Implications of Ignoring the Uncertainty in Control Totals for Generalized Regression Estimators. Calibration Estimators

Transcription

1 Implications of Ignoring the Uncertainty in Control Totals for Generalized Regression Estimators Jill A. Dever, RTI Richard Valliant, JPSM & ISR is a trade name of Research Triangle Institute. Calibration Estimators Calibrate auxiliary estimates (X) to corresponding population control totals (Deville & Särndal 1992; Särndal, Swensson, & Wretman 1992) Can produce efficient estimates (Särndal & Lundström 2005; Kott 2006) Reduce errors linked to sampling frame and nonresponse Reduce variation in weights Regression approach (Sarndal 2007) = working prediction model (Kott 2006) to closely represent the relationship between y and X Specification of the model is important for precision 1

2 The GREG, tˆ Generalized Regression (Calibration) Estimators ( ˆ ) = tˆ + t t Bˆ y R A y x A x A 1 = 1 ( ˆ ) d d + t t x x x y k A s l sa x A x l l l k k k Working prediction model: 2 ( ) = x B, ( ) = E y V a r y ε k k ε k σ k Note: When x k is substituted for y k, then by definition t ˆ A = x t x GREG vs. Estimated Control (EC) GREG t ˆ ( ˆ ) ˆ = t ˆ + t t B GREG: y R A y x A x A t ˆ ( ˆ ˆ ) ˆ = t ˆ + t t B EC-GREG: y R A y B x A x A w h t e= r x e ; ˆ ; = t d x x jj AU xk ks k and tˆ = B x j s j j B w x A 2

3 Scope of the Research Analytic survey (requires calibration) stratified multi-stage design, PSUs selected WR negligible nonresponse Bernoulli distribution used for random undercoverage error in frame Benchmark survey (estimated controls) independent from analytic survey probability-based sampling design Estimated-Control (EC) Calibration with Generalized Regression Estimation calculated from complete "population" with varying levels of precision Theory Theoretical and Empirical Evaluation Asymptotic bias and variance developed Simulation Study (Poststratified Estimator) Population = Subset of 2003 NHIS (N=21,664) Poststrata = Age group (8) by gender 4,000 frames = Coverage rates vary by poststrata Sensitivity analysis: Two outcome variables (y k ) Size of analytic survey sample Relative size of the benchmark survey 3

4 Biast ( ˆ yecr ) Theoretical Bias 4 components Unbiased if all conditions are met: 1) Model residuals are not correlated to coverage propensities 2) Same populations for analytic and benchmark surveys 3) Average model residual is zero 4) Unbiased estimates from the reference survey Empirical Bias for Estimate Total is Lower with EC Poststratification Adjustment RelBias* EC Poststratification 0.2 % None % * Hispanic persons who delayed medical care because of cost 4

5 EC-GREG Linearization Variance Estimator var ( ˆ yecr ) t = var ( t ˆ yr ) + f + f ( benchmark covariance matrix ) ( coverage propensity, model residuals ) EC-GREG Jackknife Variance Estimator tˆ = t + B x B x adjustment where adjustment is defined by: Fuller method [ECF2] = eigenvalue decomposition of benchmark covariance matrix (Fuller 1998) Nadimpalli et al. method [ECNJC] = controls chosen from N(0,1) with diagonal of covariance matrix (NJC 2004) Multi.Normal method [ECMV] = controls chosen from MV distribution with covariance matrix (Dever & Valliant 2010) 5

6 Average Percent Relative Bias Naive ECNJCm Other (ECTS, ECF2m, ECMV) Empirical Percent Bias of Variance Relative to MSE* * Hispanic persons only Relative Size of Benchmark Survey to Analytic Survey Summary of Research to Date Theoretical Empirical Variance* Estimators ~ Unbiased? Ratios Totals Naïve Œ Œ Œ Linearization Fuller NJC - N(0,1) Ø or Œ - Œ MV Normal Ævariation * Variation in subdomain results warrants additional research. 6

7 Future Research Thresholds Traditional variance estimators are acceptable Benchmark estimates too imprecise for use Variance adjustment for negative bias Improve precision for subdomains Constrained EC calibration Effects of nonresponse and measurement errors Two-phase designs with mix of controls More Information? Jill A. Dever 7