Discussing Effects of Different MAR-Settings

Transcription

1 Discussing Effects of Different MAR-Settings Research Seminar, Department of Statistics, LMU Munich Munich, Matthias Speidel Jörg Drechsler Joseph Sakshaug

2 Outline What we basically want to do: hierarchical imputation What we encounter: effects of MAR-settings Questions: explanation of these effect Discussing Effects of Different MAR-Settings 2

3 Our research DFG-Project Reason for hierarchical imputation: Item-Nonresponse nonresponse bias Imputation reduce or unmake nonresponse bias Hierarchical data hierarchical imputation Aim for unbiased linear mixed model parameter estimates Which imputation method works well, which not? And why? Focus on dummy variables imputation Discussing Effects of Different MAR-Settings 3

4 Notation Hierarchical data in our case: y j,i = β 0 + b 0j + (β 1 + b 1j ) x j,i + ε j,i i = 1,..., n j j = 1,..., m n j size of cluster j m number of clusters ( b 0 j b 1j ) N (( 0 0 ), Σ b = ( σ2 b 0 σ b1b 0 σ b0b 1 σ 2 b 1 )) ε j,i N(0, σ 2 ε) β j = β + b j Discussing Effects of Different MAR-Settings 4

5 Dummy variables imputation Dummy imputation only hierarchical imputation method implemented in SAS, SPSS and Stata. Estimates cluster specific intercept and slope imputation parameters (assumed fixed not random) Under which data and missing generating setting biased results? Explanations for the bias? Discussing Effects of Different MAR-Settings 5

6 Shortly: the main results cluster size = 10 cluster size = 25 cluster size = σ b0 2 σ b1 2 σ b0b1 ρ missing rate relative bias Figure: Relative bias after dummy imputation for different settings. Discussing Effects of Different MAR-Settings 6

7 Side results Not only data setting important, but also shape of missing function. Don t mean MCAR vs. MAR vs. MNAR (Rubin (1976)). Mean different missing functions within MAR. Discussing Effects of Different MAR-Settings 7

8 General missing function Missing function for standardized X i, desired missing rate MR 0.5 and 1 s 1: P(Y i = NA X i, MR, s) = MR (1 s) + 2 MR s logit(x i ) 1 (1) s controls strength and direction of X s influence on the missing probabilities. MCAR if s = 0 Missing probabilities on average equal to MR if X symmetrically distributed around 0 (proof missing). Discussing Effects of Different MAR-Settings 8

9 Illustrations missing mechanism: varying MR missing functions MR = 0.5 MR = 0.25 MR = 0.10 MR = 0.05 P(Y = NA) standardized X Figure: Illustration of missing probabilities for different missing rates Discussing Effects of Different MAR-Settings 9

10 Illustrations missing mechanism: varying s missing functions s = 1.0 s = 0.5 s = 0.0 s = 0.5 s = 1.0 P(Y = NA) standardized X Figure: Five different shapes of missing functions (s = 1, 0.5, 0, 0.5, 1). All lead to MAR, except MCAR of course. Discussing Effects of Different MAR-Settings 10

11 Results when estimating Σ b after dummy imputation σ b0 2 s σ b1 2 s σ b0b1 s ρ s Figure: Relative empirical biases under different missing functions. Discussing Effects of Different MAR-Settings 11

12 Questions 1. Explanation for differences in parameter estimates for different MAR settings? 2. Sufficient to look at variance matrix from imputation parameters (instead of variances after imputation)? Sufficient to look at variance matrices of a single cluster? 3. If bias in var( ˆβ j ) fix not clearly visible, sufficient to compare it to the unbiased var( ˆβ j ) rand? How to say matrix A is larger than matrix B? Elementwise comparison? Multiplicative or additive matrix? Interpretation? Discussing Effects of Different MAR-Settings 12

13 Imputation parameters variances Main interest in Var( ˆβ j β, Σ b, σ 2 ε) fix (readability)var( ˆβ j ) fix = σ 2 ε (X T X) 1 (2) Maybe compare it to what can be found in (Goldstein, 2011: p. 69) Var( ˆβ j β, Σ b, σ 2 ε) rand Var( ˆβ j ) rand = σ 2 ε (X T X + σ 2 ε Σ 1 b ) 1 (3) Beside elementwise comparison, multiplicative and additive term γ Var( ˆβ j ) rand = Var( ˆβ j ) fix (4) δ + Var( ˆβ j ) rand = Var( ˆβ j ) fix (5) Discussing Effects of Different MAR-Settings 13

14 Multiplicative disparity Properties and interpretation of γ? γ Var( ˆβ j ) rand = Var( ˆβ j ) fix γ σε 2 (X T X + σε 2 Σ 1 b ) 1 = σε 2 (X T X) 1 γ = (X T X) 1 (X T X + σε 2 Σ 1 b ) = (X T X) 1 X T X + (X T X) 1 σ 2 ε Σ 1 b = I + (X T X) 1 σ 2 ε Σ 1 b = I + Var( ˆβ j ) fix Σ 1 b (6) Discussing Effects of Different MAR-Settings 14

15 Additive disparity Properties and interpretation of δ? γ Var( ˆβ j ) rand = Var( ˆβ j ) fix (I + Var( ˆβ j ) fix Σ 1 b ) Var( ˆβ j ) rand = Var( ˆβ j ) fix Var( ˆβ j ) rand + Var( ˆβ j ) fix Σ 1 b Var( ˆβ j ) rand = Var( ˆβ j ) fix (7) Now, we have the matrix δ: δ = Var( ˆβ j ) fix Σ 1 b Var( ˆβ j ) rand (8) δ = (X T X) 1 σε 2 Σ 1 b σε 2 (X T X + σε 2 Σ 1 b ) 1 Discussing Effects of Different MAR-Settings 15

16 References I Goldstein, Harvey (2011): Multilevel Statistical Models. Chichester (UK): Wiley, 4 ed.. Rubin, Donald B. (1976): Inference and Missing Data. In: Biometrika, Vol. 63, No. 3, p , URL Discussing Effects of Different MAR-Settings 16

17 Additional material Discussing Effects of Different MAR-Settings 17

18 Details of imputation parameters variance Dummy imputation: Var( ˆβ j ) fix = σ 2 ε n x 2 i ( x i ) 2 ( x2 i x i x i n ) (9) σ 2 ε = n 2 var(x) ( x2 i x i x i n ) (10) Mixed effects imputation: Var( ˆβ j ) rand = σε 2 (( n x i σ 2 ε x i xi 2 ) + σ0 2 σ2 1 σ2 01 ( σ2 1 1 σ 01 σ 01 σ0 2 )) (11) To shorten the expression, I define a = σ 2 ε. σ0 2 σ2 1 σ2 01 Var( ˆβ j ) rand = σε 2 ( n + a σ2 1 1 x i a σ 01 x i a σ 01 xi 2 + a σ0 2 ) (12) Discussing Effects of Different MAR-Settings 18

19 Details of imputation parameters variance (cont.) Var( ˆβ j ) rand = (n + a σ1 2) ( x2 i + a σ0 2) ( x i a σ 01 ) 2 σ 2 ε ( x2 i + a σ0 2 x i + a σ 01 x i + a σ 01 n + a σ1 2 ) (13) Discussing Effects of Different MAR-Settings 19

20 Details on delta δ =Var( ˆβ j ) fix Σ 1 b Var( ˆβ j ) rand σ 2 ε = n 2 var(x) ( x2 i x i x i n ) 1 σ0 2 σ2 1 σ2 01 ( σ2 1 σ 01 σ 01 σ0 2 ) (14) σ 2 ε (n + a σ 2 1 ) ( x2 i + a σ 2 0 ) ( x i a σ 01 ) 2 ( x2 i + a σ0 2 x i + a σ 01 x i + a σ 01 n + a σ1 2 ) Discussing Effects of Different MAR-Settings 20

21 Our simulation parameters Hierarchical data in our case: y j,i = b 0j + (0.8 + b 1j ) x j,i + ε j,i ( b 0 j ) N (( 0 b 1j 0 ), Σ b = ( )) ε j,i N(0, 92.2) x j,i N(0, 81) n j dependent on setting 10, 25, or 50 m = 100 Individuals class affiliation known and variable in data set. Parameters from NEPS data (and similar to CILS4EU data). Discussing Effects of Different MAR-Settings 21

22 Matthias Speidel Jörg Drechsler Joseph Sakshaug