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1 A. Demnati and J. N. K. Rao Statistics Canada / Carleton University A Presentation at the Third International Conference on Establishment Surveys June 18-21, 2007 Montréal, Québec, Canada June 20, 2007 Linearization Variance Estimators for Survey Data: Some Recent Work

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2 Situation looking for a method of variance estimation that is simple is widely applicable has good properties provides unique choice for estimators of nonlinear finite population parameters defined explicitly or implicitly using calibration weights under missing data using repeated survey of model parameters SM, 2004 JSM, 2002 and JMS, 2002 FCSM, 2003 Symposium, 2005 of dual frames JSM, 2007

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3 Demnati –Rao Approach General formulation Finite population parameters Model parameters Estimator for both parameters Variance estimators associated with and are different

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4 Demnati –Rao Approach ( Survey Methodology, 2004 ) Write the estimator of a finite population parameter as with if element k is not in sample s; if element k is in sample s;

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5 Demnati –Rao Approach ( Survey Methodology, 2004 ) with A linearization sampling variance estimator is given by : variance estimator of the H-T estimator of the total is a (N×1) vector of arbitrary number

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6 Demnati –Rao Approach ( Survey Methodology, 2004 ) Example – Ratio estimator of For SRSand

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7 Demnati –Rao Approach ( Survey Methodology, 2004 ) Example – Ratio estimator of is a better choice over customary Royall and Cumberland (1981) Särndal et al. (1989) Valliant (1993) Binder (1996) Skinner (2004)

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8 Demnati –Rao Approach Also in Survey Methodology, 2004: Calibration Estimators: Two-Phase Sampling the GREG Estimator the Optimal Regression Estimator the Generalized Raking Estimator New Extensions: Wilcoxon Rank-Sum Test Cox Proportional Hazards Model

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9 Model parameters (Symposium, 2005) Finite-population assumed to be generated from a superpopulation model : model expectation and variance where f is the sampling fraction. For multistage sampling, the psu sampling fraction plays the role of f. Inference on model parameter Total variance of : : design expectation and variance i)if f 0 then ii)if f 1 then In case i),

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10 Example: Ratio estimator when y is assumed to be random for where A d is a 2×N matrix of random variables with k th column: Define We have We get where A b is a 2×N matrix of arbitrary real numbers with k th column: where is an estimator of the total variance of

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11 Estimator of the total variance of with when Note that is an estimator of model covariance and A variance estimator of is given by where when and when

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12 Hence where = model variance + sampling variance where and Under SRS,

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13 Under ratio model, Note: g-weight appears automatically in Note: remains valid under misspecification of Hence, and the finite population correction 1-n/N is absent in

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14 Simulation 1: Unconditional performance We generated R=2,000 finite populations, each of size N=393 from the ratio model where are independent observations generated from a N(0,1) are the number of beds for the Hospitals population studied in Valliant, Dorfman, and Royall (2000, p ) One simple random sample of specified size n is drawn from each generated population Parameter of interest:

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15 Simulation 1: Unconditional performance Ratio estimator: We calculated: Simulated and its components and

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16 Simulation 1: Unconditional performance Figure 1: Averages of variance estimates for selected sample sizes compared to simulated MSE of the ratio estimator.

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17 Simulation 2: Conditional performance We generate R=20,000 finite populations, each of size N=393 from the ratio model using the number of beds as One simple random sample of size n=100 is drawn from each generated population Parameter of interest: We arranged the 20,000 samples in ascending order of -values and then grouped them into 20 groups each of size 1,000

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18 Simulation 2: Conditional performance Figure 2: Conditional relative bias of the expansion and ratio estimators of

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19 Simulation 2: Conditional performance Figure 3: Conditional relative bias of variance estimators

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20 Simulation 2: Conditional performance Figure 4: Conditional coverage rates of normal theory confidence intervals based on, and for nominal level of 95%

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21 g-weighted estimating functions: model parameter Generalized Linear Model Linear Regression Model Logistic Regression Model is the solution of weighted estimating equation: is solution Special case: (GREG)

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22 Simulation 3: Estimating equations We generated R=10,000 finite populations, each of size N=393 from the model One simple random sample of size n=30 is drawn from each generated population Parameter of interest: Population units are grouped into two classes with 271 units k having x =350 in class 2 Using the number of beds as leads to an average of about 60% for z Post-stratification: X=(271,122) T

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23 Simulation 3: Estimating equations Table 2: DR variance estimator ParameterNo Calibration Post- stratification Table 1: Monte Carlo Variances ParameterNo CalibrationPost-stratification Table 3: DR naïve variance estimator ParameterNo Calibration Post- stratification

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24 Multiple Weight Adjustments Weight Adjustments for Units (or complete) nonresponse Calibration Due to lack of time, not presented in the talk, but it is included in the proceeding paper

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25 Concluding Remarks We provided a method of variance estimation for estimators: The method of nonlinear model parameters using survey data is simple has good properties defined explicitly or implicitly is widely applicable provides unique choice using multiple weight adjustments under missing data Thank you Very Much

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