Small area estimation by model calibration and "hybrid" calibration. Risto Lehtonen, University of Helsinki Ari Veijanen, Statistics Finland

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1 Small area estimation by model calibration and "hybrid" calibration Risto Lehtonen, University of Helsinki Ari Veijanen, Statistics Finland NTTS Conference, Brussels, March 2015

2 Lehtonen R. and Veijanen A. (2015) Design-based methods to small area estimation and calibration approach. In: Pratesi M. (Ed.) Analysis of Poverty Data by Small Area Estimation. Chichester: Wiley. (Forthcoming) 2

3 Outline Framework Estimators for area (domain) means HT estimator Model-free calibration Model calibration Hybrid calibration Assisting model Monte Carlo experiment Summary Literature 3

4 Calibration methods in survey sampling (with reference to domain estimation) 4

5 Modelling & calibration in domain estimation with MC and HC Modelling in MC and HC Generalized linear fixed-effects models Generalized linear mixed models Calibration in model calibration MC Calibration of estimated total of predictions to the population level, domain level or intermediate level Calibration in hybrid calibration HC Calibration of estimated total of predictions and estimated total auxiliary of x-variables to population level, domain level or intermediate level NOTE: In HC, the x-variables in the modelling phase and in the calibration phase can overlap or they can be separate variables 5

6 Estimators for domain means Domain means y t / N y / N HT estimator d d d ku k d yˆ tˆ / N a y / N dht dht d ks k k d Calibration estimators yˆ tˆ / N w y / N d d d ks k k d s s U, a 1/ are design weights w d d k k k d are (method-specific) calibration weights d d 6

7 Model-free calibration equation w z z N, x,..., x ( MFC ) i i i d 1i pi is iu iu iu d d d d where z (1, x,..., x ), s s U i 1i pi d d NOTE: Multi-purpose weighting No explicit model statement Calibration of x-variables at the domain level 7

8 Model calibration equation ( MC ) w ˆ i zi zi Nd, yi is iu iu d d d where z (1, yˆ ), s s U i i d d NOTE: Single-purpose weighting Common model formulation for all domains Calibration of y-predictions at the domain level 8

9 MC: Technical treatment The calibration weights minimize ( MC ) w i ai ( MC ) λ w i zi zi is a d i isd iu d where s s U, a 1/ z 2 d d i i (1, yˆ ), yˆ are predictions from the assisting model i i i The calibrated weights are defined in: w a ( MC ) k k k (1 λz ), where λ is the Lagrange coefficient λ z az az z i i i i i i iu d isd isd 1 9

10 Hybrid calibration equation w z z N, x,..., x, yˆ ( HC ) i i i d 1i pi i is iu iu iu iu d d d d d where z (1, x,..., x, yˆ ), s s U i 1i pi i d d NOTE: Single-purpose weighting Common model formulation for all domains Calibration of y-predictions and x-variables at the domain level 10

11 Assisting model in MC and HC Linear fixed-effects model y xβ k k k x β (1, x,..., x ) vector of auxiliary variables for k k 1k pk (,,..., ) fixed effects 0 1 p Estimate parameter vector β from the data by WLS Calculate predictions yˆ xβˆ for all k U k k 11

12 Monte Carlo experiments Synthetic finite population of 1,000,000 elements Continuous study variable y Domain structure D = 40 domains K = 1000 SRSWOR samples Sample size n = 2000 Two continuous x-variables corr( y, x ) 0.5 corr( y, x ) One categorical x-variable Assisting model Linear fixed-effects model No domain-specific terms Estimators of domain means (1) Direct HT estimator (2) Direct MFC estimator No model statement Calibration: Both x-variables (3) Semi-direct MC estimator Linear fixed-effects model Model: Both x-variables Calibration: Fitted y-values (4) Semi-direct HC estimator Linear fixed-effects model Model: Both x-variables Calibration: Fitted y-values x-variable x 1 12

13 Quality measure of estimators Accuracy Relative root mean squared error RRMSE (%) Median calculated over domain sample size classes (Minor Medium Major) 1 K 2 RRMSE( ˆ ˆ d ) ( d ( sk ) d ) / d K k 1 NOTE: All methods considered are (nearly) design unbiased 13

14 Table 1. Median relative root mean squared error (RRMSE) (%) of Horvitz- Thompson, model-free calibration, model calibration and hybrid calibration estimators of domain means over domain size classes. Expected domain sample size Minor <20 Medium Major >85 All domains Direct estimators Horvitz-Thompson Model-free calibration z k x1 k x2k (1,, ) Semi-direct model-assisted estimators Model: y x x Model calibration z k (1, yˆ ) Hybrid calibration z k 0 1 1k 2 2k k k (1, x ˆ 1, y ) k k k

15 Summary Model-free calibration Outperforms HT in accuracy Coherence property holds for both x-variables Model calibration Outperforms HT, model-free calibration and hybrid calibration Coherence property for x-variables not met Hybrid calibration Clearly outperforms HT but not MFC or MC Coherence property holds for one x-variable (but not for the other) Drawback: Increased variation in weights 15

16 References Deville J.-C. and Särndal C.-E. (1992) Calibration estimators in survey sampling. JASA 87, Estevao V. M. and Särndal C.-E. (1999) The use of auxiliary information in design-based estimation for domains. Survey Methodology 2, Lehtonen R., Särndal C.-E. and Veijanen A. (2009) Model calibration and generalized regression estimation for domains and small areas. SAE 2009 Conference, Elche, June Lehtonen R. and Veijanen A. (2009) Design-based methods of estimation for domains and small areas. Chapter 31 in Rao C. R. and Pfeffermann D. (Eds.) Handbook of Statistics Vol. 29B. Sample Surveys. Inference and Analysis. Amsterdam: Elsevier, Lehtonen R. and Veijanen A. (2012) Small area poverty estimation by model calibration. Journal of the Indian Society of Agricultural Statistics 66, Lehtonen R. and Veijanen A. (2014) Small area estimation of poverty rate by model calibration and "hybrid" calibration. NORDSTAT Conference, Turku, June 2014 Lehtonen R. and Veijanen A. (2015) Design-based methods to small area estimation and calibration approach. In: Pratesi M. (Ed.) Analysis of Poverty Data by Small Area Estimation. Chichester: Wiley. (Forthcoming) Montanari G. E. and Ranalli M. G. (2005) Nonparametric model calibration estimation in survey sampling. JASA 100, Montanari G.E. and Ranalli M.G. (2009) Multiple and ridge model calibration. Proceedings of Workshop on Calibration and Estimation in Surveys Statistics Canada. Särndal C.-E. (2007) The calibration approach in survey theory and practice. Survey Methodology 33, Wu C. and Sitter R.R. (2001) A model-calibration approach to using complete auxiliary information from survey data. JASA 96,

17 Thank you for your attention 17