Variance Estimation When Donor Imputation is Used to

Variance Estimation When Donor Imputation is Used to Fill in Missing Values Jean-François Beaumont and Cynthia Bocci Statistics Canada Third International Conference on Establishment Surveys Montréal, June 18 -21, 2007

Overview n Context n Donor imputation n Variance estimation n Simulation study n Conclusion 2

Context n Population parameter to be estimated : n n Domain total: Estimator in the case of full response: n Calibration estimator n Horvitz-Thompson estimator 3

Donor Imputation n Imputed estimator : with n With donor imputation, the imputed value is n A variety of methods can be considered in order to find a donor l(k) for the recipient k 4

Donor Imputation n n Two simple examples: n Random Hot-Deck Imputation Within Classes n Nearest-neighbour imputation Practical considerations that add some complexity to the imputation process: n Post-imputation edit rules n hierarchical imputation classes 5

Imputation Model n Most imputation methods can be justified by an imputation model: n The donor imputed estimator is assumed to be approximately unbiased under the model: 6

Current. Variance Estimation Methods n n Assuming negligible sampling fractions n Chen and Shao (2000, JOS) for NN imputation n Resampling methods Our method is closely related to: n Rancourt, Särndal and Lee (1994, proc. SRMS): Assumes a ratio model holds n Brick, Kalton and Kim (2004, SM): Condition on the selected donors 7

Imputation Model Approach n Variance decomposition of Särndal (1992, SM): n For any donor imputation method, we have: 8

Estimation of the nonresponse variance n The estimation of the nonresponse variance is achieved by estimating n Noting that the nonresponse error is: n Then, the nonresponse variance estimator is: 9

Estimation of the mixed component n Similarly, the estimation of the mixed component is achieved by estimating n The mixed component estimator is: n This component can be either positive or negative and may not always be negligible 10

Estimation of the sampling variance n Let n The strategy consists of n be the full response variance est. n Estimating n Replace by their estimates the unknown This leads to the sampling variance estimator: 11

Estimation of the sampling variance n n This strategy is essentially equivalent to n Randomly imputing the missing values using the imputation model n Computing the full response sampling variance estimator by treating these imputed values as true values n Repeating this process a large number of times and taking the average of the sampling variance estimates Similar to multiple imputation sampling variance estimator 12

Simulation study n Generated a population of size 1000 n Two y-variables: n n n LIN: Linear relationship between y and x n NLIN: Nonlinear relationship between y and x Two different sample sizes: n Small sampling fraction: n=50 n Large sampling fraction: n=500 Response probability depends on x with an average of 0. 5 13

Simulation study n Imputation: Nearest-Neighbour imputation using x as the matching variable n Estimation of n LIN: Linear model in perfect agreement with the LIN y-variable n NPAR: Nonparametric estimation using the procedure TPSPLINE of SAS 14

Simulation study n Two objectives: n Compare the two ways of estimating n LIN and NPAR n Compare three nonparametric methods: n NPAR_Naïve: NPAR with the sampling variance being estimated by the naïve sampling variance Kalton and Kim, 2004) n CS : method of Chen and (Brick, Shao (2000) 15

Results: Large sampling fraction Method Relative Bias in % RRMSE in % y-LIN y-NLIN LIN -2. 4 358. 4 15. 7 514. 1 NPAR -0. 3 -18. 8 21. 5 54. 6 16

Results: Small sampling fraction Method Relative Bias in % y-LIN y-NLIN RRMSE in % y-LIN y-NLIN NPAR -4. 9 -13. 3 41. 8 245. 4 NPAR_ Naïve -5. 9 -10. 4 42. 1 265. 8 CS -9. 1 -9. 4 52. 8 257. 8 17

Results: Large sampling fraction Method Relative Bias in % RRMSE in % y-LIN y-NLIN NPAR -0. 3 -18. 8 21. 5 54. 6 NPAR_ Naïve -0. 3 -12. 0 21. 8 69. 1 CS 33. 9 59. 6 53. 7 118. 7 18

Conclusion n Nonparametric estimation of seems beneficial (robust) with Nearest-Neighbour imputation n Our proposed method is valid even for large sampling fractions n It seems to be slightly better to use our sampling variance estimator instead of the naïve sampling variance estimator 19

Conclusion n Work done in the context of developing a variance estimation system (SEVANI) n Methodology implemented in the next version 2. 0 of SEVANI n Estimation of : n Linear model n Nonparametric estimation 20

Thanks - Merci For more information please contact Pour plus d’information, veuillez contacter Jean-François Beaumont Jean-Francois. Beaumont@statcan. ca Cynthia Bocci Cynthia. Bocci@statcan. ca 21