Multivariate area level models for small area estimation. a

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1 Multivariate area level models for small area estimation. a a In collaboration with Roberto Benavent Domingo Morales González d.morales@umh.es Universidad Miguel Hernández de Elche Multivariate area level models for small area estimation p. 1/

2 Index. 1. Small Area Estimation. 2. Multivariate Fay-Herriot models. 3. Application to Spanish Living Condition Survey data. Multivariate area level models for small area estimation p. 2/

3 Small Area Estimation Official surveys are designed to obtain reliable estimates in planned domains. For example, The Spanish Living Condition Survey (SLCS) has sufficiently large sample sizes in the autonomous communities (planned domains). Then, the direct estimators have acceptably small mean squared errors in the autonomous communities. However, the SLCS sample sizes are too small within provinces (unplanned domains or small areas) and therefore the direct estimates are not reliable in these domains. Small Area Estimation is a branch of Statistics that gives procedures to improve the direct estimates in unplanned domains. We introduce small area estimators based on Multivariate area-level mixed models. Multivariate area level models for small area estimation p. 3/

4 Multivariate Fay-Herriot models Let U be a finite population partitioned intod domains U 1,...,U D. Let µ d = (µ d1,...µ dr ) be a vector of characteristics of interest in the domaind. Let y d = (y d1,...y dr ) be a vector of direct estimators ofµ d. The multivariate Fay-Herriot model is defined in two stages. The sampling model is y d = µ d +e d, d = 1,...,D, (1) the vectors e d N (0,V ed ) are independent, ther R covariance matrices V ed are known. The linking model assumes that the µ dr s are linearly related to x dr = (x dr1,...,x drpr ) withp r explanatory variables. x d = diag(x d1,...,x dr ) R p withp = R r=1 p r. β = (β 1,...,β r) p 1. Multivariate area level models for small area estimation p. 4/

5 Multivariate Fay-Herriot models González-Manteiga et al. (2008b) considered the linking model µ d = x d β +1 R v d, v d ind N 1 (0,σ 2 v), d = 1,...,D, (2) where 1 n is the n 1vector with all elements equal to 1. We introduce a multivariate Fay-Herriot model by assuming (1) and substituting the condition (2) by the more realistic linking model µ d = x d β +u d, u d ind N R (0,V ud ), d = 1,...,D, (3) the vectors u d s are independent of the vectors e d s. The R R covariance matrices V ud depend onmunknown parameters, θ 1,...,θ m, with1 m R(R 1) 2 +R. Multivariate area level models for small area estimation p. 5/

6 Multivariate Fay-Herriot models We consider four particularizations of model (3). Model 0 is the product of independent marginal models that assumes (1), (3) and takes V ed = diag (σedr), 2 V ud = diag (σur), 2 d = 1,...,D, (4) 1 r R 1 r R the sampling error variances σedr 2 s are known, m = R and θ r = σur,r 2 = 1,...,R. The components ofe d and u d are independent under Model 0. Model 1 assumes (1) and (3), with a known but not necessarily diagonal matrixv e, and independent components ofu d, i.e. V ud = diag (σur), 2 d = 1,...,D, (5) 1 r R m = R and θ r = σ 2 ur,r = 1,...,R. Model 0 is Model 1 with V e diagonal. Multivariate area level models for small area estimation p. 6/

7 Multivariate Fay-Herriot models Model 2 assumes (1), (3) with AR(1)-correlated u d, i.e. V ud = σ 2 uω d (ρ), Ω d (ρ) = 1 1 ρ 2 m = 2,θ 1 = σ 2 u,θ 2 = ρ. 1 ρ ρ R 1 ρ 1 ρ R 2... ρ R 1 ρ R 2 1 Model 3 assumes (1), (3) with HAR(1)-correlated u d, i.e., (6) u dr = ρu dr 1 +a dr, u d0 N ( 0,σ0 2 ) ind, adr N ( 0,σr) 2, r = 1,...,R, (7) σ0 2 = 1, u d0, a dr, r = 1,...,R, are independent, m = R+1 and θ 1 = σ 2 1,...,θ R = σ 2 R,θ R+1 = ρ. Multivariate area level models for small area estimation p. 7/

8 Application to real data We are interested in estimating small area poverty proportions and gaps by using data from the 2006 Spanish Living Condition Survey (SLCS). We calculate EBLUPs based on multivariate Fay-Herriot models. The target domains are the 52 Spanish provinces crossed by sex (D = 104). The target indicators are the poverty proportion (α = 0) and gap (α = 1), Ȳ αd = 1 N d N d j=1 y αdj, y αdj = ( z Edj z ) α I(E dj < z), z is the poverty line and E dj is the equivalised net income of individual j within domaind, j = 1,...,N d,d = 1,...,D. s is the global sample and s d is the sample of domain d The sample sizes are n andn d respectively, so that s = D d=1 s d and n = D d=1 n d. Multivariate area level models for small area estimation p. 8/

9 Application to real data The direct estimator of the domain total Y dr = N d j=1 y drj is Ŷ dir dr = j s d w dj y drj, where w dj s are the official calibrated sampling weights. The estimated domain size is ˆN dir d = j s d w dj. A direct estimator of the domain mean Ȳ dr isȳ dr = Ŷ dir dr / ˆN dir d. Theȳ dr s are the responses in the area-level model. The design-based covariances of these estimators are approximated by ĉov π (Ŷ dir dr 1,Ŷ dir dr 2 ) = j s d w dj (w dj 1)(y dr1 j ȳ dr1 )(y dr2 j ȳ dr2 ), σ π,d,r1,r 2 = ĉov π (ȳ dr1,ȳ dr2 ) = ĉov π (Ŷ dir dr 1,Ŷ dir dr 2 )/ ˆN 2 d. We take the σ π,d,r1,r 2 s as the known elements of the matrixv ed in the multivariate Fay-Herriot models. Multivariate area level models for small area estimation p. 9/

10 Application to real data The available auxiliary variables are the domain proportions of people in the categories of the following classification variables: Age (age1: 15,age2: 16 24,age3: 25 49, age4: 50 64, age5: 65), Education (edu0: less than primary, edu1: primary, edu2: secondary, edu3: university), Citizenship (cit1: Spanish,cit2: not Spanish), Labor situation (lab0: 15, lab1: employed, lab2: unemployed, lab3: inactive). As the proportions of people in the categories of a given variable sum up to one, we take the reference categories out of the auxiliary data file. The reference categories are age5, edu3,cit2 and lab3. We present two applications. The first application jointly estimates 2006 poverty proportions and gaps for provinces crossed by sex. The second application jointly estimates 2005 and 2006 poverty proportions for provinces crossed by sex. Multivariate area level models for small area estimation p. 10/

11 Application 1 For jointly estimating 2006 poverty proportions (α = 0) and gaps (α = 1), we fit Model 3 to a subset of auxiliary variables. Variables constant age1 age2 edu1 cit1 lab2 β p-value Table 1. Regression parameters and p-values for Model 3,α = 0, Variables constant edu0 edu1 edu2 cit1 lab1 β p-value Table 2. Regression parameters and p-values for Model 3,α = 1, By observing the signs of the regression parameters we conclude that provinces having larger proportions of population in categories age1, age2, edu1, cit1 and lab2 have greater poverty proportion. On the other side, provinces having larger proportions of population in categories edu0, edu1, edu2, and cit1 and smaller proportions of population in the category lab1 have greater poverty gaps. Multivariate area level models for small area estimation p. 11/

12 Application 1 The estimates of the variance component parameters are ˆσ 2 u1 = , ˆσ 2 u2 = and ρ = We test H 0 : σ 2 u1 = σ 2 u2. The test statistics is T 12 = σ 2 u1 σ 2 u2 ν11 +ν 22 2ν 12 = , ν ij,i,j = 1,2,3 are the elements of the inverse of the REML Fisher information matrix of Model 3 evaluated at ˆθ = (ˆσ 2 1,ˆσ 2 2, ˆρ). AsT 12 asym N(0,1) under H 0, thep-value is We conclude that random effects variances are different and we prefer Model 3 instead of Model 2. We also test H 0 : ρ = 0. The test statistics ist ρ = ρ ν33 = AsT ρ asym N(0,1) under H 0, the p-value is We conclude that both components (poverty proportion and gap) are positively correlated and we prefer Model 3 instead of Model 1. Multivariate area level models for small area estimation p. 12/

13 Application 1 Poverty Proportion Men 2006 pd<10 10<pd<20 20<pd<30 pd>30 Poverty Gap Men 2006 gd<3 3<gd<6 6<gd<10 gd>10 Poverty Proportion Women 2006 pd<10 10<pd<20 20<pd<30 pd>30 Poverty Gap Women 2006 gd<3 3<gd<6 6<gd<10 gd>10 Figure 1. Poverty proportions (top) and gaps (bottom) for men (left) and women (right) in Spanish provinces during Multivariate area level models for small area estimation p. 13/21

14 Application 1 Root mse poverty proportion men 2006 Root mse poverty gap men DIR EBLUP DIR EBLUP Figure 2. Root-MSEs of direct and EBLUP (under Model 3) estimators of poverty proportions (left) and gaps (right) in Spanish provinces during Multivariate area level models for small area estimation p. 14/

15 Application 2 For jointly estimating 2005 and 2006 poverty proportions (α = 0), we fit Model 3 to a subset of auxiliary variables. Variables constant age1 age2 edu1 cit1 lab2 β p-value Table 3. Regression parameters and p-values for Model 3,α = 0, Variables constant age1 age2 edu1 cit1 lab2 β p-value Table 4. Regression parameters and p-values for Model 3,α = 0, By observing the signs of the regression parameters we conclude that provinces having larger proportions of population in categories age1, age2, edu1, cit1 and lab2 have greater poverty proportion in 2005 and Multivariate area level models for small area estimation p. 15/

16 Application 2 The estimates of the variance component parameters are ˆσ 2 u1 = , ˆσ 2 u2 = and ρ = We test H 0 : σ 2 u1 = σ 2 u2. The test statistics is T 12 = σ 2 u1 σ 2 u2 ν11 +ν 22 2ν 12 = , whereν ij,i,j = 1,2,3 are the elements of the inverse of the REML Fisher information matrix of Model 3 evaluated at ˆθ = (ˆσ 2 1,ˆσ 2 2, ˆρ). AsT 12 asym N(0,1) under H 0, thep-value is We cannot conclude that random effects variances are different and we prefer Model 2 instead of Model 3. Therefore, we fit Model 2 to the subset of auxiliary variables. Multivariate area level models for small area estimation p. 16/

17 Application 2 Variables constant age1 age2 edu1 cit1 lab2 β p-value β p-value Table 5. Regression parameters andp-values for Model 2 andα = 0. We test H 0 : ρ = 0 under model 2. The test statistics is T ρ = ρ ν22 = , ν ij,i,j = 1,2 are the elements of the inverse of the REML Fisher information matrix of Model 2 evaluated at ˆθ = (ˆσ 2, ˆρ). AsT ρ asym N(0,1) under H 0, the p-value is We conclude that both components (2005 and 2006 poverty proportions) are positively correlated and we prefer Model 2 instead of Model 1. Multivariate area level models for small area estimation p. 17/

18 Application 2 Poverty Proportion Men 2005 pd<10 10<pd<20 20<pd<30 pd>30 Poverty proportion Men 2006 pd<10 10<pd<20 20<pd<30 pd>30 Poverty Proportion Women 2005 pd<10 10<pd<20 20<pd<30 pd>30 Poverty proportion Women 2006 pd<10 10<pd<20 20<pd<30 pd>30 Figure 3. Poverty proportions in 2005 (top) and 2006 (bottom) for men (left) and women (right) in Spanish provinces during Multivariate area level models for small area estimation p. 18/21

19 Application Root mse poverty proportion men 2006 DIR EBLUP Root mse poverty proportion men 2005 DIR EBLUP Figure 4. Root-MSEs of direct and EBLUP (under Model 2) estimators of poverty proportions for 2006 (left) and 2005 (right) in Spanish provinces. Multivariate area level models for small area estimation p. 19/

20 References Benavent R., Morales D. (2016). Multivariate Fay-Herriot models for small area estimation. Computational Statistics and Data Analysis, 94, Esteban, M.D., Morales, D., Pérez, A., Santamaría, L., Two Area-Level Time Models for Estimating Small Area Poverty Indicators. Journal of the Indian Society of Agricultural Statistics, 66, 11, Esteban, M.D., Morales, D., Pérez, A., Santamaría, L., Small area estimation of poverty proportions under area-level time models. Computational Statistics and Data Analysis, 56, Fay, R.E., Herriot, R.A., Estimates of income for small places: An application of James-Stein procedures to census data. Journal of the American Statistical Association 74, González-Manteiga, W., Lombardía, M.J., Molina, I., Morales, D., Santamaría, L., 2008b. Analytic and Bootstrap Approximations of Prediction Errors under a Multivariate Fay-Herriot Model. Computational Statistics and Data Analysis, 52 (12), Särndal C.E., Swensson B., Wretman J., Model assisted survey sampling. Springer-Verlag. Multivariate area level models for small area estimation p. 20/

21 Thank you Thank you for your attention Multivariate area level models for small area estimation p. 21/