Nonresponse weighting adjustment using estimated response probability

Transcription

1 Nonresponse weighting adjustment using estimated response probability Jae-kwang Kim Yonsei University, Seoul, Korea December 26, 2006

2 Introduction Nonresponse Unit nonresponse Item nonresponse Basic strategy for nonresponse Unit nonresponse : Call-back, Nonresponse weighting adjustment Item nonresponse : Imputation 2

3 Basic Setup Stratum Pop. Size Mean Sample Size Respondents N R Ȳ R n R Nonrespondents N M Ȳ M n M Entire population N Ȳ n SRS from the entire population, but observe only on the respondents. Use ȳ R (respondent mean) to estimate the population mean. Bias (ȳ R ) V ar (ȳ R ). = N R ) (ȲR Ȳ M N. = SR 2 n R 3

4 Two problems : Biased : Ȳ R Ȳ M Large variance due to n R < n

5 Nonresponse weighting adjustment (NWA) method Under no missing data: Ŷ HT = i A π i y i π i = P r (i A): first-order inclusion probability of unit i A: index set of the intended sample Response indicator function: R i = { if unit i responds, 0 if unit i does not respond. 4

6 Idea : Use two-phase sampling approach Population (U) P hase Sample (A) P hase2 Respondents (A R ) Estimation: Let φ i A = P r (R i = A). If φ i A were known, then Ŷ φ = R i y i i A π i φ i A would be conditionally unbiased. In practice, we use an estimator ˆφ i A of φ i A. The NWA estimator is Ŷ NW A = R i y i i A π i ˆφ i A 5

7 Example : Logistic regression model for φ i A Model : φ (x i, α) = exp ( x i α) + exp ( x i α) Estimation of α 0 by the maximum likelihood method : Solve S (α) [R i φ (x i, α)] x i = 0 i A for α. An iterative method can be used to solve the nonlinear equation : α (t+) α (t) ( S/ α) S ( α (t)) 6

8 More generally, the score equation can be weighted: for some weight k i. S k (α) i A k i [R i φ (x i, α)] x i = 0 Should we use weights or not? Optimal k i? 7

9 Asymptotic Properties of NWA estimator Assumptions about the population and the sample [A.] Sequence of finite populations with bounded fourth moments. [A.2] No extreme weights dominate the others. [A.3] n-consistency holds for the mean-type estimators 8

10 Assumptions about the response probability [B.] φ i A does not depend on the value of others. φ i A = φ i ) (i.e. [B.2] The responses are independent: Cov ( { ) φi ( φ R i, R j = i ) if i = j 0 otherwise. [B.3] The response probability is parametrically modelled. φ i = φ ( x i ; α 0), for some known smooth function φ (x i ; ) of parameter α evaluated at α = α 0. [B.4] φ i is uniformly bounded. 9

11 Estimation of φ i Estimation of α 0 : Use weighted score equation α i A k i [R i ln (φ i ) + ( R i ) ln ( φ i )] = 0, where k i is the weight of unit i in the score equation for α. Alternative representation: S k (α) (let) = i A k i (R i φ (x i ; α)) h i (α) = 0, where h i (α) = {logit (φ i )} / α. Use ˆφ i = φ (x i ; ˆα) where ˆα is the solution to S k (α) = 0. 0

12 Basic Idea (for deriving the asymptotic properties) Write the NWA estimator as a function of ˆα: Ŷ NW A (ˆα) i A π i φ i (ˆα) R iy i Taking a Taylor expansion of Ŷ NW A (ˆα) around α = α 0 : Ŷ NW A (ˆα). = Ŷ NW A ( α 0 ) + [ ŶNW A α ( α 0 )] (ˆα α 0 ) The second term in RHS does not contribute to the expectation, but does contribute to the variance.

13 Basic Idea - continued Taking a Taylor expansion of S k (ˆα) = 0: S k (ˆα). = S k ( α 0 ) + [ Sk α ( α 0 )] (ˆα α 0 ) Combine the two expansions: Ŷ NW A (ˆα). = Ŷ NW A ( α 0 ) [ ŶNW A α. = Ŷ NW A ( α 0 ) γ N S k ( α 0 ) ( α 0 )] [ Sk α ( α 0 )] S k ( α 0 ) ( where Ŷ NW A α 0 ) = i A πi φ ( i R i y i and S k α 0 ) = i A k i (R i φ i ) h i0. 2

14 Main Result Linearization Ŷ NW A. = i A π i [ π i φ i k i h i0 γ N + R i φ i ( yi π i φ i k i h i0 γ N ) ] Conditional expectation E ( Ŷ NW A A ). = i A π i y i Conditional variance V ( Ŷ NW A A ). = i A π 2 i φ i φ i ( yi π i φ i k i h i0 γ N ) 2 3

15 Main Result -continued The NWA estimator is asymptotically unbiased regardless of the choice of k i. The variance of the NWA estimator is minimized for k i = y i π i φ i h i0 γ If we don t have any prior information about the distribution of y i, then k i = π i φ i seems to be a reasonable choice for optimal NWA estimation. 4

16 Main Result -continued Estimate α 0 using k i = π i φ i : S k (α) = i A k i (R i φ (x i ; α)) h i (α) = i A π i φ i (R i φ (x i ; α)) h i (α) S k (ˆα) = 0 is equivalent to i A π i R i ˆφ i h i (ˆα) = i A π i h i (ˆα). Thus, optimal score equation = calibration equation. 5

17 Back to Example - Logistic regression model for φ i Under the logistic regression model, logit (φ i ) = x i α0 and S k (α) i A k i [R i φ (x i, α)] x i = 0 Optimal score equation i A π i R i ˆφ i x i = i A π i x i. Optimal NWA estimator applied to x = Complete sample estimator applied to x. 6

18 Simulation Study An artificial bivariate population of size N = 0, 000: ( ) [( ) ( )] yi i.i.d. 2 ρ N,, i =, 2,, N 2 ρ x i Unequal probability sampling (by stratified sampling) Generate missing data using a logistic regression model of R i on x i. About 30% missing data. 7

19 Simulation Study - continued Four estimators. Two-phase estimator : NWA estimator using true φ i 2. Unweighted NWA estimator : NWA estimator using k i = 3. Weighted NWA estimator : NWA estimator using k i = /π i 4. Optimal NWA estimator : NWA estimator using k i = / (π i φ i ) 0,000 Monte Carlo samples of size n = 00 and n = 400 are generated repeatedly from the fixed population. 8

20 Monte Carlo standardized variances of the NWA estimators, based on 0,000 samples. n Estimator Standardized variance ρ = 0.0 ρ = 0.3 ρ = 0.6 Two-phase Unweighted NWA Weighted NWA Optimal NWA Two-phase Unweighted NWA Weighted NWA Optimal NWA

21 Variance estimation Linearization : Ŷ NW A. = i A π i η i where η i = π i φ i k i h i0 γ N + R i φ i ( yi π i φ i k i h i0 γ N ) Extended definition of R i : { if unit i responds if sampled R i = 0 if unit i does not respond if sampled, for i =, 2,, N. 20

22 Variance estimation - continued Classical two-phase approach: Population (U) P hase Sample (A) P hase2 Respondents (A R ) Reverse approach: Population (U) Responding Population (U R ) Respondents (A R ) 2

23 Variance decomposition under the reverse approach: where V E V 2 V V E V ( Ŷ NW A ). = V + V 2 i A π i i A π i η i R, R 2,, R N η i R, R 2,, R N Variance component estimation ˆV = i A ˆV 2 = j A ˆφ i i A R π ij π i π j ˆη i ˆη j π ij π i π j (ˆφ i ) ( ) 2 y i π iˆφ i k iˆγ N 22

24 Extension : Nonresponse cell method A special case of NWA method. Commonly used. Partition the sample into G cells : A = A A 2 A G Assume that the response rates φ i are constant in a cell. For i A g, use ˆφ i = i A g πi R i i A g πi 23

25 Extension - Continued NWA cell : Two cell formation criteria Qausi-randomzation approach : Equal response probability assumption Model-based approach : Homogeneous study-item-value assumption Cross-classification of two dimensions of cells is not feasible : collapse the cells in an ad-hoc manner 24

26 Extension - Continued Previously we used Ŷ NW A = G g= i A π g i i A g πi R i y i i A g πi R i Here, the cells are formed to have equal response probability. Directly use ˆφ i in the cell-weighting estimator Ŷ NW A2 = G g= i A π g i i A g πi ˆφ i R i y i i A g πi ˆφ i R i 25

27 Extension - Continued Taylor expansion Ŷ NW A2 = ŶHT + G g= i A g π i ( ) Ri (y i ȳ g ) φ i Variance V ar ( Ŷ NW A2 ) = V ar (ŶHT ) +E G g= i A g π 2 i ( ) φ i (y i ȳ g ) 2 The variance will be smaller if the cells are formed with homogeneous y s. 26

28 Conclusion Even if you know the true response probability, it s better to use the estimated response probability for the NWA estimation. Maximum likelihood method may be optimal for estimating α 0, but not optimal for NWA estimation. Standard practice of calibration is indeed an optimal procedure. Variance estimation is possible using the reverse approach. 27

29 In the cell-weighting NWA method, we can Use the estimated response probability to control the bias. Use the weighting cell to control the variance.

30 Thank You! 28