Variance estimation in EU-SILC survey

Transcription

1 Varance estmaton n EU-SILC survey Mārtņš Lberts* * Mathematcal Support Dvson, Central Statstcal Bureau of Latva (Martns.Lberts@csb.gov.lv) bstract: The mplementaton of varance estmaton methods resamplng methods (dependent random group method and jacnfe) and lnearaton n EU-SILC survey are dscussed n the paper. The paper focuses on estmaton of varance for totals, ratos of two totals and Gn coeffcent. However developed methodology could be used also for other statstcs. The developed program (n SPSS ) for estmaton of varance for arbtrary sample wth possbltes of adjustng the parameters of methods appled wll be also presented. Introducton Measurement of accuracy s mportant part n producton of statstcs based on survey samplng. The most common measure of accuracy s samplng error. The tas of study s to develop methodology for estmaton of samplng errors for complex (nonlnear) statstcs and to apply t to Household Budget Survey (HBS) and EU- SILC (SILC). 2 Populaton Parameters Three types of populaton parameters wll be consdered n the paper total, the rato of two totals and Gn coeffcent. Parameter Populaton parameter Estmate of parameter Total The rato of two totals Gn ndex X N x X R Y N 2 xr X 00 G NX R j ascendng by x, Ran of unt f sorted Xˆ n x w Xˆ Rˆ Yˆ 2 G 00 n x ˆ ˆ w R X, NXˆ ˆ Rˆ w Estmate of ran of j unt f sorted ascendng by x

2 3 Desgn of Surveys Both surveys consdered n the study share smlar desgn. Households and ndvduals are survey unts. Two-stage samplng s used for households; two-stage cluster sample s used for ndvduals. Stratfed systematc pps (samplng wth probablty proportonal to se) sample of populaton census (2000) areas s used at the frst stage. Stratfcaton s made by degree of urbansaton Rga, 6 other largest ctes, towns and rural areas (four strata). PSUs are selected by several startng ponts (6 or 3 for HBS, 4 for SILC). Smple random samplng of households s used at second stage. ll ndvduals from selected households are sampled so households form clusters of ndvduals. 4 Estmaton of Samplng Errors It s hard to fnd drect estmators of samplng errors for estmates of complex statstcs especally n case of samplng desgn descrbed n prevous secton. The approxmaton methods are used as alternatve. Re-samplng methods (dependent random groups and jacnfe) and lnearaton methods are consdered n the paper. 4. Dependent Random Groups The sample s from populaton U s dvded n non-overlappng subgroups s, K, s. The sample s should be dvded so that all subgroups preserve the same samplng desgn as the sample s. The estmate of populaton parameter θ could be estmated as θ ˆ ˆ, K, θ. It s possble to estmate a varance of θˆ by Vˆ DRG2 ( ) ˆ ( a ˆ θ ) 2 θ () a 4.2 Jacnfe Smlarly to dependent random groups technque the sample s dvded n nonoverlappng sub-samples. The parameter θ s estmated from the sample s by deletng one of sub-sample for each a, K,. The resultng estmates θˆ ( a) are used to estmate the varance of θˆ by ( ˆ ) ˆ ( θ( ) ˆ α θ ) Vˆ θ (2) α 2

3 4.3 Lnearaton The dea of lnearaton s to estmate a varance of complex statstcs usng the same estmator of varance as for totals. The goal of lnearaton s to fnd for each unt n the sample so that varance of ˆ θ could be approxmated by Vˆ (3) ( ˆ θ ) Vˆ s π Dfferentable parameters can be lneared by expanson n Taylor seres. For rato of Y R two totals X can be expressed n form ( y Rx ) (4) X Broader class of parameters can be lneared usng extended theory by J. C. Devlle (999). For example for Gn coeffcent can be expressed n form 2x U ( x x ) + 2 x ( x x ) x ( Gn + ) U N U x U x + Nx (5) 5 Software for Estmaton of Samplng Errors To apply the theory descrbed n prevous secton software n SPSS macro language has been developed. 5. Possbltes of the Software It s possble to use the software for both sngle stage and mult-stage samplng. In case of mult-stage samplng errors are estmated at the level of PSUs. Stratfcaton s allowed at the frst stage. Desgn weghts should be avalable for software. For estmaton desgn weghts are ncreased proportonally to rato of full sample se and sub-sample se. It s possble to apply non-response correcton for user defned response homogenety groups and post-stratfcaton by one varable. It s possble to estmate samplng errors for totals (SUM), rato of two totals (RTIO) and Gn coeffcent (GINI). Lnearaton of RTIO and GINI s possble to speedup the executon of software.

4 It s possble to use two re-samplng methods for estmatng of samplng errors jacnfe and dependent random groups technque. Methods are appled at the level of PSUs. Correcton of fnte populaton s appled at level of PSUs. User can freely choose the number of sub-samples and how sub-samples are created. PSUs could be sub-grouped n random or user defned order. The groupng of subgroups and sub-samplng of these groups s possble. Sample unts can be dvded n sub-unts by applyng parameters of sample unt to correspondng sub-unts. For example Gn coeffcent has to be estmated at ndvdual level by applyng to each ndvdual equalsed ncome. The ncome of household s dvded by equalsed household se (accordng to modfed OECD scale) and the result s appled to all household members. Household s sample unt and ndvduals are sub-unts. 5.2 Base of the Software The software s wrtten n SPSS syntax usng macro commands. Currently t s based on sx macro commands:!lnrat lnearaton of rato;!lngn lnearaton of Gn coeffcent;!estm estmator of ndcator;!weght weghtng of sub-sample;!e_ton estmaton of ndcator usng estmator and weghts;!proc estmaton of samplng error;!proc_u man procedure. User can control the software usng several parameters. For example: Fle survey data fle (n SPSS format); Strata varable of stratfcatons; Psu varable of PSUs; D_sv varable of desgn weghts; Meth method of resamplng dependent random groups or jacnfe; E_tor estmator; Ln lnearaton (Yes/No); Dv number of sub-samples; nd other parameters.

5 Example of executon of the software:!proc_u dr "C:\Darbs\Stocholm\DRG\fles\SILC" fle "C:\Darbs\Stocholm\DRG\Data\SILC\SILC2005_data_ver02.sav" p_fle "C:\Darbs\Stocholm\DRG\Data\SILC\dem_nfo.sav" strataprl / psuat ecr / pop_psu4263 hh_ddb030 / per_sper_s d_svd_sv respresp resp_grat ecr / p_grprl / p_varper_s p_toted_s methdrg JCK / rorder0 / repeat psu_grsel_nr / ordersel_nr / dv4 / e_torrtio / ln0 / levelh / eqscaleper_s / varhh07n hs3n / fast. The software s good tool for research. It s possble to test dfferent methods and parameters of methods for estmaton of samplng error. The software has been used for estmaton of samplng errors n EU-SILC and HBS surveys. It has been tested on dfferent SPSS versons SPSS.5, SPSS 2 and SPSS 4. 6 Results The software has been used for estmaton of samplng errors n EU-SILC 2005 survey. The next table shows results of samplng errors of two ndcators Lowest monthly ncome to mae ends meet (X) and Total housng cost (Y). Table Estmates of samplng errors n EU-SILC survey Method Estmator Estmaton Dependent Random Groups Jacnfe Estmaton of varance Coeffcent of Varaton (%) SUM(X) SUM(Y) SUM(X)/SUM(Y) SUM(Y)/SUM(X) GINI(X) GINI(Y) SUM(X) SUM(Y) SUM(X)/SUM(Y) SUM(Y)/SUM(X) GINI(X) GINI(Y)

6 Study about the lnearaton shows that t could be used to gat faster estmates. In ths case the estmates of samplng error are almost the same comparng estmates wth and wthout lnearaton. Table 2 Estmates of samplng errors usng lnearaton for Gn coeffcent Method Estmator Number of subsamples Estmate of CV wthout lnearaton Estmate of CV wth lnearaton Comparson DRG GINI % DRG GINI % DRG GINI % DRG GINI % JCK GINI % JCK GINI % JCK GINI % JCK GINI % Estmates of samplng error are dependent on methodology of creatng sub-samples (number of sub-samples, order of PSUs). The estmates of CV by dfferent subsamplng are varyng. It can be seen n next table. Table 3 Estmates of samplng errors by dfferent sub-samplng Nr Method Estmator Number of subsamples Estmate of CV DRG GINI JCK GINI JCK GINI DRG GINI DRG GINI JCK RTIO JCK RTIO DRG RTIO JCK RTIO DRG RTIO Conclusons The software created durng the research s a good tool for usng dfferent methods of estmaton of samplng errors. The software can be upgraded wth addtonal methods or estmators of ndcators. nalyss of lnearaton method shows that lnearaton s useful method n estmaton of samplng errors. The analyss about the results of the survey wll be contnued.

7 References Publcaton Central Statstcal Bureau of Latva (2005), Mājsamnecības budžets gadā, Rīga. J. C. Devlle (999), Varance Estmaton for Complex Statstcs and Estmators: Lnearaton and Resdual Technques, Survey Methodology, Statstcs Canada, Vol. 25, No. 2, Publcaton European Commsson, Eurostat, The SS macro for lnearng EU-SILC complex ncome ndcators, User Gude, Drectorate F: Socal Statstcs and Informaton Socety, Unt F-3: Lvng condtons and socal protecton statstcs. Journal rtcle J. Lapņš, E. Vass, Z. Prede, S. Bālņa (2002), Household Sample Surveys n Latva, Statstcs n Transton Journal of the Polsh Statstcal ssocaton, Volume 5, Number 4. n Unpublshed Paper M. Lberts (2005), Ilases apseojumu teorja (Survey Samplng), LU, Rīga. n Unpublshed Paper M. Lberts (2004), Prases darba atsate, LU, Rīga. S. L. Lohr (999), Samplng: Desgn and nalyss, Broos/Cole Publshng Company, Pacfc Grove, Calf. Journal rtcle. Sandström, J. H. Wretman, B. Waldén (988), Varance Estmators of the Gn Coeffcent Probablty Samplng, Journal of Busness & Economc Statstcs, Vol. 6, No., mercan Statstcal ssocaton. SPSS Inc (2002), SPSS Syntax Reference Gude. C.-E. Särndal, B. Swensson, J. Wretman (992), Model sssted Survey Samplng, Sprnger-Verlag, New Yor Webste Wpeda, K. M. Wolter (985), Introducton to Varance Estmaton, Sprnger-Verlag