untitled

Size: px

Start display at page:

Download "untitled"

かずまさあんさい
5 years ago
Views:

1 c Särndal, 2005 Groves et al.,

2 Covariate 3 Taylor et al., 2001;, 2007; Schonlau et al., 2009, 2009, Heckman Heckman, 1974 Heckman, 2009 Ishwaran and James,

3 5 4 Blocked Gibbs sampler N N 1 N 2(= N N 1) y z =1 z =0 z x 1 z =1 y z =0 y y E(y) (2.1) Ê(y)= 1 N N i=1 y i (2.2) ȳ obs = N i=1 ziyi N i=1 zi y z i i y 1.

4 Selection on observables Selection on unobservables (2.3) p(z y,x)=p(z x) x y Little and Rubin, 2002 y Missing at randon 2.3 (2.4) p(y x,z=1)=p(y x,z=0)=p(y x) (2.5) E(y z =1, x)=e(y x) (2.6) E(y)=E x[e(y x)] = E x[e(y z =1,x)] = E(y z =1, x)p(x)dx E a a p(x) z =1 y x g(x β) β (2.7) (2.8) 1 N 1 N N g(x i β) i=1 N z iy i +(1 z i)g(x i β) i=1 β

5 Post-stratification Deming and Stephan, 1940; Ireland and Kullback, 1968 Iterative proportional fitting Deville and Särndal, , x x z 2.3 z =1 2.4 (3.1) E x {E ( z E(z x) y x )} { ( ) } z = E x E x E(y x) = E(y) E(z x) i e i = p(z i =1 x) IPW Inverse probability weighting (3.2) N z i i=1 N i=1 y i e i z i e i g(x β),

6 , 2009 Taylor et al., 2001;, 2007;, Brookhartetal Strongly Ignorable Treatment Assignment , , 2006

7 9 Särndal and Lundström, y i i y x yi y (3.3) y i = x t yiβ y + ɛ yi β y x y δ i i x δi (3.4) δ i = x t δiβ δ + ɛ δi δ i > 0 y i 3.4 ɛ y ɛ δ 2 ɛ δ 1 (3.5) ( ɛ y ɛ δ ) (( N 0 0 ) (, σ 2 y ρσ y ρσ y 1 β y Heckman Heckman, L (3.6) L = Pr(δ i 0) [Pr(δ i > 0 y i)pr(y i)] i:δ i 0 i:δ i 0 i:δ i >0 = [1 Φ ( [ ( { }) x t ) 1 δiβ δ ] Φ x t δiβ δ + ρ (y i x t yiβ 1 ρ 2 y) σ y i:δ i >0 σ y )) ( )] 1 yi x t yiβ y φ σ y φ( ) Φ( )

8 z y z =1 z =0 2 ( { }) 1 (3.7) p(z =0 y,x y,x δ )=p(δ 0 y,x y,x δ )=1 Φ x t δβ δ + ρ (y x t 1 ρ 2 σ yβ y) y ɛ y ɛ δ ρ 0 δ 0 m =0 y y , y (4.1) Pr(z =1 y,x)=e(y,x,α) (4.2) N { p(yi x i,θ)p(z i =1 y } { } 1 zi z i i,x i,α) p(y i x i,θ)p(z i =0 y i,x i,α)dy i i=1 α θ p(y x) p(z y,x) y y y J y = j (4.3) p(y = j)= p(y = j x, θ)p(x)dx θ y Lee and Berger 2001 p(z y,x) Sethuraman, 1994 x y

9 11 Lee and Berger 2001 Lee and Berger 2001 y z f Hidden covariate, 2009 Imbens 2003 f x f x 3.3 y y u (4.4) u = B t x + λf + ɛ y u 1 0 y ɛ 4.4 (4.5) u = λf + ɛ ɛ ɛ (4.6) ɛ π k N(B k x,σ k ) k=1 B k,σ k k B,Σ z δ δ>0 z =1 δ 0 z =0 3.4 (4.7) δ = γf + η η (4.8) η π k N(α t kx,φ 2 k) k=1 α k,φ 2 k k α,φ 2 γ 4.2 N (4.9) p(y i f i,x i,θ)p(z i =1 f i,x i,α)p(f i)df i z i =1 N p(y i f i,x i,θ)p(z i =0 f i,x i,α)p(f i)df idy i z i =0

10 N = p(y i f i,x i,θ)p(z i =1 f i,x i,α)p(f i)df i z i =1 N p(z i =0 f i,x i,α)p(f i)df i z i =0 5. Blocked Gibbs sampler 5.1 Ishwaran and Zarepour Y L DP L(a,G 0) (5.1) L y p l f( θ l ) l=1 L Ishwaran and James 2001 Blocked Gibbs sampler Blocked Gibbs sampler Miyazaki and Hoshino 2009 Blocked Gibbs sampler y i = g(u i) z =1(δ>0) u i f i,x i,θ,k p(u i f i,x i,θ ki ), δ i f i,x i,θ,k p(δ i f i,x i,θ ki ) (i =1,...,N), (5.2) L k i κ κ l 1 l ( ) l=1 κ p(κ), θ p(θ τ ), τ p(τ) g y u N i k i i l k i = l 1 l ( ) k i = l 1 0 θ ki i k i θ θ B Σ α κ =(κ 1,...,κ L) Stick-breaking (5.3) l 1 κ l = V l (1 V m) m=1 V l Beta(a l,b l ) b l = L m=l+1 am a l = ν/l ν

11 13 κ l Blocked Gibbs sampler 5.2 Blocked Gibbs sampler (5.4) f N(0,1), λ N(µ λ,σλi), 2 vec(b) N(µ B,σBI) 2 α N(µ α,σαi), 2 φ 2 χ 2 (n φ 2,c), Σ W 1 (n Σ,D) χ 2 W 1 vec(a) A Blocked Gibbs sampler 1. u u y Albert and Chib λ,σ,b,α,φ iteration L L θ θ iteration m k m {k 1,...,k m} L m θ p(θ τ ) m θ (5.5) p(θ k j k,u,δ,f,x) p(θ k j τ ) {i:k i =k j } p(u i k i,f i,x i,θ ki )p(δ i k i,f i,x i,θ ki ) (j =1,...,m) λ 4.5 f Hoshino 2001 (( N ) 1 [ (5.6) λ k j N σ 2 λ I + N fi 2 Σ 1 k j σ 2 λ Iµ λ + fi 2 Σ 1 k j (ui B k ], j xi) i:k i =k j ( N σ 2 λ I + i:k i =k j f 2 i Σ 1 k j ) 1 ) i:k i =k j Σ B u Gelman et al., 2003 α φ 2 δ 3. f f i i

12 (5.7) p(f i k i,u i,δ i,x i,θ ki ) p(u i k i,f i,x i,θ ki ) p(δ i k i,f i,x i,θ ki ) p(f i) (i =1,...,N) p(u i k i,f i,x i,θ ki ) p(δ i k i,f i,x i,θ ki ) f i (5.8) f (( i N λ t k i Σ 1 k i λ t k i + γk 2 i φ 2 k i ( λ t ki Σ 1 k i λ t k i + γ 2 k i φ 2 k i +1 ) 1) +1 ) 1[ λ t ki Σ 1 k i (u i B ki x i)+γ ki φ 2 k i (δ i α t k i x i) ], 4. k k L k=1 p kiδ k ( ) (i =1,...,N) p ki (5.9) p ki = 5. κ κ k κ k p(u i k i,f i,x i,θ ki )p(δ i k i,f i,x i,θ ki ) L k=1 κ kp(u i k i,f i,x i,θ ki )p(δ i k i,f i,x i,θ ki ) k 1 κ k = V k (1 V m) m=1 ( V k Beta a k + M k,b k + L m=k+1 M m ) M k k 5.3 Blocked Gibbs sampler E(y) (5.10) E(y)= (5.11) Ê(y)= 1 M m=1 L l=1 M 1 N p l p(y k = l,f,x,θ l )p(f)p(x)df dx N L ( p(yi ki m = l,fi m,x i,θ m l ) ) 1(ki m =l) i=1 l=1 1(ki m = l) m iteration k i l 1 0 f δ γ ν γ ν

13 K D.K , 2009, 63.8 Cox & Snell IPW 3 L =1 4 L =20 4 IPW γ 0.2, 0.4, γ 0.2, 0.4, 0.7 ν 1, 3, 10 3 γ f γ =0.2, 0.4, 0.7 4, 16, γ =0.7 ν

14 iteration 35,000 5,000 Burn-in Phase 30,000 ν ν 10 ν = ν =1 5 4 ν =3 6 4 ν = γ 0.7 ν IPW 3.

15 17 4. γ = γ = γ =0.7

16 γ =0.7 ν =10 1 ˆp 1 4 γ =0.7 ν =10 ˆp 4 ξ (6.1) ζ = ˆp 1 ˆp ξ ˆp Silverman Rule-of-thumb 1.06 ˆσˆp 1 N 1/5 N γ =0.4 4 γ =0.7 ν =10 D.K. General Social Survey; GSS, Davis, Smith & Marsden IPW

17 19 8. ζ

18 K = =73.1 N = 1729 ( ) ( ) ( ) (7.1) , (0.2481, ) Copas and Eguchi

19 21, 2010 NEDO 19 JST Albert,J.H.andChib,S Bayesian analysis of binary and polychotomous response data, Journal of the American Statistical Association, 88, Brookhart, M. A., Schneeweiss, S., Rothmanm, K. J., Glynn, R. J., Avorn, J. and Stürmer, T Variable selection for propensity score models, American Journal of Epidemiology, 163, Copas, J. B. and Eguchi, S Local model uncertainty and incomplete data bias with discussion Journal of the Royal Statistical Society, Series B, 63, Deming, W. E. and Stephan, F. F On a least squares adjustment of a sampled frequency table when the expected marginal totals are known, Annals of Mathematical Statistics, 11, Deville, J. C. and Särndal, C. E Calibration estimators in survey sampling, Journal of the American Statistical Association, 88, Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B Bayesian Data Analysis, Chapman & Hall, New York. Groves, R. M., Dilman, D. A., Eltinge, J. L. and Little, R. A. J Surveys Nonresponse, Wiley, New York. Heckman, J. J Shadow prices, market wages and labor supply, Econometrica, 42, Heckman, J. J Sample selection bias as a specification error, Econometrica, 47, Hoshino, T Bayesian inference for finite mixtures in confirmatory factor analysis, Behaviormetrika, 28, Web 27 59

20 Imbens, G. W Sensitivity to exogeneity assumption in program evaluation, American Economic Review, 93, Ireland, C. T. and Kullback, S Contingency tables with given marginals, Biometrika, 55, Ishwaran, H. and James, L. F Gibbs sampling methods for stick-breaking priors, Journal of the American Statistical Association, 96, Ishwaran, H. and Zarepour, M Markov chain Monte Carlo in approximate Dirichlet and beta two-parameter process hierarchical models, Biometrika, 87, Lee, J. and Berger, J. O Semiparametric Bayesian analysis of selection models, Journal of the American Statistical Association, 96, Little, R. J. A. and Rubin, D. B Statistical Analysis with Missing Data, 2nd ed., Wiley, New York. Miyazaki, K. and Hoshino, T A Bayesian semiparametric item response model with Dirichlet process priors, Psychometrika, 74, Särndal, C-E. and Lundström, S Estimation in Surveys with Nonresponse, Wiley, New York. Schonlau, M., van Soest, A., Kapteyn, A. and Couper, M Selection bias in web surveys and the use of propensity scores, Sociological Methods & Research, 37, Sethuraman, J A constructive definition of Dirichlet priors, Statistica Sinica, 4, Taylor, H., Bremer, J., Overmeyer, C., Siegel, J. W. and Terhanian, G The record of internetbased opinion polls in predicting the results of 72 races in the November 2000 U.S. elections, International Journal of Market Research, 43, No

21 Proceedings of the Institute of Statistical Mathematics Vol. 58, No. 1, 3 23 (2010) 23 Semiparametric Estimation under Nonresponse in Survey and Sensitivity Analysis: Application to the 12th Survey of the Japanese National Character Takahiro Hoshino Department of Economics, Nagoya University In recent years, the collection rate for conventional types of surveys such as visit survey with random sampling has been declining. Therefore, a solution for bias due to nonresponse needs to be developed. We formulate the bias due to nonresponse in a survey as selection bias in econometrics, and I point out the problems in applying covariate adjustment methods to nonresponse in surveys. In this paper, we propose a semiparametric Bayes model where a latent hidden covariate affect both the response variables and the indicator of nonresponse using Dirichlet process mixtures. By changing some portion of parameters, we can conduct a sensitivity analysis to investigate how much the confidence interval is under a high rate of nonresponse. We apply the proposed method to the 12th survey of the Japanese National Character, and found that the method provides more reasonable confidence intervals, compared to that calculated without any model assumption. Key words: Dirichlet process mixture model, propensity score, hidden covariate, sociological surveys, covariate adjustment, selection bias.

dvi

dvi 2017 65 2 217 234 2017 Covariate Balancing Propensity Score 1 2 2017 1 15 4 30 8 28 Covariate Balancing Propensity Score CBPS, Imai and Ratkovic, 2014 1 0 1 2 Covariate Balancing Propensity Score CBPS