## Zolemba masamu

Muzowonjezereka izi, ndikufotokozera zina mwazigawo zomwe zili m'mutuwu mu mawonekedwe pang'ono a masamu. Cholinga apa ndi kukuthandizani kuti mukhale omasuka ndi zolemba ndi masamu omwe amagwiritsidwa ntchito ndi akatswiri ofufuza kuti mutha kusintha zina mwazinthu zamakono zolembedwa pamituyi. Ndikuyamba poyesa zitsanzo zopezeka, ndikusunthira ku sampuli ndi osayanjanitsika, ndipo potsiriza, osakhala ndi zitsanzo.

Zotsatira zothekera

Monga chitsanzo, tiyeni tione cholinga cha kuyerekezera chiwerengero cha umphawi ku United States. Lolani $$U = \{1, \ldots, k, \ldots, N\}$$ akhale chiwerengero cha anthu ndipo alole $$y_k$$ phindu la zotsatira zosinthika kwa munthuyo $$k$$ . Mu chitsanzo ichi $$y_k$$ ndikuti munthu $$k$$ alibe ntchito. Potsirizira pake, lolani $$F = \{1, \ldots, k, \ldots, N\}$$

Chitsanzo choyambirira cha sampuli ndi sampuli yopanda phindu popanda kusinthidwa. Pankhaniyi, munthu aliyense ali ndi mwayi wophatikizidwa mu chitsanzo $$s = \{1, \ldots, i, \ldots, n\}$$ . Pamene deta ikusonkhanitsidwa ndi kapangidwe kameneka, ochita kafukufuku akhoza kulingalira kuti chiŵerengero cha kusowa kwa ntchito ndi chitsanzo chimatanthauza:

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

kumene $$\bar{y}$$ ndi chiwerengero cha kusowa kwa ntchito pakati pa anthu ndipo $$\hat{\bar{y}}$$ ndikulingalira kwa chiwerengero cha kusowa kwa ntchito ( $$\hat{ }$$ kawirikawiri amagwiritsira ntchito kusonyeza kulingalira).

Ndipotu, kafukufuku sagwiritsa ntchito sampuli mosavuta. Pa zifukwa zosiyanasiyana (chimodzi mwa zomwe ine nditi ndifotokoze mu mphindi), ofufuza nthawi zambiri amapanga zitsanzo zosagwirizana zofanana. Mwachitsanzo, ochita kafukufuku angasankhe anthu ku Florida ali ndi mwayi waukulu wophatikizapo kuphatikizapo anthu ku California. Pachifukwa ichi, chitsanzocho chimatanthauza (eq. 3.1) sichingakhoze kukhala cholingalira chabwino. M'malo mwake, ngati pali zosayerekezereka zowonjezera, ofufuza amagwiritsa ntchito

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

kumene $$\hat{\bar{y}}$$ ndikulingalira kwa chiwerengero cha kusowa ntchito ndipo $$\pi_i$$ ndi mwayi wa munthu $$i$$ ' $$i$$ wa kuphatikizidwa. Potsatira ndondomeko yoyenera, ndiyitanitsa woyimilira mu eq. 3.2 Wowonetsera Horvitz-Thompson. Werenganitsa wa Horvitz-Thompson ndiwothandiza kwambiri chifukwa umapangitsa kulingalira kosaganizira kuti mwina zingapangidwe bwanji (Horvitz and Thompson 1952) . Chifukwa chakuti woyang'anira Horvitz-Thompson akubwera mobwerezabwereza, ndi zothandiza kuzindikira kuti zikhoza kulembedwa kachiwiri

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

kumene $$w_i = 1 / \pi_i$$ . Monga aq. 3.3 amavomereza, woyerekeza wa Horvitz-Thompson ndi chitsanzo cholemetsa chimatanthauza pamene zolemerazo zimagwirizana kwambiri ndi mwayi wosankhidwa. Mwa kuyankhula kwina, mwina munthu sangakhale nawo m'chitsanzocho, kulemera kwake komwe munthu ayenera kulowa muyeso.

Monga tafotokozera poyamba, kafukufuku amawonetsa anthu omwe ali ndi zifukwa zosiyana zowonjezera. Chitsanzo chimodzi cha mapangidwe omwe angapangitse kusalinganizana koyenera kukhala nawo ndizoyimira zitsanzo , zomwe ndizofunika kumvetsetsa chifukwa zimagwirizana kwambiri ndi ndondomeko yoyesa kutchedwa stratification . Sampangidwe yosakanizidwa, wofufuzira amawononga chiwerengero cha anthu kuti akhale $$H$$ osiyana ndi omwe ali osiyana. Magulu awa amatchedwa strata ndipo amasonyezedwa ngati $$U_1, \ldots, U_h, \ldots, U_H$$ . Mu chitsanzo ichi, mndandanda uli ndi mayiko. Kukula kwa magulu kukuwonetsedwa monga $$N_1, \ldots, N_h, \ldots, N_H$$ . Wofusayo angafune kugwiritsa ntchito sampuli yokhala ndi cholinga kuti atsimikizire kuti ali ndi anthu okwanira mu boma lililonse kuti apange kulingalira kwa msinkhu wa umphawi.

Kamodzi anthu chagawikana nakwera strata, amaganiza kuti kafukufuku wa amasankha mtengo mwachisawawa chitsanzo yosavuta popanda m'malo a kukula $$n_h$$ , paokha lililonse strata. Komanso, ganizirani kuti aliyense amene wasankhidwa mu chitsanzo amakhala wofunsayo (Ndikugwira ntchito yosayankha mu gawo lotsatira). Pankhaniyi, mwayi wokhalapo ndi

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

Chifukwa chakuti izi zikhoza kukhala zosiyana pakati pa munthu ndi munthu, pakupanga chiwerengero cha zojambulazo, ochita kafukufuku amafunika kulemetsa aliyense amene akumuyankhayo mwachindunji cha mwayi wawo wophatikizapo pogwiritsa ntchito woyang'anira Horvitz-Thompson (tsamba 3.2).

Ngakhale kuti Woweruza wa Horvitz-Thompson alibe tsankho, ofufuza akhoza kufotokoza molondola (kutanthauza, kusiyana kwakukulu) kulingalira mwa kuphatikiza chitsanzo ndi chithandizo chothandizira . Anthu ena amaona kuti n'zosadabwitsa kuti izi ndi zoona ngakhale pamene pali zitsanzo zenizeni zowonongeka. Njirazi zogwiritsira ntchito zothandizira othandizira ndizofunikira kwambiri chifukwa, monga momwe ndisonyezera panthawi ina, mfundo zothandiza ndizofunikira pakupanga zowerengera kuti mwina zingapangidwe ndi osapereka komanso zosakhala zotsalira.

Njira imodzi yodziwiritsira ntchito zothandizira othandizira ndi yopangidwira . Mwachitsanzo, tangoganizirani kuti wofufuza amazindikira chiwerengero cha amuna ndi akazi mu zigawo 50; tikhoza kutanthauzira kukula kwa gulu ngati $$N_1, N_2, \ldots, N_{100}$$ . Kuphatikiza mfundo izi zothandizira ndi zitsanzo, wofufuzirayo akhoza kupatulira chitsanzochi m'magulu a $$H$$ (pankhaniyi 100), pangani kulingalira kwa gulu lirilonse, ndiyeno pangani chiwerengero chokwanira cha gululi amatanthauza:

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

Zambiri, woyimilira mu eq. 3.5 akhoza kukhala olondola kwambiri chifukwa amagwiritsira ntchito chidziwitso cha anthu - $$N_h$$ -kuti chiwerengero cholondola ngati chithunzi chosasinthika chimachitika. Njira imodzi yoganizira za izi ndikuti chingwe chotsatira chithunzichi chikufanana ndi momwe zimakhalira pambuyo poti deta yasonkhanitsidwa kale.

Pomalizira, gawo lino lafotokoza zojambula zochepa chabe: zitsanzo zophweka zopanda malire, zitsanzo zopanda malire, ndi zitsanzo zotsatizana. Wafotokozanso mfundo zikuluzikulu ziwiri zokhudza kulingalira: Wowonetsera Horvitz-Thompson ndi pambuyo pake. Kuti mudziwe tsatanetsatane wa zojambulazo, onani mutu 2 wa Särndal, Swensson, and Wretman (2003) . Kuti mupeze chithandizo chokwanira komanso chokwanira cha sampuli, onani gawo 3.7 la Särndal, Swensson, and Wretman (2003) . Kuti mudziwe zambiri zokhudza malo owonetsera Horvitz-Thompson, onani Horvitz and Thompson (1952) , Overton and Stehman (1995) , kapena gawo 2.8 la @ sarndal_model_2003. Kuti mupeze chithandizo chowonjezereka cholemba chithunzi, onani Holt and Smith (1979) , Smith (1991) , Little (1993) , kapena 7.6 ya Särndal, Swensson, and Wretman (2003) .

Zomwe zingatheke kusinthana ndi zosayenera

Pafupifupi onse kufufuza kwenikweni alibe; ndiko kuti, si onse omwe ali chitsanzo cha anthu akuyankha funso lirilonse. Pali mitundu iwiri ikuluikulu yopanda malire: chinthu chosagwirizana ndi china chilichonse . Mu chinthu chopanda ulemu, ena omwe amafunsidwa samayankha zinthu zina (mwachitsanzo, nthawi zina anthu omwe amafunsidwa sakufuna kuyankha mafunso omwe amawaganizira kuti ndi ofunika). Mu bungwe losavomerezeka, anthu ena omwe amasankhidwa kuti akhale chitsanzo cha anthu sagwirizane ndi kafukufuku konse. Zifukwa ziwiri zomwe zimapangitsa kuti munthu asapitirize kutengapo mbali ndi kuti munthu sangathenso kulankhulana ndipo munthu woyesedwayo amalembedwa koma amakana kutenga nawo mbali. Mu gawo ili, ine ndikuyang'ana pa unit nonresponse; Owerenga chidwi ndi chinthu chopanda pake ayenera kuona Little ndi Rubin (2002) .

Ofufuza kawirikawiri amaganiza za kufufuza ndi gawo lopanda kuyankha ngati ndondomeko ya magawo awiri. Pachiyambi choyamba, wofufuzirayo amasankha chitsanzo $$s$$ chomwe munthu aliyense ali nacho chophatikizidwa $$\pi_i$$ (kumene $$0 < \pi_i \leq 1$$ ). Ndiye, mu gawo lachiwiri, anthu omwe asankhidwa mu chitsanzo amavomereza ndizotheka $$\phi_i$$ (kumene $$0 < \phi_i \leq 1$$ ). Ndondomeko ya magawo awiriwa amachititsa omaliza omaliza a $$r$$ . Kusiyana kwakukulu pakati pa magawo awiriwa ndikuti ochita kafukufuku amawongolera ndondomeko yosankha zitsanzo, koma sadziwa kuti ndi ndani mwa iwo omwe atengedwa. Kuyika njira ziwiri izi palimodzi, mwinamwake kuti wina akhale woyankha ndi

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

Kuti ndikhale wophweka, ndimaganizira za momwe zowonongeka zowonongeka ndizosawerengeka mosavuta. Ngati wofufuzira amasankha zitsanzo za kukula $$n_s$$ zomwe zimapereka $$n_r$$ , ndipo ngati wofufuzirayo amanyalanyaza osayankha ndipo amagwiritsa ntchito tanthawuzo la omwe afunsidwa, ndiye kuti chiwerengero cha kulingalira chidzakhala:

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

$$cor(\phi, y)$$ ndi mgwirizano wa anthu pakati pa mayankho ndi zotsatira zake (mwachitsanzo, udindo wa umphawi), $$S(y)$$ chiwerengero cha anthu (mwachitsanzo, kusowa ntchito chikhalidwe), $$S(\phi)$$ ndiyomwe anthu amasiya kutembenuka, ndipo $$\bar{\phi}$$ ndizofunika zowonjezera anthu (Bethlehem, Cobben, and Schouten 2011, sec. 2.2.4) .

Eq. 3.7 akusonyeza kuti kulemekeza sikungayambe kusonyeza kuti pali zosayenera ngati zilipo izi:

• Palibe kusiyana pakati pa ulova wa ntchito $$(S(y) = 0)$$ .
• Palibe kusiyana pakati pazidziwitso zowonongeka $$(S(\phi) = 0)$$ .
• Palibe mgwirizano pakati pa kuyankhidwa kwachangu ndi udindo wosagwira ntchito $$(cor(\phi, y) = 0)$$ .

Tsoka ilo, palibe chimodzi mwa zinthu izi zikuwoneka. Zikuwoneka kuti sizingatheke kuti sipadzakhalanso kusintha kwa ntchito kapena kuti sipadzakhala kusiyana pakati pa zowonongeka. Choncho, mawu ofunika mu eq. 3.7 ndi mgwirizano: $$cor(\phi, y)$$ . Mwachitsanzo, ngati anthu omwe sagwira ntchito angathe kuwayankha, ndiye kuti chiwerengero cha ntchito chidzasinthidwa mmwamba.

Chizoloŵezi chopanga kulingalira pamene kulibe kudziletsa ndiko kugwiritsa ntchito chithandizo chothandizira. Mwachitsanzo, njira imodzi yomwe mungagwiritsire ntchito zothandizira othandizira ndizochitsulo (kumbukirani tsamba 3.5 kuchokera pamwamba). Izi zikusonyeza kuti chiyanjano cha kulingalira kwa-post-stratification ndi:

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

" $$cor(\phi, y)^{(h)}$$ , $$S(y)^{(h)}$$ , $$S(\phi)^{(h)}$$ , ndipo $$\bar{\phi}^{(h)}$$ amatanthauzidwa ngati pamwambapa koma amalembedwa kwa anthu mu gulu $$h$$ (Bethlehem, Cobben, and Schouten 2011, sec. 8.2.1) . Choncho, chisankho chonse chidzakhala chaching'ono ngati gulu lirilonse likuchepa. Pali njira ziwiri zomwe ndimakonda kuganizira zokhala ndi zochepa mu gulu lililonse lamasewera. Choyamba, mukufuna kuyesa kupanga magulu osiyana omwe pali kusiyana kwakukulu koyankhidwa ( $$S(\phi)^{(h)} \approx 0$$ ) ndi zotsatira ( $$S(y)^{(h)} \approx 0$$ ). Chachiwiri, mukufuna kupanga magulu omwe anthu omwe mumawawona ali ngati anthu omwe simukuwawona ( $$cor(\phi, y)^{(h)} \approx 0$$ ). Kuyerekeza aq. 3.7 ndi eq. 3.8 kumathandiza kumvetsetsa pamene chingwechi chitha kuchepetsa ubwino chifukwa cha kusalankhula.

Pomalizira, gawo ili lapereka chitsanzo cha zitsanzo zotsatila ndi zosayankhidwa ndikuwonetsera chisankho chomwe kusalongosola kungathe kufotokozera zonse popanda komanso kusinthika kwazithunzi. Bethlehem (1988) imatulutsa chisokonezo chomwe sichimapangidwa chifukwa chosaganizira zojambula zambiri. Kuti mudziwe zambiri pogwiritsa ntchito chingwechi kuti musinthe, onani Smith (1991) ndi Gelman and Carlin (2002) . Mndandanda wa stratification ndi mbali ya njira zambiri zomwe zimatchulidwa kuti ziwerengero zowonongeka, onani Zhang (2000) chithandizo cha kutalika kwa nkhani ndi Särndal and Lundström (2005) kuti Särndal and Lundström (2005) chithandizo cha nthawi yaitali. Kuti mudziwe zambiri pazinthu zina zolemetsa kuti musinthe, onani Kalton and Flores-Cervantes (2003) , Brick (2013) , ndi Särndal and Lundström (2005) .

Zosatheka kukhala zitsanzo

Zomwe sizingatheke zimaphatikizapo zojambula zosiyanasiyana (Baker et al. 2013) . Poganizira mozama za zitsanzo za Xbox zomwe Wang ndi anzake adagwiritsa ntchito (W. Wang et al. 2015) , mukhoza kuganizira za mtundu umenewu ngati imodzi yomwe gawo lofunika la kapangidwe ka sampuli si $$\pi_i$$ ( mwayi wotsatiridwa wotsatiridwa wa kuphatikizidwa) koma $$\phi_i$$ (zomwe zimayankhidwa ndi anthu oyankha). Mwachibadwa, izi siziri bwino chifukwa $$\phi_i$$ sadziwika. Koma, monga Wang ndi anzake adasonyezera, mtundu uwu wothandizira-ngakhale kuchokera pa chithunzi chokhala ndi zolakwika zazikulu-sikuyenera kukhala zoopsa ngati wofufuzayo ali ndi mfundo zabwino zothandizira komanso chitsanzo chabwino kuti afotokoze mavutowa.

Bethlehem (2010) imapereka zochuluka zopezeka pamwambapa zazomwe zimatchulidwa pambuyo pa stratification kuti ziphatikize zolakwika zonse zopanda malire. Kuphatikizira pazinthu zowonjezera, njira zina zogwirira ntchito ndi zosakhala zosatheka-ndipo mwinamwake zingaphatikizepo ndi zolakwika zowonongeka ndi zina zomwe (Ansolabehere and Rivers 2013; ??? ) , propensity score weighting (Lee 2006; Schonlau et al. 2009) , ndi kuyerekezera (Lee and Valliant 2009) . Mutu umodzi wodziwika pakati pa njirazi ndi kugwiritsa ntchito mauthenga othandizira.