Multiple Regression Analysis with Data from Complex. Survey

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1 Secto o Survey Research Methods JSM 0 Multple Regresso Aalyss wth Data from Complex Survey Esher Hsu, Chu-Hu Lee, Che-Mg Che 3 Assocate Professor, Departmet of Statstcs, Natoal ape Uversty, ape, awa Graduate Studet, Departmet of Statstcs, Natoal ape Uversty, ape, awa 3 Graduate Studet, Departmet of Statstcs, Natoal ape Uversty, ape, awa Abstract hs study explores multple regresso aalyss wth complex survey data. Four methods of multple regresso aalyss, amely, ordary least squares, weghted least squares, probablty weghted least squares, ad Quas-Atke probablty weghted least squares are proposed for comparso by Mote Carlo approach to compare ther effcecy based upo bas, varace, ad MSE. he data from "awa Socal Chage Survey 007" collected uder a stratfed uequal probablty samplg were used for emprcal aalyss to compare four proposed methods based upo the estmated regresso coeffcets ad RMSE. he smulato results show that probablty weghted least squares estmator ad Quas-Atke weghted least square estmator perform better tha others uder the uequal probablty desg. he emprcal results cosst wth the smulato results. he emprcal results show that the educato years of respodets awa has sgfcat egatve relatoshp wth ther age but has postve relatoshp wth ther parets educato years. Keywords: Multple Regresso Aalyss; Stratfed weghted least squares estmator; Probablty weghted least squares estmator; Quas-Atke weghted least square estmator; Complex Survey; Socal Chage. Itroducto he samplg desg s gettg more mportat alog wth the creasg demad of precse data for makg a better decso whch thus has boosted the use of complex 308

2 Secto o Survey Research Methods JSM 0 survey practce ad has also rased the mportace of uequal probablty samplg as well. I prcple, the statstcal aalyss has to be adjusted alog wth the samplg desg to obta a better statstcal ferece. I order to smplfy the process of statstcal ferece, the mechasm of the samplg desg s usually gored. hat may cause based estmato or obta a wrog cocluso. It has occurred frequetly, such as, the estmator wth smple radom samplg used for the data collected uder uequal probablty samplg. Recetly, regresso aalyss wth complex surveys has become popular. For regresso aalyss, tradtoal estmators, such as least squares estmator, used wth data collected uder complex survey may reduce the accuracy of the statstcal aalyss. Fuller ad Wu (005) proposed a regresso aalyss wth survey samples. Fuller ad Wu (005a) proposed a estmato of regresso coeffcets wth uequal probablty samples. he study results of Fuller ad Wu (005a) show that the least squares method would obta a based estmator wth uequal probablty samples as the varace s ot homogeety. Hot, Smth ad Wter (980) proposed a weghted least squares method wth complex survey uder equal probablty samplg. DuMouchel ad Duca (983) proposed a weghted least squares estmator for multple regresso aalyses of stratfed samples. he weghted least squares estmator could reduce the bas, but elarge the varace of estmato. Cragg (989) proposed a Quas-Atke weghted least square estmator to reduce the varace of estmato. Whte (980) proposed a Ecker-whte varace-covarace estmator (E-W VCE) to solve the estmator cosstecy uder heteroskedastcty of varace. hs paper ams to compare the estmators of regresso coeffcets uder stratfed samplg wth uequal probablty based upo a Mote Carlo approach ad proposed proper estmators for a further emprcal study. Four methods of multple regresso aalyss proposed by ths study, amely, ordary least squares (OLS), weghted least squares (WLS), probablty weghted least squares (PWLS), ad Quas-Atke probablty weghted least squares (Q-A PWLS) are used ths study for comparso aalyss to see ther performace uder data collected wth stratfed uequal probablty samplg. A emprcal study s coducted to see how the estmators work practce. 309

3 Secto o Survey Research Methods JSM 0. Methodology I ths study, a Mote Carlo smulato expermet s coducted to compare the performace of the estmators of regresso coeffcets uder stratfed samplg wth uequal probablty based upo ther bases ad varaces. he estmators clude ordary least squares, weghted least squares, probablty weghted least squares, ad Quas-Atke probablty weghted least squares. he smulato study follow the steps: () geeratg a stratfed populato, () from the geerated stratfed populato repeatedly draw 0,000 stratfed uequal probablty samples, (3) obtag the regresso coeffcets for each sample by four proposed estmators, ad (4) comparg the bas ad varace of the estmators.. Populato ad Uequal probablty samplg I stratfed samplg the populato of N uts s dvded to k strata (subpopulato) of N, N, N k uts, respectvely. For each stratum, t s assumed that the fte subpopulato of N h uts s a smple radom sample of sze N h from a fte subpopulato. he populato value obtaed for the j th ut wth th stratum s deoted by (x, y ), =,,,k; j=,,,n, where x s a depedet varable, ad y s a depedet varable. he populato data are characterzed by a regresso model of the form Y β ε where E ε ) 0 for all, (), N N N ( N N where ( ) YN y, y,, yk wth y ( y,y,,yn ) ; ( x, x,, x ) wth x ( x, x,, x ), x (, x,x,,x ) ; N k N q ε (,,, ) wth ε (,,, ) ; β ( 0,,, q). N N N he sample s draw by a uequal probablty samplg. he j th ut wth th stratum (x, y ) s assged depedetly a probablty of eterg the sample. A sample cosstg of K uequal probablty samples of,,, k uts are sampled from k strata (subpopulato) of N, N, N k uts, respectvely. he stratfed uequal probablty samplg s repeated for 0,000 tmes to obta 0,000 samples.. Estmators of regresso coeffcets For each sample, regresso coeffcets are estmated by the ordary least squares, 30

4 Secto o Survey Research Methods JSM 0 weghted least squares, probablty weghted least squares, ad Quas-Atke probablty weghted least squares estmator... Ordary least squares estmator he estmator βˆ LSE ad ts varace-covarace matrx V( β ˆ LSE ) of the ordary least squares (OLS) estmator are as follows: βˆ (x x ) x y, LSE V ˆ ( β LSE ) (x x ) x Σx (x x ), () where Σ dag(,,, ). For error term wth homogeety, the V ( β ˆ LSE ) ca be cosstetly estmated by followg equato: ˆ ˆ V( β ) σ(x ˆ x ), where LSE (y x βˆ ) LSE σ ˆ, (q ) (3) For error term wth heteroscedastcty, the V ( β ˆ LSE ) ca be cosstetly estmated by Ecker-Whte varace-covarace matrx V ˆ (ˆ ) as follows: EW β LSE ˆ ˆ ˆ VEW ( β LSE ) (x x ) x ΣEW x(xx), where Σˆ dag ( ˆ, ˆ,, ˆ ), ˆ ( y x βˆ ), EW LSE (4).. Weghted least squares estmator Stratfed samplg wth samplg fracto P ad sample weght W of th stratum expressed as follows: N P, W,,,,k, N P (5) the regresso model s expressed as followg: / / / Wstr y Wstr xβ Wstr ε, where str dag ( str _, str _,, str _ k ), str _ dag( W,W,,W ) W W W W W (6) he weghted least squares estmator matrx Vˆ (ˆ ) are expressed as β str βˆ str ad ts estmated varace-covarace Vˆ (ˆ β βˆ str (x Wstrx ) x Wstry, W x ) x W Dˆ W x x str ) ( x str str e, str str ( Wstrx ), (7) 3

5 Secto o Survey Research Methods JSM 0 where D ˆ dag ( ˆ e, ˆ e,, ˆ e ), ˆ e dag( e ˆ,e ˆ,,e ˆ ), e,str,str,str k,str,str,str,str,str ê y x β ˆ,,,,k; j,,,.,str str..3 Probablty weghted least squares estmator Assume that the j th ut wth th stratum (x, y ) s assged depedetly a probablty of eterg the sample. he regresso model s expressed as followg: / / / W y W xβ W ε, (8) where W dag ( w, w,, wk ), w dag( w,w,,w ), w,,,,k; j,,,. he probablty weghted least squares estmator βˆ PW ad ts estmated varace-covarace matrx Vˆ (ˆ ) are expressed as β PW Vˆ (ˆ β βˆ PW (x Wx ) x Wy, W x ) x W Dˆ W x PW ) ( x PW PW e, PW PW ( x WPW x ), (9) where ˆ ( ˆ ˆ ˆ De,PW dag e,pw, e,pw,, ek,pw ), ˆe,PW dag( e ˆ,PW,e ˆ ˆ,PW,,e ),PW, ê y x β ˆ,,,,k; j,,,.,pw PW..4 Quas-Atke probablty weghted least squares estmator he Quas-Atke probablty weghted least squares estmator s proposed by Magee (998) to reduce the varace of the probablty weghted least squares estmator. he regresso model s expressed as followg: A / W / / / / / y A W xβ A W ε, (0) where A dag ( A, A,, Ak ), W dag ( w, w,, wk ), A dag( exp ( z ),exp ( z ),,exp ( z ), exp ( z )), k w dag( w,w,,w ), w, p x x,,,k; j,,,. z, Var( x ) he Quas-Atke probablty weghted least squares estmator varace-covarace matrx Vˆ(ˆ β ) are expressed as QA βˆ QA ad ts estmated ˆ(ˆ βˆ (x x AWy, QA AWx ) V βqa) ( x AWQAx ) x AWQADe, QAWQAAx ( x AWQAx ), ˆ () 3

6 Secto o Survey Research Methods JSM 0 where ˆ ( ˆ ˆ ˆ De,QA dag e,qa, e,qa,, ek,qa ), ˆe,QA dag( e ˆ,QA,e ˆ ˆ,QA,,e ),QA, ê y x β ˆ,,,,k; j,,,.,qa QA 3. Smulato study 3. Sample geerato he smulato s coducted by MALAB program ths study. I the smulato study, a populato cosstg of three strata wth sze N =400, N =500, ad N 3 =600 was depedetly geerated from subpopulatos of,, ad 3, respectvely. Let,, ad 3 three subpopulatos are dstrbuted as followg: : U Ω ; : U Ω ; : U Ω, () where U ~U ( 70, 30), Ω ~ N (0, ), U ~U (70, 30), Ω ~ N (0, ), U ~U (70, 330), Ω ~ N (0, ). here are two cases are take for stadard devato. Case I: 0, for =,,3; Case II: 0, 0, Y gve s geerated as Y 50. (3) where, ~N (0, 5 ), ~N (0, ). I order to see whether the varablty of varace of error flect the performace of the estmators of regresso coeffcets, two cases are take for : () =5 for =,,3, () 5, 0, 3 5. Moreover, three cases are take for : (), () exp( ), (3) exp( ) to see how the homogeety of varace error term flect the performace of the estmators of regresso coeffcets. wo kds of varaces depedet varable mxed wth sx kds of varace error term results to twelve populato settgs for smulato study. he dstrbutos of the twelve are dsplayed Fgure A. Appedx A. 33

7 Secto o Survey Research Methods JSM 0 he sample cluso probabltes for elemet x, y ) are geerated by ( (4) [ exp( )], where ~ N(0, 0 ) 7.5 Ad obta sample cluso probablty N,,,3, j,,,. I order to compare the effect of sample sze, three cases are used for smulato: () wth small equal sze, = = 3 =5, () wth proportoal allocato, =0, =5, ad 3 =30, (3) wth large equal sze, = = 3 =35. he expectato, varace, ad mea squares of error are calculated based upo the smulato results for those four estmators as follows j 0000 ˆ E( ˆ), ˆ ( E( ˆ)) var( ˆ), ˆ ( ) MSE( ˆ) (5) 3. Smulato results welve populato regresso models carred from the twelve populato settgs are descrbed able. he smulato results for dfferet sample szes are show able. As we expect, for the case of homogeety of varace both ad error term amog three strata, model (), the bas of the four estmator are all small; whle the MSE of OLS ad WLS estmators are smaller tha that of PWLS, QA-PWLS. For the case of heteroscedastcty the error term, model () ad (), the error term depeds o, the estmator of OLS ad WLS have larger bas tha others, the bas s sgfcat o the case of small sample sze; whle the MSE for all estmators are all small. For the cases of model (), (), ad (), the varace of amog three strata are dfferet. he bases of OLS ad WLS estmators are larger tha that of PWLS ad QA-PWLS. he bas s sgfcat o small sample sze. he MSE of OLS ad WLS are smaller tha others as the error term s depedet of ; whle the MSE of OLS ad WLS are larger tha others as the error term depeds o. For the case of model 3(), 3(), ad 3(), the varaces of amog three strata are same, but varaces of error tem are dfferet. he bases of OLS ad WLS estmators are larger tha that of PWLS ad QA-PWLS. he MSE of OLS s smaller tha others for the case of small sample sze, but the MSE are smlar amog the four estmators larger sample sze. hat shows that the QA-PWLS ca reduce the varace for large sample sze. For 34

8 Secto o Survey Research Methods JSM 0 the case of model 4(), 4(), ad 4(), both of the varaces of ad varace of error term are dfferet amog three strata. he bases of OLS ad WLS estmators are larger tha that of PWLS ad QA-PWLS. he MSE of OLS s smaller tha others for the case of small sample sze, but the MSE are smlar amog the four estmators larger sample sze. he MSE of OLS ad WLS are smaller tha others as the error term s depedet of ; whle the MSE of OLS ad WLS are larger tha others as the error term depeds o. able : Specfcato of populato regresso models for smulato Model Populato regresso Stadard devato Stadard devato of No. model of () Y , for =,,3 5,,, 3 () Y exp( ) () Y exp( ) () Y , 0, ,,, 3 () Y exp( ) () Y exp( ) 3() Y , for =,,3 5, 0, 3 5 3() Y exp( ) 3() Y () Y () Y () Y exp( ) 0, 0, , 0, 3 5 exp( ) exp( ) I summary, the estmators of OLS ad WLS are based uder stratfed uequal probablty samplg as the varaces of amog three strata are dfferet or the error 35

9 Secto o Survey Research Methods JSM 0 term depeds o. he MSE of OLS s smaller tha others o small sample sze. For large sample sze, the QA-PWLS ca reduce varace ad obta smaller varace tha PWLS. he smulato results show that PWLS ad QA-PWLS perform better tha OLS ad WLS terms of bas uder stratfed uequal probablty samplg; but PWLS has larger varace. able : Smulato results of ˆ Model No. () () (),, ) Case Case Case 3 Case Case Case 3 Case Case Case 3 ( 3 OLS bas MSE WLS bas MSE bas PWLS MSE QA-PWLS bas E (trace) MSE QA-PWLS bas E (det) MSE Note: Case : (,, 3)=(5,5,5) ; Case : (,, 3)=(0,5,30) ; Case 3: (,, 3)=(35,35,35). able : Smulato results of ˆ (Cotue a) Model No. () () (),, ) Case Case Case 3 Case Case Case 3 Case Case Case 3 ( 3 OLS WLS bas MSE bas MSE PWLS bas MSE QA-PWLS bas (trace) MSE QA-PWLS bas (det) MSE

10 Secto o Survey Research Methods JSM 0 able : Smulato results of ˆ (Cotue b) Model No. 3() 3() 3(),, ) Case Case Case 3 Case Case Case 3 Case Case Case 3 ( 3 OLS WLS bas MSE bas MSE PWLS bas MSE QA-PWLS bas (trace) MSE QA-PWLS bas (det) MSE able : Smulato results of ˆ (Cotue c) Model No. 4() 4() 4(),, ) Case Case Case 3 Case Case Case 3 Case Case Case 3 ( 3 OLS WLS bas MSE bas MSE PWLS bas MSE QA-PWLS bas (trace) MSE QA-PWLS bas (det) MSE Mote Carlo approach s used ths paper to compare the effcecy of the four estmators of regresso coeffcets based upo bas, varace, ad MSE. he smulato results show that probablty weghted least squares estmator ad Quas-Atke weghted least square estmator are ubased estmators of regresso coeffcets. he smulato results also fd that the Quas-Atke weghted least square estmator has a smaller asymptotc varace tha least squares estmator. Smulato results show that the 37

11 Secto o Survey Research Methods JSM 0 ordary least squares estmator s based uder the data collected uder the uequal probablty desg; whle uder the equal probablty desg the weghted least squares estmator s better tha ordary least squares, but uder the uequal probablty desg weghted least squares estmator may have a larger varace. 4. Emprcal study o exame the results carred out by smulato study prevous secto. hs study uses the real data set of "awa Socal Chage Survey 007, Phase 5, Wave 3," collected uder a stratfed uequal probablty samplg by the Isttute of Socology Academa Sca for emprcal comparso of the three methods, amely, OLS, PWLS, ad Q-A PWLS va comparg the estmates of regresso coeffcets, RMSE, ad R. 4. Samplg desg he real data set of awa Socal Chage Survey s collected by a complex survey, stratfed mult-stage cluster samplg, whch cludes,989 observatos. he populato s stratfed to sx strata (rego), each rego wth people. I each rego, N tows are selected wth probablty proportoal to the tow s populatoc. vllages are selected wth probablty proportoal to the vllage s populatov from each selected tow. he m people are selected from each selected vllage. he probablty of the perso j the th stratum cluded sample ad ts weght wu are show as follows. C V m Nm ( N.)(.( ) ) ; C V wu Nm. (6) I order to crease the precso of estmato, recursve rakg wth sex, age, ad stratum s used ths study to reach the cosstecy of the dstrbutos of frequecy betwee sample ad populato. he weght used for rakg s wt wt N,, 6, N. (7) 4. Varables used for regresso aalyss Four varables are used for regresso aalyss to see the relatoshp betwee 38

12 Secto o Survey Research Methods JSM 0 respodets total educato years ad hs (her) parets total educato years. he varables are descrbed as follows. Depedet varable Y (edu): total years of educato. Idepedet varable (age): respodet s age. Idepedet varable (f-edu): total years of educato of respodet s father. Idepedet varable 3 (m-edu): total years of educato of respodet s mother. he sample statstcs ad the test for equalty of mea ad equalty of varace over sx strata are show able 3. he hypothess test for mea equalty ( H0: p 6p) shows that all the varables have sgfcat dffereces amog sx regos. he Bartlett s test for homogeety of varace ( H0: p 6p) all shows that all the varables have heterogeety of varace amog sx regos. able 3: Sample statstcs of the varables Varables Stratum (rego) Core ctes Geeral ctes New ctes radtoal coutes Rural coutes Seor coutes P-value mea p<0.000 (age) s.d p= mea p<0.000 (f_edu) s.d p= mea p<0.000 (m_edu) s.d p<0.000 Y mea p<0.000 edu s.d p=0.006 Note: p-values the last colum are from AVOVA test for mea equalty ad Bartlett s test for homogeety of varace, respectvely. 4.3 Regresso aalyss hree estmators, OLS, PWLS, ad Q-A WPLS, are used to estmate the regresso coeffcets, whch the OLS estmator s take from equato () ad ts varace estmator s from equato (4), PWLS estmator ad ts varace estmator s take from equato (9) ad QA-PWLS estmator ad ts varace estmator s take from equato 39

13 Secto o Survey Research Methods JSM 0 (). he weght w for PWLS ad QA-PWLS s calculated as followg w wt wu wt,,,...,6; j,,...,. (8) he estmated coeffcets are show able 4. he emprcal results cosst wth prevous studes. he results show that there s o bg dfferece amog the estmated parameters of those three methods. he results also show that the educato years of respodets have sgfcat egatve relatoshp wth ther ages but have postve relatoshp wth ther parets educato years. able 4: Estmated regresso models OLS WPLS Q-A WPLS Estmate St. Error Estmate St. Error Estmate St. Error 0 3 R-Square RMSE (0.07) (0.).8397 (0.050) (0.0000) 0.36 (0.0000) (0.0000) (0.000) 0.00 (0.000) (0.000) -0.9 (0.0000) -0.6 (0.0000) (0.0000) Cocluso he samplg desg s gettg more complex to comply wth a varety of socal evromet ad to crease the precso of samplg survey as well. he tradtoal estmators used wth complex survey may lower the accuracy of the statstcal aalyss. hs study explores the methods of regresso aalyss o survey data obtaed uder a complex samplg. Four methods of multple regresso aalyss proposed by ths study, amely, ordary least squares, weghted least squares, probablty weghted least squares ad Quas-Atke probablty weghted least squares are used ths study for comparso aalyss. Mote Carlo approach s used ths paper to compare the effcecy of the four estmators of regresso coeffcets based upo bas, varace, ad MSE. he smulato results show that probablty weghted least squares estmator ad Quas-Atke probablty weghted least squares estmator perform better tha ordary least squares estmator ad weghted least squares estmatos terms of bas, but probablty weghted least squares estmator has a larger varace for estmatg regresso 30

14 Secto o Survey Research Methods JSM 0 coeffcets uder complex survey. Quas-Atke probablty weghted least squares estmator performs better tha other estmator terms of bas ad MSE as the error term ad depedet varables have heterogeety of varace amog strata. he smulato results also fd that the Quas-Atke weghted least square estmator has a smaller asymptotc varace tha least squares estmator o the cases of larger sample sze. hs study uses the data of "awa Socal Chage Survey 007, Phase 5, Wave 3," collected uder a stratfed uequal probablty samplg by the Isttute of Socology Academa Sca for emprcal comparso of those three methods va comparg the estmates of regresso coeffcets, RMSE, ad R. he emprcal results cosst wth prevous studes ad the smulato results ths study. he results show that there s o bg dfferece amog the estmated parameters of those three methods. he results also show that the educato year of respodets has sgfcat egatve relatoshp wth ther age but has postve relatoshp wth ther parets educato year. Refereces Ajma, V. B Appled Ecoometrcs Usg the SAS System. Cochra, W. G Samplg echques, 3 rd edto. Cragg, J. G. 99. Quas-Atke estmato of heteroskedastcty of ukow form.j.ecoometr.,54,79-0. DuMouchel, W. H. ad G. J. Duca Usg sample survey weghts multple regresso aalyses of stratfed samples. Joural of the Amerca Statstcal Assocato, 78, 383, Graubard, B. I. ad E. L. Kor. 00. Iferece for superpopulato parameters usg sample surveys. Statstcal Scece.7,, Hase, M. H. ad W. N. Hurwtz O the theory of samplg from fte populatos. he Aals of Mathematcal Statstcs, 4, Holt, D.,. M. F. Smth, ad P. D.Wter Regresso aalyss of data from complex surveys. Joural of the Royal Statstcal Socety, 43, 4, Kuter, M. H., C. J. Nachtshem, J. Neter, ad W. L Appled Lear Statstcal th Models, 5 edto. Magee, L Improvg Survey-Weghted Least Squares Regresso. Joural of Royal Statstcal Socety, 60,, 5-6. Rutemller, H.C., ad D.A. Bowers Estmato a Heteroscedastc Regresso Model. Joural of the Amerca Statstcal Assocato, 63, 3,

Secto o Survey Research Methods JSM 0 Whte, H. 980.

heteroskedastcty. Ecoometrca, 48, 87-838. Wu, Y.Y. ad A. Fuller. 005.

I Proceedgs of the Survey Research Method Secto, Amerca Statstcal Assocato,

Estmato of regresso coeffcets wth uequal probablty samples. 389-3899.

15 Secto o Survey Research Methods JSM 0 Whte, H A heteroskedastcty-cosstet covarace matrx estmator ad a drect test for heteroskedastcty. Ecoometrca, 48, Wu, Y.Y. ad A. Fuller Prelmary estg Procedures for regresso wth survey samples. I Proceedgs of the Survey Research Method Secto, Amerca Statstcal Assocato, Wu, Y.Y. ad A. Fuller. 005a. Estmato of regresso coeffcets wth uequal probablty samples. I Proceedgs of the Survey Research Method Secto, Amerca Statstcal Assocato, Appedx A Populato ()~() Populato ()~() Populato 3()~() Populato 4()~() Fgure A.: Scatter dagram of populatos 3

Application of Calibration Approach for Regression Coefficient Estimation under Two-stage Sampling Design

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