## Amanqaku eMathematika

Kule sihlomelo, ndiza kuchaza ezinye zeengcamango ezivela kwisahluko kwimodi encinane yemathematika. Injongo apha kukukunceda ufumane ukhululekile kunye nobunqamlezo kunye nesakhelo seemathematika esisetyenziswe ngabaphandi abaphando ukuze wenze utshintsho kwezinye izinto ezibhaliweyo ezibhalwe kule zihloko. Ndiza kuqala ngokuzisa isampula, mhlawumbi uye kwisampula enokwenzeka ngokungabonakaliyo, kwaye ekugqibeleni, isampampu engenakunokwenzeka.

Sampula

Njengomzekelo osebenzayo, makhe siqwalasele injongo yokuqikelela izinga lokungabikho kwemisebenzi e-United States. Yenza $$U = \{1, \ldots, k, \ldots, N\}$$ makube ngabantu abajoliswe kuyo kwaye makabe $$y_k$$ ngokubaluleka $$y_k$$ eguqukileyo kumntu $$k$$ . Kulo mzekelo $$y_k$$ nokuba umntu $$k$$ akasebenzi. Ekugqibeleni, vumela $$F = \{1, \ldots, k, \ldots, N\}$$ makube sisakhelo sabantu, ngenxa yokulula kuthiwa yinto efanayo nabantu abajoliswe kuyo.

Isampula esisisiseko sopling isampula esilula ngaphandle kokutshintshwa. Kule meko, umntu ngamnye ufanelekile ukuba afakwe kwisampuli $$s = \{1, \ldots, i, \ldots, n\}$$ . Xa idatha iqokelelwa kunye noyilo lwesampulu, abaphandi banokuqikelela ukuba inani labantu abangenasempesheni nentsingiselo lithetha ukuthi:

$\hat{\bar{y}} = \frac{\sum_{i \in s} y_i}{n} \qquad(3.1)$

apho $$\bar{y}$$ yintengo yokungabikho komsebenzi kubemi kwaye $$\hat{\bar{y}}$$ uqikelelo lwezinga lokungaqeshwa ( $$\hat{ }$$ isetyenziswe ukubonisa uqikelelo).

Enyanisweni, abaphandi badla ngokusebenzisa isampula esilula ngaphandle kokutshintshwa. Ngezizathu ezahlukeneyo (enye endiza kuyichaza ngomzuzwana), abaphandi badla ngokudala ama-sampulu ngamanqanaba angalinganiyo wokufakwa. Ngokomzekelo, abaphandi banokukhetha abantu baseFlorida banamathuba aphezulu okufakwa ngaphandle kwabantu baseCalifornia. Kule meko, isampuli ithetha (iq. 3.1) kungenzeka ukuba ingabi ngqi kelelo. Kunoko, xa kunokungalingani kwamathuba okufakwa, abaphandi basebenzisa

$\hat{\bar{y}} = \frac{1}{N} \sum_{i \in s} \frac{y_i}{\pi_i} \qquad(3.2)$

apho $$\hat{\bar{y}}$$ uqikelelo lwezinga lokungasebenzi kwaye $$\pi_i$$ ngumntu $$i$$ amathuba okufakwa. Ukulandela umgangatho oqhelekileyo, ndiza kubiza uqikelelo kwi-eq. 3.2 umlinganisi weHorvitz-Thompson. Umlinganisi weHorvitz-Thompson uyiluncedo kakhulu kuba lukhokelela ekuqikelelweni kokungabikho kokusesikweni kwanoma yimuphi umzekelo wokukhangela isampula (Horvitz and Thompson 1952) . Ngenxa yokuba umlinganisi weHorvitz-Thompson uvela ngokukhawuleza, kunceda ukuba uqaphele ukuba unokuphinda ubhale kwakhona

$\hat{\bar{y}} = \frac{1}{N} \sum_{i \in s} w_i y_i \qquad(3.3)$

apho $$w_i = 1 / \pi_i$$ . Njengo-eq. 3.3 utyhila, umlinganisi weHorvitz-Thompson uyisampuli esinexabiso elithetha ukuba izisindo zihambelana nendawo yokukhetha. Ngamanye amazwi, umntu omncinci angabandakanywa kwisampuli, ubunzima obunzima ukuba umntu angene kuwo uqikelelo.

Njengoko kuchaziwe ngaphambili, abaphandi bahlala bexubusha abantu abanamathuba angalinganiyo wokufakwa. Omnye umzekelo wokuyila ongakhokelela ekungalinganiseni kwamathuba okufakwa kufakwe isampuli , okubalulekile ukuqonda kuba kuhambelana ngokusondeleyo kwinkqubo yokuqikelela ekuthiwa yi- post-stratification . Kwi-sampling echanekileyo, umphandi uyahlula ulwabiwo lwabantu ekujoliswe kuyo kwi $$H$$ ngamaqela apheleleyo kunye namaqela apheleleyo. La maqela abizwa ngokuba yi- strata kwaye iboniswe njenge- $$U_1, \ldots, U_h, \ldots, U_H$$ . Kulo mzekelo, i-strata yimihlaba. Ubukhulu bamaqela buboniswa njenge $$N_1, \ldots, N_h, \ldots, N_H$$ . Umphandi unokufuna ukusebenzisa isampuli echanekileyo ukuze aqiniseke ukuba unabantu abaneleyo kwicandelo ngalinye ukwenza uqikelelo lwezinga lokungasebenzi.

Xa isixa sabantu sihlukaniswe kwi- strata , cinga ukuba umphandi ukhetha isampula esingahleliyo ngaphandle kokutshintshwa kobukhulu $$n_h$$ , ngokuzimela kwinqanaba ngalinye. Ngaphezulu, cinga ukuba wonke umntu okhethwe kwisampuli uba ngummangalelwa (ndiya kuthatha ingaphenduli kwinqanaba elilandelayo). Kule meko, amathuba okufakwa

$\pi_i = \frac{n_h}{N_h} \mbox{ for all } i \in h \qquad(3.4)$

Ngenxa yokuba ezi zinto zinokuthi zihluke kumntu kumntu, xa kuqikelelwa kulolu hlobo lwesakhiwo, abaphandi kufuneka balinganise ummangalelwa ngamnye ngokuchasene nawo amathuba okubandakanywa ngokusebenzisa umlinganisi weHorvitz-Thompson (u-3.2).

Nangona umlinganisi weHorvitz-Thompson engabalulekanga, abaphandi banokuvelisa okuchanekileyo (oko kukuthi, ukuhlukana okuphantsi) ngokudibanisa isampuli ngolwazi olungancedayo . Abanye abantu bayamangaliswa kukuba oku kuyinyani nangona kukho isampula esilungele ukubulawa. Olu buchule kusetyenziswa ulwazi oluncedisayo kubaluleke ngakumbi kuba, njengoko ndiza kubonisa kamva, ulwazi oluncedisayo lubaluleke kakhulu ekwenzeni uqikelelo kumasampula enokwenzeka ngokungabikho nxamnye namasampula angenakwenzeka.

Enye inqubo eqhelekileyo yokusebenzisa ulwazi oluncedisayo lusetyenziso lwe- post-stratification . Cinga, umzekelo, ukuba umphandi uyazi inani lamadoda nabasetyhini kwi-50 nganye; sinokubonisa ukuba ubukhulu beqela njenge $$N_1, N_2, \ldots, N_{100}$$ . Ukudibanisa olu lwazi oluncedisayo kunye nesampuli, umphandi angahlula isampula kumaqela e- $$H$$ (kule meko 100), yenza uqikelelo kwiqela ngalinye, kwaye udale umlinganiselo olinganisiweyo weli qela uthetha:

$\hat{\bar{y}}_{post} = \sum_{h \in H} \frac{N_h}{N} \hat{\bar{y}}_h \qquad(3.5)$

Kancinane, uqikelelo kwi-eq. I-3.5 inokuthi ichaneke ngakumbi kuba isebenzisa ulwazi $$N_h$$ oluntu-i- $$N_h$$ okuchanekileyo ukuba isampula engalinganiyo iyakhethwa. Enye indlela yokucinga ngayo kukuba ukulandelelana kwe-post kufana nokucatshungulwa kokulandelelana emva kwedatha sele iqokelelwa.

Ekugqibeleni, eli candelo lichaze iiplani ezimbalwa zokupakisha: i-sampling elula engaqhelekanga ngaphandle kokuthatha indawo, isampuli ngokungenalinganiyo, kunye nesampampu. Kuye kwachaza iingcamango ezimbini ezibalulekileyo malunga nokuqikelelwa: umlinganisi weHorvitz-Thompson kunye ne-post-stratification. Ukufumana inkcazo engaphezulu ngokusemthethweni kwindlela yokukhangela isampula, funda isahluko 2 Särndal, Swensson, and Wretman (2003) . Ukufumana unyango Särndal, Swensson, and Wretman (2003) ngqo Särndal, Swensson, and Wretman (2003) jonga kwicandelo 3.7 Särndal, Swensson, and Wretman (2003) . Ingcaciso yezobugcisa bepropati yeHorvitz-Thompson, bona Horvitz and Thompson (1952) , u- Overton and Stehman (1995) , okanye i-2.8 ye-@ sarndal_model_2003. Ukufumana unyango olungaphezulu ngokusemthethweni lwe-post-stratification, bona Holt and Smith (1979) , Smith (1991) , Little (1993) , okanye i-7.6 Särndal, Swensson, and Wretman (2003) .

Isampula esinokuthi sinokungabikho

Phantse zonke iimvavanyo zophando azikho nto; okokuthi, akubona wonke umntu kwisampuli yoluntu uphendula yonke imibuzo. Kukho iintlobo ezimbini eziphambili zokungabi nantoni: into engeyiyo neyunithi . Ngento engekho nto, abanye abaphendulayo abaphenduli izinto ezithile (umzekelo, ngamanye amaxesha abaphenduli abafuni ukuphendula imibuzo abayicinga ingqalelo). Kwiyunithi ngokungakhathaliseki, abanye abantu abakhethiweyo kwisampuli yabemi abaphendulanga kwisaveyi nonke. Izizathu ezibini eziqhelekileyo zeyunithi ngaphandle kokungqinelani kukuba umntu othemputshane akanakunxibelelana kwaye umntu othe isampula uyaqhagamshelana naye kodwa wenqaba ukuthatha inxaxheba. Kule candelo, ndiya kugxininisa kwiyunithi ngaphandle kokuphulaphula; Abafundi abanomdla kwinto ngaphandle kokunye kufuneka babone u-Little noRubin (2002) .

Abaphandi bahlala becinga malunga nophando kunye neyunithi-mpendulo njengenkqubo yesampula yesampula ezimbini. Kwisigaba sokuqala, umphandi ukhetha isampuli $$s$$ enjalo ukuba umntu ngamnye unako ukufakwa $$\pi_i$$ (apho $$0 < \pi_i \leq 1$$ ). Emva koko, kwinqanaba lesibini, abantu abakhethiweyo kwisampuli baphendule ngokusemandleni $$\phi_i$$ (apho $$0 < \phi_i \leq 1$$ ). Le nkqubo yesibini ibangela ukuba isethi yokugqibela yabaphenduli $$r$$ . Ukwahlukana okubalulekileyo phakathi kwezi zigaba zibini kukuba abaphandi balawula inkqubo yokukhetha isampuli, kodwa abayikulawula ukuba yeyiphi yalaba bantu abathintelayo abaphenduliweyo. Ukubeka ezi zimbini iinkqubo kunye, ithuba lokuba umntu uya kuba ngummangalelwa

$pr(i \in r) = \pi_i \phi_i \qquad(3.6)$

Ngenxa yokulula, ndiza kuqwalasela imeko apho isilathisi sokuqala isampula isampula esilula ngaphandle kokutshintshwa. Ukuba umphandi ukhetha isampula yobukhulu $$n_s$$ evelisa $$n_r$$ abaphenduli, kwaye ukuba umphandi uyayigatya ingaphenduli kwaye isebenzisa intsingiselo yabamphenduli, ngoko-ke i-bias of estimation iya kuba:

$\mbox{bias of sample mean} = \frac{cor(\phi, y) S(y) S(\phi)}{\bar{\phi}} \qquad(3.7)$

apho i- $$cor(\phi, y)$$ ilungelelaniso loluntu phakathi kweempembelelo zokuphendula kunye nesiphumo (umzekelo, isimo sengqesho esingenalo msebenzi), $$S(y)$$ ukuphambuka komgangatho wesibalo (umzekelo, ukungasebenzi isimo), $$S(\phi)$$ ukuphambuka komgangatho wabemi wokuphendulwa kwempendulo, kwaye $$\bar{\phi}$$ yindlela yokuphendula impendulo yabantu (Bethlehem, Cobben, and Schouten 2011, sec. 2.2.4) .

Eq. I-3.7 ibonisa ukuba ukungabi nantoni akuyi kuzisa i-bias ukuba kukho na iimeko ezilandelayo:

• Akukho ukuhluka kwimeko yokungabikho kwemisebenzi $$(S(y) = 0)$$ .
• Akukho ukuhluka kwindlela yokuphendula ngokufanelekileyo $$(S(\phi) = 0)$$ .
• Akukho ukulungelelaniswa phakathi kwempendulo yokuphendula kunye nesimo sokungabikho kwengqesho $$(cor(\phi, y) = 0)$$ .

Ngelishwa, akukho nanye yale miqathango ibonakala ngathi. Kubonakala kungenakwenzeka ukuba akukho kuhluka kwimo engqesho okanye ukuba ayiyi kuba nantlukwano kwimpembelelo yokuphendula. Ngaloo ndlela, igama eliphambili kwi-eq. 3.7 lulungelelaniso: $$cor(\phi, y)$$ . Ngokomzekelo, ukuba abantu abangaqeshwayo banokuthi baphendule, ngoko ke izinga lokuqingqwa kwemisebenzi liya kuhlaselwa phezulu.

Iqhinga lokwenza uqikelelo xa kukho ukungabi nantoni kukusebenzisa ulwazi oluxhasayo. Ngokomzekelo, enye indlela ongayisebenzisa ngayo ulwazi oluxhasayo yinkcazo yokuposa (khumbula u-3.5 ukusuka phezulu). Kubonakala ukuba i-bias ye-post-stratification estimator is:

$bias(\hat{\bar{y}}_{post}) = \frac{1}{N} \sum_{h=1}^H \frac{N_h cor(\phi, y)^{(h)} S(y)^{(h)} S(\phi)^{(h)}}{\bar{\phi}^{(h)}} \qquad(3.8)$

apho $$cor(\phi, y)^{(h)}$$ , $$S(y)^{(h)}$$ , $$S(\phi)^{(h)}$$ , kunye $$\bar{\phi}^{(h)}$$ bachazwa njengentla ngasentla kodwa banqatshelwe kubantu kwiqela $$h$$ (Bethlehem, Cobben, and Schouten 2011, sec. 8.2.1) . Ngaloo ndlela, i-bias jikelele iya kuba yincinci ukuba i-bias kwiqela ngalinye le-post-stratification lincinci. Kukho iindlela ezimbini endikuthanda ukuzenza malunga nokwenza i-bias encinane kwinqanaba ngalinye le-post-stratification. Okokuqala, ufuna ukuzama ukwakha amaqela axhamlayo apho kukho ukutshintshana okuncinane kwindlela yokuphendula ( $$S(\phi)^{(h)} \approx 0$$ ) kunye nesiphumo ( $$S(y)^{(h)} \approx 0$$ ). Okwesibini, ufuna ukwakha amaqela apho abantu obubonayo banjengabantu ongaboniyo ( $$cor(\phi, y)^{(h)} \approx 0$$ ). Ukuthelekisa iq. 3.7 kunye neq. 3.8 unceda ukucacisa xa i-post-stratification inokunciphisa i-biased ngenxa yokungabi nantoni.

Ekugqibeleni, eli candelo linikeze imodeli yokuba kungenzeka ukuba isampuli singaphenduli kwaye ibonise i-bias ukuba ngaphandle kokungeniswa kungeniswa kokubili ngaphandle kunye kunye nokulungiswa kwe-post-stratification. Bethlehem (1988) inikezela ngokubaluleka kwezinto ezibangelwa ukungabi naluphi uhlobo lwezinto zokucwangcisa eziqhelekileyo. Ngolunye ulwazi ekusebenziseni i-post-stratification ukulungelelanisa ukungabi nxamnye, bona Smith (1991) kunye Gelman and Carlin (2002) . I-post-stratification inxalenye yentsapho epheleleyo yezobugcisa ezibizwa ngokuba yi-estimation calibration, bona iZhang (2000) yonyango Särndal and Lundström (2005) -length kunye ne- Särndal and Lundström (2005) ukwenzela unyango Särndal and Lundström (2005) elide. Ngolunye ulwazi kwezinye iindlela zokulinganisela ukulungelelanisa ukungabi nantoni, jonga Kalton and Flores-Cervantes (2003) , Brick (2013) , kunye Särndal and Lundström (2005) .

Isampula esingenakwenzeka

Isampula esingenakwenzekayo kubandakanya iindidi ezahlukeneyo zoyilo (Baker et al. 2013) . Ukugxininisa ngokukodwa kwisampuli se-Xbox abasebenzisi baka-Wang kunye nabo basebenzelana nabo (W. Wang et al. 2015) , unokucinga ngaloo hlobo lwesampula njengenye apho inxalenye eyintloko yokwakha isampula akuyiyo $$\pi_i$$ ( $$\phi_i$$ ngumphandi wokufakwa) kodwa i- $$\phi_i$$ (ukuphendula ngokuphendula ngokuphendula). Ngokwemvelo, oku akunjalo kuba i $$\phi_i$$ ayingaziwa. Kodwa, njengoko uWang kunye nabalingane bakhe babonisa, le ndlela yokukhetha isampula-ukususela kwisakhelo sampulu kunye nephutha elikhulu lokufihla-akumele libe yintlekele ukuba umphandi unolwazi oluncedisayo kunye nesimo esihle samanani ukuphendula ngale ngxaki.

Bethlehem (2010) iphakamisa ezininzi iziphumo ezikhankanywe ngasentla malunga ne-post-stratification ukuba zibandakanye zombini ukungabikho kokungaziphathi kakuhle kunye neziphene zokufikelela. Ukongeza kwi-post-stratification, ezinye iindlela zokusebenza kunye (Ansolabehere and Rivers 2013; ??? ) nantoni-zibandakanya ukulinganisa isampula (Ansolabehere and Rivers 2013; ??? ) , ukulinganisa amanqaku okulinganisa (Lee 2006; Schonlau et al. 2009) , kunye nokulinganiswa (Lee and Valliant 2009) . Omnye umxholo oqhelekileyo phakathi kwezi ndlela kukusetyenziswa kolwazi oluncedisayo.