 2-Level Bootstrap (Carpenter, Goldstein and Rasbash, 2003) SBB Simple semi-parametric Block bootstrap using empirical Level 1 and Level 2 residuals SBB.Post SBB with post-bootstrap tilting & tethering adjustments SBB.Prior SBB using internally rescaled empirical residuals SBB.Prior.Adj SBB using internally rescaled residuals with post-bootstrap adjustment to recover REML estimate of the variance of the estimated variance components 16

Simulation Design • D = 50 , 100 Total number of clusters: • n i = 5, 20 Uniform cluster sample sizes: • n = 250,500,1000, 2000 • 1000 simulations • 1000 Bootstrap samples per method per simulation Model • y ij = β 0 + β 1 x ij + u i + e ij , j = 1,..., n i ; i = 1,.., D • x ij  U (0,1) IID IID • (0, σ u (0, σ e u i ⊥ e ij | x ij 2 ) , 2 ) u i  e ij  2 = 0.04 and 2 = 0.16 • β 0 = 1 , β 1 = 2 , σ u σ e • u i and e ij generated using four scenarios ... 17

Set A - Normal Scenario 2 = 0.04) and 2 = 0.16) e ij  N (0, σ e • u i  N (0, σ u Set B - Chi-Square Scenario ( ) / ( ) / 2 − 1 2 − 1 ⎡ ⎤ ⎡ ⎤ • χ 1 χ 1 u i  0.2 2 e ij  0.4 2 ⎦ ⎦ and ⎣ ⎣ 18