## Mathematical notes

Muchidimbu ichi, ndicharondedzera mamwe mazano kubva muchitsauko mune fomu zvishoma zvemasvomhu. Chinangwa pano ndechekukubatsira kuti uwane rusununguko nehuwandu hwemasvomhu uye masvomhu inoshandiswa nevanotsvakurudza tsvakurudzo kuitira kuti iwe ugone kuchinja kune zvimwe zvezvinyorwa zvekugadzira zvakanyorwa pamisoro idzi. Ini ndichatanga nokuisa mukana wokuenzanisira, uye enda kune imwe nguva sampling nekusava nehanya, uye pakupedzisira, kwete-zvingave zvigadziriswa sampling.

Probability sampling

Semuenzaniso unofambisa, ngatitarisei chinangwa chekufungidzira kushaya kwemabasa muUnited States. Rega $$U = \{1, \ldots, k, \ldots, N\}$$ vave vanhu $$y_k$$ uye regai $$y_k$$ nekukosha kwechigumisiro chinokonzerwa nemunhu $$k$$ . Mumuenzaniso uyu $$y_k$$ ndewokuti munhu $$k$$ haashandi. Pakupedzisira, regai $$F = \{1, \ldots, k, \ldots, N\}$$ vave chimiro chevanhu, izvo nekuda kwekufungidzirwa zvinofungidzirwa kuti zvakafanana nechinangwa chevanhu.

Shanduro yakakosha yakagadziriswa iri nyore shanduro sampuli isina kushandurwa. Munyaya iyi, munhu mumwe nomumwe anokwanisa kunge akabatanidzwa mumuenzaniso $$s = \{1, \ldots, i, \ldots, n\}$$ . Apo dheta iri kuunganidzwa nemuenzaniso wekugadzirisa, vatsvakurudzi vanogona kufungidzira kuwanda kwehuwandu hwekushaya kwehuwandu hwemabasa nemuenzaniso hunoreva:

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

apo $$\bar{y}$$ isiri yehuwandu hwemabasa muhuwandu uye $$\hat{\bar{y}}$$ kuenzanisa kwekushaya kwehuwandu hwebasa ( $$\hat{ }$$ is commonly yakashandiswa kuratidza muongorori).

Muzvokwadi, vanotsvakurudza havawanzoshandisa shanduro dzisinganzwisisiki pasina kushandurwa. Nokuda kwezvikonzero zvakasiyana-siyana (rimwe randicharondedzera mune imwe nguva), vatsvakurudzi vanowanzogadzira zvidzitiro zvisinganzwisisiki zvingaita zvekubatanidzwa. Somuenzaniso, vatsvakurudzi vangasarudza vanhu muFlorida vane mikana yakawanda yekubatanidzwa kunze kwevanhu muCalifornia. Muchiitiko ichi, samuenzaniso inoreve (iq. 3.1) inogona kunge isingave yakakosha. Pane kudaro, kana pane kukwana kusinganzwisisiki kwekuiswa, vatsvakurudzi vanoshandisa

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

apo $$\hat{\bar{y}}$$ kuenzanisa kwekushaya kwehuwandu hwemabasa uye $$\pi_i$$ munhu $$i$$ s probability of inclusion. Kutevedzera tsika yakazara, ini ndichatiidza muongorori muq. 3.2 muHorvitz-Thompson anofungidzira. Muongorori weHorvitz-Thompson unonyanya kukosha nekuti unotungamirira kusinganzwisisiki kusinganzwisisiki kwehupi zvingangodaro sampling yekugadzira (Horvitz and Thompson 1952) . Nemhaka yokuti muongorori weHorvitz-Thompson anouya kazhinji, zvinobatsira kuona kuti rinogona kunyorwa zvakare se

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

apo $$w_i = 1 / \pi_i$$ . As eq. 3.3 inoratidzira, muongorori weHorvitz-Thompson muenzaniso unorema unoreve kuti zviyero zvinopesana zvakadini nekukwanisa kusarudzwa. Mune mamwe mazwi, zvisingabviri kuti munhu ave akabatanidzwa mumuenzaniso, kuwedzera kukura kwokuti munhu anofanira kuwanikwa mumutengo.

Sezvakatsanangurwa kare, vatsvakurudzi vanowanzotarisa vanhu vane maitiro asina kukodzera ekubatanidzwa. Chimwe chimiro chekugadzirwa kunogona kutungamirira kusina kufanana kwezvakaitika zvekuiswa kune zvakakoserwa shanduro , iyo inokosha kunzwisisa nokuti inowirirana zvikuru nekuenzanisira nzira inonzi post-stratification . Mukati rakagadzirirwa sampling, mumwe mutsvakurudzi anoparadzanisa vanhu vanofungidzirwa mu $$H$$ pamwe chete nemapoka akazara. Aya mapoka anonzi chidimbu uye anoratidzwa se $$U_1, \ldots, U_h, \ldots, U_H$$ . Mumuenzaniso uyu, chidimbu ndeche nyika. Mazana emapoka anoratidzwa se $$N_1, \ldots, N_h, \ldots, N_H$$ . Mumwe mutsvakurudzi angave achida kushandisa shanduro yakagadziriswa kuitira kuti ave nechokwadi chokuti ane vanhu vakakwana muhurumende imwe neimwe kuti vaite maonero ehurumende ekushaya basa.

Kana imwe nguva vagari vakaparadzaniswa vachiita chidimbu , fungidzira kuti muongorori anosarudza soro risina kushandiswa pasina kushandiswa kwemazana $$n_h$$ , pasina chimiro kubva kune rumwe rutivi. Uyezve, fungidzira kuti munhu wose akasarudzwa mumuenzaniso anova muteereri (Ini ndichabata kwete mhinduro mune chikamu chinotevera). Munyaya iyi, mukana wokubatanidzwa ndewe

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

Nemhaka yokuti izvi zvinokonzerwa zvinogona kusiyana kubva kumunhu kusvika kune munhu, paanenge achienzanisa kubva kune shanduro iyi yakagadzirwa, vanotsvakurudza vanoda kurerutsa munhu wese anopindura nemutsara wehutano hwavo hwokubatanidzwa vachishandisa Horvitz-Thompson estimator (ndima 3.2).

Kunyange zvazvo muongorori weHorvitz-Thompson asingasaruri, vatsvakurudzi vanogona kubudisa zvakanyatsoruramisa (kureva,, kusiyana-siyana) kuenzanisa nekubatanidza samuenzaniso nemashoko anobatsira . Vamwe vanhu vanozviona zvisingashamisi kuti izvi ndezvechokwadi kunyange kana paine kunyatsokwanisa kuurawa sampling. Iyi nzira yekushandisa ruzivo rwebetsero inonyanya kukosha nokuti, sezvandichazozviratidza gare gare, ruzivo rwebetsero inokosha pakugadzirisa kuenzanisa kubva zvichida kuenzanisa pasina kusateerera uye kubva kune zvisingabviri zvidzidzo.

Imwe nzira inowanzoshandiswa yekushandisa ruzivo rwekubatsira ndeyekuita-stratification . Fungidzira, somuenzaniso, kuti muongorori anoziva nhamba yevarume nevakadzi mune imwe neimwe ye50 inotaura; tinogona kureva mazana eboka aya se $$N_1, N_2, \ldots, N_{100}$$ . Kubatanidza ruzivo rwekubatsira nemuenzaniso, muongorori anogona kuparadzanisa sampuli muzvikwata zve $$H$$ (munyaya ino 100), ita chiyero cheboka rimwe nerimwe, uye ezvo ugoita chiyero chakarehwa cheboka iri zvinoreva:

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

Zvichida, muongorori muq. 3.5 inogona kunge iri yakarurama kwazvo nokuti inoshandisa ruzivo rwehuwandu hwevanhu-iyo $$N_h$$ -kuti yakarongororwa inofanirwa kana shanduro isina kukodzera inosarudzwa kusarudzwa. Imwe nzira yekufunga pamusoro payo ndeyekuti kuiswa kwe-post-stratification kwakafanana nekufungidzira kushambadzira mushure mekunge data yave yatotorwa.

Mukupedzisa, chikamu chino chakatsanangura zvishomanana zvinyorwa zvakagadziriswa: shanduro dzisinganzwisisiki dzisingashanduri, shanduro isina kuenzana, uye stratified sampling. Yakatsanangurawo mazano makuru maviri pamusoro pekufungidzira: Horvitz-Thompson estimator uye post-stratification. Kuti uwane tsanangudzo yakawanda yezvigadziriswe sampling zvirongwa, ona chitsauko 2 Särndal, Swensson, and Wretman (2003) . Kuti uwane kurapwa kwakakwana uye kwakakwana kwekatevedzwa sarudzo, ona chikamu 3.7 Särndal, Swensson, and Wretman (2003) . Nokuda kwetsananguro yezvakagadzirwa zvezvinhu zveHorvitz-Thompson estimator, ona Horvitz and Thompson (1952) , Overton and Stehman (1995) , kana chikamu 2.8 che @ sarndal_model_2003. Kuti uwane hutano hwakanyanya hwemashure-stratification, ona Holt and Smith (1979) , Smith (1991) , Little (1993) , kana chikamu 7.6 Särndal, Swensson, and Wretman (2003) .

Zvichida sampling nekusaremekedza

Inenge yose inotsvakurudza chaiyo isina hanya; iyo ichiti, havasi vose vari mumuenzaniso wevanhu vanopindura mibvunzo yese. Kune mhando miviri mikuru yekusaremekedza: chinhu chisina kuremekedza uye chikwata chisingatauri . Mune chinhu chisingaremekedzi, vamwe vanopindura havapinduri zvimwe zvinhu (semuenzaniso, pane dzimwe nguva vanenge vapindura havadi kupindura mibvunzo yavanofunga kuti vanofunga). Muchikoro chisina kuremekedza, vamwe vanhu vakasarudzwa nokuda kwemuenzaniso wevanhu havabvumi kuongororo zvachose. Izvo zvikonzero zviviri zvakajairika zveununiti kusaremekedza ndezvokuti munhu akashandurwa haakwanisi kuonana uye munhu wekuenzanisa anongobvunzwa asi anoramba kubatanidzwa. Muchikamu chino, ini ndichaisa pfungwa pane chimwe chinhu chisina kuremekedza; vaverengi vanofarira chinhu chisingaremekedzi vanofanira kuona Little naRubin (2002) .

Vanotsvakurudza kazhinji vanofunga nezvekuongorora pamwe neununiti kwete-mhinduro sematanho maviri-shanduro yekuita. Mutsara wokutanga, muongorori anosarudza sampu $$s$$ yakadai kuti munhu mumwe nomumwe ane mukana wokubatanidzwa $$\pi_i$$ (apo $$0 < \pi_i \leq 1$$ ). Zvadaro, muchikamu chechipiri, vanhu vakasarudzwa mumuenzaniso vanopindura nevanokwanisa $$\phi_i$$ (apo $$0 < \phi_i \leq 1$$ ). Iyi nhanho yematanho maviri inoguma mugadziro yekupedzisira yevakapindura $$r$$ . Chimwe chinhu chakakosha pakati pezvikamu izvi zviviri ndezvokuti vatsvakurudzi vanodzora nzira yekusarudza samuenzaniso, asi havatauri kuti ndeupi wevanhu vakadhaniwa vanove vapindura. Kuisa zvirongwa zviviri izvi pamwe chete, mukana wekuti mumwe munhu achava mhinduro ndeye

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

Nokuda kwekuita nyore, ini ndichafunga nyaya iyo yakasikwa shanduro yakagadzirwa isinganzwisisiki shanduro pasina kushandurwa. Kana muongorori achisarudza muenzaniso wekukura $$n_s$$ iyo inobereka $$n_r$$ vapindura, uye kana muongorori akanganwa kwete mhinduro uye anoshandisa zvinorehwa nevakabvunzwa, ipapo kusarura kwekufungidzira kuchave:

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

uko $$cor(\phi, y)$$ kuwirirana kwevanhu pakati pemhinduro yekuita uye chigumisiro (semuenzaniso, kutadza kushanda kwemabasa), $$S(y)$$ inharaunda yakasiyana-siyana yevanhu (zvichida, kusarasha chimiro), $$S(\phi)$$ inharaunda yevanhu inokanganisa kupindura kwemaitiro, uye $$\bar{\phi}$$ ndiyo inoshandiswa nehuwandu hwehupfumi hwemhinduro (Bethlehem, Cobben, and Schouten 2011, sec. 2.2.4) .

Eq. 3.7 inoratidza kuti kusaremekedza hakungatauri kukanganisa kana chimwe chezviitiko zvinotevera zvichisangana:

• Hakuchina kuchinja kwehutano hwemabasa $$(S(y) = 0)$$ .
• Hakuchina kuchinja mukupindurwa kwemagetsi $$(S(\phi) = 0)$$ .
• Hapana kuwirirana pakati pekupindura kwemaitiro uye kushaiwa kwemaitiro $$(cor(\phi, y) = 0)$$ .

Zvinosuruvarisa, hapana chimwe chezviitiko izvi chinoratidzika. Zvinoratidzika kusinganzwisisiki kuti hakuzovi nekusiyana kwehutano hwemabasa kana kuti hakuchazovi nekusiyana mumhinduro yekupindura. Nokudaro, izwi rechirevo muq. 3.7 ndiko kuwirirana: $$cor(\phi, y)$$ . Semuenzaniso, kana vanhu vasina basa vasingakwanisi kupindura, ipapo inotaridzirwa yehuwandu hwemabasa ichave yakasarudzwa kumusoro.

Nzira yekuita kuongororwa kana pasina kusaremekedza ndeyokushandisa ruzivo rubatsiro. Somuenzaniso, imwe nzira yaungashandisa nayo rubatsiro rwekubatsira ndeyekunyora-stratification (yeuka eq 3.5 kubva kumusoro). Icho chinotarisa kuti kushamwaridzana kwe-post-stratification estimator ndeyokuti:

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

$$cor(\phi, y)^{(h)}$$ , $$S(y)^{(h)}$$ , $$S(\phi)^{(h)}$$ , uye $$\bar{\phi}^{(h)}$$ zvinotsanangurwa sepamusoro asi zvinongoratidzwa kuvanhu vari muboka $$h$$ (Bethlehem, Cobben, and Schouten 2011, sec. 8.2.1) . Nokudaro, kukashamwaridzana kwese kuchave kuduku kana kushamiswa kwega rimwe nerimwe kwema-stratification boka ridiki. Pane nzira mbiri dzandinoda kufunga pamusoro pekuita sarudzo shoma pane imwe neimwe ye-post-stratification group. Chokutanga, unoda kuedza kuumba mapoka akasiyana-siyana apo pane kuchinja kwakanyanya mukupindura kwepanyika ( $$S(\phi)^{(h)} \approx 0$$ ) uye mugumisiro ( $$S(y)^{(h)} \approx 0$$ ). Chechipiri, unoda kuumba mapoka apo vanhu vaunoona vakaita sevanhu vamusingaoni ( $$cor(\phi, y)^{(h)} \approx 0$$ ). Kuenzanisa eq. 3.7 uye eq. 3.8 rubatsiro runotsanangura kana kushambadzira kwemashure kunogona kuderedza kusarura kunokonzerwa nekusaremekedza.

Mukupedzisa, chikamu chino chakapa muenzaniso wekugadzirisa shanduro nekusava nemhinduro uye kuratidzira kusarura kwekuti kusaremekedza kunogona kuisa zvose pasina uye nekugadziriswa kwemashure. Bethlehem (1988) inopa kubviswa kwechisarudzo chinokonzerwa nekusaona mamwe maitiro ekugadzira shanduro. Nokuda kwekushandisa kushandisa-stratification kugadzirisa kusava nehanya, ona Smith (1991) Gelman and Carlin (2002) . Post-stratification chikamu chemhuri yakawanda yehutano Särndal and Lundström (2005) , ona Zhang (2000) yehutano-urefu urefu uye Särndal and Lundström (2005) yehutano hwehurefu hwebhuku. Nokuda kwezvimwewo dzimwe nzira dzekuyera dzekugadzirisa kusava nehanya, ona Kalton and Flores-Cervantes (2003) , Brick (2013) , uye Särndal and Lundström (2005) .

Zvisiri-mikana sampling

Zvisingatenderwi sampling zvinosanganisira zvakasiyana-siyana zvezvigadzirwa (Baker et al. 2013) . Kutarisa zvakananga pamusoro pechinyorwa che Xbox chekushandisa naWang uye vashandi (W. Wang et al. 2015) , unogona kufungidzira nezvechimiro chechimiro seimwe iyo chikamu chikuru chekugadzira sampuli haisi $$\pi_i$$ ( uyo anotsvakurudza-anotungamirirwa nemukana wekubatanidzwa) asi $$\phi_i$$ (inopindurwa-inotungamirirwa nemhinduro yekuita). Zvinonzwisisika, izvi hazvisi zvakanaka nokuti $$\phi_i$$ hazvizivikanwi. Asi, sezvakaita Wang uye avo vaaishanda navo, rudzi urwu rwekusarura-kunyangwe-kubva pachigadziro chemuenzaniso uye kukanganisa kwakawanda-hakufaniri kuva nengozi kana muongorori ane ruzivo rwebetsero hwakanaka uye nhamba yakanaka yekuenzanisira kuti aite zvematambudziko aya.

Bethlehem (2010) inowedzera zvakawanda zvezvinyorwa zvataurwa pamusoro pamusoro pekuisa-stratification kusanganisira zvose zvisingatauri uye zvikanganiso zvekugovera. Mukuwedzera kune-post-stratification, mamwe maitiro ekushanda nevasingaiti-zvidzidzo-uye zvichida samples nezvikanganiso zvekuvhara uye zvisina kureva-inosanganisira kuenzanisa (Ansolabehere and Rivers 2013; ??? ) , propensity score weighting (Lee 2006; Schonlau et al. 2009) , uye kuenzaniswa (Lee and Valliant 2009) . Chimwe chinhu chinowanzozivikanwa pakati pezvinhu izvi ndezvekushandiswa kwemashoko anobatsira.