SURVEY SAMPLING IN ECONOMIC AND SOCIAL RESEARCH

Transcription

1 SURVEY SAMPLIG I ECOOMIC AD SOCIAL RESEARCH

2 Studa Eonomczne ZESZYTY AUKOWE WYDZIAŁOWE UIWERSYTETU EKOOMICZEGO W KATOWICACH

3 SURVEY SAMPLIG I ECOOMIC AD SOCIAL RESEARCH Edted b Janusz L. Wwał and Tomasz Żądło Katowce 0

4 Komtet Redacn Kstna Lseca (pzewodncząca, Anna Lebda-Wbona (seetaz, Floan Kuźn, Maa Mchałowsa, Anton edelńs, Iena Pa, Stansław Swadźba, Tadeusz Tzasal, Janusz Wwał, Teesa Żabńsa Komtet Redacn Wdzału Zaządzana Janusz Wwał (edato naczeln, Wocech Gamot (seetaz, Teesa Żabńsa, Jace Szołtse, Włodzmez Rudn Rada Pogamowa Loenzo Fatton, Mao Glow, Mloš Kál, Bonsław Mcheda, Zdeně Moláš, Maan oga, Gwo-Hsu Tzeng Redato Elżbeta Spadzńsa-Ża Copght b Wdawnctwo Unwestetu Eonomcznego w Katowcach 0 ISB ISS Wesą pewotną Studów Eonomcznch est wesa papeowa Wszele pawa zastzeżone. Każda epoduca lub adaptaca całośc bądź częśc nnesze publac, nezależne od zastosowane techn epoduc, wmaga psemne zgod Wdawc WYDAWICTWO UIWERSYTETU EKOOMICZEGO W KATOWICACH ul. Maa 50, Katowce, tel , fa e-mal: [email protected]

5 COTET ITRODUCTIO... 7 Czesław Domańs FIRST ASSOCIATIOS OF POLISH STATISTICIAS... Steszczene... 7 Wocech Gamot O POOL-ADJACET-VIOLATORS ALGORITHM AD ITS PERFORMACE FOR O-IDEPEDET VARIABLES... 8 Steszczene... 9 Anna Imołe, Janusz Gołaszews, Dausz Załus, Zbgnew asals PRACTICAL STATISTICAL AD ECOOMIC ASPECTS OF USIG SURVEY STUDIES FOR IDETIFICATIO OF THE KEY PLAT CULTIVATIO TECHOLOGY FACTORS... 3 Steszczene Aadusz Kozłows THE USEFULESS OF PAST DATA I SAMPLIG DESIG FOR EXIT POLL SURVEYS Steszczene Jan Kubac, Alna Jędzecza THE COMPARISO OF GEERALIZED VARIACE FUCTIO WITH OTHER METHODS OF PRECISIO ESTIMATIO FOR POLISH HOUSEHOLD BUDGET SURVEY Steszczene Doota Raczewcz SOME ASPECTS OF POST EUMERATIO SURVEYS I POPULATIO CESUSES I POLAD AD GERMAY Steszczene... 76

6 Ondře Vlus OPTIMIZATIO OF SAMPLE SIZE AD UMBER OF TASKS PER RESPODET I COJOIT STUDIES USIG SIMULATED DATASETS Steszczene Janusz L. Wwał O LIMIT DISTRIBUTIO OF HORVITZ-THOMPSO STATISTIC UDER THE REJECTIVE SAMPLIG Steszczene Tomasz Żądło O ACCURACY OF TWO PREDICTORS FOR SPATIALLY AD TEMPORALLY CORRELATED LOGITUDIAL DATA Steszczene AUTHORS... 06

7 ITRODUCTIO Sample suves povde one of the most challengng felds fo applng the statstcal methodolog. The confont the eseache wth a vast dvest of unque pactcal poblems encounteed n the couse of studng populatons. The nclude, but ae not lmted to: non-samplng eos, specfc populaton stuctues, contamnated dstbutons of stud vaables, non-satsfacto sample szes, ncopoaton of the aula nfomaton avalable on man levels, smultaneous estmaton of chaactestcs n vaous subpopulatons, ntegaton of data fom man waves o phases of the suve and ncompletel specfed samplng pocedues. Omnpesent constants on tme and cost addtonall complcate the pocess of desgnng a suve. Dealng wth such condtons bngs about the need fo fomulatng sophstcated statstcal pocedues dedcated to specfc condtons of a sample suve. It gves bth to wde vaet of appoaches, methodologes and pocedues boowng the stength fom vtuall all banches of statstcs. Ths monogaph was pepaed on the bass of the papes that wee pesented dung the seventh confeence Suve Samplng n Economc and Socal Reseach that too place on 8-0 Septembe 0 n Katowce, Poland. The chaptes ae etended and mpoved vesons of the confeence papes. The authos deal wth vaous theoetcal and pactcal ssues. The common motve of all papes s the elaton to sample suves. The pape of Czesław Domańs s devoted to the fst thee most valuable achevements of Polsh statstcans and the nfluence on the development of ntenatonal statstcs. Fstl, the Autho dscusses the ole of Tadeusz Plat ( pofesso of statstcs and admnstaton at the Lvov Unvest as a co-founde of the Intenatonal Statstcal Insttute. Secondl, the dea of the fst Polsh census conducted n 789 s pesented, whch was devsed and eecuted b a membe of Palament Fde Józef eal Moszńs ( Thdl, the establshment of the fst Polsh Cha of Statstcs n the Wasaw School of Law and Admnstaton s dscussed, whch was headed b Wawznec Suowec ( pofesso of statstcs and economcs. Wocech Gamot concentates on the Pool-Adacent-Volatos algothm sometmes abbevated as PAVA. The ognal algothm s fomulated unde the assumpton of ndependence between andom vaables whose epectatons ae to be estmated. Seveal modfcatons of ths pocedue wee developed n the lteatue but unde ndependence. Hence, n ths pape a smulaton stud s caed out to assess popetes of PAVA-based odeed pobablt estmates unde coelaton.

8 8 ITRODUCTIO Anna Imołe, Janusz Gołaszews, Dausz Załus and Zbgnew asals dscuss pactcal statstcal and economc aspects of usng suve studes fo dentfcaton of the e plant cultvaton technolog factos. The consde suve stud caed out n 008 n ode to detemne the e elements n a plant poducton technolog and to calculate unt poducton costs of gowng wnte e (Secale ceeale L. fo gan. The suves coveed e gan poduces n notheasten Poland, who gow e on an aceage of ove ha. The economc analss was pefomed based on dect outlas on poducton; unt costs and dect magn wee calculated and the stuctue of costs as well as poftablt of wnte e poducton wee detemned. Aadusz Kozłows studes usefulness of past data n samplng desgn fo et poll suves. The man stess s put on the use of wdel avalable databases contanng detals of past electons esults. B means of smulaton epements the effectveness of technque of connectng the selecton of new sample wth past esults (ted sample pocedue s evaluated and optmal paametes fo ths technque ae ndcated. A modfcaton of the pocedue s also poposed. The best esults ae obtaned fo statfed samplng wth the use of elements of ted sample pocedue. The possbltes of cost educton of suves wthout peudce to the effectveness b means of the ght selecton of solel lage pecncts ae also ndcated. Jan Kubac and Alna Jędzecza compae Genealzed Vaance Functon wth othe methods of pecson estmaton fo Polsh Household Budget Suve. A statng pont was the estmaton of Balanced Repeated Replcaton vaances o bootstap vaances n the stuaton whee usng BRR was not applcable. To evaluate the GVF model the hpebolc functon was used. The computaton was done usng WesVAR and SPSS softwae and some specal pocedues pepaed fo R-poect envonment. The assessment of estmates consstenc fo countes was also conducted b means of small aea models. Doota Raczewcz pesents some aspects of post enumeaton suves n populaton censuses n Poland and Geman. The Autho begns wth compason of populaton censuses n Poland and Geman. et attenton s pad to data qualt and potental eos n populaton censuses. Compason s made of pncples of post-enumeaton suves n censuses n Poland and Geman. What s moe, ntenatonal ecommendaton on qualt assessment of populaton censuses accodng to the U and EUROSTAT s pesented. Ondře Vlus dcussed optmzaton of sample sze and numbe of tass pe espondent n conont studes usng smulated datasets. The Autho pesents an appoach based on analzng batches of smulated datasets wth gven chaactestcs. The atcle ncludes ovevew of the esults fo choce-based conont studes wth usual level of complet. Seach fo an optmal combnaton

9 ITRODUCTIO 9 of sample sze and numbe of tass pe espondent that allows us to acheve equed accuac of ou outputs wth optmal cost s of man focus but senstvt of the ecommendatons wth espect to changes n fed paametes of the datasets s also ncluded. Janusz L. Wwał studes lmt dstbuton of Hovtz-Thompson statstc unde the eectve samplng. On the bass of the papes b Bege and Snne (005 and Háe (964 he consdes the lmt dstbuton of H-T statstc standadzed b ts sample vaance. Moeove, the vaance of the H-T estmato s consdeed unde the assumpton that the aula vaable value s the obsevaton of the vaable unde stud but wth measung eo. Tomasz Żądło compaes accuac of two pedctos fo spatall and tempoall coelated longtudnal data based on Monte Calo smulaton stud usng R pacage. The fst pedcto unde stud s the empcal best lnea unbased pedcto (EBLUP deved fo some specal case of the Geneal Lnea Med Model whee spatal and tempoal coelatons ae taen nto account. The second pedcto s EBLUP deved unde the assumpton of lac of spatal and tempoal coelaton.

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11 Czesław Domańs FIRST ASSOCIATIOS OF POLISH STATISTICIAS. Intal Remas The ea 0 wll wtness the celebatons of the 00th annvesa of foundng the Polsh Statstcal Socet the fst assocaton of statstcans n Poland. The pesent confeence offes an ecellent oppotunt to pesent the contbuton made b Polsh statstcans towads an oveall development of statstcs as a dscplne of scence and ddactcs. The confeence wll also commemoate the ublee of the Cha of Statstcs whch was establshed 60 eas ago, fst as a pat of the Hghe School of Economcs, then the Academ of Economcs and pesentl the Economc Unvest n Katowce. It s woth pesentng hee thee most valuable achevements of Polsh statstcans and the nfluence on the development of ntenatonal statstcs, long befoe the Polsh Statstcal Socet came nto estence.. Conductng n 789 the fst-eve natonal census, whch was devsed and eecuted b a membe of Palament Fde Józef eal Moszńs ( Establshng n 8 the Cha of Statstcs n the Wasaw School of Law and Admnstaton, whch was headed b Wawznec Suowec ( pofesso of statstcs and economcs. 3. Co-foundng the Intenatonal Statstcal Insttute b pofesso of statstcs and admnstaton at the Lvov Unvest Tadeusz Plat ( Statstcal congesses.. Statstcal Congesses Befoe the Intenatonal Statstcal Insttute was founded n 840 t became customa fo scholas epesentng one banch of scence to meet at congesses whose man am was develop methods of posng and solvng mpotant poblems. The fst dea of statstcal congesses was conceved n London n 85 b Lambet Adolphe Jacques Quételet ( , the

12 Czesław Domańs chaman of the cental Belgan Statststcal Commsson. The congesses wee held n co-opeaton wth the govenment of the espectve count whee the net sesson was scheduled. Standadzng of admnstatve statstcs and adoptng a geneal methodolog of conductng censuses wee the man poblems dscussed dung subsequent statstcal congesses. Altogethe nne sessons of such congesses wee convened n the followng ctes: Bussels (853, Pas (855, Venna (857, London (860, Beln (863, Floence (867, the Hague (869, St. Petesbug (87, Budapest (876. At St. Petesbug Congess n 87 a commsson was set up n ode to supevse wos on the unfcaton of the ntenatonal statstcs. In 878 the commsson poposed that a specal ntenatonal bod, fosteng coopeaton between natonal statstcal offces should be fomed. Howeve, the poposal was eected b Geman and Swtzeland who wee of the opnon that such a bod would unnecessal ntefee wth the ntenal affas of ndvdual countes. As a esult of ths opposton congesses ceased to be convened, et the dea of mang statstcs an ntenatonal dscplne was not abandoned. Followng a sees of meetngs n Pas and London, a conventon was held on June 4, 885 n London, whee the Intenatonal Statstcal Insttute was bought nto beng. The conventon, attended b patcpants, was called sesson and fom that tme onwads all the egula meetngs of membes have become nown as sessons of the Intenatonal Statstcal Insttute (ISI. Polsh statstcans wee among those statstcans who too an actve pat n foundng the Intenatonal Statstcal Insttute and wee the Insttute membes. The most emnent epesentatve of the Polsh statstcs of the tme was August Ceszows ( A. Ceszows, who attended the Statstcal Congess n Pas n 855, was the man speae of one of the sessons of the Congess. The name of Tadeusz Zgmunt Plat should also be bought hee, as the fst Polsh membe of the Insttute among ts 00 statuto epesentatves. Tadeusz Plat ( gaduated fom the Unvest of Lvov and and contnued hs educaton n Beln whee he specalzed n statstcs unde the scentfc supevson of a enowned schola d Edmund Engel. In 867 he eaned hs docto s degee n law on the bass of the doctoal dssetaton Pactce n all poltcal and legal slls. Two eas late Plat obtaned a postdoctoal degee havng wtten the habltaton thess enttled Uebe den Begff des wtschaftlchen Wethes and became a pvate assocate pofesso n the Cha of Socal Economcs of the Lvov Unvest. He was confeed hs second postdoctoal degee n admnstatve law (870 and developed futhe hs nowledge of statstcs dung hs studes n the Pussan Statstcal Offce n Beln (87. In 87 Tadeusz Plat was apponted assstant pofesso of Unvest of Lvov and became the head of the Cha of Statstcs

13 FIRST ASSOCIATIOS OF POLISH STATISTICIAS 3 and Admnstaton, and a few eas late (878 he became a full pofesso. He was elected to be the dean of the Facult of Law and Admnstaton (fou tems of offce.e. 880/88, 884/885, 889/890, 900/90, the Pesdent of the Unvest (886/887, the Vce-Pesdent (887/888. Snce 909 he was an honoa pofesso. In the eas Plat seved seveal tems of offce as a deput fo the Galca Palament. He headed the Statstcal Offce of the atonal Depatment n Lvov fo ove fou decades (874-90, and woed as a deput mashal of the atonal Depatment ( In 888 he was apponted a coespondng membe of the Academ of Leanng n Cacow, and snce 98 a membe of the Polsh Academ of Leanng. He was also a membe of the Boad ( and vce-pesdent of the Galcan Economc Socet, a coespondng membe of the Cental Statstcal Commsson n Venna (snce 876, and a full membe of the Scentfc Socet n Lvov (snce 90. The etensve scentfc output of the Autho ncludes nte ala: Methods of Collectng Data fo Havest Statstcs (87, On Muncpal Statstcal Offces (87, Composton of Commune Repesentaton n Ctes and Towns of Galca (874, Statstcal Pesentaton of Communal Stuctue n Galca and Results of Local Electons (874, The Tetboo of Statstcs of Galca (900, and Statstcs (93. In hs studes Plat pesented methodologcal poblems, focusng on agcultual statstcs. Among methods used n statstcs of vegetable poducton he poposed estmaton methods as an mpotant souce of statstcal nfomaton. 3. Fst Geneal Census n Poland The begnnngs of statstcal actvtes on the Polsh tetoes concde wth the poceedngs of the so called Fou Yea Palament Sesson.e. the eas The Palament adopted a esoluton on cang out n 789 the fst natonal populaton census combned wth smoe egstaton. The Census esults wee to help the Palament to pass a bll on mposng a new ta, whch was supposed to povde mone towads epenses on pemanent, one-hunded- -thousand am. The autho of the statstcal tables of the 789 Census and a statstcal method of the mlta ta calculaton was a deput Fde Józef eal Moszńs ( It s woth notng that although the populaton and smoe egstaton of 789 was the fst state census, numeous othe egstes, nventoes and censuses appeaed n Poland as eal as 6th centu (e.g. populaton census of the Cacow Docese ( and the Ploc Docese of the eas 773, 776 and 778 and the wee conducted fo ta, economc, mlta and chuch easons. The numecal data contaned n these egstes

14 4 Czesław Domańs eman valuable souce of statstcal nfomaton fo all nds of estmatons and analses. Moszńs ponted out that the wealth of the state cannot be measued b the affluence of seveal astocatc famles and a couple of thousands of ch ctzens, but t athe should be measued b settlements, the wealth of townspeople and countmen, pospeous tade and floushng cafts. The statstcal method of the mlta ta assessment poposed b Moszns n the Palament was of absolutel unque chaacte and t was used nowhee else ethe befoe o afte that tme. In ode to measue the value of land and popet n a gven povat (dstct n an obectve wa he poposed a statstcal method based on the followng data: value of land and popet n the povat assessed on the bass of deeds of sales fo the peod of last eas as ecoded n dstct boos; t was seen as epesentatve enough fo mang calculatons, numbe of smoes obtaned fom teasue taffs; both alenated fo the peod of last eas and those whch wee not subect to puchase o sale. The obtaned nfomaton allowed the Teasue Commssons to mae calculatons based on the value of alenated goods, thans to smoe numbes, and assess pecsel the value of popetes n a gven povat tang nto account the popoton between smoes of alenated goods and the total. Fde Józef Moszńs suppoted the dea of the so-called co- -equaton that s a fa fscal sstem whch made the gent and the chuch pa taes based on pofts deved fom the land and popet. As a esult of Moszńs s ntensve effots the Fou Yea Palament ( passed a legslatve act of geat mpotance and etemel nteestng fo the hsto of statstcs n Poland a consttuton of June, 789 nown as Smoe Inspecton and Populaton Regste whch was n fact the fst populaton census n the hsto of Poland. The census ncluded the ual and the uban populaton and ecluded the gent and the cleg. It encompassed the followng categoes: se, occupaton and socal status obsevng the dffeence between sons (n two age goups up to 5 eas and above 5 eas and daughtes. The fst estmatons of the populaton numbe n Poland wee poduced b some of the above-mentoned statstcans. Józef Wbc povded n 777 the estmated numbe of populaton of people; Alesande Buchng n 77 gave the numbe of 8.5 mllon; Stansław Staszc n 785 estmated the populaton at the level of 6 mllon; Fde Moszńs, afte the fst census of 789 poduced the numbe of people; the fgue dd not nclude the gent and the cleg, who wee not the subect of the census, et the numbe was estmated at the level of thousand.

15 FIRST ASSOCIATIOS OF POLISH STATISTICIAS 5 4. Fst Cha of Statstcs The gowth of nteest n statstcs n Poland was maed b the foundaton of the fst cente of statstcal nowledge the Cha of Statstcs at the Wasaw School of Law and Admnstaton n 8. Headng the Cha was entusted to a statstcan and economst Wawznec Suowec ( Afte the fst Cha of Statstcs had been establshed, thee was an upsuge n the nteest n statstcs as a sepaate banch of scence and not as a tool to be used n admnstaton. It s woth tang a close loo at the pofle of the fst Polsh pofesso of statstcs. He was bon n Gnezno povnce n a gent faml of modeate means. Havng completed hs studes Suowec stated pofessonal caee as a pvate tuto what helped hm to get to now academc centes of Venna and Desden. In 807 he became a membe of the Wasaw Socet of Fends of Scences and at the tme he alead bult hs eputaton as an epet n socal ssues. Although he onl gave lectues n statstcs fo one ea, he was engaged n educatonal and scentfc actvtes fo most of hs lfe. In 8 he was apponted as a seceta geneal n the Mnst of Educaton and esgned fom pedagogcal dutes. In the Congess Kngdom of Poland he too the post n the Councl fo admnstatve affas and educatonal funds. The lst of most mpotant studes of W. Suowec s qute long and ncludes among othes: On the fall of ndust and towns n the old Poland (80, On ves and floatng of the Gand Duch of Wasaw (80, On statstcs of the Gand Duch of Wasaw (8-83. Suowec also too nteest n the populaton poblems and eamned dffeent easons fo populaton development. As a statstcan he peceved was, llteac, non-poductve calamtes and maladmnstaton as factos whch advesel affected populaton development. Wawznec Suowec togethe wth Ignac Stawas and Domn Ksńs wee the fst Polsh scholas to defne the subect and tass of statstcs. I. Stawas and D. Ksńs epessed the vews at the meetngs of the Wasaw Socet of Fends of Scences. The fome voced hs opnons n Septembe 809 but hs pesentaton was publshed n the Annals of the Socet n 8 whle the latte pesented them n Apl 84. W. Suowec not onl ceated a boad famewo fo statstcs but also woed to develop ths elatvel new feld of scence b gvng lectues n the Academ of Law and Admnstaton n the academc ea 8/8.

16 6 Czesław Domańs 5. Fnal Remas Two sessons of the Intenatonal Statstcal Insttute wee held n Poland. In August 99 Wasaw was the host of the 8th Sesson of the Intenatonal Statstcal Insttute. The fact that Poland was the oganze of the sesson epessed espect of the ntenatonal statstcal communt fo the achevements of Polsh statstcs. S Polsh epesentatves gave pesentatons n the couse of the Sesson: E. Sztum de Sztem: Statstcal method fo eamnaton of ndces of economc development; E. Lpńs: Remas on wong methods of the Polsh Insttute of Economc and Pce Reseach; S. Rzepewcz: On possblt of compang cme statstcs n dffeent countes; S. Szulc: On the so-called standadzaton o mpovng coeffcents; J. eman: Contbuton to the theo of elablt of statstcal hpotheses; J. Peałewcz: Ependtue and evenue of publc-legal assocatons. In Septembe 975 the 40th Sesson of the Intenatonal Statstcal Insttute was oganzed n Wasaw. Dung the Sesson Polsh statstcans pesented the followng papes: W. Maceews, W. Welfe: Foecastng models fo the natonal econom plannng and the elevance of the natonal nfomaton sstem; K. Zagós: Soco-demogaphc statstcs n the sstem of cental socoeconomc plannng; R. Batoszńs: A model fo s of abes; T. Walcza: The ole of moden statstcal nfomaton sstem fo management and plannng; E. Kzeczowsa: An ntegated sstem of ntenatonal statstcal compasons; J. Kudca: The possbltes of applng nput-output elatons to the econometc macomodels. Refeences Domańs, Cz. (0 Setna ocznca powstana Polsego Towazstwa Statstcznego (00 eas of the Polsh Statstcal Socet. Wadomośc Statstczne n 9(604, wzeseń 0, -0. Domańs, Cz. (004 Jubleusz Polsego Towazstwa Statstcznego Tadce obecne zadana statst w Polsce (Jublee of the Polsh Statstcal Socet. Tadton and the pesent-da tass of statstcs n Poland, ed. A. Zelaś, Wdawnctwo AE Kaów.

17 FIRST ASSOCIATIOS OF POLISH STATISTICIAS 7 Kleczńs, J. (886 Mędznaodow Insttut Statstczn (Intenatonal Statstcal Insttute. Pzegląd Pols oczn XI, t. II, Łuaszewcz, J. (995 Polsa hstoogafa a statsta (Polsh hstoogaph and statstcs. Bblotea Wadomośc Statstcznch, t. 46, Romanu, K. (975 Udzał Pols w pacach Mędznaodowego Insttutu Statstcznego (Contbuton of Poland to the wo of the Intenatonal Statstcal Insttute. Wadomośc Statstczne n 8. PIERWSZE ZRZESZEIE POLSKICH STATYSTYKÓW Steszczene W 0 ou pzpada 00. ocznca powstana pewszego zzeszena statstów polsch Polsego Towazstwa Statstcznego. Wato na te onfeenc zaznaczć wład statstów polsch w ozwó statst ao dscpln nauowe ddatczne. Konfeenca ta ma ówneż chaate ubleuszow, zwązan z 60-lecem Kated Statst, dzałaące wcześne w amach Wższe Szoł Eonomczne, potem Aadem Eonomczne, a obecne Unwestetu Eonomcznego w Katowcach. Wspomnm edne o tzech osągnęcach polsch statstów, tóe maą oddzałwane mędznaodowe:. Pzepowadzene w 789 ou pewszego spsu państwowego, tóego pomsłotwócą głównm ealzatoem bł poseł h. Fde Józef Moszńs ( Powołane w 8 ou Kated Statst w Szole Pawa Admnstac w Waszawe, tóe eownctwo powezono pofesoow statst eonom Wawzńcow Suowecemu ( Współudzał w twozenu Mędznaodowego Insttutu Statstcznego pofesoa statst admnstac Unwestetu Lwowsego Tadeusza Plata (

18 Wocech Gamot O POOL-ADJACET-VIOLATORS ALGORITHM AD ITS PERFORMACE FOR O-IDEPEDET VARIABLES. Intoducton Estmaton of odeed epectatons s a poblem that has attacted attenton of eseaches fo moe than fft eas. Ths nteest s eflected b a wde lteatue statng wth papes of Ae at al (955, Bun (955 and van Eeden (956, 957, 958, Katz (963 as well as Hanson et al (973, Sacowtz and Stawdeman (974 and then developed among othes b Sacowtz (98, Lee (983, Best and Chaavat (990, Chaas and van Eeden (99, Qan (99, Bloc et al (994, Ahua and Oln (00, Budaov et al (004, Jewel and Kalbflesch (004 and Hansohm (007. A good summa of the state of nowledge s pesented n monogaphs of Robetson et al. (988 and van Eeden (006. Somehow, t appeas that appoaches of all these authos shae a common featue. amel, t s alwas assumed that andom vaables fo whch one deses to compute estmates satsfng odeng constants ae ndependent. Ths autho s not awae of an estmaton pocedue that accounts fo possble coelaton between such vaables. Hence two possble choces ae: constuctng a pocedue dedcated to coelated data o nvestgatng the popetes of estng estmaton stateges when ndependence assumpton s dopped. In ths pape the ognal Pool-Adacent-Volatos algothm (PAVA of Ae et al (955 s evsted and ts popetes n the case of non-zeo coelaton ae assessed n a smulaton stud.. Pool-Adacent-Volatos algothm Let,,..., n be unnown pobabltes satsfng a smple ode:... n (

19 O POOL-ADJACET-VIOLATORS ALGORITHM 9 Let ndependent tals be made of an event wth pobablt fo,...,n. Let denote the numbe of successes n the -th tal and let * p / fo,...,n. The PAVA pocedue computes estmates p,..., p n of,..., n satsfng ( b teatvel goupng (megng ntal estmates * * p,..., p n nto blocs and aveagng them wthn each bloc. The pocedue wos though epeatng followng steps (see Hädle (99, Ae et al (955 and de Leeuw et al (009. Assgn each component fo,...,n to a sepaate goup so ntall (0 (0 n goups G,..., G n est. Set ntal estmate of mean pobablt n each -th (0 * goup to ~ p g p fo g,...,n g Whle thee est some goups n the -th step of algothm such that assocated estmates of mean pobablt volate the odeng constant, fnd ( ( ( mamum-length sequence of such goups (sa G g, Gg+,..., Gh and mege them (+ ( ( ( (+ ( nto a sngle goup G g G g G g+... G h whle G G + hg + fo >g and assgn a mean pobablt estmate ~ ( p to the goup ( G + g ( ( whle ~ + p ~ p fo >g + hg g ( + ( + G g G g 3 When teaton stops afte the last sa K-th step (whee K {0,,...}, wth H goups emanng assgn a mean pobablt estmate computed fo a goup to each of ts membe components so that the fnal estmate fo the component s ( p ~ p g fo G g, g,,...,h If,..., n ae ndependent, ths pocedue leads to a vecto of estcted mamum lelhood estmates fo pobabltes,,..., n. We wll now abandon the ndependence assumpton and allow fo some coelaton among s. 3. A smple coelaton model To nvestgate popetes of PAVA estmato n the case when vaables ae coelated we wll assume a smple model statng that coelaton coeffcent between ndvdual bna vaables s the same fo all pas of subsequent vaables: (,, (, 3, ( 3, 4,...,( -,. Hence a pocedue geneatng bna andom vectos n the fom [,..., ] satsfng E( m and Cov(, - / V 0.5 ( V 0.5 ( - fo,..., and some abtal chosen

20 Wocech Gamot 0 m (0,, (0, s needed. Let a vecto U [U,...,U ] consst of ndependent components: U ~Unf(0, and denote: p (m m - (-m (-m m m - ( p 0 m p (3 The fst component of ma be geneated as J (m and subsequent components ae obtaned accodng to the fomula: 0 fo m p J fo m p J 0 (4 whee < a U fo 0 a U fo (a J (5 fo,..., so that E(J (a a. Such a smple pocedue elds a vecto satsfng desed constants snce: m p (m p p p m ( m p m m p m ( m p J E m m p J E 0 0 P( E( P( E( E( and ( V ( V m ( - m ( - m (m m m p m m m m p m m P( P( m m, P( E( E( E(, Cov ( The pocedue depends on the ablt to geneate pseudo-andom numbes U,...,U mtatng ndependent andom vaables havng unfom dstbuton on (0,. Man such geneatos ae wdel avalable ncludng the fast mplementaton of Mesenne-Twste algothm b Matsumoto and shmua (998 mplemented n the R pacage. Ths geneato wll be used n

21 O POOL-ADJACET-VIOLATORS ALGORITHM ou stud. Some sample output of the poposed pocedue wll now be pesented. Fo m [/40, /40, 3/40,...,] and 0 we got a tpcal ealzaton of a bna andom vecto: [ ] Fo m [0.5,...,0.5] and 0 we got a tpcal ealzaton: [ ] Fo m [0.5,...,0.5] and 0.8 we got a tpcal ealzaton: 3 [ ] Fo m [0.5,...,0.5] and -0.8 we got a tpcal ealzaton: 4 [ ] 4. Smulaton esults A smulaton stud was caed out n ode to assess how the bas and mean squae eo of PAVA estmates fo odeed pobabltes depend on the sample sze when vaables ae coelated. Thee smulaton epements wee caed out. In each epement the sequence of n 0,40,...,00 bna vectos was geneated ndependentl h tmes usng the pocedue descbed n pevous secton. All the epements wee caed out usng scpts n R (R Development Coe Team (0. PAVA estmates wee computed usng the gpava functon mplemented n the R sotone pacage (see de Leeuw et al (009 fo a descpton. In the fst epement magnal pobabltes wee set to: m [0.48, 0.49, 0.5, 0.5,0.5] wth 0.0, 0., 0.5, 0.8. In the second epement the wee set to m [0.33, 0.33, 0.35, 0.37,0.37] wth 0.0, 0., 0.5, 0.8. In the thd epement magnal pobabltes amounted to: m 3 m [0.48, 0.49, 0.5, 0.5,0.5]

22 Wocech Gamot wth 0.0, -0., -0.5, Magnal pobabltes wee chosen close to each othe n ode to mae the effects of coectng beached constants b PAVA cleal vsble. The bas and mean squae eo obseved n the fst epement ae shown n fgues and. The bas and mean squae eo obseved n the second epement ae shown n fgues 3 and 4. The bas and mean squae eo obseved n the thd epement ae shown n fgues 5 and Bas( Bas( Bas( Bas( p p p 3 p 4 p Fg.. The bas of PAVA estmates fo m m and 0.0, 0., 0.5, 0.8 In all thee epements the scope of obseved bas depends on the poston of a vaable n the smple ode (. The bas fo estmates p 4 and p 5 of ghtmost pobabltes 4 and 5 tends to be postve whle fo leftmost pobabltes and the bas of estmatos p and p tends to be negatve.

23 O POOL-ADJACET-VIOLATORS ALGORITHM 3 Meanwhle, estmato p 3 of the nnemost pobablt 3 seems to be appomatel unbased. The ntoducton of stong postve coelaton n the fst and second epement seems to educe the bas to some etent. The effect of negatve coelaton assessed n the thd epement s moe comple: estmates of outemost vaables and 5 seem to be unaffected whle the bas of estmates fo and 4 s slghtl educed. Anwa, n all epements and fo all paametes,..., 5 the bas of estmates appaentl tends to zeo when sample sze n nceases. Hence thee s no evdence that the asmptotc unbasedness of PAVA estmates whch was poven b Ae et al (955 unde assumpton of ndependence ceases to hold n the pesence of coelaton MSE(... 5e-04 e-03 e-0 MSE(... 5e-04 e-03 e-0 p p p 3 p 4 p MSE(... 5e-04 e-03 e-0 MSE(... 5e-04 e-03 e Fg.. The MSE of PAVA estmates fo m m and 0.0, 0., 0.5, 0.8

24 4 Wocech Gamot 0 0. Bas( Bas( Bas( Bas( p p p 3 p 4 p Fg. 3. The bas of PAVA estmates fo m m and 0.0, 0., 0.5, 0.8 The mean squae eo of estmates depends on a poston of a vaable n the ode ( as well. In all thee epements the MSE fo estmatos p and p 5 s cleal the hghest of all the fve and onl n the second epement the obseved dffeence between these two s moe ponounced (wth p beng somehow moe accuate than p 5. Anwa, n all epements the MSE of all estmatos p,...,p 5 appaentl tends to zeo wth gowng sample sze whch suggests that consstenc s etaned unde coelaton.

25 O POOL-ADJACET-VIOLATORS ALGORITHM MSE(... 5e-04 e-03 e-0 MSE(... 5e-04 e-03 e-0 p p p 3 p 4 p MSE(... 5e-04 e-03 e-0 MSE(... 5e-04 e-03 e Fg. 4. The MSE of PAVA estmates fo m m and 0.0, 0., 0.5, 0.8

26 6 Wocech Gamot 0-0. Bas( Bas( Bas( Bas( p p p 3 p 4 p Fg. 5. The bas of PAVA estmates fo m m 3 and 0.0, -0., -0.5, -0.8

27 O POOL-ADJACET-VIOLATORS ALGORITHM MSE(... e-04 e-03 e-0 MSE(... e-04 e-03 e MSE(... e-04 e-03 e-0 MSE(... e-04 e-03 e-0 p p p 3 p 4 p Fg. 6. The MSE of PAVA estmates fo m m 3 and 0.0, -0., -0.5, Concluson Smulaton epements caed out dung ths stud coveed seveal multvaate dstbutons of a bna vecto, nvolvng dependences between ts ndvdual components. Even fo ve stong coelatons, no evdence of an depatues fom the consstenc popet was found. Hence, pesented esults suggest that PAVA estmates ma etan consstenc n the stuaton when bna vaables ae coelated. Obvousl, those pomsng smulaton esults do not consttute a fomal poof of consstenc as the cove onl a few of nfntel man possble combnatons of paametes. Howeve the ustf theoetcal effots amed at establshng popetes of PAVA-based estmates unde coelaton. Such effots ma sgnfcantl wden the ange of possble applcatons fo the PAVA pocedue.

28 8 Wocech Gamot Refeences Ahua, R.K., Oln, J.B. (00 A fast scalng algothm fo mnmzng sepaable conve functons subect to chan constants. Opeatons Reseach 49, Ae, M., Bun, H.D., Ewng, G.M., Red, W.T., Slveman, E. (955 An empcal Dstbuton functon fo Samplng wth Incomplete Infomaton. The Annals of Mathematcal Statstcs 6(4, Best, M.J., Chaavat,. (990 Actve Set Algothms fo Isotonc Regesson; A Unfng Famewo. Mathematcal Pogammng 47, Bloc, H., Qan, S., Sampson, A. (994 Jounal of Computatonal and Gaphcal Statstcs 3(3, Bun, H.B., (955 Mamum lelhood estmates of monotone paametes. The Annals of Mathematcal Statstcs 6, Budaov, O., Gmwall, A., Hussan, M. (004 A Genealzed PAV Algothm fo Monotonc Regesson n Seveal Vaables. COMPSTAT Poceedngs n Computatonal Statstcs. Phsca-Velag/Spnge, Hedelbeg, Chaas, A., van Eeden, C. (99 Baes and admssblt popetes of estmatos n tuncated paamete spaces. Canadan Jounal of Statstcs 9, -34. van Eeden, C. (956 Mamum lelhood estmaton of odeed pobabltes. Poc. Kon. edel. Aad. Wetensch. Se. A. 60, van Eeden, C. (957 Mamum lelhood estmaton of patall o completel odeed pobabltes. Poc. Kon. edel. Aad. Wetensch. Se. A. 59, van Eeden, C. (958 Testng and estmatng odeed paametes of pobablt dstbutons. Ph.D. thess, Unvest of Amstedam. van Eeden, C. (006 Restcted Paamete Space Estmaton Poblems: Admssblt and Mnmat Results. Spnge. ew Yo. de Leeuw, J., Hon, K., Ma, P. (009 Isotone optmzaton n R: Pooladacent-volatos algothm (PAVA and actve set methods. Jounal of Statstcal Softwae 3(5, -4. Hansohm, J. (007 Algothms and eo estmatons fo monotone egesson on patall peodeed sets. Jounal of Multvaate Analss 98,

29 O POOL-ADJACET-VIOLATORS ALGORITHM 9 Hanson, D.L., Pledge G.., Wght F.T. (973 On Consstenc n Monotonc Regesson. The Annals of Statstcs (3, Hädle W. (99 Appled onpaametc Regesson. Cambdge Unvest Pess. Jewel,.P., Kalbflesch, J.D. (004 Mamum lelhood estmaton of odeed multnomal pobabltes, Bometcs 5(, Katz, M.W. (963 Estmatng odeed pobabltes. Annals of Mathematcal Statstcs 34, Lee, C.C. (983 The mn-ma algothm and sotonc egesson. Annals of Statstcs, Matsumoto, M., shmua, T. (998 Mesenne twste: a 63-dmensonall equdstbuted unfom pseudo-andom numbe geneato. ACM Tansactons on Modelng and Compute Smulaton 8 (, Qan, S. (99 Mnmum lowe sets algothm fo sotonc egesson. Statstcal Pobablt Lettes 5, R Development Coe Team (00 A language and envonment fo statstcal computng. R Foundaton fo Statstcal Computng, Venna. Robetson, T., Wght, F.T., Dsta, R.L. (988 Ode estcted statstcal nfeence. Wle, ew Yo. Sacowtz, H. (98 Pocedues fo mpovng the MLE fo odeed bnomal paametes. Jounal of Statstcal Plannng and Infeence 6, Sacowtz, H., Stawdeman, W. (974 On the admssblt of the M.L.E. fo odeed bnomal paametes. Annals of Statstcs, O WŁASOŚCIACH ALGORYTMU PAVA DLA ZMIEYCH ZALEŻYCH Steszczene Algotm PAVA (od ang. Pool-Adacent-Volatos Algothm est populanm nazędzem estmac wozstwanm do szacowana watośc oczewanch cągu zmennch losowch w stuac, gd dostępna nfomaca dodatowa pozwala stwedzć, że mędz tm watoścam oczewanm zachodz elaca poządu. Uzsane za pomocą tego algotmu oszacowana masmalzuą (waunowo funcę waogodnośc pz założenu, że elaca ta

30 30 Wocech Gamot est spełnona oaz poszczególne zmenne są nezależne. Wdae sę, że żadna z pzedstawonch w lteatuze pzedmotu modfac te pocedu estmac ne uwzględna możlwośc wstąpena zależnośc pomędz poszczególnm zmennm. W nneszm atule pzedstawono ezultat espementów smulacnch tóch celem bło zbadane własnośc oszacowań uzsanch za pomocą te pocedu gd zmenne są soelowane.

31 Anna Imołe Janusz Gołaszews Dausz Załus Zbgnew asals PRACTICAL STATISTICAL AD ECOOMIC ASPECTS OF USIG SURVEY STUDIES FOR IDETIFICATIO OF THE KEY PLAT CULTIVATIO TECHOLOGY FACTORS. Intoducton Suve studes ae a eseach method that s wdespead n socal scences but less common n ago-techncal studes. Among the publcatons whch have appeaed n Poland, manl concened wth the methodolog of usng suves fo evaluaton of plant cultvaton technologes and ago-techncal factos, notewoth ae papes wtten b Kzmus (98, Kzmus et al.(995, Laudańs et al. (007a, 007b and Imołe et al. (00. Plant poducton s govened b cetan, well-defned cultvaton ecommendatons, especall mpotant when qualt standads mposed b contact ageements ae to be met. Due to techncal and economc condtons, a fame s not alwas able to adhee to such ecommendatons n pactce, but at the same tme changes on the fam poduce maet enfoce poduces to ethe change o modf a poducton technolog. Selectng an adequate combnaton of ago-techncal factos depends on the qualtatve and quanttatve paametes of a maet poduct (eld, but the decson s also shaped b such oganzaton of plant poducton whch enables the fame to mnmze poducton costs and mamze the poft. The volume and qualt of eld ae a poduct of man factos, whch compse elements of plant ago-technolog and andom events. A geneal poblem n all eseach methods s the dentfcaton of factos whch can be named as the e ones n a gven technolog. In the pesent stud, t has been assumed that ceatng a new cultvaton technolog o modfng an estng begn though the ecognton of the technologcal foundatons of

32 3 Anna Imołe, Janusz Gołaszews, Dausz Załus, Zbgnew asals poducton. In espect of the methodolog, a decson to use suves has been made. Because the eseach coveed a lage aea, t was athe dffcult to have the suve completed b all agcultual poduces. Suve studes ae sgnfcantl affected b the tme whch elapses fom events whch a suve nvestgates to the tme when espondents ae ntevewed and the fom of questons (Conad et al Wnte cops cultvaton s chaactezed b elatvel long duaton. Fo the espondents eples to be elable, a suve should be completed as soon as possble afte the temnaton of a poducton pocess and befoe a new ccle begns. Anothe dffcult n suve-based studes s the geneal unwllngness of agcultual poduces to eveal detaled nfomaton about the ago-techncal factos of the poducton the conduct (ecept stuatons when montong poducton on a gven fam s compulso. Theefoe, the esults of suves, even when appled to a epesentatve sample, can be budened wth an eo and although the ae a valuable mateal fo scentfc eseach, the should not be used fo mang poducton ecommendatons. When plannng ths suve stud, the authos pesumed that t should geneate a geneal vew of the confguaton of factos nvolved n e cultvaton technolog and enable economc evaluaton of the poducton as well as selecton of factos fo futhe eamnaton n stct epements. Ths pape s theefoe an attempt at usng suve data fo evaluaton of a technolog of e cultvaton, mang an economc evaluaton and selectng e ago-techncal factos.. Methodolog of suve studes The pesent suve on the technolog of gowng wnte e coveed the aea of notheasten Poland, the povnces of Wama and Mazu, Podlase and Mazowsze. In ode to eflect the cuent economc status of agcultual poduces, ou selecton of espondents was ntentonal and the sze of a wnte e plantaton ove ha was the selecton cteon. Most of the suveed fams gew e unde contacts wth e pocessng plants (manl mlls. Dung dect ntevews at fams, suve questonnaes wee completed. The questonnae was dvded nto fou goups of questons, whch wee to detemne the value of a plantaton, pe-sowng teatments, qualt of seed mateal, ago-techncal aspects of gan sowng, plantaton teatments and havest. Statstcal analss of the suves. The pelmna stage of statstcal analss of the data povded b the questonnaes conssted of codng the data. The factos wee dvded nto natual categoes (e.g. foecops o class anges (e.g. levels of ntogen fetlzaton accodng to the technologcal gudelnes fo e cultvaton gven n the efeences.

33 PRACTICAL STATISTICAL AD ECOOMIC ASPECTS OF USIG SURVEY STUDIES 33 The net step n ou analss conssted of ceatng a lnea model and analzng gan elds pe ha fo the whole sample populaton and dvded nto bologcal foms of cultvas,.e. hbd and populaton. Fo the patcula tpes of cultvas, the model ncluded onl such ago-techncal factos that wee nvolved n a technolog of gowng those cultvas. Fo assessment of the man effects of the factos and de-composton of the contbuton of patcula poducton factos nto the vaablt of gan eld, tpe III sums of squaes wee used and the coeffcent η (eta-squae was detemned, whch eflected the elatve contbuton of an eamned poducton facto to the volume of eld. η SS Ef SS. / Og. whee: SS s the sum of squaes of the vaablt of a gven effect, Ef. and SS s the sum of squaes of the geneal vaablt of a model. Og. In the late pat of statstcal analss, a heach of the cultvaton technolog factos was establshed (evaluaton of the mpotance of factos va an applcaton of classfcaton tees analses wee made fo the whole populaton and dvded nto cultva foms. The classfcaton tees wee constucted fom a leanng set, whch conssted of the uppe and lowe quatle of the populaton, coespondng, espectvel, to low and hgh elds. The C&RT (Classfcaton and Regesson Tees method was appled to constuctng a tee that ehausted the seach fo one-dmensonal dvsons. Ths method vefes all possble dvsons fo each pedctve vaable n ode to fnd out a dvson fo whch the best mpovement of the goodness of ft (o else the hghest educton n the lac of ft appeas. The goodness of ft was detemned wth the Gn coeffcent, whch eaches the value 0 when onl one class appeas n a gven node. Fo stoppng the dvson, the opton cut at an eo of wong classfcaton was chosen, so that a tee was dvded untl the moment when all the nodes wee clea (contanng obects fom onl one class o havng no moe than a specfed mamum numbe of obects. Ths numbe was set as 5. The sze of a tee was set accodng to V-fold coss-valdaton. All statstcal analses wee pefomed wth the ad of the compute softwae STATISTICA 9.0. The economc analss of the esults. The nvento of teatments and appled equpment was used fo detemnaton of labou, tactve powe and technologcal devces as well as mateal outlas used fo cultvaton of e. The costs of eplotaton of techncal means wee computed wth the method suggested b the Insttute of Economcs and Agcultual machne Eplotaton, the Insttute of Cvl Engneeng and Agcultual Machne n Wasaw (Muzalews 00. The mateal costs (e.g. mneal fetlzes, plant

34 34 Anna Imołe, Janusz Gołaszews, Dausz Załus, Zbgnew asals potecton chemcals wee detemned as a poduct of the use and pce pe unt. Fo the calculatons, the maet pces as of June 00 wee taen. The pat ate pe hou of labou was computed accodng to the aveage pa n the whole Polsh econom ( assumng that as the EUROSTAT clams peson can wo no moe than annual wo unt (AWU, even f the actuall wo longe. The annual wo unt (AWU s an equvalent of the tme taen to pefom the wo done b peson emploed on a full-tme bass at a fam. In Poland, t s assumed that,0 hous of wo pe ea ae an equvalent of a full-tme ob n agcultue ( The value of outlas ognatng fom own poducton (seed mateal was estmated wth the own costs method. The cost of mneal fetlzes was assessed b the compaatve method, tansfeng the aveage maet value of the fetlze s mneal components onto the analogous components found n FYM, tang nto consdeaton the amount of ntogen applcable n a gven ea. The dect costs also nclude the suchage of ndect costs. The poducton poftablt nde, undestood as a ato of the value of poducton whch s a potental commodt to the total costs of the poducton outlas, was appled as a snthetc economc measue whch egaded the effectveness of the outlas (asals et al The costs have been pesented n a functonal patten, dstngushng patcula outlas elated wth a gven teatment,.e. pe-sowng sol tllage, sowng, fetlzaton, applcaton of plant potecton chemcals. The costs of the teatments nclude the outlas on eplotaton of machnes, labou outlas and ependtues on mateal poducton means. 3. The esults The suve stud encompassed 73 vllages n ten admnstatve dstcts lng n thee povnce: Wama and Mazu, Podlase and Mazowsze. Dung face-to-face ntevews, 0 questonnaes wee flled n; the coveed envonmentall dffeent vaants of e poducton on 53 fams, whch had at least ha of wnte e gown fo gan n the stuctue of cops sown n 007/008. When the data fom all the plantatons wee collected, the analss of vaance of the e gan elds demonstated the sgnfcance of all the man effects, ecept pe-sowng tllage and pe-sowng fetlzaton. In tun, the analss of the poducton technolog appled to hbd cultvas poved that the pe-sowng tllage, seed dessng and weed and fungus contol teatments wee non-sgnfcant, but when populaton e was gown, the non-sgnfcant factos ncluded pe-sowng tllage, seed cetfcaton gade, sowng technque, ow spacng, fungal contol and applcaton of a etadant.

35 PRACTICAL STATISTICAL AD ECOOMIC ASPECTS OF USIG SURVEY STUDIES 35 Fg.. The aea coveed b the suves and numbe of suves n the admnstatve dstcts of notheasten Poland Analss of vaance of gan eld of e Table Specfcaton Hbd cultvas Populaton cultvas df SS III p df SS III p Oganc fetlzaton Pe-sowng cultvaton Pe-sowng fetlzaton Cultvas 3 a a Seed cetfcaton Seed dessng Sowng technque Date of sowng Sowng ate Row spacng Depth of sowng

36 36 Anna Imołe, Janusz Gołaszews, Dausz Załus, Zbgnew asals Top dessng Mechancal cultvaton b Fungcde applcaton Hebcde applcaton Retadant applcaton Date of havest Eo In total a when analzed cultvas wee a facto tested b the analss coveng tpes of cultvas b the facto was absent fom the technolog The de-composton of contbuton of patcula poducton factos to the total vaablt demonstated that the andom factos made up the lagest contbuton to the vaablt of e gan elds, especall n the case of hbd e (6% (fg.. The mao factos detemnng the eld of e gans wee the date and paametes of sowng. In espect of the tpe of cultvas, lage dffeentaton was dscoveed. When gowng populaton cultvas, the factos elated to seed qualt and plant cultvaton teatments wee found to domnate, wheeas n the cultvaton of hbd vaetes, whee the seed mateal s echanged on 66% of the analzed plantatons, the domnant effect was poduced b the ago-techncal factos connected wth seed sowng. Fg.. Decomposton of vaablt of factos n wnte e poducton

37 PRACTICAL STATISTICAL AD ECOOMIC ASPECTS OF USIG SURVEY STUDIES 37 Dependng on the fom of e, the volume of aveage gan elds vaed b.05 t. Thee classfcaton tees wee constucted fo the total e populaton and fo the two e foms: populaton and hbd vaetes. On each occason, the leanng set conssted of the uppe and lowe quatle of elds. The mao facto dscmnatng e poducton (based on the esults fom all the e plantatons was the sowng ate. Hgh elds classfed ntall as low ones wee dscmnated b the sol class and sol comple, as well as ntogen dessng. Fo the populaton cultvas, the moment the cultvaton technolog factos had been ncluded, the mao detemnant was the applcaton of seed dessng (fg. 3. Hgh elds ntall classfed as low ones wee detemned b ntogen top dessng, followed b the date of sowng, ow spacng and plant potecton measues such as the applcaton of a hebcde. Fg. 3. Classfcaton tee of low and hgh elds of wnte e populaton cultvas In espect of hbd foms, hgh elds wee obtanable at a low sowng ate (n accod wth the cultvaton ecommendatons pepaed fo hbd e and applcaton of a hebcde (fg. 4. Hgh elds ntall classfed as low ones wee detemned b the paametes defnng the qualt of sowng seeds and the date of sowng. Fg. 4. Classfcaton tee of low and hgh elds of wnte e hbd cultvas

38 38 Anna Imołe, Janusz Gołaszews, Dausz Załus, Zbgnew asals Among the analzed poducton factos, ans acheved fo hbd vaetes wee much dffeent fom the ones fo populaton foms (fg. 5. The hghest ans wee acheved b the paametes elated to sowng, ow spacng 79% fo pe-sowng tllage and 76% fo the sowng ate. In contast, fo populaton cultvas the hghest an was scoed b weed contol measues, followed b paametes connected wth the qualt of seed mateal, cultvas and ntogen top dessng. Fg. 5. Ranng of the mpotance of some vaables fo populaton and hbd cultvas [%] The costs calculaton n agcultual pactce should be used as a souce of nfomaton useful when mang stategc decsons as well as opeatonal ones. Economc analss enables fames to optmze the poducton stuctue and to mae a moe atonal use of patcula technques. When ceeal pces ae unstable, one of the ve few chances to mpove the economc output on fams whch gow ceeals as a commodt s the vefcaton of outlas and costs (asals et al The volume and qualt of e gan eld wee sgnfcantl shaped b fetlzaton. As demonstated b the conducted suve, 40% of the total dect costs wee ncued b fetlzaton opeatons alongsde the ependtue on the puchase of fetlzes (table. The costs elated to chemcal potecton of e plants wee etemel vaed, dependng on the sze of a fam. On fams whch had up to 7 ha of aable land, the hadl eached %, but went up to % on fams whch had ove 00 ha of aceage. The outlas on sol tllage befoe sowng wee lowe n lage fams, whch possess moe effcent machnes and can aggegate the pefomed opeatons.

39 PRACTICAL STATISTICAL AD ECOOMIC ASPECTS OF USIG SURVEY STUDIES 39 Stuctue of dect costs [ ha - ] Table Sze of a fam [ha] < >00 In total Sol cultvaton Sowng Fetlzaton Plant potecton Havest In the costs stuctue of all the suveed fams, the costs connected wth the ago-techncal opeatons made up on aveage 6.%, ncludng 3.9% of the outlas on pe-sowng teatments (fg. 6. The outlas on such poducton means as seed mateal, plant potecton chemcals and fetlzes consttuted 38.9% of all the dect costs. The hghest ependtue was ncued b the puchase of fetlzes (8%. Fg. 6. Stuctue of dect costs of wnte e cultvaton on fams n notheasten Poland [%] In the poducton pocess, the total sum of the ncued costs onl weal coesponded to the evaluated poftablt of a gven technolog. It s not untl we compae the costs wth the volume of poduced eld that the effectveness of a technolog n queston can be seen. The unt poducton cost (UPC s a poduct of the total costs and the volume of poducton. Fgue 7 shows the aveage unt poducton cost fo decton [dt] of e dependng on the sze of a fam. The aveage UPC anged fom 9.70 to 6.0 fo decton. In the goup

40 40 Anna Imołe, Janusz Gołaszews, Dausz Załus, Zbgnew asals of small fams,.e. up to 7 ha of aceage, t hghl vaed: fom 6.4 to 9.49 fo decton; a smlal hgh devaton was ecoded fo the fams between ha of aceage (6.55 /dt. On the emanng fams, the AUPC dffeed b /dt. Fg. 7. Unt poducton cost of gowng wnte e on fams n notheasten Poland [ ha - ] The feld opeatons, as alead mentoned, consttuted 60% of the dect costs. Aggegatng such opeatons mpoved the qualt of a feld pepaed fo e plantaton and, consequentl, the volume of eld. It also educed the use of fuel, whch meant lowe dect costs of e poducton. Attanng hghe elds and deceasng the outlas esulted n lowe unt poducton costs (fg. 8. Fg. 8. Value of unt costs dependng on sol tllage educton [ dt].

41 PRACTICAL STATISTICAL AD ECOOMIC ASPECTS OF USIG SURVEY STUDIES 4 Fetlzaton s one of the most mpotant factos whch affect the volume of elds. Fo hbd cultvas, the lowest poducton costs pe unt wee obtaned when ates povdng g/ha of fetlzes wee appled. Dvdng ths dose of fetlzaton s not effectve, as t ses the outlas. The hghest eld was obtaned when the fetlzaton ate was ove 90 g/ha, dvded nto two doses, whch was the most effectve fetlzaton vaant (the lowest unt costs (fg. 9. Fo populaton cultvas, dvson of fetlzaton ates nceased the elds. onetheless, fetlzaton wth a sngle ate of up to 60 g/ha was the onl economcall vable fetlzaton vaant. Fg. 9. Unt poducton costs and eld volume dependng on fetlzaton ates fo the foms of wnte e [ dt] A snthetc economc measue whch taes nto account the effectveness of the outlas s the dect magn effectveness nde, whch s a ato, epessed n pe cent, of the dect magn to the goss commodt poducton. A negatve value of ths nde was found fo a populaton cultva gown on small fams wth poo famng pactce (fg. 0. As the volume of elds nceased, the magn ate contnued to se, eachng the value of 70% fo hbd foms. Fg. 0. Dect magn ate [%] dependng on the fom of e and eld volume

42 4 Anna Imołe, Janusz Gołaszews, Dausz Załus, Zbgnew asals Poducton opeatons ae unde the nfluence of some endogenous factos: the poducton potental of each fam,.e. land, labou and captal esouces, the qualt and usablt, and eogenous ones, poduced b some etenal nfluence poduced on agcultue (Sażńsa 00. Maet pces fo e ae hghl unstable, vang fom ea to ea between 80.5 and 0. euo pe dt - ( Besde the ndect ncome obtaned fom sellng gan, fames also eceve subsdes, whch lagel affect the poft obtaned fom ths tpe of poducton, especall on small, low- -poductvt fams, whch would geneate loss was t not fo the subsdes the ae pad (fg.. Low elds obtaned on oganc fams ae compensated fo onl b hghe subsdes. Fg.. Poft obtaned fom e poducton dependng on a e fom and eld volume [ ha - ], b tpe of fam management 4. Conclusons A suve stud, caed out n the fom of face-to-face ntevews, can be a souce of valuable data on the cuentl appled plant poducton technologes ove an aea coveed b that eseach, and the AOVA methods, ncludng tpe III sums of squaes and classfcaton tees, can popel dscmnate among the analzed factos, allowng eseaches to dentf the e factos necessa to obtan hgh elds. When dffeent tpes of agcultual cultvas ae gown, the esults of suves and of the subsequent analses wll enable us to captue these poducton factos whch ae unvesal n chaacte and the ones whch ae specfc fo a gven tpe of cops. In e cultvaton, 39% of the gan eld vaablt s attbutable to the qualt of a cultvaton feld. The poducton technques sgnfcantl dffeentate

43 PRACTICAL STATISTICAL AD ECOOMIC ASPECTS OF USIG SURVEY STUDIES 43 between the tpes of e hbd and populaton vaetes. Bette eldng hbd cultvas ae hghl vaable n tems of the factos connected wth the qualt of seed mateal and plant cultvaton teatments, wheeas the goup of most sgnfcant ago-techncal factos n cultvaton of populaton cultvas compses the seed sowng technques. It s among these goups of e gan poducton factos that we should dentf the ones fo testng n stct and poducton epements. The economc analss of the esults of ou suve shows what costs e cultvaton ncus, whch can change due to gowng pces of dffeent aw mateals but whch ae also dependent on the appled technolog, cultvaton aceage, labou potental o the avalable machnes and tools. The economc costs calculaton, whch was completed n a ea when gan sellng pces wee hgh, showed that all the analzed e plantatons geneated pofts. Howeve, those fames who eled on etensve technologes o oganc famng obtaned low elds and then the poftablt of poducton was ensued eclusvel b the eceved subsdes. It was economcall vable to use cultvaton aggegates o aggegated cultvaton and sowng machnes, because then the numbe of uns was lowe (savng on fuel and labou and good sol condtons wee mantaned, affectng the volume of elds. Refeences Conad, F., Rps, L., Fce, S. (009 Seam Effects n Quanttatve Responses. Jounal of Offcal Statstcs (5(3. Imołe A., Gołaszews J., Załus D., Stawana-Kosoe A. (009 Metodczne aspet badań nad technologam upaw ośln. XXXIX Mędznaodowe Colloquum Bometczne, Kazmez Doln, 7-0 wześna 009. Kzmus, J. (98 Ocena pognoza efetwnośc głównch cznnów plonotwóczch zbóż. Roczn au Rolnczch Sea A, 05, Kzmus, J., Laudańs, Z. (995 Waun cznn plonowana zbóż. Część II. Ocena współzależnośc wbanm metodam statstcznm. Buletn Insttutu Hodowl Almatzac Rośln 93, Laudańs, Z., Mańows, D., & Seczo, L. (007 Póba ocen technolog upaw pszenc ozme na podstawe danch anetowch gospodastw ndwdualnnch. Część II. Ocena technolog upaw. Buletn Insttutu Hodowl Almatzac Rośln 4,

44 44 Anna Imołe, Janusz Gołaszews, Dausz Załus, Zbgnew asals Laudańs, Z., Mańows, D., Seczo, L. (007 Póba ocen technolog upaw pszenc ozme na podstawe danch anetowch gospodastw ndwdualnch. Część I. Metoda wodębnana technolog upaw. Buletn Insttutu Hodowl Almatzac Rośln 44, Muzalews, A. (00 Koszt esploatac maszn. Waszawa: Falent. asals, Z., Sadows, T., Stępeń, A. (004 Poducna, eonomczna enegetczna efetwność poduc ęczmena ozmego pz óżnch pozomach nawożena azotem. Acta Scentaum Polonaum, Agcultua 3(, Sażńsa, A. (00 Koszt eonomczne wbanch dzałalnośc poduc oślnne w latach Roczn au Rolnczch Sea G, 97, z. 3, STATISTICA 9.0. StatSoft. PRAKTYCZE ASPEKTY STATYSTYCZO-EKOOMICZE WYKORZYSTAIA BADAŃ AKIETOWYCH W TYPOWAIU KLUCZOWYCH CZYIKÓW TECHOLOGII UPRAWY ROŚLI Steszczene Badana anetowe pzepowadzone w 008 ou mał na celu oeślene luczowch elementów technolog poduc oaz alulacę osztów ednostowch poduc żta ozmego (Secale ceeale L. upawanego na zano. Anetzacą obęto poducentów zana żta w północno-wschodne Polsce powadzącch upawę na aeale węszm nż ha. Kwestonausz anetow zaweał ptana połączone w gup dotczące: chaatest ogólne gospodastwa, cznnów technologcznch poduc, 3 ocen enegochłonnośc (agotechnczne oaz 4 stutu naładów. Dane o cznnach poduc stanowł pedto w ogólnm modelu lnowm, a zmenną zależną bł plon zana. W analze waanc plonu zana wozstano sum wadatów tpu III oaz oszacowano efet główne cznnów. Analzę eonomczną wonano na podstawe naładów bezpośednch ponesonch na poducę, oblczono ednostowe oszt oaz nadwżę bezpośedną, oeślono stutuę osztów oaz zsowność poduc żta ozmego.

45 Aadusz Kozłows THE USEFULESS OF PAST DATA I SAMPLIG DESIG FOR EXIT POLL SURVEYS. Intoducton Et poll s a suve conducted on the electon da n whch espondents (votes leavng the pollng staton answe,.a. on who the cast the votes. Ths suve s so popula manl thans to the televson statons, fo whch nowng the electon esults ust afte the pollng statons have been closed, espectve of the fact that the esult s onl appomate, allows them to fst comments and lve analss on the electon nght, whch guaantees a ve hgh veweshp. The dea fo ths tpe of suves was bon s the US and thee t was developed most ntensvel. As Fanov (99 sas, the fst suve on the electon da too place n 940 n Denve. The fst et poll n the fom we now toda,.e. on a lage scale and at the equest of meda, too place n 967 and was conducted fo CBS (Lev, 983. The ceaton and development of suve methodolog s ascbed to Waen Mtofs (Mooe, 003. In Poland the fst ths tpe of eseach was conducted b Ośode Badana Opn Publczne (OBOP dung the fst and second ound of pesdental electon n 990. Et poll s one of the few sample suves, the esults of whch ma be confonted wth the complete enumeaton and, what s moe, n a ve shot peod of tme. Fom the statstcal pont of vew, ths gves a possblt of the mmedate valdaton of the appled methodolog. Fo the eseach centes conductng ths tpe of suves t s a nd of a challenge because the malpactce ma cause them to lose the eputaton and tust not onl to a patcula eseach cente but to the polls n geneal. In the goup of suves elated to electon, et poll has a specal place fo a few easons. Fstl, populaton of suve does not nclude all people enttled to votng but onl people who actuall vote. Thans to that, on conta to pe-electon suves, the sceenng poblem of how to dentf lel votes does not est. Secondl, the questons n et poll ae elated to facts and not ntentons whch ma dffe

46 46 Aadusz Kozłows fom the actual electon decsons. Ths ssue s of patcula mpotance especall n case of changng poltcal pefeences a few das befoe electon (so-called late swng. As Hlme (008 emphaszes, the et poll s moe clea to espondents, an am of t s moe obvous and not aousng msgvngs whch esult n lowe non-esponse ate compaed to othe electon suves. Also the sze of the sample (fo Poland a tens of thousands s fa moe hghe than n standad suves. Wth egad to above-mentoned easons, the equements of the suve s ecpents concenng ts pecson ae hghe than the equements concenng othe electon suves. Howeve, the am of et poll s not onl pedcton of the electon esult. Ths suve delves a lot of valuable nfomaton about votes dstbuton n dffeent soco-demogaphc goups, the changes of poltcal pefeences n elaton to pevous electon, the motves of choosng a patcula pat o canddate, the motves of choosng the tme of votng etc. Ths nfomaton enables a thoough analss of the esults and wll be used untl the net electon due to the fact that cuent poltcal suves, manl of the above-mentoned easons, do not povde so detaled data wth the necessa pecson. In the less stablzed democaces, et polls ndectl pefom a functon of legtmac of electon and ts esults the offcal esults happen to be questoned f the dffe fom those obtaned fom the ndependent et poll (the eamples of such stuatons ma be found n Andeenova, 008. Unntentonal effect of et poll ma also be an nfluence on the potental votes motvaton to go to the polls f the pelmna esults ae announced befoe closng the last pollng staton. Ths poblem concens manl the US whee thee s no legal pohbton on publshng suves esults befoe all pollng statons have been closed. Ths ssue s wdel dscussed b,.a. Semou (986, Lens (008.. Statstcal aspects of et poll Et poll s a two-stage suve. Pma stage unts ae pecncts and the seconda stage unts ae votes. As long as selecton of espondents to the sample s concened thee s an ageement between theosts and pacttones that the best choce n ths case s a sstematc samplng. Ths appoach manl esults fom the uneven dstbuton of patcula pat votes dung the da, whch was the obect of stud.a. Kloman (976, Busch and Lese (985. The sgnfcant nfluence on the choce of the tme of da has an electon da, n the US t s usuall Tuesda, n the UK Thusda,.e. wong das. In Poland, as n the maot of countes, electon taes place on holda. Respondents chosen to the sample ae ntevewed b the use of self-admnsteed questonnae, whch s then put n the envelope o deposted n the specall pepaed ballot bo. Bshop and Fshe (995 poved that ths mode of data

47 THE USEFULESS OF PAST DATA I SAMPLIG DESIG 47 collecton, called secet ballot deceases tem non-esponses and socall desable esponses compaed to face-to-face ntevew, whch s eflected n moe accuate estmates. Befoe pollstes begn ntevews t s cucal to establsh net to whch pollng statons the suve wll be conducted. In Poland ove twent fve thousands of pecncts ae ceated dung the electon. The sample eflectng most fathfull natonwde esults needs to be chosen fom ths populaton. Baeto et al. (006 state: In fact, ths s the most mpotant step n et pollng. Unoffcall, accodng to the one of the eseach centes, those past eos n Polsh et poll manl esult fom the unepesentatve sample of pollng statons. The conventonal appoach towads ths ssue s a andom selecton of pecncts, howeve, ths appoach does not gve the enough guaantee of epesentatveness of sample. Moeove, as Szede (007 emphaszes, elng onl on the andom samplng means n fact that the pollste admts that he/she lacs the valuable a po nowledge about the suveed populaton. Snce such nowledge ests we should as not f but how t should be used? One of the often utlzed and wdel accepted methods s the dvson of a populaton to stata. The choce of statfng vaables and establshng the numbe of stata s not an obvous thng. Addtonall, optmal paametes ma dffe between countes and ma change wth tme. Fo eample Lev (983, whle chaactezng Amecan pactces, mentons to 6 stata ceated based on past votng behavous, geogaphc egons, uban vs. ual countes, pecent foegn stoc, tpe of votng equpment, o poll closng tmes. The analss of the dstbuton of esults n patcula stata fom past electon wll suel mae t ease to desgn stata popel. Anothe technque whch also uses the data about past electon esults s ted samplng pocedue. Ted sample means that the bass fo ceatng electon foecast s a sample of pecncts whch tuned out to be the most epesentatve one dung the past electon (Hofchte, 999. Ths conssts n samplng a cetan amount of samples (fom a statstcal pont of vew each of them s of the same value and choosng the sample whch eflected the patcula past electon esults chosen as a efeence pont. Of couse, ths technque eques the complete data about electon esults on the level of pecncts. If ths data s unavalable, fo eample n the UK, the same pecncts ma be suveed n the followng electons and the own data collected n pevous eas ma be used to coect the esults. The successful applance of ths method n 005 s descbed b Cutce & Fth (008. As fa as the numbe of sampled pollng statons s concened, t s the esult of a compomse between budget estctons and the statstcal theo. Fewe pollng statons n a sample and moe espondents fom one pecnct ncease samplng eo, wheeas moe pollng statons n a sample eques moe

48 48 Aadusz Kozłows pollstes, whch nceases the costs. umbe of pollng statons n a sample n Polsh et polls oscllates usuall fom fve hunded to one thousand. The am of ths pape s the empcal vefcaton of the usefulness of a po data, manl nfomaton about past electon esults, n ode to ncease the qualt,.e. epesentatveness of pollng statons sample n et poll. 3. Data Data about electon esults snce pesdental electon 000 on the level of pecncts s wdel avalable on the Państwowa Komsa Wbocza (PKW web stes. Ths data s geat fo smulatve analss of the pocess of samplng to et poll. Howeve, compang the electon esults fo patcula pecncts between two electons ma cause some fomal and substantve dffcultes. Accodng to votng sstem (Act fom Jul, 6 th 998 the pecncts ae ceated b authot of muncpalt n the wa that the nclude fom fve hunded to thee thousand ctzens. Between the successve electons the dvson of muncpalt to pecncts ma change, and n fact ths happens ve often, due to the change of muncpalt s bodes, numbe of ctzens n muncpalt o pecnct, the change of the numbe of councllos n town councl o the change n the dvson of muncpalt to electoal dstcts. The lac of the cental supevson ove ceatng pecncts and the above-mentoned changes esult n the lac of an unequvocal e to dentf pecncts between electons. Substantve dffcultes esult fom the natual demogaphc changes (eachng votng age, deaths, mgatons, votng outsde vote s dstct (ths phenomenon was ve sgnfcant dung the second ound of pesdental electon 00 due to untpcal electon da and the wdespead nfomaton about such possblt and changes on the poltcal scene. Databases shaed b PKW nclude the followng nfomaton: Tetoal dentfcaton of unt (names and codes of vovodeshps, countes and muncpaltes; Pecnct numbe (numeaton appled wthn muncpalt; Pecnct addess (locaton of boad of electons; Tpe of tetoal unt (ct, uban aea on the uban-ual aea, ual aea on the uban-ual aea, vllage, dstcts of captal ct Wasaw; umbe of people enttled to vote; umbe of ballots dstbuted (tunout; Vald votes; umbe of votes cast on a patcula commttees/canddates. Based on the compason of the pecnct numbe, addess and the numbe of people enttled to vote, the pecncts whch have not changed dung successve electons ma be dentfed. Fom techncal pont of vew, ths

49 THE USEFULESS OF PAST DATA I SAMPLIG DESIG 49 eques a scupulous and had wo because due to the fact that thee s ncoheence n notng some of the vaables (manl addess t s mpossble to appl one algothm that would pa pecncts fom two electons. Tang that nto consdeaton, decson was made to dentf pecncts onl between two electons,.e. pesdental electon 00 (fst votng (WP0 and palamenta electon to the Sem 007 (WS07. The obect of analss s settng such a samplng plan that wll mamze pobablt of choosng the best sample of pecncts fo estmaton of esults of WP0 b usng detaled esults fom WS07. In ths analss onl the egula pecncts wee taen nto consdeaton, ecludng the pecncts ceated n hosptals, psons, detenton centes, on shps, socal welfae centes and aboad. In WS07, the commttees whch dd not have canddates n all pecncts wee ecluded. Addtonal nfomaton about combned dstcts s pesented n table. Table umbe of pecncts and people enttled to vote n all and combned pecncts umbe of pecncts umbe of people enttled to vote total combned total combned WP (89,% (9,8% WS (90,% (93,3% 4. Analss The obect of analss s the fst stage of et poll,.e. samplng of the pecnct. Due to ths fact, the vaablt of the esults occung on the second stage,.e. esultng fom andom samplng of espondents, s not taen nto account. Whle calculatng the esult, the actual esults n the sampled pecncts wee taen nto consdeaton. As a measue evaluatng the smlat of the sample to the whole populaton, aveage Manhattan dstance has been used: n AMD p P 00 % ( n whee: AMD metcs fo th sample, p elatve esult of th commttee/canddate n th sample, P elatve esult of th commttee/canddate n the whole count, n sze of sample (one hunded.

50 50 Aadusz Kozłows The same measue has also been used n calculaton of the dffeence between the natonwde esult and patcula pecncts esults. In ode to mae the esults compaable, fo eve tested technque the sze of sample s one hunded pecncts. Fstl, t was decded to epementall chec how ted sample pocedue affects the effectveness of samplng technque compaed to smple andom samplng. Fo ths pupose, a followng smulaton was desgned: m ndependent samples wee dawn (samplng wthout eplacement, fom m samples the one whch had the least AMD value n the palamenta electon to the Sem 007 was chosen, 3 fo the chosen sample the AMD was calculated fo the pesdental electon 00, 4 ponts -3 wee epeated one thousand tmes fo fve dffeent m values. In table the esults of the above-mentoned smulaton ae pesented,.e. basc descptve statstcs fo the dstbuton of one thousand AMDs n elaton to WP0, fo dffeent values of m paamete. The fst case, n whch numbe of geneated samples equals one, s de facto smple andom samplng. AMDs fo ted sample pocedue and wth dffeent m values Table m Mn. st Q Medan Mean 3d Q Ma. 0,043 0,95 0,9 0,336 0,438,55 5 0,033 0,9 0,8 0,94 0,44 0, ,035 0,8 0,6 0,7 0,0 0, ,07 0,0 0,37 0,43 0,75 0, ,09 0,094 0,8 0,36 0,70 0,38 It easl to notce that wth nceasng numbe of geneated samples (m, out of whch the best sample n tems of WS07 s chosen, the bette samples ae obtaned n tems of WP0. Both aveage levels of AMDs and the dspeson of dstbuton ae educed. Ted sample pocedue s thus an effectve technque nceasng the qualt of the sample of pecncts. Moeove, the geat mpovement of the esults n elaton to SRS s obtaned ust afte 5 geneated samples and b nceasng m value to the level of 000 the mpovement s stll notceable but s not so sgnfcant. The numbe of commttees whch had canddates n the whole count n WS07 s seven whle the numbe of canddates n WP0 s ten. Howeve, the vast maot of votes was obtaned b thee pates/canddates (apat fom the fouth n the ode n WS07 Polsm Stonnctwem Ludowm (PSL wth the esult of 8,9%, commttees/canddates outsde top thee obtaned less than 3%

51 THE USEFULESS OF PAST DATA I SAMPLIG DESIG 5 of the votes. Tang that nto consdeaton, n the futhe analss the estmaton of the esults onl of the thee most popula canddates s unde focus. In table 3 the esults of the above descbed smulaton ae pesented, havng egad onl to the thee hghest esults. AMDs fo ted sample pocedue and wth dffeent m values, onl the thee hghest esults Table 3 m Mn. st Q Medan Mean 3d Q Ma. 0,053 0,54 0,835 0,95,65 3, ,08 0,7 0,45 0,47 0,635,50 0 0,0 0,54 0,389 0,47 0,537, ,06 0,00 0,35 0,335 0,438, ,0 0,95 0,30 0,39 0,43,067 In case of the 3 most popula canddates, the smla dependences est as n the case of all canddates,.e. the smalle AMD values fo the hghe numbe of geneated samples and the deceasng mpovement of effectveness. The futhe ncease of m n the above smulaton pocedue would sgnfcantl ncease the calculaton needs and smulaton eecuton tme, so, n ode to chec f futhe ncease of m value leads to the mpovement of the esults, the followng pocedue was poposed: ndependent samples wee dawn (samplng wthout eplacement; fo eve sample the AMD was calculated n WS07; 3 00 samples wth the smallest AMD wee chosen and fo all of the them the AMD n WP0 was calculated (table 4. Table 4 AMDs fo the best 00 samples n WS07 Mn. st Q Medan Mean 3d Q Ma. fo mn(ws ,040 0,3 0,344 0,37 0,53 0,830 0, ,06 0,09 0,34 0,350 0,466 0,806 0, ,060 0,95 0,30 0,334 0,434 0,846 0, ,060 0,93 0,307 0,34 0,43 0,97 0,060 Tang nto consdeaton aveage values and quatles, the esults ae gettng bette (onl the aveage fo s wose than pevous one, howeve the dffeences ae elatvel small. The computatonal capabltes of moden computes enables geneatng mllons of samples wthout an poblem, howeve, t seems that geneatng moe than ten thousand samples does not mae much sense. Ths follows fom the fact that ve good samples ma be

52 5 Aadusz Kozłows obtaned alead wth a seveal thousand daws, howeve, these samples ae not necessal the most epesentatve ones n the pevous electon. In the last column of table 4 the AMDs fo the best samples n WS07 ae pesented. In two cases out of fou the values ae hghe than the aveage. Based on that, the concluson ma be dawn that the choce of the best sample out of geneated not alwas s the best soluton. Ths stuaton s llustated b gaph and. 4 3 WP0 0 Gaph. AMDs fo samples WS07 As t s cleal shown n gaph, thee s a dependence between AMD n WS07 and AMD n WP0. Howeve, the close to the coodnate sstem s ogn, the weae the dependence. Theefoe, the samples whch ae the closest to X-as (AMD n WS07 ae not alwas the closest to Y-as (AMD n WP0. The smla thng ma be notced n gaph, whee the best fve hunded samples n WS07, soted n the non-deceasng ode, and the AMD n WP0 ae pesented. Wth the ncease of the numbe of geneated samples ( the bette samples n tems of the smlat to the geneal esults n WP07 ae obtaned, howeve, the smlat of those samples to the geneal esults n WP0 emans moe o less the same and the values ae pett much dspesed. Theefoe, t seems ustfed to ntoduce modfcaton to the ted sample pocedue whch would mean that not the best sample n tems of epesentatveness n the past electon would be chosen to the suve but one out of one hunded to fve hunded best samples would be selected. The moe samples ae geneated ( the moe ustfed the modfcaton seems to be. The ssue that needs to be futhe analzed s how ths one sample should be chosen. Pehaps, thee ae attbutes whch wll allow to sepaate samples whch wll eman epesentatve n the subsequent electons fom those whch

53 THE USEFULESS OF PAST DATA I SAMPLIG DESIG 53 epesentatveness wll sgnfcantl deteoate. The autho compaed tunout, the numbe of people enttled to vote and the vaablt of the esults of patcula commttees n the best samples, howeve, no sgnfcant dffeences wee dentfed AMD AMD sample sample AMD AMD sample sample sample Gaph. AMDs fo the best fve hunded samples (soted out of geneated ( - WP0, - WS07 Anothe method to ncease the epesentatveness of sample s applng statfed samplng. Based on the analss of the dffeentaton of esults n WS07, eght stata wee dentfed usng followng featues: tetoal dvson of the count (two vaants: noth-westen aea ncludng nne vovodeshps and south-easten aea ncludng seven vovodeshps and the tpe of subdvson (fou vaants: ctes muncpaltes wth the numbe of people enttled to vote ove eght thousand, towns, othe uban aea, ual aea. Allocaton popotonal to the aveage of elatve shae of two featues: numbe of people enttled to vote and numbe of pecncts n a statum was appled (table 5.

54 54 Aadusz Kozłows Allocaton of the sample Table 5 Ctes Towns Othe uban aea Rual aea oth-west South-east Subsequentl, the statfed samplng was compaed b smulaton wth unestcted samplng usng also the elaton to the past esults. Fo ths pupose ten thousand samples wee dawn n accodance wth eve scheme and descptve statstcs wth AMDs wee calculated n the wa t was done n the pevous smulatons (table 6. On aveage, thans to statfed samplng moe epesentatve samples wee obtaned compaed to unestcted samplng. The same advantage of statfed samplng occus when the elements of ted sample pocedue ae ntoduced,.e. out of the pevousl geneated ten thousand samples, the one hunded most epesentatve n WS07 ae chosen. The samples obtaned b applng ths method, wth gven, tuned out to be, on aveage, the most epesentatve ones. Compason of the esults of smple andom samplng smulaton and statfed samplng smulaton fo samples Table 6 Mn. st Q Medan Mean 3d Q Ma. All samples SRS 0,03 0,483 0,8 0,940,77 4,576 Statfed 0,036 0,47 0,783 0,859,80 3,08 The best 00 samples n tems of WS07 SRS 0,06 0,09 0,34 0,350 0,466 0,806 Statfed 0,073 0,07 0,9 0,30 0,40 0,860 The last aspect of et poll s an ntentonal omsson of the smallest pecncts n samplng pocedue. Such behavou fom the eseach nsttuton s pont of vew s acceptable due to fnancal benefts because whle suveng the smalle numbe of lage pecncts, the same sample of espondents ma be obtaned wth lowe costs. Obvousl, ntentonal ecluson of some of the unts out of the samplng populaton affects the estmates. Ths fault, howeve, can be elmnated f ted sample pocedue s appled, because the sample s chosen n such wa that t eflects the total esult, whch also ncludes the pecncts omtted n samplng. In table 7 the chaactestcs of the best one hunded (n tems of accuac n WS07 out of ten thousand geneated samples ae pesented, the numbe of samplng populaton was educed n the subsequent ows b the pecncts wth the numbe of people enttled to vote smalle than.

55 THE USEFULESS OF PAST DATA I SAMPLIG DESIG 55 AMDs fo the best 00 samples out of 0 000, wth the omsson of pecncts smalle than Table 7 Mn. st Q Medan Mean 3d Q Ma ,048 0,6 0,34 0,348 0,465 0, ,04 0,3 0,39 0,338 0,434 0, ,05 0,97 0,77 0,36 0,409 0, ,06 0,9 0,3 0,345 0,437 0, ,08 0,8 0,345 0,366 0,49 0, ,09 0,86 0,307 0,3 0,405 0, ,059 0,7 0,349 0,368 0,465 0, ,068 0,7 0,344 0,365 0,477, ,073 0,3 0,40 0,43 0,560, ,098 0,39 0,454 0,473 0,584, ,09 0,336 0,474 0,534 0,696, 000 0,070,090,86,69,487,834 The above esults show that n case of WP0 the omsson of pecncts smalle than s hunded to eght hunded people enttled to vote would not educe the epesentatveness of sample but, n fact, t would ncease t. Even lmtng the samplng populaton onl to pecncts lage than one thousand people enttled to vote would not nvolve the loss of qualt of the sample f the ted sample pocedue s appled. 5. Conclusons The detaled data about the past electons esults s a valuable souce of addtonal nfomaton enablng the mpovement of samplng n et poll. Based on the full esults of the pesdental electon 00 and the palamenta electon to the Sem 007 t was poven b means of smulaton epements that applng ted sample pocedue sgnfcantl mpoves the epesentatveness of the sample. Ths beneft nceases along wth the gowth of the numbe of geneated samples. Howeve, the moe samples ae geneated the moe advsable t s to modf the pocedue n a wa that nstead of choosng the best sample fom the past electon, the choce s made fom the fst seveal hunded samples. The method of ths choce eques futhe analss. The mpovement was obtaned b applng statfed samplng n whch the populaton was dvded to eght stata based on two vaables: geogaphc dvson and the tpe of tetoal unt. It was also poven that b applng ted sample pocedue the smallest pecncts ma be omtted n the suve, whch s benefcal fom fnancal and oganzatonal pont of vew and does not futhe affect the esults.

56 56 Aadusz Kozłows Acnowledgements The autho would le to than Pof. Mosław Szede fo the nspaton to conductng the eseach and valuable comments when wtng the atcle and Mosław Bogdanowcz fom Kaowe Buo Wbocze fo shang the data n the useful fomat. Refeences Andeenova, A., Moeno, A. (008 Usng et polls to do moe than poect outcomes: the ole and functons of et polls n advanced and new democaces. 3MC Confeence Poceedngs, Beln. Baeto, M.A., et al. (006 Contoveses n et poll. Poltcal Scence and Poltcs Vol. 39, o. 3, Bshop, G.F., Fshe, B.S. (995 Secet ballots and self-epots n an et-poll epement. Publc Opnon Quatel Vol. 59, o. 4, Busch, R.J., Lese, J.A. (985 Does tme of votng affect et poll esults? Publc Opnon Quatel Vol. 49, o., Cutce, J., Fth, D. (008 Et Pollng n a Cold Clmate: The BBC-ITV Epeence n Btan n 005 [wth Dscusson]. Jounal of the Roal Statstcal Socet. Sees A (Statstcs n Socet Vol. 7, o. 3, Fanovc, K.A. (99 Technolog and the Changng Landscape of Meda Polls, n: T.E. Mann, G.R. Oen (eds Meda Polls n Amecan Poltcs. Boongs Insttuton, Washngton, DC. Hlme, R. (008 Et polls a lot moe than ust a tool fo electon foecasts, n: M. Caballo, U. Helma (eds. Publc opnon pollng n a globalzed Wold. Spnge, Beln. Hofchte, J. (999 Et polls and electons campagns, n: B.I. ewman, (ed. Handboo of poltcal maetng. Thousand Oas, Sage Publcatons. Kloman, R. (976 What Tme Do People Vote? The Publc Opnon Quatel Vol. 40, o., Lens, J. (008 ew methodologcal Issues n conductng et polls. 3MC Confeence Poceedngs, Beln. Lev, M.R. (983 The methodolog and pefomance of electon da polls. Publc Opnon Quatel Vol. 47, o.,

57 THE USEFULESS OF PAST DATA I SAMPLIG DESIG 57 Mooe, D.W. (003 ew Et Poll Consotum Vndcaton fo Et Poll Invento Insde the polls, Gallup, ( Semou, S. (986 Do et polls nfluence votng behavo. Publc Opnon Quatel Vol. 50, o. 3, Szede, M. (007 O ol nfomac spoza pób w badanach sondażowch. Pzegląd Socologczn LVI/, Ustawa z dna 6 lpca 998., Odnaca wbocza do ad gmn, ad powatów semów woewództw. Dz. U. 998, n 95, poz. 60, at. 30. UŻYTECZOŚĆ DAYCH Z PRZESZŁOŚCI DLA PLAU LOSOWAIA W BADAIACH TYPU EXIT POLL Steszczene Głównm zadanem et poll est pedca wnu wboczego tuż po zamnęcu loal wboczch. e mne ważnm celem badana est oszacowane ozładów głosów w óżnch pzeoach społeczno-demogafcznch. Kluczową westą dla aośc tch oszacowań est wbó odpowedne pób obwodów głosowana. W atule poddane został analze altenatwne do losowana postego metod dobou pób obwodów. Główn nacs położono na wozstane powszechne dostępnch baz danch ze szczegółowm wnam pzeszłch wboów. Za pomocą espementów smulacnch ocenono efetwność techn powązana wbou nowe pób z pzeszłm wnam (ted sample pocedue oaz wsazano optmalne dla ne paamet, a taże zapoponowano pewną modfacę pocedu. alepsze wn uzsano dla losowana wastwowego z zastosowanem elementów pocedu ted sample. Wsazano ówneż możlwość educ osztów badana bez stat na efetwnośc popzez odpowedn dobó włączne dużch obwodów.

58 Jan Kubac Alna Jędzecza THE COMPARISO OF GEERALIZED VARIACE FUCTIO WITH OTHER METHODS OF PRECISIO ESTIMATIO FOR POLISH HOUSEHOLD BUDGET SURVEY. Intoducton In the ecent eas, the demand fo elable small aea estmates has sgnfcantl nceased all ove the wold. It s manl due to ts gowng use n fomulatng polces, the allocaton of govenment funds and n the egonal plannng. Ceatng elable estmates fo small aeas, whee sample sze s substantall lowe than fo the whole count (fo eample fo countes n Poland UTS4 s a geat challenge fo statstcans. The poblems can ase not onl wth the elable dect estmates fo small aeas but wth the assessment of the pecson as well. Even when admnstatve data ae avalable that can be used as aula nfomaton to ncease the pecson of dect estmates, thee mght not be the sample szes lage enough fo patcula unts to enable the elable estmaton of standad eos. In such a case, an appomate technque fo vaance estmaton called Genealzed Vaance Functon (GVF can be a good choce. In the Polsh lteatue thee ae elatvel few publcatons that dscuss GVF appoach n the contet of small aea estmaton. Howeve, dung the last two decades, some publcatons concenng estmaton fo countes wee pesented. Hee we can menton the papes b Bacha, Lednc and Weczoows (003, 004, Gołata (004, Kodos (004 and Kubac (999, 006 that summaze the esults of estmaton fo countes obtaned on the bass of Polsh Labo Foce Suve. Anothe mpotant wo n ths feld was the epot enttled Socal Ecluson and Integaton n Poland: An Indcatos-based Appoach pepaed fo UDP (006, whee data fom Polsh Household Budget Suve (also fo countes wee used. The concept and the esults pesented n the epot can be consdeed ponee n Polsh lteatue and ma be useful n pactce wheneve the estmates fo countes ae needed.

59 THE COMPARISO OF GEERALIZED VARIACE FUCTIO 59 Genealzed Vaance Functon appoach s wdel nown n the subect lteatue. Hee we can menton the papes of Cho et al. (00 and of Johnson and Kng (987, U.S. Depatment of Educaton, Offce of Educatonal Reseach and Impovement (995, and the tetboos b Loh (999 and Wolte (985. The man advantage of the method s ts smplct and the educton of publcaton costs (also n the small aea case, when the amount of estmates ma be lage. The appomatons of samplng eos ae smplfed fo a vaet of estmates that ae tpcall geneated fom a suve wth a lage numbe of vaables. Anothe advantage of GVF appoach s ts stablt that can educe the unelable values fo some cases (the method can poduce moe stable samplng eo estmates b aveagng ove tme and genealzng n some wa. In the pape we pesent n detal the esults of the applcaton of Genealzed Vaance Functon to the small aea estmaton n Poland, compang wth othe dect vaance estmaton methods.. Outlne of dect vaance estmaton methods appled n the pape The basc concept of the eplcaton appoach appled to the vaance estmaton s epeatable selecton of subsamples fom the whole sample. The vaance of a statstc based on the whole sample can be obtaned usng the vaablt of the estmates that ase fom the subsamples. Replcaton technques ae often used n applcatons concenng comple suve desgns, complcated estmatos and when comple weght stuctue s used. An mpotant advantage of eplcaton methods s the smplct. It s connected wth the smplct of pocedues whch can be appled to the estmaton of means, popotons, totals, coelatons and so on. Anothe advantage s the possblt of modfcaton of desgn weghts caused b nonesponse o post-statfcaton. Ths modfcaton can be easl appled compang wth some analtcal fomulas. Some dsadvantage of eplcaton technques s the computatonal complet, howeve n the case of moden compute sstems ths mpedment can be easl ovecome, especall fo data fles of medum sze, as n most sample suve cases. Thee s also a dsadvantage elated to nadequate selecton of subsamples fo comple samplng desgns. In such a case, when subsamples do not eflects the sample desgn, the esults obtaned n ths wa ma be seousl based. Replcaton technques can be mplemented usng the followng schema. At the fst step, subsamples selecton that coesponds to the sample allocaton among stata should be caed out. Often, as n the BRR and Fa method, thee should be two subsamples n each statum. Ths equement s

60 60 Jan Kubac, Alna Jędzecza not alwas met fo small aeas (n patcula fo countes, what maes the vaance estmaton moe dffcult. In such a case, othe estmaton technques should be used. At the second step, the eplcaton weghts ae computed, usng the methods smla to the ones used fo the whole sample. At the thd step, the estmaton fo the subsamples s conducted usng the same method as fo the whole sample. At the fnal step, the vaance fo the whole sample s computed, usng the whole sample and subsamples esults. The vaance of θˆ s computed usng the followng fomula: whee: θ Is the paamete of nteest, A V( ˆ θ c ˆ θ - ˆ α θ ( α θˆ s the paamete estmate θ fo the whole sample, θˆ a s the paamete estmate θ fo the eplcate a, A s the numbe of eplcates, c I s constant that depends on the eplcaton method. Values of the paamete c fo dffeent vaants of eplcaton methods ae summazed n the table. Replcaton methods and the value of paamete c Table Method Abbevaton Value of paamete c Balanced Repeated Replcaton BRR /A Fa s method FAY /(A(- Jacnfe JK Bootstap - /(A- Balanced Repeated Replcaton Balanced Repeated Replcaton s the method most often used fo multstage and mult-stata samplng desgns. In the smplest veson the whole populaton s dvded nto L stata, and fo each statum two subsamples ae chosen. Each eplcate half-sample estmate s fomed b selectng one of the two subsamples fom each statum based on a Hadamad mat and then usng onl the selected subsample to estmate the paamete of nteest. In the case of

61 THE COMPARISO OF GEERALIZED VARIACE FUCTIO 6 the estmaton of pe-capta means, the estmates of nomnato and denomnato ae calculated fst usng the followng fomula: a L [ P + ( P ] ( h ah whee : P ah Hadamad mat element dependent on eplcaton numbe and stata numbe. Elements of that mat tae values 0 o. L numbe of stata a et we calculate a (3 and fnall h a a ah A V ( ( a (4 A h Jacnfe method The samplng desgn n the case of Jacnfe method s dentcal to the BRR case. Hee the selecton of two subsamples wth eplacement fo each statum s assumed. The basc dffeence between BRR and JK s the method of fomng the eplcates, afte goupng the subsamples n pas. In the case of JK the weght of the fst subsample s doubled fo a patcula statum, and the weght of the second one s multpled b zeo whle the weghts of the othe elements ae not modfed. Ths pocess s epeated fo each stata. The fst and the second half-samples ae detemned b the sot ode n the data fle detemned b dentfe of a subsample. If thee ae L stata, L eplcates should be ceated. Smlal to BRR, f we cannot povde the estence of two halfsamples fo each statum, some othe pocedues ae equed. Fa s method The Fa s method s nown as a BRR vaant and has popetes smla to the Jacnfe method. The basc dea of the Fa s method s the modfcaton of suve weghts less than n BRR method, whee one half-sample s zeoweghted and the second half-sample has the weght. As a esult, the fst halfsample has the weght loweed b a facto, and the second half has the weght multpled b compensatng facto -. Fo eample, f 0.9, the weghts wll be loweed n the fst half-sample b 0 pecent and nceased b 0 pecent n second half-sample. The c facto estng n the fomula ( taes the fom /(A(-.

62 6 Jan Kubac, Alna Jędzecza Ovevew of bootstap method As the acnfe, the bootstap method s connected wth a boade ange of ssues than suve samplng. It s also appled n hpothess testng, and the constucton of classfcaton and egesson models. Ths method ma be pelmnal descbed assumng that a sample was dawn b means of a smple andom samplng wth eplacement. In such a case an ognal sample s teated as a populaton and the samples (called bootstap samples ae selected fom t. B ceatng subsamples fom the ognal sample usng the same samplng desgn we can epect them to mtate the popetes of the whole populaton. Such an appoach was fst devsed b Efon (979, but the detaled descpton of the method can be found n Efon and Tbshan (993. The bootstap method was appled n the pape because fo some samplng unts onl one statum was chosen and the samplng desgn was smla to the smple andom samplng. Fo lage unts, the BRR method was used, that natuall mmcs the ognal samplng desgn. Ths s vald n patcula fo lage ctes. In the smplest case detemnng the samplng vaance b means of the bootstap method can be descbed as follows: Daw ndependentl A 500 smple andom samples (wth eplacement fom the whole sample of sze n. Usng the selected bootstap samples, calculate the values of the paamete ˆ*a θ usng the same fomula as fo the whole sample. Detemne the estmate of the estmato vaance usng the followng fomula A a Vˆ * ( ˆ θ ˆ ( θ ˆ θ (5 A a Genealzed Vaance Functon appoach The method assumes that a smple model s constucted that allows us to detemne the pecson estmate on the bass of the paamete estmate tself. The pocedue can be descbed as follows. Usng eplcaton o anothe vaance estmaton method, the vaances of paametes, descbed tˆ tˆ tˆ,,..., as should be obtaned. Assumng v be the vaance of the estmato of the fom: v ˆ ˆ (6 V ( t / t CV (ˆ t one can popose a model, that lns the values of v and. In most tˆ cases the model s as follows: β v α + (7 tˆ

63 THE COMPARISO OF GEERALIZED VARIACE FUCTIO 63 Usng well nown egesson technques, the egesson coeffcents α and β can be estmated. The last step s to detemne the standad estmaton eo on the bass of egesson equaton. 3. Results of Compason of Genealzed Vaance Functon wth othe Vaance Estmaton Methods As t has been seen n the table, the models obtaned fo dffeent ncome vaables have elatvel good statstcal popetes, howeve some of them fts the data slghtl less. Ths s patculal due to outle estence (as n the avalable ncome case see table 3 and due to the natue of vaables that descbe ae attbutes (as fo othe ncome. Table 3 ncludes GVF estmates fo a model wthout constant. Such a model was chosen because of some dffcultes wth fttng the values usng the model wth constant. In such a case GFV values fo hghe dect estmate values ae oveestmated, and sometmes the dependence has evesed chaacte (fo lowe estmate values, lowe GVF values ae obtaned, what s undesable. Table Summa of GVF models fo dffeent ncome vaables on the bass of count estmates fom Polsh Household Budget Suve Vaable R-squae F-statstc Avalable ncome 0,77 33,34 Income fom hed wo 0,554 43,3 Income fom self-emploment 0,343 74,0 Income fom socal secut benefts 0,437 69,76 Retement pas 0, ,07 Pensons esultng fom nablt to wo 0,67 560,4 Faml pensons 0, ,50 Income fom othe assstance benefts 0,53 367,3 Unemploment benefts 0, ,08 Othe ncome 0,47 4,7 Othe ncome of whch gfts 0,59,63 The vaance estmates obtaned b means of GVF ae usuall geate than the coespondng eplcate estmates and also n most cases geate than the bootstap estmates. Howeve, loong at the eample pesented n the table 3 one can easl notce that thee ae some dect values that ae geate than the coespondng GVF estmates. Ths confms the stablt of GVF estmates whch ma be patculal useful when vaance estmates fo small aea models ae needed. Please note, that fo some countes, thee ae no values of vaance

64 64 Jan Kubac, Alna Jędzecza estmates obtaned usng vaous eplcaton technques. It comes b the fact, that n some stata thee s onl one PSU. Analzng n detal the esults pesented n the table 3, one can also obseve sgnfcantl lowe CV values fo eplcaton and bootstap methods that wee obtaned fo Wasaw (ge hghlghted. Ths obsevaton suggests applng Geneal Vaance Functon fo lage ctes sepaatel fom the emanng pat of the sample. Table 3 Estmated values of CV fo avalable ncome b countes n mazowece egon usng Balanced Repeated Replcaton (BRR, Jacnfe (JK, Fa, Bootstap and GVF Count Dect avalable ncome estmate BRR Jac nfe (JK Coeffcent of vaaton Fa Bootstap Bootstap wth deff Bałobzes 389, ,0 36,34 4,98 Cechanows 69,0 5,34 0,37 5,68 6,64 0,39,88 Gawolńs 494,7 5,7 5,7 5,65 5,84 9,5 3,9 Gostnńs 504, ,83 8,53 3,6 Godzs 785,4,65,94,59,76 8,4 0,55 Góec 769,03-35,67 7,07 0,7 6,09 0,66 Legonows 63, ,7,47 8,67 Lps 506, ,99 3,88 3,4 Łosc 56, ,37 30,33 3,0 Maows 490,83,43 6,9 8,8 9,38 4,69 3,34 Mńs 636,7-8,00 6,43 7,6,93,7 Mławs 56,8 5,86 6,84 5,56 8,,7 3,0 owodwos 635,86 7,45 3,08, 5,9 8,9,7 Ostows 55, ,5 9,9,57 Otwoc 690,89 7,47 4,83 5,94 4,04 6,3,5 Paseczńs 835,07 0,87,34,3 8,0,68 0,3 Płoc 543,07 7,4 4,03 8,89 7,78,9,68 Płońs 53,08 8,99 8,89 8,99 9,49 4,86 3,05 Puszows 875,50 7,49 5,9 6,54 5,5 8,6 9,99 Pzsus 499, ,50,7 3,3 Pułtus 59, ,85,70,84 Radoms 49,58 5,7 5,47 3,57 4,5 6,49 3,3 Sedlec 536, ,,7,76 Sepec 58, ,5 8,03,86 Sochaczews 756,45 4,37 5,8,43,5 8,0 0,75 Soołows 45, ,7 3,06 3,89 Szdłowec 550,45 4,30 4,60 6,5 5,57 8,7,60 Waszaws 3,6,94 3,0,99,8,84 8,6 Waszaws zachodn 356, ,4 3,70 8,0 Węgows 66,87 3,04 3,0 3,04,78 0,0,90 Wołomńs 694,99 7,78 6,7 5,98 7,0,3, wszows 555, ,8 6,,54 Zwoleńs 7, ,04 5,,00 Żuomńs 480, ,8,0 3,48 Żadows 733,43,0,4,53 4,58 7,8 0,9 m. Ostołęa 74, ,93,4 0,85 m. Płoc 77, ,9 0,83 0,64 m. Radom 68,53,6,75,8 3,79 5,94,79 m. Sedlce 77,0 3,80 8,79 8,9 5,90 9,3 0,65 GVF

65 THE COMPARISO OF GEERALIZED VARIACE FUCTIO 65 It s woth mentonng that the ntal values fo GVF models wee combned usng eplcaton estmates (BRR and smple bootstap estmates (wthout deff coecton. Ths was done manl because of the fact that fo small countes smple bootstap eflects onl the vaablt due to smple andom samplng and no statfcaton s avalable. Such an appoach ma also be vald when the dstbutons of eplcate and bootstap vaance estmates ae compaed (see fg.. Hee, fo man cases, the eplcaton estmates ae lowe than the coespondng bootstap estmates, what ma ndcate that usng such an appoach one can avod the undeestmaton of vaances. It can also be notced that the fequenc at the modal value of CV dstbuton obtaned b means of the bootstap method s vsbl hghe than fo the othe eplcaton technques. At the same tme the bootstap CV dstbuton seems hghl concentated aound ts mean. On the othe hand, the CV dstbutons obtaned usng BRR o Jacnfe methods pesent moe flat pattens manl due to moe fequent outlng values. That ma eplan that egulat that medan CV value obtaned usng bootstap was slghtl lowe than the medan obtaned b the othe eplcaton technques (see fg.. evetheless, the values of medan fo both: bootstap and the othe methods do not dffe much, what ma ndcate that thee s no undeestmaton usng smple bootstap n combned estmates. In the case of evaluatng the qualt of estmaton, the queston about the bas of estmates fo small aeas should also be consdeed. Below, the model fo ncome fom hed wo obtaned fo countes usng the data fom HBS suve and Polsh Ta Regste (POLTAX s pesented gaphcall. Constuctng the model some ncome-elated vaables wee nvolved. (see fg. 3. Fg.. Dstbuton of CV of avalable ncome estmatos

66 66 Jan Kubac, Alna Jędzecza Fg.. Dstbuton of estmato coeffcent of vaance obtaned usng eplcaton technques, bootstap technques and GVF fo avalable ncome fom Polsh HBS n 003 The POLTAX vaables contan, among othes, ncome fom hed wo that s defned smlal to the coespondng HBS vaable. The model was obtaned usng the data fom household budget suve and selected POLTAX vaables, that s the vaables obtaned b aggegaton of selected POLTAX ncome vaables and dvdng them b populaton sze fo countes. The model pesented n the fg. 3 was estmated usng SAE pacage fo R-poect envonment and EBLUP technques usng REML method. Moe detals of ths method can be found at Kubac, Gancow and Jędzecza (009 pape. Fg. 3. Model of ncome fom hed wo fo countes obtaned usng dect estmates fom HBS and vaous POLTAX ncome-elated vaables computed wth applcaton of EBLUP

67 THE COMPARISO OF GEERALIZED VARIACE FUCTIO 67 Gaphcal eamnaton of the esults eveals that the estmates obtaned b means of the methods pesented above pesent elatvel good consstenc wth the populaton data and ae not seousl based. Ths ma ndcate that the method ma also be used n the futue. 4. Conclusons The esults pesented n the pape confm that t s possble to obtan elable ncome-elated estmates fo countes, usng the data fom Household Budget Suve. The Genealzed Vaance Functon appoach allows us not onl to obtan elatvel stable estmates of estmaton eos but also maes t possble to povde the estmaton fo countes that ae not pesent n HBS sample and can onl be teated usng small aea models. Howeve, a specal cae should be taen, when the analss s pefomed fo moe detaled vaables. It would theefoe be advsable to use a combned method of vaance estmaton n that case. Futhe eamnaton of othe dect methods of pecson estmaton, as Jacnfe Repeated Replcaton o Samplng wth ove-eplacement, usng as an nput the vaances that ase fom GVF models, ma be consdeed n the futue studes. Also a moe detaled dscusson of small aea models usng pecson values obtaned b means of GVF technque should be conducted n the futue. Refeences Bacha C., Lednc B., Weczoows R. (003 Estmaton of Data fom the Polsh Labou Foce Suves b povats (countes n (n Polsh. GUS, Waszawa. Bacha C., Lednc B., Weczoows R. (004 Applcaton of comple estmaton methods to the dsaggegaton of data fom Polsh Labou Foce Suve n 003 (n Polsh. GUS, Waszawa. Cho, M.J., Eltnge, J.L., Geshunsaa, J., Huff, L. (00 Evaluaton of Genealzed Vaance Functon Estmatos fo the U.S. Cuent Emploment Suve. Poceedngs of the Amecan Statstcal Assocaton, Suve Reseach Methods Secton, Efon, B., (979 Bootstap methods: anothe loo at the acnfe. Annals of Statstcs 7, -6. Efon, B. and Tbshan, R.J. (993 An Intoducton to the Bootstap. Chapman & Hall, ew Yo.

68 68 Jan Kubac, Alna Jędzecza Gołata, E. (004a Poblem of Estmatng Unemploment fo Small Domans n Poland. Statstcs n Tanston 6, Johnson, E.G., and Kng, B.F. (987 Genealzed vaance functons fo a comple sample suve. Jounal of Offcal Statstcs 3, Kodos, J. (005 Some Aspects of Small Aea Statstcs and Data Qualt. Statstcs n Tanston Vol. 7, o., Kubac, J. (999 Evaluaton of Some Small Aea Methods fo Polsh Labou Foce Suve n one Regon of Poland. Poceedngs of the IASS Satellte Confeence on Small Aea Estmaton, Rga, Latva, Kubac, J. (006 Remas on the Polsh LFS and Populaton Census Data fo Unemploment Estmaton b Count. Statstcs n Tanston Vol. 7, o. 4, Kubac J., Gancow B., Jędzecza A. (009 An eample of empcal best lnea unbased pedcto (EBLUP applcaton fo small aea estmaton n Polsh Household Budget Suve, 9-0, Poceedngs fom confeence Suve Samplng n Economc and Socal Reseach oganzed b Unvest of Economcs n Katowce n 008. Loh, S. (999 Samplng: Desgn and Analss. Dubu Pess. Rao, P.S.R.S. (00 Samplng Methodologes wth Applcatons. Chapman and Hall/CRC. UDP Poland (006 Socal Ecluson and Integaton n Poland: An Indcatos-based Appoach. Wasaw. U.S. Depatment of Educaton, Offce of Educatonal Reseach and Impovement (995 Desgn Effects and Genealzed Vaance Functons fo the Schools and Staffng Suve (SASS, Febua 995. Wolte, K.M. (985 Intoducton to Vaance Estmaton. Spnge-Velag, ew Yo. PORÓWAIE METODY UOGÓLIOEJ FUKCJI WARIACYJEJ Z IYMI METODAMI SZACOWAIA PRECYZJI DLA BADAIA BUDŻETÓW GOSPODARSTW DOMOWYCH Steszczene Waanca lub współcznn zmennośc estmatoa mogą bć pzedstawone ao funce ego watośc oczewane z użcem zależnośc znane

69 THE COMPARISO OF GEERALIZED VARIACE FUCTIO 69 ao uogólnona funca waancna. W atule pzedstawono ezultat oblczeń dla szacunów pecz obemuącch óżne ategoe dochodu wznaczone dla powatów. Podstawą oblczeń bł dane pochodzące z Badana Budżetów Gospodastw Domowch. Rezultat otzmane metodą uogólnone func waancne został poównane z nnm uposzczonm metodam szacowana waanc. Jao punt wśca pzęto metodę szacowana waanc z użcem zównoważonch półpób eplacnch (tzw. BRR oaz metodę bootstapową, gd zastosowane metod BRR bło nemożlwe. W celu oeślena modelu dla uogólnone func waancne użto func hpebolczne. Oblczena pzepowadzono, stosuąc pogam WesVAR oaz SPSS, a ówneż własne pocedu oblczenowe pzgotowane dla paetu R-poect. Oszacowano ówneż zgodność szacunów dla powatów z użcem model dla małch obszaów oaz danch admnstacnch.

70 Doota Raczewcz SOME ASPECTS OF POST EUMERATIO SURVEYS I POPULATIO CESUSES I POLAD AD GERMAY. Intoducton Populaton census s valuable data souce of populaton and ts stuctue. Although as eve statstcal suve, a census s not pefect qualt and s based of eos. Eos n the census esults ae classfed nto two geneal categoes: coveage eos and content eos. Coveage eos ae the eos that ase due to omssons o duplcatons of pesons o housng unts n the census enumeaton. Content eos ae eos that ase n the ncoect epotng o ecodng of the chaactestcs of pesons, households and housng unts enumeated n the census. Both nds of eos can be assess b post enumeaton suve. Populaton censuses n Poland and Geman conducted n ths ea have the smla methodolog, both censuses ae egste based and andom sample suveed. Also both countes conducted post enumeaton suves seveal wees afte the man census. Thee ae ve detaled ecommendaton how to ca out the populaton census and the post enumeaton suves, poposed b EUROSTAT and U. Poland and Geman compl wth these ecommendaton n dffeent degee. The am of ths pape s to compae post enumeaton suves and populaton censuses n Poland and Geman, to pesent poblem wth data qualt and potental eos of censuses and to pont at ntenatonal ecommendaton on qualt assessment of populaton censuses accodng to the U and EUROSTAT.. Compason of populaton censuses n Poland and Geman The populaton census povdes the most detaled nfomaton about populaton and ts tetoal dstbuton, demogaphc stuctue, socal and pofessonal, as well as soco-economc chaactestcs of households and famles,

71 SOME ASPECTS OF POST EUMERATIO SURVEYS I POPULATIO CESUSES 7 and the esouces, and housng condtons at all levels of tetoal dvson of the count: natonal, egonal and local level. Populaton censuses ae conducted n Poland and n the wold about eve ten eas. Populaton censuses ae the lagest statstcal poects n all countes. In ths ea populaton censuses wee conducted both n Poland and n Geman. In both countes these censuses has the specal capact. In Poland because t s the fst census snce Poland became an EU membe state. Ths mples a sees of commtments, ncludng need to povde nfomaton n the socal-demogaphc and soco-economc, to the etent and tme lmts set b the Euopean Commsson. In Geman because ths s the fst populaton census afte the Beln Wall fell and unfcaton of two pats of Geman, man people moved fom east to west and Euopean ntegaton has pogessed apdl. In the fome teto of the Fedeal Republc of Geman, the last complete enumeaton was held n 987, n the GDR a populaton census was taen n 98. Eve snce, the offcal numbe of populaton has been detemned usng a statstcal method called ntecensal populaton updates. In both countes populaton censuses wee pefomed a egste-based censuses. owadas because of hgh technologcal pogess a lot of data ae stoed n man admnstatve egstes, data bases and state epots of some nsttuton. These souces of data contan compaable nfomaton. In both censuses thee ae collected nfomaton on buldngs and housng, on households and on people. In Geman thee ae two suves: the Census of Buldngs and Housng and the Household Suves. In the Census of Buldngs and Housng, all 7.5 mllon ownes of esdental popet eceve a questonnae b post b the efeence date. The questonnae wll equest nfomaton on esdental buldngs and dwellngs, such as the ea of constucton, tpe of buldng, equpment, floo space and numbe of ooms. In the Household Suves n Geman, 0 pecent of the populaton wee ntevewed b ntevewes on o afte 9 Ma 0. The espondents n Geman wee selected andoml. In Poland 0 pecent wee also selected andoml, usng statfed samplng, stata wee ceated on the bass of nfomaton fom admnstatve egstes and state tetoal egste (TERYT. In both censuses people wee equested to povde nfomaton also on the educaton and tanng, emploment and mgaton. Both countes used a med-mode method fo data collecton fom multple souces and combnes a complete enumeaton wth sample suves. Intevew b: CAPI Compute Asssted Pesonal Intevew; Self egstaton b Intenet: CAII Compute Asssted Intenet Intevew; CATI Compute Asssted Telephone Intevew. Both the Cental Statstcal Offce n Poland and the Fedeal Statstcal Offce n Geman povde ndvdual data potecton.

72 7 Doota Raczewcz 3. Data qualt and potental eos n populaton censuses Populaton census such as othe statstcal suve s not pefect qualt and eos can and do occu at all stages of the census opeaton. Statstcal data ae hgh qualt f the ae accessble and clafed, elevant, on tme and punctual, compaable, coheent and accuate. Accessblt and clat means that data ae easl accessble and avalable n foms sutable to the uses, uses now how to get the data, and the data povde help uses n ntepetng the esults and gves data n pope fomats etc. Relevance s the degee to whch statstcs meet cuent and potental uses needs. Tmelness and punctualt means that data ae avalable n shot tme. Compaablt s then t s possbl to compae data between countes and dffeent tme of peods. To ensue compaablt of the esults at Euopean level, all membe states have to povde nfomaton on a specfed ange of vaables wth unfed defntons and classfcatons. Coheence s when data ae adequate to be elabl combned n dffeent was and fo vaous uses. Accuac s the most mpotant featue of statstcal data and epesses the closeness of the tue value. Tue value, whch n pactce s not nown, s a value that s obtaned that, f the data wee collected and analsed wthout an eos fo all unts of the taget populaton. The data would be accuate f a full-scale suve wee pefectl conducted and all elevant condtons of the suve wee pefect. The pefect condtons depend on specfcaton of the stud obect, soco-poltcal stuaton of the count, methods of data collecton, used statstcal pocedues, ncludng methods of estmaton, qualfcatons of pesonnel nvolved n the stud (ntevewes, contolles, people that codng and nputtng data, eseach peod, teatment of the questons and eplanatons, used defntons etc.. The data accuac depends on samplng and non-samplng eos. Pevous censuses wee full scale suves wth no pobablt mechansm, full populaton was ntevewed, no sample was selected, so no samplng eos aose. The stuaton s dffeent n ths ea censuses n Poland and Geman, whee espectvel 0 and 0 pecent samples wee dawn fom populaton. And samplng eos must be taen nto consdeaton. In addtonal, t appea the poblem wth combnng data fom two souces: suve data and egste data. Regstes can be also naccuate and the ae a souce of some nd of eos. on-samplng eos occu n eve suve, both census and sample suve and the can ase n patcula stages of the suve and census, begnnng fom plannng the suve and endng on mang the suve esults avalable to uses. Thee ae two nds of non-samplng eos: completeness eos and content eos. The completeness eos ae coveage eos (omsson,

73 SOME ASPECTS OF POST EUMERATIO SURVEYS I POPULATIO CESUSES 73 duplcaton and eoneous ncluson of unt and non esponse (not locaton, absence at home, no contact, efusals, lost questonnaes, questonnaes dscaded dung the nspecton. Eos of contents ae esponse eos (caused b espondent, ntevewe, dung ecodng and copng data, data pocessng eos (dung codng, nput and edtng, table ceaton and calculatons, eos of analss and pesentaton of esults (ncoect methods of analss, msntepetaton, eoneous statstcal nfeence, ncoect pesentaton of the esults, (Kodos, Post enumeaton suves n populaton censuses n Poland and Geman Thee ae a lot of methods to assess the qualt of populaton census: Qualt contol technques such as ntenal consstenc checs; Compasons of esults wth othe data souces ncludng pevous censuses, cuent household suves, and/o admnstatve ecods; Recod-checng, n whch ndvdual census ecods ae matched aganst altenatve souces and specfc data tems ae checed fo accuac; Some evaluatons analze, ntepet, and snthesze the effectveness of census components and the mpact on data qualt o census coveage; Post-enumeaton suves ae used to estmate census coveage eo; Post-census suves desgned to measue content eo ae usuall nown as e-ntevew suves; Ethnogaphc and socal netwo methods povde a wa to stud the effects of moblt on census coveage o to measue census coveage of specfc subpopulatons. In ths pape specal attenton s pad to post-enumeaton suve, conducted afte 0 populaton census both n Poland and Geman. In the post-enumeaton suve n 0 populaton census n Geman, 5% of all patcpants of the suve households ae ased a few wees afte the fst ntevew a second tme b othe ntevewes then n the household suve. Such checs on suve esults ae ntenatonall accepted and allow an assessment of the qualt of the esults fom the suve households. In pncple, agan the same ssues as the suve households ae made, but thee ae less questons. The suve follows the same pocedue as the households suve, onl wth dffeent ntevewes. It s also esponsble fo ths suve n some povnces opposed to collecton agenc, but the State Statstcal Offce. Even wth follow-up suve, we natuall offe the oppotunt to complete ethe the questonnae alone and etuned to the State Statstcal Offce o the Desgnated collecton agenc o to epot the nfomaton onlne. Fo easons of data potecton, a tansfe of the questonnae b e-mal s not allowed.

74 74 Doota Raczewcz In the post-enumeaton suve n the 0 populaton census n Poland, 5% of patcpants ownng phones of the 0% suve sample wee ased onl b phone agan a few wees afte the fst ntevew. Samplng was statfed as n the fst ntevew. The esults of ths post enumeaton suve have been not et publshed, moeove also the esults of the post-enumeaton suve n the 00 populaton census n Poland have been not et publshed. It was conducted thee wees afte the man census. A pma samplng unt was Census Enumeaton Aea (CEA. Out of all 77,59 CEAs fo PCES 903 CEAs wee selected usng statfed samplng desgn b egon wth popotonal allocaton. Altogethe 60,09 dwellngs wee selected. 7 census tems wee checed. Fst post-enumeaton suve n the populaton census n Poland on samplng bass was appled n 978 fo the 978 Populaton Census (Zasępa, 993. Qute easonable sze of samples and samplng desgns wee also used fo post-enumeaton suve n the 988 populaton census n Poland (owa, 998 and n the Mco-census 995 (GUS, 996; Szabłows et al, Some ntenatonal ecommendaton on qualt assessment of populaton census The ecommendatons of EUROSTAT and Unted atons on qualt assessment of populaton census ae (EUROSTAT, 006, 007, 009; U, 006, 00: An evaluaton/assessment of undecoveage and ovecoveage. A descpton of methods used to coect fo undecoveage and ovecoveage. An evaluaton/assessment of measuement and classfcaton eos. An evaluaton/assessment of pocessng eos, especall whee manual codng of data n fee tet fomat s used. Fo the post-enumeaton suve to be useful n measung coveage and content eos, t must be well planned and mplemented. It s suggested (Kodos, 007, that effots should be made to: evaluate the PES caed out afte the 00 Populaton Census and Housng as a fst step fo the net census pepaaton; develop good aea fames, wth well-defned and mutuall eclusve enumeaton aeas; desgn plausble pobablt samples to facltate obectve genealzaton of PES esults to elevant domans; consde applcaton of dual estmaton sstem; pepae a pogamme fo checng qualt of egstes f the ae to be used n the census opeaton; consde applcaton of small aea estmaton methods; adopt effcent but ealstc matchng ules;

75 SOME ASPECTS OF POST EUMERATIO SURVEYS I POPULATIO CESUSES 75 hamonze defntons and concepts used n both the census and the PES; ensue that tems ncluded n the PES fo matchng puposes ae elevant and useful; nvolve well-taned and qualfed feld staff; tan e staff, nvolved n the desgn of PES samples, n suve samplng methods; ca out pe-tests fo the PES pocess and feld econclaton; allocate adequate funds to the PES wthn the famewo of the census; eep the PES as smple as possble and stc to obectves that ae attanable; publsh all methodolog of the PES. 6. Concludng emas Dung last ten eas the Cental Statstcal Offce of Poland mpoved qualt of statstcal data evdentl. A numbe of suves ae ve well pepaed and mplemented. Impovement ma be seen n statstcal publcatons and analss. Qualt Repots fo Euostat ae pepaed popel. Howeve, some pats of qualt epots ma be publshed n offcal statstcal ounal, such as Statstcs n Tanston new sees o Wadomosc Statstczne. Post-enumeaton suves ae woth conductng f the ae caefull planned and functon wthn opeatonal and statstcal constants. Coopeaton of the dffeent nd of epets nvolved n pepaaton, mplementaton, pocessng and publcaton of populaton census s ve mpotant fo the qualt of census esults. Whle ndependence between the census and the PES s a fundamental equement, n pactce opeatonal ndependence seems to suffce because t s not possble to mae all the vaous aspects of the census and PES opeatons mutuall eclusve. Methodolog fo the census pepaaton and the data assessment esults should be publshed. Refeences EUROSTAT (007 Handboo on Data Qualt Assessment: Methods and Tools. EUROSTAT (009 Handboo fo Qualt Repots. EUROSTAT (009 Standad fo Qualt Repot. Wong Goup on Assessment of Qualt n Statstcs, Luemboug, Apl 4-5. GUS (998 Methodolog and Ogansaton of Mcocensuses (n Polsh. Statsta w patce, Waszawa. Kodos, J. (988 Qualt of Statstcal Data (n Polsh. PWE, Wasaw.

76 76 Doota Raczewcz Kodos, J. (007 Some Aspects of Post-Enumeaton Suves n Poland. Statstcs n Tanston new sees, Decembe 007, Vol. 8, o. 3, owa, L. (998 Qualt of Census Data, In: Tendences of Changes n Stuctue of Populaton, Households and Famles n (n Polsh. GUS, Wasaw, -3. Szabłows, J., Wesołows, J. and Weczoows, R. (996 Inde of Fttng as a Measue of Data Qualt on bass of the Post-enumeaton Suve of Mcocensus 995 (In Polsh. Wadomośc Statstczne o. 4, U (006 Recommendatons fo the 00 Censuses of Populaton and Housng, n coopeaton wth EUROSTAT. ECE, ew Yo, Geneva. U (00 Post Enumeaton Suves, Opeatonal gudelnes, Techncal Repot. ew Yo. Zasępa, R. (993 Use of Samplng Methods n Populaton Censuses n Poland. Statstcs n Tanston Vol., o., 6. WYBRAE ASPEKTY BADAŃ KOTROLYCH DO SPISÓW LUDOŚCI W POLSCE I IEMCZECH Steszczene Opacowane dotcz aośc danch potencalnch błędów, ae poawaą sę w spsach ludnośc. Doonano w nm taże poównana spsów ludnośc w Polsce emczech, a taże zasad metod pzepowadzena badań ontolnch do spsów ludnośc w tch dwóch aach. Omówono taże mędz-naodowe zalecena w spawe ocen aośc spsów ludnośc, pzgotowane pzez OZ Euostat.

77 Ondře Vlus OPTIMIZATIO OF SAMPLE SIZE AD UMBER OF TASKS PER RESPODET I COJOIT STUDIES USIG SIMULATED DATASETS. Intoducton Because of the ncease n computng speed and avalablt of eas-to-use commecal softwae fo estmatng heachcal Baesan (HB models applcaton of these methods became a standad n analzng choce-based conont data. Toda maet eseaches can estmate models of complet that was not attanable o would eque geat amount of esouces few decades ago. On the othe hand t mght mae man pacttones foget that even the capabltes of HB models ae not unlmted and that thee ae man factos that have a negatve mpact on the success of the stud. Ths atcle emphaszes the need to caefull desgn the stud wth espect to the sample sze and numbe of tass shown to the espondents used and ecommends checng suffcenc of the setup wth use of smulated datasets. These should eflect possble scenaos and mpact of vaous factos.. Conont analss and use of Baesan models Eal applcatons of conont analss n maetng eseach snce (Geen and Rao, 97 wee based on atng seveal full-pofle concepts. These wee based on the othogonal desgn to allow fo modelng each espondent s pefeences. Fo studes whee lage numbe of attbutes needed to be studed, patal pofle methods based on tade-off matces o hbd methods such as ACA (Johnson, 987 combnng patal-pofle tass wth self-eplanato secton have been developed. Whle these appoaches wee found useful n undestandng consumes pefeences, Louvee and Woodwoth (983 commented on the low

78 78 Ondře Vlus applcablt fo foecastng choces n compettve stuatons and suggested dffeent appoach. Ths appoach was based on multple choce models whose fundamentals wee lad eale b McFadden (974. Dscete choce tass wee much moe ealstc fo the espondents than anng o atng a set of full-pofle concepts. And too was the dscete choce model moe appopate fo pedcton of consumes demand and ceatng maet smulatos. Unfotunatel, the benefts of the choce-based conont ddn t come fo fee. The nfomaton povded b the espondent n the choce tass s onl of odnal tpe and s collected n a qute neffcent wa. The espondent needs to udge seveal concepts befoe mang hs choce, whch taes moe tme and effot. Wth lttle nfomaton collected pe espondent t became mpossble to estmate the choce model fo each ndvdual and the onl wa to analze the data was use of aggegate models. Aggegate models ae facng seous poblems when appled on the eallfe data. Most of the ssues stem fom the heteogenet of the populaton n tems of the pefeences whch maes pedctons based on such models spuous. Thee wee seveal appoaches used to ovecome these obstacles such as use of segmentaton of espondents o latent-class models (see Vens, Wedel (996 fo an ovevew. But t wasn t befoe the ntoducton of heachcal Baesan models snce papes of Allenb et al. (995 and Len et al (996 when the choce-cased conont became the most popula method. HB methods allow fo estmatng easonabl accuate ndvdual level pefeences b modelng heteogenet of the populaton as f the ndvdual pat-woths wee sampled fom a common dstbuton. Paametes of ths dstbuton ae estmated smultaneousl wth the ndvdual level paametes. Whle ths appoach povdes valuable mpovements of the pedcton accuac of ou models t maes assessment of epected model accuac befoe the eseach s actuall done moe complcated. 3. Sample sze detemnaton Sample sze and numbe of tass shown to each espondent (dectl affectng the length of the questonnae ae not onl the e factos of conont model accuac but also mao dves of cost of the stud. It s theefoe necessa to be able to come up wth easonable estmates of these two paametes befoe the stud s done. Snce equements egadng the numbe of attbutes to be tested and numbe of levels fo each attbute ae often subect of dscussons wth the clent t s also of geat mpotance to be able to assess the mpact of these changes on ethe the sample sze needed o on the epected accuac of the model.

79 OPTIMIZATIO OF SAMPLE SIZE 79 Unfotunatel thee s no magc fomula that would gve us accuate estmaton of what sample sze wll be needed to fulfll the goals of the stud wth hgh degee of confdence gven the paametes of the poblem. Accodng to Tang (006 sample sze ecommendatons ae mostl based on two followng appoaches: elng on past epeence wth smla studes and geneal ules of the thumb o geneatng snthetc datasets and checng fo sample eos of ou pat-woth estmates. The pobabl most nown ule of a thumb to estmate necessa sample sze fo a choce-based conont stud (Ome, 998 assumes that: havng espondents complete moe tass s appomatel as good as havng moe espondents, wth nceasng numbe of attbutes numbe of paametes to be estmated gows but nfomaton that s ganed n each tas gows at the same ate. Based on these assumptons the sample sze n should satsf nequalt c n 500, ( MT whee M s the numbe of altenatves pe choce tas, T s the numbe of tass to be shown n each tas and c s a constant epesentng complet of the setup (mamum numbe of levels pe attbute. Tang (006 suggests moe geneal veson of the heustc, namel p(- d n I, ( MT whee d s the pecentage of cases whee espondents choose a none altenatve, I s the nde epesentng heteogenet of the sample and p s the numbe of paametes to be estmated pe espondent. Howeve, none of the two heustcs mentoned tae nto consdeaton what the goals of the stud ae and what s the accuac needed to acheve them. 4. Compason wth esults based on smulated datasets To test fo suffcenc of gven sample sze to each the goals set I have desgned smulato geneatng pat-woth utltes fo atfcal populaton chaactezed b followng lst of paametes: attbute mpotance vaablt (v, numbe of clustes (C, cluste level vaablt (α, ndvdual level vaablt (β, level of nose n decsons (λ, mnmum/mamum eecton ate b none opton ( mn / ma, shae of andoml answeng espondents (e.

80 80 Ondře Vlus Conta to the tadtonall used geneatos based on mtues of multvaate nomal dstbutons of pat-woths such as Vens et al. (996 o moe ecentl Wth (00, sepaate pat-woth fo a cente, cluste cente and each ndvdual wee geneated, then aveaged wth espect to the weghts α and β and n the fnal step escaled so that the mpotance of each attbute s matchng the geneated mpotance fo each ndvdual. Known, unde contol Unnown Suve paametes Populaton paametes Model accuac Conont tass Atfcal espondents Estmated paametes Eta tass Smulated esponse data Gaph. Smulated datasets appoach scheme Geneated pat-woths seved n the assessment of model accuac that was done accodng to the scheme depcted n gaph. Smulated esponse data wee used fo choce model estmaton followed b compason of eal and estmated pefeence shaes fo 00 eta tass. In the eample, mean absolute eo of the pedcted shae Pˆ fo K holdout tass m K Pˆ m Pm MAE K, (3 was used as a measue of accuac. In total 635 models pe scenao wth sample sze angng fom 5 to,600 and wth 5 to 5 tass pe espondent used fo estmaton wee estmated and compaed. Results fo a scenao based on tpcal stud paametes A 6 attbutes, L a 6 levels pe attbute, M 4, C, β 0.4, e 0 and no none opton ae shown n the gaph.

81 OPTIMIZATIO OF SAMPLE SIZE 8 0,00 0,50 T MAE 0,00 0, , n Gaph. Results fo a scenao based on tpcal stud paametes A 6, La 6, M 4, C, β 0.4, e 0 and no none opton wth T n the ange of 5 to 5. Whle the ule fom Tang (006 assumes that we can altenatvel use n 900 and T 5, n 450 and T 0, n 300 and T 5, n 5 and T 0 o n 80 and T 5 and each smla accuac we can see n the Gaph. that ths s not tue. Whle nceasng the sample sze ove 400 hadl mpoves ou esults, gven that the espondents wll be able to fll moe tass we can each suffcent accuac wth elatvel small sample. 5. Conclusons The esults have shown that we ma n some cases optmze the sample sze and/o numbe of tass pe espondent of ou stud wth espect to eachng the specfc goal wth needed level of accuac f we smulate the accuac befoe the stud s done. Howeve asde fom the tme complet of the smulaton (whch could be mpoved on n the futue wth nceased speed of the hadwae and potentall moe effcent algothms some othe dawbacs have been found. amel we should stll eep n mnd that: f we use andom geneaton of the pat-woths we need to have at least seveal hundeds of smulatons to get easonable estmates, (pehaps less andom appoach mght lead to hghe consstenc,

82 8 Ondře Vlus f we ae not cetan wth some aspects of the populaton we need to do a senstvt analss. (ths mght be solved b dong moe smulatons to bette lean how changes n an facto nfluences the accuac. Refeences Allenb, G.M., Aoa,., Gnte, J.L. (995 Incopoatng po nowledge nto the analss of conont studes. Jounal of Maetng Reseach 3, 5-6. Geen, P.E., Rao, V.R. (97 Conont measuement fo quantfng udgmental data. Jounal of Maetng Reseach 8, Johnson, R.M. (987 Adaptve conont analss. Sawtooth Softwae Inc. 987 Sawtooth Softwae Confeence Poceedngs, Len, P.J., DeSabo, W.S., Geen, P.E., Young, M.R. (996 Heachcal Baes conont analss: ecove of patwoth heteogenet fom educed epemental desgns. Maetng Scence 5 (, Louvee, J.J., Woodwoth, G. (983 Desgn and analss of smulated consume choce o allocaton epements: an appoach based on aggegate data. Jounal of Maetng Reseach 0, McFadden, D. (974 Condtonal logt analss of qualtatve choce behavo. Fontes n Econometcs (, Tang, J., Vandale, W., Wene, J. (006 Sample plannng fo CBC models: ou epeence. Sawtooth Softwae Inc., 006 Sawtooth Softwae Confeence, -6. Vens, M., Wedel, M., Wlms, T. (996 Metc conont segmentaton methods: a Monte Calo compason. Jounal of Maetng Reseach 33 (, Wth, R. (00 HB-CBC, HB-Best-wost-CBC o no HB at all? Sawtooth Softwae Inc., 00 Sawtooth Softwae Confeence Poceedngs,

83 OPTIMIZATIO OF SAMPLE SIZE 83 OPTYMALIZACJA LICZEBOŚCI PRÓBY I LICZBY ZADAŃ A RESPODETA W AALIZIE COJOIT Z WYKORZYSTAIEM DAYCH SZTUCZYCH Steszczene Szesze wozstane heachcznch model baeowsch pozwala na uzsane doładnch oszacowań zachowań espondenta bez onecznośc uwzględnana welu scenausz. Z duge ston tudne est oszacowane pzed pzepowadzenem badana lczb atbutów lczb ch pozomów dla dane lczebnośc pób z uwzględnenem osztów zman. W opacowanu pezentowane est podeśce bazuące na pzetwazanu wsadowm danch smulowanch o pewnch chaatestach. Głównm celem est poszuwane optmalne ombnac lczebnośc pób lczb zadań na espondenta, tóa pozwala na uzsane zadane doładnośc optmalne watośc osztów, ale taże badane ważlwośc poponowane eomendac ze względu na zman watośc ustalonch paametów.

84 Janusz L. Wwał O LIMIT DISTRIBUTIO OF HORVITZ-THOMPSO STATISTIC UDER THE REJECTIVE SAMPLIG. Samplng desgn Let U (,,..., be a fed populaton of the sze. An obsevaton of a vaable unde stud (a postve valued aula vaable attached to the -th populaton element wll be denoted b (,,...,. Moeove, let,,,. The paametes of the vaables ae: U U U,,,, u ( ( U η uz (,, z v0 γ, v0 γ, The sample of sze n, dawn wthout eplacement fom the populaton, wll be denoted b s. The samplng desgn s denoted b P(s and ncluson pobabltes of the fst and second odes b, fo,..., and,t fo t,,..., t,..., espectvel. Let S be the sample space of the samples of sze n, dawn wthout eplacement. The samplng desgns of smple samples dawn wthout eplacement s: P 0 ( s fo all s S. The followng n eectve samplng desgn s consdeed: q n n (

85 O LIMIT DISTRIBUTIO OF HORVITZ-THOMPSO STATISTIC 85 whee U,,,. Let a (a 0 f the -th populaton element s (not selected to the sample s. So, E(a, E(a a h h. The eectve samplng algothm s as follows. Fstl, the sample s of sze n s selected wth eplacement and pobabltes popotonal to q fom the populaton U. If n the ust selected sample thee ae not eplcatons of the populaton elements, then the samplng algothm s stopped. When n the sample s thee s epetton of at least one populaton element, then the new sample s of sze n s selected wth eplacement fom the populaton U wth pobabltes popotonal to q. The selecton pocess s eplcated untl the sample s fee of an eplcaton s obtaned. Let d q q ( q O( n d O n ( and ( ( Háe (964, pp. 508, poved that d q d and q, U, when d q. Moeove, let us note that d q < n < n and d < n < n. So, f d q, n and -n.. Vaance of the Hovtz-Thompson estmato The well nown Hovtz-Thompson (95 statstc s as follows. a ˆ s, (3 n Let U ( q ~. (4 ( q Let the sample be selected accodng to the eectve samplng. Háe (964, pp. 5-53, wote that when d q, the statstc ŷ s s desgn unbased estmato of and unde the assumpton that ~ the vaance v( ˆ s s:

86 Janusz L. Wwał 86 ( ( s q v ~ ˆ (5 o ( ( s q q n v ~ ˆ (6 o ( ( ( q s q q d v, ˆ. (7 Háe (964 wote that the epesson (7 has been deved fom the well nown geneal Yates-Gund (953 fomula: ( ( s v, ˆ (8 Háe (964, pp. 5, showed that unde the eectve sample and d q d ( ( (9 ( ( ( n O d (0 Hence, the epesson (9 let us ewte the (8 one as follows. ( s d v ( ( ˆ ( In the Append the above epesson has been tansfomed to the followng foms: ( ( ( 0, (, (, ( ˆ + + n O n v s η η η ( o ( ( ( (,, ˆ n O n v s η η (3

87 O LIMIT DISTRIBUTIO OF HORVITZ-THOMPSO STATISTIC 87 η. The equalt 0 s possble when the ont dstbuton of the vaable unde stud and the aula one s smmetc. because (, v ( ˆ * s n τ ( + η ( + ( η (, η (, η (,, , + O ( n (4 whee the smbols γ, η ut (.,., ae eplaned at the begnnng of the pape and τ η ( (,, (, η (, η (, η (, 0 (, 40 0 τ. 4. Vaance of the Hovtz-Thompson estmato unde estng measung eos Let us assume that values of the vaable have to be measued. The values of the vaable ae obsevatons of whch can be based b measung eo. The measue eos ae denoted b e, U, and the fst addtve model of geneatng the values of the vaable s as follows. o equvalentl b +e,,,. -e,,, and the epesson (3 taes the followng fom v s η n e 0 + ( ˆ η ( h, + ( h, h O( n whee h h U and teated as the value of the vaable h. e h s the elatve measung eo and t s

88 Janusz L. Wwał 88 The net model of geneatng the values of the vaable s multplcatve and t s defned b c,,, o equvalentl b c,,, ow, the epesson (3 taes the followng fom ( ( ( (,, ˆ 0 * + + n O c c c n v s η η 5. Estmatos of the vaance Háe (964, pp. 50, poposed to estmate the vaance ( s v ˆ b means of the followng statstcs ( ( s s s a q n n v ~ ( ˆ (5 whee ( ( s a q a q ~. (6 The well nown Goud and Yates estmato of the vaance s as follows: ( s s a a v, ˆ (7 whee q,,,.

89 O LIMIT DISTRIBUTIO OF HORVITZ-THOMPSO STATISTIC 89 On the bass of the epesson (9 we have d ( d ( ( and ( (. (8 d ( ( The above esults let us tansfom the epesson (7 to the fom: vˆ s ( ˆ s a a ( (, d ( ( (9 So, E( v ( ˆ v( ˆ ˆ, when d. s s s Theoem. Let the mean value of the aula vaable fulfl the nequaltes 0 M, v 0 <, v 0 <, v 40 <. Unde the eectve vˆ ˆ samplng desgn fo q,,..,, gven b the epesson ( the statstc s( s s the desgn consstent estmato of the vaance v( ˆ s, when d. The poof s ncluded n the append. 6. Lmt dstbuton t s Háe (964, pp. 54-5, poved the followng (see Bege (99, too. Theoem. If the eectve samplng desgn s mplemented the statstc ˆ s v ( ˆ s has the standad nomal dstbuton, so t s t ~ (0,, f and onl f d q and ξ 0, whee ξ {ε: L(ε}, ε}, q L( ε z, v q ( ˆ s { : z > εq v( ˆ } s

90 90 Janusz L. Wwał z ~ and ~ s defned b the epesson (4. We ae gong to pove the smla theoem but about convegence to nomalt of the statstcs: ˆ s tˆ s (0 vˆ ˆ s ( s whee the sample vaance v ( ˆ ˆ s defned b the epesson (9. s s Theoem 3. Let the mean value of the aula vaable fulfl the nequaltes 0 M, v 0 <, v 0 <, v 40 <. Unde the eectve samplng desgn fo q,,..,, gven b the epesson ( the statstc tˆ s ~ (0,, when n and -n. Moeove, unde the addtonal assumpton that 0, t s ~ (0,. Poof. The theoem esults staghtfowad fom the theoems and and fom the well nown theoem of Sludz, see Bege and Snne (005, too. 7. Concluson Asmptotc nomalt of the Hovtz-Thompson statstc s ve mpotant fom pactcal pont of vew because t let us constuct confdence nteval fo the populaton mean as well as testng statstcal hpothess on mean value. Fo nstance, such hpotheses ae consdeed n fnancal audt, because thee s fequentl consdeed eectng samplng desgn as a patcula case of so called dolla samplng. Acnowledgement The eseach was suppoted b the gant numbe fom the Polsh Mnst of Scence and Hghe Educaton. Append The devaton of the epesson (.

91 O LIMIT DISTRIBUTIO OF HORVITZ-THOMPSO STATISTIC 9 ( s d n v ( ( ˆ ( ( ( d n ( ( ( ( ( ( d n ( ( ( d n ( ( ( ( d n ( ( ( ( ( ( ( d n So, ( ( ( 3 ( ( ˆ s n n v ( ( ( n n ( ( ( ( ( + n n ( (, (, (, (, ( + + n O n n n η η η η ( et, we have:

92 Janusz L. Wwał 9 ( ( ( ( ( + n n n +, (, (, ( 0 n n n η η η ( ( + + +, (, (, (, (, ( 0 0 n n η η η η η ( ( 0, (, ( + O n n η η Fnnal, the obtaned esult and the esults ( and ( lead to the epesson (. Append Poof of the theoem. Háe (964, pp. 508, poved that fo all,,, q when d, whee d s gven b the equaton (3. Let us emnd ouself the followng defnton the smbol O(. eplaned e.g. b Lea (977. Defnton. Let h(u and t(u be two functons defned n a neghbouhood of value u 0. The smbol O(t means that thee ests such the neghbouhood 0< u-u 0 <δ and the value M that M u t u h ( ( The Defnton and the epesson ( lead to the followng. Let g h ( and t( -. ow, we have

93 O LIMIT DISTRIBUTIO OF HORVITZ-THOMPSO STATISTIC 93 < M t h M 0 ( (,,,. So, g O( - because the above nequalt s tue fo all >0. ( ( ( ( ( ( ( ( ( s s s s s s s s s v V v E v E v v E ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( ( ( s s s v v E ˆ ˆ ˆ ( ( ( + s s d v E, 4 4 ( ( ( ( ˆ ˆ + h h h ( ( ( h h h a a a a E d d ( ( ( ( ( ( ( ( It s well nown that ( ( ( ( ( l l l l a a E a a E a a E a a E a a a a E ( ( ( + s s d v E, 4 4 ( ( ( ( ˆ ˆ + h h h ( ( h h h d d ( ( ( ( ( ( ( (

94 Janusz L. Wwał 94 ( + d d, 4 4 ( ( ( ( ( ( + h h h h h ( + d d, 4 4 ( ( ( ( ( ( + h h h h h ( + d d, 4 4 ( ( ( ( ( + h Hence, ( ( ( ( s s s v d d v E ˆ ( ( ( ( ˆ ˆ, ( ( ( ( s s s d d v v E, 4 4 ( ( ( ( ˆ ˆ ˆ ( ( ( ( O n O n n O,

95 O LIMIT DISTRIBUTIO OF HORVITZ-THOMPSO STATISTIC 95 O ( n ( O ( n ( v v , So, E 4 ( vˆ ( ˆ v( ˆ O( n s 0 P s s Hence, on the bass of the well nown Tchebshev nequalt we have: ( vˆ ( ˆ v( ˆ s -n. s s ( ˆ s ( ˆ ( vˆ ( ˆ s s v( ˆ ˆ vs V ς P δ O( n. v s δ s So, the estmato vˆ ( ˆ conveges to the vaance ( s s v ˆ when n and s Refeences Bege, Y.G. (998 Rate of convegence to nomal dstbuton fo Hovtz- Thompson estmato. Jounal of Statstcal Plannng and Infeence 67, Bege, Y.G., Snne, C.J. (005 A acnfe vaance estmato fo unequal pobablt samplng. Jounal of the Roal Statstcal Socet B 67, Háe, J. (964 Asmptotc theo of eectve samplng wth vang pobabltes fom a fnte populaton. Annals of Mathematcal Statstcs 35, Hovtz, D.G., Thompson, D.J. (95 A genealzaton of samplng wthout eplacement fom fnte unvese. Jounal of the Amecan Statstcal Assocaton Vol. 47, Lea, F. (977 Dffeence and Integal Calculus (n Polsh. PW, Waszawa. Tllé, Y. (006 Samplng Algothms. Spnge. Yates, F., Gund, P.M. (953 Selecton wthout eplacement fom wthn stata wth pobablt popotonal to sze. Jounal of the Roal Statstcal Socet, Sees B, 5: 35-6.

96 96 Janusz L. Wwał O ROZKŁADZIE GRAICZYM STATYSTYKI HORVITZA-THOMPSOA Z PRÓBY DOBIERAEJ ZA POMOCĄ SCHEMATU LOSOWAIA ZWROTEGO ODRZUCAJĄCEGO PRÓBY Z POWTÓRZEIAMI ELEMETÓW Steszczene Rozważan est poblem wnosowana o watośc pzecętne w populac sończone ustalone na podstawe pób, do tóe są losowane element populac z pawdopodobeństwam popoconalnm do watośc cech dodatowe. Losowane zwotne pób o zadane z gó lczebnośc est powtazane ta długo, aż uzsam taą, w tóe element populac ne powtazaą sę. Hae wazał, że statsta Hovtza-Thompsona dla obsewac zmenne w ta losowane póbe ma ganczne ozład nomaln m.n. pod waunam, że ozma pób populac wzastaą w sposób neoganczon oaz waanca statst Hovtza-Thompsona est znana. Teść nnesze pac est neznacznm uogólnenem te własnośc ganczne ozładu pawdopodobeństwa statst Hovtza-Thompsona na pzpade, gd e waanca est ocenana za pomocą znanego estmatoa Yatesa Gund ego.

97 Tomasz Żądło O ACCURACY OF TWO PREDICTORS FOR SPATIALLY AD TEMPORALLY CORRELATED LOGITUDIAL DATA. Basc notatons Longtudnal data fo peods t,...,m ae consdeed. In the peod t the populaton of sze t s denoted b Ω t. The populaton n the peod t s dvded nto D dsont subpopulatons (domans Ω dt of sze dt, whee d,...,d. Let the set of populaton elements fo whch obsevatons ae avalable n the peod t be denoted b s t and ts sze b n t. The set of subpopulaton elements fo whch obsevatons ae avalable n the peod t s denoted b s dt and ts sze b n dt. Let: Ω dt Ωdt sdt, dt dt ndt. Let M d denotes the numbe of peods when the -th populaton element belongs to the d-th doman. Let us denote the numbe of peods when the -th populaton element (whch belongs to the d-th doman s obseved b m d. Let md Md md. It s assumed that the populaton ma change n tme and that one populaton element ma change ts doman afflaton n tme (fom techncal pont of vew obsevatons of some populaton element whch change ts doman afflaton ae teated as obsevatons of new populaton element. It means that and t completel dentf doman afflaton but addtonal subscpt d wll be needed as well. Moe about ths assumptons wll be wtten at the end of the net secton. The set of elements whch belong at least n one of peods t,...,m to sets Ωt s denoted b Ω and ts sze b. Smlal, sets Ω d, s, s d, Ω d of szes d, n, n d, d espectvel ae defned as sets of elements whch belong at least n one of peods t,...,m to sets Ω dt, s t, s dt, Ω dt espectvel. The d*-th doman of nteest n the peod of nteest t * wll be denoted b Ω d** t, and

98 98 Tomasz Żądło the set of elements whch belong at least n one of peods t,...,m to sets Ω d** t wll be denoted b Ω d*. The ntoduced notatons allow to assume that the doman afflatons of populaton elements change n tme.. Fst pedcto Supepopulaton models used fo longtudnal data (compae Vebee and Molenbeghs, 000; Hedee and Gbbons, 006 ae consdeed whch ae what s mpotant fo futhe consdeatons specal cases of the Geneal Lnea Model (GLM and the Geneal Lnea Med Model (GLMM. We popose the followng model: Yd Xdβd + Zv d d + e, d ( whee Yd col ( Y d, whee Yd s a andom vecto, called pofle, of sze d M d, and Y d (d,...,d ae assumed to be ndependent, Xd col ( X d, whee X d s nown mat of sze M d p, d Zd dag ( Z d, whee Z d s nown vecto of sze M d, d v d col ( v d d (d,...,d ae assumed to be ndependent,, whee v d s a pofle-specfc andom component and v d ed col ( e d, whee e d s a andom component vecto of sze M d and e d (,...,; d,...,d ae assumed to be ndependent, v d and e d ae assumed to be ndependent. What s moe, t s assumed that vecto of andom components v d obe assumptons of smultaneousl spatal autoegessve (SAR pocess: v ρ W v + u, ( d ( sp d d d whee Wd s the spatal weght mat fo pofles Y d, d d ~(, σ u d u 0 I. Hence, v ~ 0, R, (3 d whee ( d - T Rd σ u Cd and d ( ρ( sp ( ρ d ( sp d d d C I W I W. Moeove, elements of e d obe assumptons of autoegessve pocess AR(: e ρ e + ε. (4 d ( t d d Hence,

99 O ACCURACY OF TWO PREDICTORS FOR SPATIALLY 99 e d ( d ~ 0, Σ, (5 whee elements of Σ ae gven b l ( σ ρ ρ d ε. ( t ( t Unde the model, based on the theoem pesented b Roall (976, the best lnea unbased pedcto s gven b: ˆ θ BLU ( ˆ d** t Yd** t % d*t* βd* sd** t + + ( dag σ * * * ( * *( ˆ * * + γ Z C Z + Σ V Y X β, (6 T - T d* u d d sd s d ss d sd sd d* d * whee % d*t* s a p vecto of totals of aula vaables n Ω d** t, ˆ T - T - β X V X X V Y, ( d* sd* ss d* sd* sd* ss d* sd* - T ( σ u dag ( n V Z C Z + Σ, X sd* s nown ss d* sd* d* sd* ss d* d* mat of aula vaables, Y sd* s a Y d, γ d* s a n d * m d* p n d * md* vecto of andom vaables n d * M d* vecto of one s fo obsevatons n peod t * (n Ω d** t and zeo othewse, Z sd and Z d ae submatces of Z d obtaned b deletng ows fo unsampled and sampled elements espectvel, Σ ss d s a submat obtaned fom Σ d b deletng ows and columns fo unsampled obsevatons., whee Σ s d s a submat obtaned fom Σ d b deletng ows fo sampled obsevatons and columns fo unsampled obsevatons. 3. Second pedcto Let us assume model ( wth ( and (4 whee ρ ( sp 0 and ρ ( t 0 what means that elements of e d and elements v d ae uncoelated. Based on the model pedcto (6 smplfes to the fomula (see Żądło 0: ˆ θ BLU ( d* t* ** ** m d t d t d * ˆ ˆ Y ** ** * * * * * d t σv b Y d t d d d d d s ** d t + β + ( β, (7

100 00 Tomasz Żądło whee b * σe + σvm *, d d d d... d dp n * n * d d ˆ T T β * b * * * b * * * X X d X Y and X * s m * p nown d sd sd d sd sd sd d mat of aula vaables. 4. Smulaton stud Lmted model-based smulaton stud pepaed usng R (R Development Coe Team 0 s based on atfcal data. Populaton of sze 00 elements s dvded nto D 0 domans of szes {5, 5, 5, 0, 0, 0, 0, 5, 5, 5}. umbe of peods M 3 and balanced panel sample s studed n each peod the same n d 5 elements fom each doman ae obseved n the sample (oveall sample sze n each peod s n 50. The pupose of the stud s to pedct D 0 doman totals fo the last peod. Data ae geneated based on model ( whee, z, β β and fo abta chosen values of paametes β 00, σ, d d σ u. Fo these assumptons and balanced panel sample pedcto (7 smplfes to (Żądło (00: ˆ θ ˆ BLU Yd ** t + d* tμ (8 whee ˆ μ sd* t D n d m n m Y. d d In the smulaton the followng pedctos ae consdeed: pedcto gven b (6 assumng that vaance-covaance paametes ae nown, denoted b BLUP, pedcto gven b (6 whee vaance-covaance paametes ae estmated usng Restcted Mamum Lelhood Estmatos, denoted b EBLUP, pedcto gven b (8, denoted b SIMPLE. In the smulaton the followng values of ρ ( sp and ρ ( t ae consdeed: 0,8; 0,3; -0,3 and -0,8 what gves steen pas of these coelaton coeffcents (these pas ae pesented on -as. Realzatons of andom components ae geneated usng multvaate nomal dstbuton. The followng thee gaphs allow to compae MSEs of the consdeed pedctos. d d d d ε

101 O ACCURACY OF TWO PREDICTORS FOR SPATIALLY (.8,.8 (.8,.3 (.8,-.3 (.8,-.8 (.3,.8 (.3,.3 (.3,-.3 (.3,-.8 (-.3,.8 (-.3,.3 (-.3,-.3 (-.3,-.8 (-.8,.8 (-.8,.3 (-.8,-.3 (-.8,-.8 Gaph. Values of MSE(EBLUP/MSE(BLUP (.8,.8 (.8,.3 (.8,-.3 (.8,-.8 (.3,.8 (.3,.3 (.3,-.3 (.3,-.8 (-.3,.8 (-.3,.3 (-.3,-.3 (-.3,-.8 (-.8,.8 (-.8,.3 (-.8,-.3 (-.8,-.8 Gaph. Values of MSE(SIMPLE/MSE(BLUP

102 0 Tomasz Żądło (.8,.8 (.8,.3 (.8,-.3 (.8,-.8 (.3,.8 (.3,.3 (.3,-.3 (.3,-.8 (-.3,.8 (-.3,.3 (-.3,-.3 (-.3,-.8 (-.8,.8 (-.8,.3 (-.8,-.3 (-.8,-.8 Gaph 3. Values of MSE(SIMPLE/MSE(EBLUP In the gaph values of the atos of the MSE of the EBLUP and MSE of the BLUP. The mamum value fo the consdeed cases equals, what means that the mamum ncease of the MSE due to the estmaton of paametes of vaance-covaance mat s,%. The mean and medan values fo the consdeed cases equal,05 and,009 espectvel. In the gaph values of the atos of the MSE of the SIMPLE and MSE of the BLUP. The mamum value fo the consdeed cases equals,89, mean and medan ae,08 and,043 espectvel. In the gaph 3 esults of the compason between SIMPLE and EBLUP ae pesented. The compason s ve mpotant fom pactcal pont of vew two pedctos whch can be used n pactce ae compaed. At the gaph atos of the MSE of the SIMPLE and MSE of the EBLUP ae pesented. It s woth notng that n 30,6% of the consdeed cases values of the atos ae smalle than what means that the decease of the accuac due to the model msspecfcaton mabe smalle than the decease of the accuac due to the estmaton of the paametes of the coectl specfed model. What s mpotant, the mean and medan of the atos fo the consdeed cases equal,056 and,07 what means the aveage dffeences between these two pedctos ae small. Based on the gaph 3, t can be notced that fo the consdeed cases the atos ae small especall when the absolute values of the spatal coelaton coeffcents ae small.

103 O ACCURACY OF TWO PREDICTORS FOR SPATIALLY 03 Let us consde MSE estmato of (8 unde model model ( wth ( and (4 whee ρ ( sp 0 and ρ ( t 0, assumed n the smulaton dd, dzd, dβd β and balanced panel samples. Unde these assumptons and REML estmatos of δ σ e σ u denoted b δ ˆ ˆ σ ˆ e σ u the MSE estmato s gven b (see Żądło 00 MSE ˆ $ ( ˆ ˆ ˆ ξ ( θ BLU δ g( δ + g( δ (9 whee ˆ g ( ( ˆ σ + ˆ σ δ, ˆ ( ˆ σ ˆ e + σ d* t e v g ( δ vm d* tm n, Estmato (9 s appomatel unbased fo the smplfed model (assumng nte ala ρ ( sp 0 and ρ ( t 0 but s based unde the model ( (.e. unde assumpton of spatal and tempoal coelaton of elements of andom components vectos whch s studed n the smulaton stud (.8,.8 (.8,.3 (.8,-.3 (.8,-.8 (.3,.8 (.3,.3 (.3,-.3 (.3,-.8 (-.3,.8 (-.3,.3 (-.3,-.3 (-.3,-.8 (-.8,.8 (-.8,.3 (-.8,-.3 (-.8,-.8 Gaph 4. Values of MSE estmato of SIMPLE Summazng esults pesented n the gaph 3 and n the gaph 4 t ma be notced that usage of the pedcto unde assumpton of lac of spatal and tempoal coelaton ma be good alteatve compang wth moe complcated

104 04 Tomasz Żądło pedcto unde assumpton of nonzeo spatal and tempoal coelaton. But n ths case coect estmato of MSE should be used. 5. Summa Based on the smulaton stud two pedctos wee compaed fo spatall and tempoall coelated longtudnal data. The fst one unde the coectl specfed model whee unnown paametes ae estmated usng REML and the second pedcto unde the msspecfed model (unde assumpton of the lac of spatal and tempoal coelaton. It was shown, especall fo small values of spatal coelaton, that the second, smple pedcto can be a good altenatve to the fst one. Refeences Chanda, H., Salvat,., Chambes, R. (007 Small aea estmaton fo spatall coelated populatons a compason of dect and ndect model-based methods. Statstcs n Tanston 8(, Hedee, D., Gbbons, R.D. (006 Longtudnal Data Analss. John Wle and Sons, ew Jese. Hendeson, C.R. (950 Estmaton of genetc paametes (Abstact. Annals of Mathematcal Statstcs, Molna, I., Salvat,., Pates, M. (009 Bootstap fo estmatng the MSE of the Spatal EBLUP. Computatonal Statstcs 4, Petucc, A., Salvat,. (006 Small aea estmaton fo spatal coelaton n wateshed eoson assessment. J Agc Bol Envon Stat, Pates, M., Salvat,. (008 Small aea estmaton: the EBLUP estmato based on spatall coelated andom aea effects. Stat Methods Appl 7, 3-4. Petucc, A., Pates, M., Salvat,. (005 Geogaphc nfomaton n small aea estmaton: small aea models and spatall coelated andom aea effects. Statstcs n Tanston 7(3, Rao, J..K (003 Small aea estmaton. John Wle and Sons, ew Jese. Rao, J..K, Yu, M. (994 Small-Aea Estmaton b Combnng Tme-Sees and Coss-Sectonal Data. The Canadan Jounal of Statstcs (4, 5-58.

105 O ACCURACY OF TWO PREDICTORS FOR SPATIALLY 05 R Development Coe Team (0 A language and envonment fo statstcal computng. R Foundaton fo Statstcal Computng, Venna. Roall, R.M. (976 The lnea least squaes pedcton appoach to two- stage samplng. JASA, 7, Salvat,., Pates, M., Tzavds,., Chambes, R. (009 Spatal M-quantle models fo small aea estmaton. Statstcs n Tanston 0(, Sae, A., Chambes, R. (003 Small aea estmaton unde lnea and genealzed lnea med models wth tme an aea effects. S3RI Methodolog Wong Pape M03/5, Unvest of Southampton. Vebee, G., Molenbeghs, G.(000 Lnea Med Models fo Longtudnal Data. Spnge-Velag, ew Yo. Żadło, T. (00 On pedcton of doman total based on balanced panel data. Acta Unvestats Lodzenss, Fola Oeconomca 35, Żądło, T. (0 On some poblems of pedcton of doman total n longtudnal suves when aula nfomaton s avalable, submtted to Studa Eonomczne. O DOKŁADOŚCI DWÓCH PREDYKTORÓW DLA SKORELOWAYCH DAYCH PRZEKROJOWO-CZASOWYCH Steszczene Rozważn est model dla danch pzeoowo-czasowch uwzględnaąc dwa sładn losowe spełnaące odpowedno założena pzestzennego modelu autoegesnego oaz modelu autoegesnego w czase. W pac ozważane są dwa pedto watośc globalne w domene. Pewsz z nch est empcznm nalepszm lnowm neobcążonm pedtoem wpowadzonm pz założenu wspomnanego modelu. Dug est nalepszm lnowm neobcążonm pedtoem pz założenu modelu meszanego, w tóm element sładnów losowch są nezależne. Analza została wspata badanam smulacnm.

106 AUTHORS Czesław Domańs Depatment of Statstcs Methods, Unvest of Lodz, Poland Wocech Gamot Depatment of Statstcs, Katowce Unvest of Economcs, Poland Janusz Gołaszews Depatment of Plant Beedng and Seed Poducton, Unvest of Wama and Mazu n Olsztn, Poland Anna Imołe Depatment of Plant Beedng and Seed Poducton, Unvest of Wama and Mazu n Olsztn, Poland Alna Jędzecza Cente fo Mathematcal Statstcs, Statstcal Offce n Lodz Cha of Statstcal Methods, Insttute of Econometcs and Statstcs, Unvest of Lodz, Poland Aadusz Kozłows Depatment of Statstcs, Unvest of Gdans, Poland Jan Kubac Cente fo Mathematcal Statstcs, Statstcal Offce n Lodz, Poland Zbgnew asals Depatment of Entepse Economcs, Unvest of Wama and Mazu n Olsztn, Poland Doota Raczewcz Insttute of Statstcs and Demogaph, Wasaw School of Economcs, Poland Janusz L.Wwał Depatment of Statstcs, Katowce Unvest of Economcs, Poland Dausz Załus Depatment of Plant Beedng and Seed Poducton, Unvest of Wama and Mazu n Olsztn, Poland Tomasz Żądło Depatment of Statstcs, Katowce Unvest of Economcs, Poland Ondře Vlus Unvest of Economcs, Pague, Czech Republc