A General Approach to Variance Estimation under Imputation for Missing Survey Data

A General Approach to Variance Estimation under Imputation for Missing Survey Data J.N.K. Rao Carleton University Ottawa, Canada 1 2 1 Joint work with J.K. Kim at Iowa State University. 2 Workshop on Survey Sampling in honor of Jean-Claude Deville, Neuchâtel, Switzerland, June 24-26, 2009

Outline Item Nonresponse Deterministic imputation: Population model approach Imputed estimator Linearization variance estimator Examples: Domain estimation, Composite imputation Stochastic imputation: variance estimation Examples: Multiple imputation, binary response Simulation results Doubly robust approach Extensions

Survey Data Design features: clustering, stratification, unequal probability of selection Source of error: 1. Sampling errors 2. Non-sampling errors: Nonresponse (missing data) Noncoverage Measurement errors

Types of nonresponse Unit (or total) nonresponse: refusal, not-at-home Remedy: weight adjustment within classes Item nonresponse: sensitive item, answer not known, inconsistent answer Remedy: imputation (fill in missing data)

Advantages of imputation Complete data file: standard complete data methods Different analyses consistent with each other Reduce nonresponse bias Auxiliary x observed can be used to get good imputed values Same survey weight for all items

Commonly used imputation methods Marginal imputation methods: 1. Business surveys: Ratio, Regression, Nearest neighbor (NN) 2. Socio-economic surveys: Random donor (within classes), Stochastic ratio or regression, Fractional imputation (FI), Multiple imputation (MI)

Complete response set-up Population total: θ N = N i=1 y i NHT estimator: ˆθ n = i s d i y i where d i = π 1 i : design weight π i = inclusion probability = Pr (i s) Variance estimator: ˆV n = i s Ω ij y i y j j s Ω ij depends on joint inclusion probabilities π ij > 0

Deterministic imputation Population model approach (Deville and Särndal, 1994): E ζ (y i x i ) = m (x i, β 0 ) a i = 1 if y i observed when i s = 0 otherwise for i U = {1, 2,, N} MAR: Distribution of a i depends only on x i Imputed value: ŷ i = m(x i, ˆβ) ˆβ: unique solution of EE Û (β) = d i a i {y i m (x i, β)} h (x i, β) = 0 i s

Model specification Further model specification: Var ζ (y i x i ) = σ 2 q (x i, β 0 ) h (x i, β) = ṁ (x i, β) /q (x i, β) h i Examples: commonly used imputations 1. Ratio imputation: h i = 1 E ζ (y i x i ) = β 0 x i, Var ζ (y i x i ) = σ 2 x i 2. Linear regression imputation: h i = x i E ζ (y i x i ) = x i β 0, Var ζ (y i x i ) = σ 2 3. Logistic regression imputation (y i = 0 or 1): h i = x i log {m i / (1 m i )} = x i β 0, Var ζ (y i x i ) = m i (1 m i ) where m i = E ζ (y i x i )

Imputed estimator Imputed estimator of total θ N : ˆθ Id = } d i {a i y i + (1 a i ) m(x i, ˆβ) i s i s d i ỹ i Examples 1. Ratio imputation: m(x i, ˆβ) = x i ˆβ where ˆβ = ( i s d ) 1 ia i x i i s d ia i y i 2. Linear regression imputation: m(x i, ˆβ) = x i ˆβ where ˆβ = ( i s d ) ia i x i x i 1 i s d ia i x i y i 3. Logistic regression imputation: ˆβ is the solution to i s d i {y i m (x i, β)} x i = 0 Imputed estimator of domain total θ z = N i=1 z iy i : ˆθ I,z = i s d i z i ỹ i where z i = 1 if i D; z i = 0 otherwise.

Variance estimation Treating imputed values as if observed: Underestimation if ỹ i used in ˆV n for y i Methods that account for imputation: Adjusted jackknife: Rao and Shao (1992) Linearization (Pop. model): Deville and Särndal (1994) Fractional imputation method: Fuller and Kim (2005) Bootstrap: Shao and Sitter (1996) Reverse approach: Shao and Steel (1999)

Variance estimation (Cont d) Linearization method: Theorem 1 (Kim and Rao, 2009): Under regularity conditions, n 1/2 N 1 (ˆθId θ Id ) = o p (1) where θ Id = i s w i η i { η i = m (x i ; β 0 ) + a i 1 + c } h i {yi m (x i ; β 0 )}, { N } 1 N c = a i ṁ (x i ; β 0 ) h i (1 a i ) ṁ (x i ; β 0 ). i=1 Reference distribution: Joint distribution of population model and sampling mechanism, conditional on realized (x i, a i ) in the population. i=1

Variance estimation (Cont d) Reverse approach: 1. ˆV 1d = Ω ij ˆη i ˆη j i s j s 2. where ˆη i = η i ( ˆβ). ˆV 2d = i s ( ) 2 ( )} d i a i 1 + ĉ ĥ i {y i m x i ; ˆβ ˆV 2d valid even if V ζ (y i x i ) is misspecified. 3. Variance estimator of ˆθ Id ( θ Id ): ˆV d = ˆV 1d + ˆV 2d ˆV d approximately design-model unbiased. If the overall sampling rate negligible: ˆV d = ˆV1d

Variance estimation (Cont d) Domain estimation: 1. ˆθ I,z : design-model unbiased for θ z 2. Use ˆV 1d = Ω ij ˆη iz ˆη jz i s j s where ˆη iz = z i m(x i ; ˆβ) + a i {z i + ĉ zh i } { } y i m(x i ; ˆβ), ĉ z = { } 1 d i a i ṁ(x i ; ˆβ)ĥ i d i z i (1 a i ) ṁ(x i ; ˆβ) i s i s

Composite imputation x, y, z: z always observed Imputation model: s = s RR s RM s MR s MM θ N = N i=1 y i s RM : x observed and y missing s MM : x and y missing E ζ (y i x i, z i ) = β y x x i E ζ (x i z i ) = β x z x i Imputed estimator: ˆθ Id = ( ) d i y i + d i ˆβy x x i + i s +R i s RM i s MM d i ( ˆβy x ˆβx z z i )

Composite imputation (Cont d) ˆβ y x and ˆβ x z solutions of estimation equations: ( ) ( ) Û 1 βy x = yi β y x x i = 0 i S RR d i Û 2 ( βx z ) = i S R+ d i ( xi β x z z i ) = 0 Taylor linearization of the imputed estimator: ˆθ Id ( ˆβ) = ˆθ Id (β) ( ˆθ Id β ) ( Û β where Û = (Û1, Û 2 ) and β = ( βy x, β x z ). ) 1 Û (β)

Stochastic imputation y i = imputed value of y i such that Imputed estimator of θ N : E I (y i ) = m(x i, ˆβ) ˆm i ˆθ I = i s d i {a i y i + (1 a i ) y i } Variance estimator of ˆθ I : E I (ˆθI ) = ˆθ Id ˆV I = ˆV d + ˆV where ˆV = i s d 2 i (1 a i ) (y i ˆm i ) 2

Multiple imputation: Rubin y (1) i,..., y (M) i = imputed values of y i (M 2) ˆθ (k) I Imputed estimator = i s Rubin s variance estimator: { } d i a i y i + (1 a i ) y (k) ˆθ MI = M 1 M k=1 ˆθ (k) I ˆV R = W M + M + 1 M B M where W M is the average of M naive variance estimators and B M = (M 1) 1 M k=1 (ˆθ(k) I ˆθ ) 2 MI i

Multiple imputation (Cont d) ˆV R theoretically justified when ) ) V (ˆθId = V (ˆθn + V (ˆθId ˆθ ) n (A) (Congenialty assumption) ˆVR seriously biased if assumption (A) violated. (A) not satisfied for domain estimation when domains not specified at the imputation stage. Our proposal: ˆV MI = ˆV d + M 1 B M ˆVMI valid for ˆθ Id as well as ˆθ I,z without (A).

Binary response Model: y i x i Bernoulli {m i = m (x i, β 0 )} logit (m i ) = x i β; q (x i, β 0 ) = m i (1 m i ) q i ( ˆm i = m x i, ˆβ ) where ˆβ is the solution to d i a i {y i m (x i, β)} x i = 0 i s Stochastic hot deck imputation { yi 1 with prob ˆmi = 0 with prob 1 ˆm i ˆη i = ˆm i + a i (1 + ĉ x i ) (y i ˆm i ) ĉ = { i s d ia i ˆq i x i x i } 1 i s d i (1 a i ) ˆq i x i.

Binary response (Cont d) Fractional imputation (FI): Eliminate imputation variance V by FI M = 2 fractions: impute { yi 1 with fractional weight ˆmi = 0 with fractional weight 1 ˆm i Data file reports real values 1 and 0 with associated fractions ˆm i and 1 ˆm i. ˆθ FI = ˆθ Id : V eliminated Estimation of domain total and mean: ˆθ FI,z, ( i s d iz i ) 1 ˆθ FI,z

Binary response (Cont d) Multiple imputation (MI): { 1. Generate β N ( ˆβ, i s a ) } i ˆq i x i x i 1 2. Generate yi Bernoulli (mi ) with m i = m (x i, β ) 3. Repeat steps 1 and 2 independently M times.

Simulation Study : Binary response Finite population of size N = 10, 000 from x i N (3, 1) y i x i Bernoulli (m i ), where logit (m i ) = 0.5x i 2 z i Bernoulli (0.4) (z i : Domain indicator) SRS of size n = 100 x i and z i : always observed. y i subject to missing. Missing response mechanism a i Bernoulli (π i ) ; logit (π i ) = φ 0 + φ 1 (x i 3) + φ 2 x i 3 (a) φ 1 = 0, φ 2 = 0; (b) φ 1 = 1, φ 2 = 0; (c) φ 1 = 0, φ 2 = 1 φ 0 is determined to achieve 70% response rate. Two variance estimates of multiple imputation are computed.

Simulation Study (Cont d) Table: Relative bias (RB) of the Rubin s variance estimator (R) and proposed variance estimator (KR) for multiple imputation Parameter Response RB (%) Mechanism R KR Case 1 1.07 2.90 Population Case 2-0.29 1.42 Mean Case 3-3.96-2.09 Case 1 34.25 2.37 Domain Case 2 31.08 2.28 Mean Case 3 27.55-3.41 Conclusion: 1. KR has small RB in all cases 2. R leads to large RB in the case of domain mean: 28% to 34%

Doubly robust method Case 1: p i known (p i = probability of response) Let β be the solution to Û (β) = ( ) 1 d i a i 1 {y i m (x i, β)} h (x i, β) = 0 p i i s Imputed estimator: θ Id = i s d i {a i y i + (1 a i ) m(x i, β) } If 1 is an element of h i, then θ Id = { ( ai d i y i + 1 a ) } i m(x i, β) p i p i i s

Doubly robust method (Cont d) Properties of θ Id : 1. Under the assumed response model, E R ( θ Id ) = ˆθ n regardless of the choice of m(x i, β). 2. Under the imputation model, E ζ ( θ Id ˆθ n ) = 0. (1) and (2) imply that θ Id is doubly robust.

Doubly robust method (Cont d) Case 2: p i unknown (p i = p i (α)) Linearization variance estimator: Haziza and Rao (2006): linear regression imputation Deville (1999), Demnati and Rao (2004) approach: general case

Extensions Calibration estimators Davison and Sardy (2007): deterministic linear regression imputation, stratified SRS Pseudo-empirical likelihood intervals Other parameters