Small area estimation for poverty indicators

Transcription

1 Small area estimation for poverty indicators Risto Lehtonen (University of Helsinki) Ari Veijanen (Statistics Finland) Mikko Myrskylä (Max Planck Institute for Demographic Research) Maria Valaste (Social Insurance Institution) BaNoCoSS-2011, Norrfällsviken, Sverige

2 Outline Background Laeken indicators Models and estimators Experimental designs Results Discussion 2

3 Official EU indicators on poverty and social exclusion (Laeken indicators) At-risk-of-poverty rate Life expectancy at birth At-risk-of-poverty threshold Income quintile share ratio Persistent at-risk-of-poverty rate Persistent at-risk-of-poverty rate (alternative threshold) Relative median at-risk-of- poverty gap Regional cohesion Long-term unemployment rate Persons living in jobless households Early school leavers not in education or training Self defined health status Dispersion around the at-riskof-poverty threshold At-risk-of-poverty rate anchored at one moment in time At-risk-of-poverty rate before cash social transfers Gini coefficient In-work at risk of poverty rate Long term unemployment share Very long term unemployment rate Risto Lehtonen and Ari Veijanen 3

4 European Union Low income by age and gender All the graphs below use the main EU measure of low income, namely a household income below 60% of the contemporary, national, median household income before deducting housing costs. All household incomes are after taxes have been deducted and after adjustment ('equivalisation') for household size and composition. This level of income is referred to as 'the poverty line' as that is the term used for it by the EU. It is the same measure of low income as that used in most of the UK-specific sections of this website except that it is before, rather than after, deducting housing costs. To improve its statistical reliability given small sample sizes, the data in all the graphs bar the first is the average for the latest three years. 4

5 AMELI - Advanced Methodology for European Laeken Indicators - DoW AMELI will focus on significant improvements of methods and models for indicators of social cohesion in order to support an optimal use of indicators as well as to undertake a high-quality impact assessment for policy support. A major output of the project will be devoted to find an improved methodology for indicators that gives a reliable decision support to policy makers. Since it is extremely difficult to quantify the complete impact of data processing and estimation on the indicators, a large simulation study will be integrated into AMELI based on realistic data. 5

6 AMELI - Advanced Methodology for European Laeken Indicators AMELI ( ) FP7 research project Consortium University of Trier (Coordinator) University of Helsinki FHNW (Switzerland) Vienna University of Technology Destatis (Germany) Statistics Austria Statistics Finland Swiss Federal Statistical Office Statistics Estonia Statistical ti ti Office of the Republic of Slovenia 6

7 Poverty indicators in AMELI At-risk-of poverty rate Quintile share ratio QSR (S20/S80 ratio) Relative median at-risk-of poverty gap Gini coefficient 7

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9 Objectives of the study Study on Standard estimators Standard estimators do not use auxiliary data Investigation of statistical properties p (bias and accuracy) Introduction of alternative estimators Use of unit-level auxiliary data in the estimation procedure Use of aggregate auxiliary data in the estimation procedure Investigation of bias and accuracy of the new estimators Experimental design Design-based simulation Two different real populations p Equal and unequal probability sampling designs Various outlier contamination schemes 9

10 Estimation approaches for poverty indicators Standard estimators Design-based direct estimators Alternative estimators Design-based model-assisted estimators - Generalized regression (GREG) family - Model calibration estimators Model-based estimators - Empirical best predictor (EBP) type estimators - Synthetic (EBLUP) type estimators - Expanded prediction estimators (SYN-EP) Composite estimators Linear combination of direct estimator and model-based estimator 10

11 Modelling framework: GLMM GLMM formulation with area - specific random terms E ( y u ) = f ( x ( β + u )), where m k r k r f (.) refers to the chosen functional form k refers to population p unit k U r refers to region Ur U x = (1, x,..., x ) auxiliary variable values k 1k ( β, β,..., β ) fixed effects β = 0 1 u r 0r pr pk p = ( u,..., u ) area-spefic random effects Predictions: yˆ = f( x ( βˆ + uˆ )), k U k k r 11

12 Estimation of model parameters Most of generalized linear mixed models were fitted by R package nlme using maximum likelihood Design information was not included Used for quintile share ratio, poverty gap and Gini Incorporation of design information into model fitting by glmer function of R package lme4 lme4 fits mixed models by a penalized iteratively reweighted least squares algorithm (Bates 2011) Used for poverty rate under unequal probability sampling Some methodological references Datta (2009), Jiang and Lahiri (2006), Rao (2003) Lehtonen, Särndal and Veijanen (2003, 2005, 2009) Lehtonen and Veijanen (2009) Särndal (2007), Särndal et al. (1992) 12

13 Outlier contamination ti schemes Outlying mechanisms OCAR - Outlying completely at random OAR - Outlying at random Contamination models CCAR - Contamination completely at random CAR - Contamination at random NCAR - Contamination not at random 13

14 Outlier contamination schemes: Pi Priorities iti (1, 2 or 3) Source: Beat Hulliger & Tobias Schoch (2010) AMELI WORKING PAPER, Outlier Contamination Models and Simulation Schemes Cases considered in Deliverable 2.2 on SAE OCAR-CCAR 1% OCAR-NCAR 1% OAR-CAR OCAR-CCAR 15% 14

15 Population data for simulation Register data Western Finland Fixed finite population of one million persons 70 NUTS3 x demographic domains PPS sampling of 5,000 persons K = 1000 samples AMELIA population EU-wide, SILC data Fixed finite population of 10 million persons 40 regional domains SRSWOR sampling of 2,000 persons K = 1000 samples 15

16 Quality measures of estimators Design bias Absolute relative bias ARB (%) Accuracy Relative root mean squared error RRMSE (%) 1 K ˆ θd ( sk ) θd / θd K k =11 K =1 1 K 2 ( ˆ θd( sk) θd) / θd K k 1 Averages over domain sample size classes (minor/medium/major) / 16

17 Highlights ht Poverty rate Direct (default) estimators Indirect estimators Generalized regression MLGREG Empirical Best Predictor (EBP) Modelling framework Logistic mixed models Q: How to account for unequal probability sampling? Quintile share ratio Direct (default) estimators Indirect estimators t Prediction (synthetic) estimators SYN Expanded prediction estimators SYN-EP Composite estimators Default with SYN-EP Frequency-calibrated predictors Modelling framework Linear mixed models Q: Robustness against outlier contamination? 17

18 Poverty rate At-risk-of poverty rate At-risk-of-poverty threshold is 60% of the median equivalized household income in the whole population U Threshold is estimated with HT People whose income is below or at the at-risk-of-poverty threshold are referred to as poor Binary study variable y for poverty rate 1: In poverty, 0: Otherwise 18

19 Estimation of poverty total for domains (a) Design-based GREG type estimators t ˆ ˆ ˆ = y a e + (1) dmlgreg k k k k U d k s d = 1,..., D, where a = 1/ π, eˆ = y yˆ d k k k k k (b) Model-based EBP type estimators tˆ ˆ debp = yk, (2 ) k U d where in (1) and (2): exp( x ˆ ˆ kβ + ur) yˆ k =, k U 1+ exp( x βˆ + uˆ ) k r 19

20 Estimation of poverty rate for domains Estimator of at-risk-of poverty rate or rˆ = tˆ / N dmlgreg dmlgreg d rˆ = ˆ / ˆ dmlgreg tdmlgreg Nd where Nˆ d = ak, ak = 1/ π k k s d rˆ similarly, d = 1,..., D debp 20

21 Monte Carlo simulation 1: FIN data Fixed finite population of 1,000,000 persons Western Finland Real register data of Statistics Finland Domain structure NUTS3 regions * Gender * Age group 70 domains of interest PPS sampling design (unplanned domains) Size variable: Socio-economic status Sample size n = 5,000 persons Expected domain sample sizes Minor: -49 units, Medium-sized: 50-99, Major: 100- K = 1000 independent samples 21

22 Monte Carlo simulation 1: FIN data Binary Y-variable Poverty indicator (1=yes, 0=no) Potential X-variables Gender Age group (5 classes) Labor force status (3 classes) Socio-economic status of HH head (6 classes) - Note special role: PPS size variable Accounting for unequal probability sampling Incorporation of design variable into model Incorporation of design weights in estimation of model parameters 22

23 Poverty rate Empirical Best Predictor 23

24 Poverty rate MLGREG estimators 24

25 Conclusions for poverty rate Model-assisted GREG Assisted by logistic mixed model Small design bias in all domain size classes Good accuracy in large domains Accuracy can be poor in small domains Model-based Empirical best predictor EBP Design biased in all domain size classes Large design bias in small domains Good accuracy also in small domains EBP gains much from inclusion of design information Poverty rate estimators are robust against outlier contamination (not shown here) Of the poverty rate estimators, EBP might be the best choice unless it is important to avoid design bias 25

26 Quintile share ratio QSR S20/S80 ratio, or quintile share ratio The ratio of average income of poorest 20% of people (first quintile) to average income of richest 20% of people (fifth quintile) To find the first quintile we sort the persons by income The first quintile q d,20 is the set of poorest people in domain d whose sum of weights is below or at 20% of the total sum of weights Similarly, the fifth quintile q d,80 is the set of domain s richest people with sum of weights above or at 80% of the total of weights 26

27 Estimation approaches for QSR Direct design-based estimator Use of unit-level e auxiliary a data Prediction estimators SYN Expanded prediction estimators SYN-EP Composite estimators COMP Use of aggregate-level g auxiliary data Frequency-calibrated predictors calculated using known domain marginal totals of auxiliary variables Composite estimators t COMP Estimation under outlier contamination 27

28 Direct QSR estimator t Direct estimator of first quintile Sˆ20 a y / a, a 1/ π = = d k k k k k k q k q d,20 d,20 Direct estimator of fifth quintile Sˆ80 a y / a = d k k k k q k q d,80 d,80 Direct quintile share estimate in domain q = ˆd Sˆ2 0d S80 ˆ80d d 28

29 Modelling framework for SYN: Linear mixed models Model formulation with area-specific random terms E ( y u ) = x ( β + u ), r = 1,..., R m k r k r where r refers to region Ur U x = (1, x,..., x ) k 1k pk ( β, β,..., β ) are fixed effects β = 0 1 0r p u = ( u,..., u ) are random effects r pr Predictions: y ˆ = x ( βˆ + u ˆ ), k U k k r 29

30 Synthetic ti QSR estimator t Quintiles q and q are defined in SYN; d,20 SYN; d,80 population domain as if the weights were constant Synthetic estimator of first quintile Sˆ 20 yˆ / Ik { q } = dsyn k SYN; d,20 k q k U SYN; d,20 Fifth quintile similarly il l Synthetic quintile share estimator in domain d qˆ dsyn = Sˆ20 ˆ80 dsyn S80 dsyn d 30

31 Transformations for SYN-EP Logarithmic transformation to correct for the skewness of the distribution of the study variable Back-transformation RAST (Ratio Adjusted by Sample Total; Chambers and Dorfman, 2003, Fabrizi et al., 2007b) type transformation Tails of distribution are important! We used a more elaborate further transformation aimed at improving the histogram of predictions In population domain, we transform predictions so that they have similar histogram as the observed values in sample domain 31

32 Composite QSR estimator t Composite quintile share estimator in domain d qˆ = ˆ λ qˆ + (1 ˆ λ ) qˆ dcomp d d d dsyn where ˆ λd is an average of MSE ˆ ( qˆ dsyn ) MSE ˆ ( qˆ ) + var( ˆ qˆ ) dsyn over a domain size class d 32

33 Variance estimation for direct estimator t Estimation of var( ˆ qˆ ) d by bootstrap: An artificial population p is generated by cloning each unit with frequency equal to the design weight Bootstrap samples are drawn with the original sampling design from the artificial population The variance of the direct estimator is then estimated by the sample variance of estimates in the bootstrap samples (Leiten and Traat 2006) 33

34 MSE estimation for SYN estimator t Estimation of MSE ˆ ( q ˆ ): dsyn 2 ˆ ˆ ˆ ˆ ˆ ˆ dsyn = dsyn d d MSE ( q ) q q var( q ) MSE ( ) Rao (2003 p. 52) and Fabrizi et al. (2007) Alternative: Simulation-based method similar to Molina and Rao (2010) 34

35 Monte Carlo simulation: AMELIA Fixed finite population of 10 million persons Generated from EU-wide SILC data Domain structure D = 40 regional domains of interest SRSWOR sampling design Sample size n = 2,000 persons Contamination schemes None OCAR-CCAR 1% K = 1000 independent samples 35

36 Table 3. Quintile share estimators with linear mixed model fitted to log(income+1) including domain level random intercepts. Sampling design: SRSWOR Auxiliary variables: Age and gender with interactions, education level, activity, degree of urbanisation Domains: Regional DIS variable (40 domains) Contamination: None Data: AMELIA population Estimator Unit-level auxiliary data Average ARB (%) Average RRMSE (%) Domain size class Minor Medium Major 100- Domain size class Minor Medium Major 100- Direct (default) Indirect SYN Indirect SYN-EP Composite Area-level auxiliary data Frequency calibration Composite

37 Table 4. Quintile share estimators with linear mixed model fitted to log(income+1) including domain level random intercepts. Sampling design: SRSWOR Auxiliary variables: Age and gender with interactions, education level, activity, degree of urbanisation Domains: Regional DIS variable (40 domains) Contamination: OCAR-CCAR (1%) Data: AMELIA population Estimator Average ARB (%) Average RRMSE (%) Domain size class Minor Medium Major 100- Domain size class Minor Medium Major 100- Unit-level auxiliary data Direct (default) Indirect SYN-EP Composite Area-level auxiliary data Frequency calibration Composite

38 Conclusions for QSR Default estimator No contamination: Small design bias Large variance Outlier contamination: i Substantial bias Large variance Expanded prediction SYN-EP Some bias in all domain size classes Much better accuracy than the default estimator in all domain size classes Robust against outlier contamination Composite estimator Offers a reasonable compromise with respect to bias and accuracy 38

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41 Thank you for your attention! 41