## Zolemba masamu

Muzowonjezereka, ndikufotokozera mwachidule malingaliro okhudzana ndi kupanga chidziwitso kuchokera ku deta yosayesera mu mawonekedwe pang'ono a masamu. Pali njira zikuluzikulu ziwiri: maziko a graf, omwe amagwirizana ndi Yereya Pearl ndi anzake, komanso zida zomwe zingatheke, zomwe zimagwirizanitsidwa ndi Donald Rubin ndi anzake. Ndidzalongosola zomwe zingatheke chifukwa chogwirizana kwambiri ndi malingaliro a masamu kumapeto kwa chaputala 3 ndi 4. Kuti mudziwe zambiri pazithunzi zamakono, ndikulangiza Pearl, Glymour, and Jewell (2016) (choyamba ) ndi Pearl (2009) (patsogolo). Kuti mupeze chithandizo cha kutalika kwa buku la causal inference yomwe imaphatikizapo zotsatira zowonjezera zomwe zingatheke komanso chithunzi cha graus, Morgan and Winship (2014) .

Cholinga chazowonjezereka ndi kukuthandizani kuti mukhale omasuka ndi malemba ndi machitidwe a zikhalidwe zomwe mungathe kuchita kuti mutha kusintha zina mwazinthu zowonjezera zomwe zalembedwa pa mutu uwu. Choyamba, ndikufotokozera zomwe zingatheke. Kenaka, ndikugwiritsa ntchito kuti tikambirane za kuyesedwa kwachilengedwe monga kwa Angrist (1990) chifukwa cha ntchito ya usilikali pamalipiro. Izi zowonjezera Imbens and Rubin (2015) kwambiri Imbens and Rubin (2015) .

Ndondomeko ya zotsatira

Zokambirana zomwe zingatheke zimakhala ndi zinthu zitatu zazikuluzikulu: magawo , mankhwala , ndi zotsatira zabwino . Pofuna kufotokoza izi, tiyeni tione yankho la funso lofotokozedwa mu Angrist (1990) : Kodi zotsatira za utumiki wa usilikali pazopindula? Pachifukwa ichi, titha kufotokozera mayunitsi kuti akhale anthu oyenerera kulemba 1970 mu United States, ndipo tikhoza kulongosola anthu awa ndi $$i = 1, \ldots, N$$ . Njira zothandizira pa nkhaniyi zingakhale "kutumikira usilikali" kapena "kusatumikila usilikali." Ndidzatcha izi mankhwala ndi maulamuliro, ndipo ndidzalemba $$W_i = 1$$ ngati munthu $$i$$ ali mu chikhalidwe cha chithandizo ndipo $$W_i = 0$$ ngati munthu $$i$$ ali mu chikhalidwe cholamulira. Potsirizira pake, zotsatira zomwe zingatheke zimakhala zovuta kwambiri chifukwa zimakhudza zotsatira zowonjezera; zinthu zomwe zikanachitika. Kwa munthu aliyense woyenera kulembedwa mu 1970, tikhoza kulingalira kuchuluka kwa zomwe akanapeza mu 1978 ngati atatumikira ku usilikali, omwe ndiwatcha $$Y_i(1)$$ , ndi ndalama zomwe akanapeza 1978 ngati iwo sanatumikire ku usilikali, zomwe ine $$Y_i(0)$$ . Muzokhazikitsa zotsatira, $$Y_i(1)$$ ndi $$Y_i(0)$$ amawonedwa ngati $$W_i$$ , pamene $$W_i$$ ndi kusintha kosasintha.

Kusankhidwa kwa mayunitsi, chithandizo, ndi zotsatira ndizofunikira chifukwa zimatanthawuza zomwe zingathe-ndipo sizikhoza kuphunziridwa kuchokera ku phunziroli. Kusankhidwa kwa mayunitsi-anthu omwe ali oyenera kulembedwa mu 1970-sizimaphatikizapo akazi, ndipo kotero popanda kulingalira kwina, phunziro ili silidzatiwuza kanthu kena ka zotsatira za utumiki wa usilikali kwa amayi. Zosankha za momwe mungatanthauzire mankhwala ndi zotsatira ndi zofunikanso. Mwachitsanzo, kodi chithandizo chofuna chidwi chiyenera kugwiritsidwa ntchito pomenyera usilikali kapena kumenyana? Kodi zotsatira za chidwi ndizopindula kapena ntchito yokhutira? Pamapeto pake, kusankha mayunitsi, mankhwala, ndi zotsatira ziyenera kutsogoleredwa ndi zolinga za sayansi ndi ndondomeko za phunziroli.

Chifukwa cha kusankha kwa magulu, machiritso, ndi zotsatira zowonjezera, zotsatira za mankhwala pa munthu $$i$$ , $$\tau_i$$ , ndi

$\tau_i = Y_i(1) - Y_i(0) \qquad(2.1)$

M'mawu ena, tiyerekezera kuti munthu $$i$$ akanapeza ndalama zochuluka bwanji atatumizira momwe munthu $$i$$ akanapindulira popanda kutumikira. Kwa ine, eq. 2.1 ndi njira yabwino kwambiri yofotokozera zotsatira zake, ndipo ngakhale ziri zophweka kwambiri, izi zimapangidwa kuti zikhale zowonjezeka mu njira zambiri zofunika komanso zosangalatsa (Imbens and Rubin 2015) .

Pogwiritsira ntchito zida zowonjezereka, nthawi zambiri ndimaona kuti ndibwino kulemba tebulo lowonetsa zotsatira zomwe zingatheke komanso zotsatira za mankhwala pa magulu onse (tebulo 2.5). Ngati simungathe kulingalira tebulo ngati ili pa phunziro lanu, ndiye kuti mungafunikirenso kutanthauzira momveka bwino malingaliro anu, mankhwala, ndi zotsatira zomwe mungathe.

Table 2.5: Mndandanda wa Zotsatira Zowoneka
Munthu Zotsatira za chikhalidwe cha mankhwala Zopindulitsa mu chikhalidwe cholamulira Chithandizo cha mankhwala
1 $$Y_1(1)$$ $$Y_1(0)$$ $$\tau_1$$
2 $$Y_2(1)$$ $$Y_2(0)$$ $$\tau_2$$
$$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$
$$N$$ $$Y_N(1)$$ $$Y_N(0)$$ $$\tau_N$$
Nenani $$\bar{Y}(1)$$ $$\bar{Y}(0)$$ $$\bar{\tau}$$

Pofotokozera zotsatira zowonongeka mwa njirayi, komabe, timatha kukhala ndi vuto. Pafupifupi nthawi zonse, sitimayesetsa kuona zomwe zingatheke. Izi zikutanthauza kuti munthu wapadera adatumikira kapena sanatumikire. Choncho, tikuwona chimodzi mwa zotsatira zomwe zingatheke- $$Y_i(1)$$ kapena $$Y_i(0)$$ -koma osati zonse. Kulephera kuwona zotsatira zonse zomwe zingatheke ndi vuto lalikulu lomwe Holland (1986) linadzitcha kuti Fundamental Problem of Causal Inference .

Mwamwayi, pamene tikufufuza, tilibe munthu mmodzi; M'malo mwake, tili ndi anthu ambiri, ndipo izi zimapereka njira yozungulira vuto lalikulu la Causal Inference. M'malo moyesera kulingalira za zotsatira za mankhwala omwe ali pamtundu wa munthu, tikhoza kulingalira momwe mankhwala amathandizira ma unit onse:

$\text{ATE} = \bar{\tau} = \frac{1}{N} \sum_{i=1}^N \tau_i \qquad(2.2)$

Kugwirizana kumeneku kumayesetsabebe ponena za $$\tau_i$$ , zomwe sizingatheke, koma ndi algebra (eq 2.8 ya Gerber and Green (2012) ), timapeza

$\text{ATE} = \frac{1}{N} \sum_{i=1}^N Y_i(1) - \frac{1}{N} \sum_{i=1}^N Y_i(0) \qquad(2.3)$

Izi zikusonyeza kuti ngati ife tingakhoze amati anthu pafupifupi zotsatira pansi chithandizo ( $$N^{-1} \sum_{i=1}^N Y_i(1)$$ ) ndi anthu pafupifupi zotsatira pansi pa ulamuliro ( $$N^{-1} \sum_{i=1}^N Y_i(1)$$ ), ndiye tikhoza kulingalira momwe mankhwala amathandizira, ngakhale popanda kuyerekezera zotsatira za mankhwala kwa munthu aliyense.

Tsopano kuti ndatanthauzira chiwerengero chathu-chinthu chomwe tikuyesera kuti tiyese-ndikuyang'ana momwe tingathe kuwerengera ndi deta. Ndipo apa ife timathamangira mwachindunji ku vuto lomwe timangoona chimodzi mwa zotsatira zomwe zingatheke kwa munthu aliyense; timayang'ana $$Y_i(0)$$ kapena $$Y_i(1)$$ (tebulo 2.6). Titha kulingalira zotsatira za mankhwala ochizira poyerekeza phindu la anthu omwe adatumizidwa ku malipiro a anthu omwe sanatumikire:

$\widehat{\text{ATE}} = \underbrace{\frac{1}{N_t} \sum_{i:W_i=1} Y_i(1)}_{\text{average earnings, treatment}} - \underbrace{\frac{1}{N_c} \sum_{i:W_i=0} Y_i(0)}_{\text{average earnings, control}} \qquad(2.4)$

kumene $$N_t$$ ndi $$N_c$$ ndi chiwerengero cha anthu omwe ali ndi chithandizo ndi zoletsa. Njirayi idzagwira ntchito bwino ngati ntchito yopatsidwa chithandizo ili yopanda zotsatira, zomwe nthawi zina zimatchedwa ignorability . Mwamwayi, pakakhala palibe kuyesa, kusazindikira sikokwanira nthawi zambiri, zomwe zikutanthawuza kuti woyesa muq. 2.4 sizingatheke kutengera bwino. Njira imodzi yoganizira izi ndi yakuti pokhapokha ngati palibe ntchito yothandizira, al. 2.4 sakufanizitsa ngati ndi; ikufanizitsa mapindu a mitundu yosiyanasiyana ya anthu. Kapena amawonetsa mosiyana, popanda ntchito yothandizira, mankhwala operekedwa mwinamwake akugwirizana ndi zotsatira zomwe zingatheke.

Mu chaputala 4, ndikufotokoza momwe kufufuza kosagwiritsidwa ntchito mosasinthika kungathandizire ochita kafukufuku kupanga ziwerengero zamakono, ndipo apa ndikufotokoza momwe ochita kafukufuku angagwiritsire ntchito mwayi wa masewero achilengedwe, monga loti lotere.

Tsamba 2.6: Mndandanda wa Zotsatira Zowoneka
Munthu Zotsatira za chikhalidwe cha mankhwala Zopindulitsa mu chikhalidwe cholamulira Chithandizo cha mankhwala
1 ? $$Y_1(0)$$ ?
2 $$Y_2(1)$$ ? ?
$$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$
$$N$$ $$Y_N(1)$$ ? ?
Nenani ? ? ?

Zotsatira zachilengedwe

Njira imodzi yopangira ziwerengero zosawerengera popanda kuyesa kuyesera ndikuyang'ana chinachake chikuchitika padziko lapansi komwe mwakhala ndikukupatsani chithandizo. Njira imeneyi imatchedwa kuyesera zachirengedwe . Muzinthu zambiri, mwatsoka, chikhalidwe sichipereka mosakayikira chithandizo chomwe mukufuna anthu omwe ali nacho chidwi. Koma nthawi zina, chilengedwe chimapereka chithandizo chofanana. Makamaka, ndikulingalira nkhaniyi kuti pali mankhwala achiwiri omwe amalimbikitsa anthu kulandira mankhwala oyamba . Mwachitsanzo, zolembazo zikhoza kuonedwa kuti ndi mankhwala opititsa patsogolo omwe analimbikitsa anthu ena kuti ayambe kulandira chithandizo choyamba, chomwe chinali kutumikira usilikali. Izi zimatchedwa kuti kulimbikitsidwa . Ndipo njira yowunika yomwe ine ndikufotokozera kuthana nayo vutoli nthawi zina imatchedwa kuti instrumental variables . Pachifukwa ichi, poganiza, ochita kafukufuku angagwiritse ntchito chilimbikitso kuti aphunzire za zotsatira za mankhwala oyambirira a gawo limodzi la magawo.

Pofuna kuthana ndi mankhwala awiriwa-kulimbikitsanso ndi chithandizo chachikulu-tikufunikira mndandanda watsopano. Tangoganizani kuti anthu ena amalembedwa mwachisawawa ( $$Z_i = 1$$ ) kapena sanalembedwe ( $$Z_i = 0$$ ); mu izi, $$Z_i$$ nthawi zina amatchedwa chida .

Ena mwa omwe adalembedwa, ena adatumikira ( $$Z_i = 1, W_i = 1$$ ) ndipo ena sanatero ( $$Z_i = 1, W_i = 0$$ ). Mofananamo, mwa iwo omwe sanalembedwe, ena adatumikira ( $$Z_i = 0, W_i = 1$$ ) ndipo ena sanatero ( $$Z_i = 0, W_i = 0$$ ). Zotsatira zomwe zingatheke kwa munthu aliyense tsopano zingathe kufalikira kuti ziwonetsere momwe zilili ndi chilimbikitso ndi chithandizo. Mwachitsanzo, mulole $$Y(1, W_i(1))$$ akhale malipiro a munthu $$i$$ ngati adalembedwa, kumene $$W_i(1)$$ ndilo utumiki wake ngati adalembedwa. Komanso, tikhoza kugawa anthu kukhala magulu anayi: makampani osungira katundu, osatengapo mbali, osokoneza, komanso ogwira ntchito nthawi zonse (tebulo 2.7).

Table 2.7: Mitundu Ina ya Anthu
Lembani Utumiki ngati udapangidwa Utumiki ngati sunalembedwe
Makampani Inde, $$W_i(Z_i=1) = 1$$ Ayi, $$W_i(Z_i=0) = 0$$
Osatengapo konse Ayi, $$W_i(Z_i=1) = 0$$ Ayi, $$W_i(Z_i=0) = 0$$
Otsutsa Ayi, $$W_i(Z_i=1) = 0$$ Inde, $$W_i(Z_i=0) = 1$$
Ogwira-nthawizonse Inde, $$W_i(Z_i=1) = 1$$ Inde, $$W_i(Z_i=0) = 1$$

Tisanayambe kukambirana momwe zotsatira za chithandizochi zimakhudzire (mwachitsanzo, ntchito ya usilikali), titha kuyamba kufotokozera zotsatira ziwiri za chilimbikitso (mwachitsanzo, kulembedwa). Choyamba, titha kufotokoza zotsatira za chilimbikitso pa chithandizo choyamba. Chachiwiri, titha kufotokozera zotsatira za chilimbikitso pa zotsatira. Zidzakhala kuti zotsatira ziwirizi zikhoza kuphatikizidwa kuti ziwonetsetse zotsatira za chithandizo pa gulu lapadera la anthu.

Choyamba, zotsatira za chilimbikitso pa chithandizo zingatanthauzidwe kwa munthu $$i$$ monga

$\text{ITT}_{W,i} = W_i(1) - W_i(0) \qquad(2.5)$

Komanso, izi zingathe kufotokozedwa pa anthu onse monga

$\text{ITT}_{W} = \frac{1}{N} \sum_{i=1}^N [W_i(1) - W_i(0)] \qquad(2.6)$

Potsiriza, tingathe kulingalira $$\text{ITT} _{W}$$ pogwiritsa ntchito deta:

$\widehat{\text{ITT}_{W}} = \bar{W}^{\text{obs}}_1 - \bar{W}^{\text{obs}}_0 \qquad(2.7)$

komwe $$\bar{W}^{\text{obs}}_1$$ ndilo mlingo wa mankhwala omwe akuwonetsedwa ndi omwe akulimbikitsidwa ndipo $$\bar{W}^{\text{obs}}_0$$ ndi kuchuluka kwa chithandizo cha mankhwala kwa omwe sanalimbikitsidwe. $$\text{ITT}_W$$ nthawi zina amatchedwanso kuchuluka kwa msinkhu .

Kenaka, zotsatira za chilimbikitso pa zotsatira zikhoza kufotokozedwa kwa munthu $$i$$ monga:

$\text{ITT}_{Y,i} = Y_i(1, W_i(1)) - Y_i(0, W_i(0)) \qquad(2.8)$

Komanso, izi zingathe kufotokozedwa pa anthu onse monga

$\text{ITT}_{Y} = \frac{1}{N} \sum_{i=1}^N [Y_i(1, W_i(1)) - Y_i(0, W_i(0))] \qquad(2.9)$

Potsiriza, tingathe kulingalira $$\text{ITT}_{Y}$$ pogwiritsa ntchito deta:

$\widehat{\text{ITT}_{Y}} = \bar{Y}^{\text{obs}}_1 - \bar{Y}^{\text{obs}}_0 \qquad(2.10)$

pamene $$\bar{Y}^{\text{obs}}_1$$ $$\bar{W}^{\text{obs}}_0$$ ndi zotsatira za omwe sanalimbikitsidwe.

Pomalizira pake, timakumbukira zotsatira za chidwi: zotsatira za mankhwala oyambirira (mwachitsanzo, ntchito ya usilikali) pamapeto (mwachitsanzo, mapindu). Tsoka ilo, zimapezeka kuti munthu sangathe, kulingalira zotsatira zake pa mayunitsi onse. Komabe, ndi zifukwa zina, ofufuza akhoza kulingalira momwe zotsatira za chithandizo cha operekera mankhwala (mwachitsanzo, anthu omwe angatumikire ngati atalembedwa ndi anthu omwe sangatumikire ngati sanalembedwe, tebulo 2.7). Ndidzatcha kuti chiwerengero chachikulu cha causal effect (CACE) (chomwe nthawi zina chimatchulidwa kuti chiwerengero cha mankhwala akuchizira, LATE):

$\text{CACE} = \frac{1}{N_{\text{co}}} \sum_{i:G_i=\text{co}} [Y(1, W_i(1)) - Y(0, W_i(0))] \qquad(2.11)$

kumene $$G_i$$ amapereka gulu la munthu $$i$$ (onani tebulo 2.7) ndi $$N_{\text{co}}$$ ndi chiwerengero cha makampani. Mwa kuyankhula kwina, eq. 2.11 ikufanizira mapepala a mapepala omwe amalembedwa $$Y_i(1, W_i(1))$$ ndipo sanalembedwe $$Y_i(0, W_i(0))$$ . Chiwerengero cha eq. 2.11 Zikuwoneka kuti n'zovuta kulingalira kuchokera ku deta yodziwika chifukwa sizingatheke kuzindikira anthu omwe amagwiritsa ntchito deta (kudziwa ngati wina akuphatikizirani kuti muwone ngati adatumikira pamene adalembedwera komanso ngati adatumikira pamene sanalembedwe).

Izi zimakhala zochititsa chidwi-kuti ngati pali makampani ogulitsa, ndiye kuti amapereka chimodzi chimapanga zifukwa zitatu zowonjezera, ndizotheka kulingalira za CACE kuchokera ku deta yowona. Choyamba, wina ayenera kuganiza kuti ntchito yopita kuchipatala ndi yophweka. Pankhani yolemba loti izi ndi zomveka. Komabe, m'madera ena omwe masewero achilengedwe samadalira mwakuthupi, lingaliro limeneli lingakhale lovuta kwambiri. Chachiwiri, wina ayenera kuganiza kuti iwo sali olakwika (lingaliro ili limatchedwanso kuti monotonicity assumption). Pogwiritsa ntchito zolembazo zikuwoneka kuti ndi zomveka kuganiza kuti pali anthu ochepa omwe sangatumikire ngati atalembedwa ndipo adzatumikira ngati sanalembedwe. Chachitatu, ndipo pomalizira pake, kumabwera chidziwitso chofunika kwambiri chomwe chimatchedwa lamulo loletsedwa . Pansi pa lamulo loletsedwa, munthu ayenera kuganiza kuti zotsatira zonse za ntchito yopatsidwa chithandizo zimadutsa kudzera kuchipatala chomwecho. M'mawu ena, munthu ayenera kuganiza kuti palibe zotsatira zenizeni za chilimbikitso pa zotsatira. Pankhani ya kukonza loti, mwachitsanzo, wina ayenera kuganiza kuti kafukufukuyu sakhala ndi zotsatirapo zapindula kupatulapo kupyolera mu usilikali (chifaniziro 2.11). Lamulo loletsedwa likhoza kuphwanyidwa ngati, mwachitsanzo, anthu omwe adalembedwera amathera nthawi yochuluka kusukulu pofuna kupeŵa ntchito kapena ngati olemba ntchito sankatha kubwereka anthu omwe adalembedwa.

Ngati izi zitatu (ntchito yosavuta kuchipatala, palibe zonyansa, ndi lamulo loletsedwa), ndiye

$\text{CACE} = \frac{\text{ITT}_Y}{\text{ITT}_W} \qquad(2.12)$

kotero tikhoza kulingalira CACE:

$\widehat{\text{CACE}} = \frac{\widehat{\text{ITT}_Y}}{\widehat{\text{ITT}_W}} \qquad(2.13)$

Njira imodzi yoganizira za CACE ndikuti ndi kusiyana kwa zotsatira pakati pa omwe analimbikitsidwa ndi omwe sanalimbikitsidwe, akukakamizidwa ndi kuchuluka kwa chiwerengero.

Pali zikhomo ziwiri zofunika kuzikumbukira. Choyamba, chiletso chochotseratu ndi chidziwitso cholimba, ndipo chiyenera kukhala cholungamitsidwa pazomwe zimakhazikitsidwa ndi vuto, zomwe kawirikawiri zimafuna luso la malo. Lamulo lochotseratu sizingakhale loyenera ndi kulimbikitsana kwina kulimbikitsa. Chachiwiri, vuto lodziwika bwino lomwe limagwiritsidwa ntchito poyesa kusintha mosiyanasiyana kumabwera pamene chilimbikitso $$\text{ITT}_W$$ chithandizo (pamene $$\text{ITT}_W$$ n'chaching'ono). Izi zimatchedwa chida chofooka , ndipo chimabweretsa mavuto osiyanasiyana (Imbens and Rosenbaum 2005; Murray 2006) . Njira imodzi yoganizira za vuto ndi zida zofooka ndizoti $$\widehat{\text{CACE}}$$ akhoza kumvetsetsa zochepa zapakati pa $$\widehat{\text{ITT}_Y}$$ Kuphwanya kwa lamulo lochotseratu-chifukwa zotsutsanazi zikutukulidwa ndi zing'onozing'ono $$\widehat{\text{ITT}_W}$$ (onani tsamba 2.13). Mwinamwake, ngati chithandizo chimene chilengedwe chimapereka sichikhudza kwambiri chithandizo chomwe mumachidera, ndiye kuti mudzakhala ovuta kuphunzira za chithandizo chomwe mumachidera.

Onani mutu 23 ndi 24 wa Imbens and Rubin (2015) kuti mudziwe zambiri zokhudza zokambiranazi. Njira yowonjezera ndalama yopangira zida zankhondo nthawi zambiri imayesedwa potsata kulinganirana, osati zotsatira zabwino. Kuti mudziwe zowonjezera, onani Angrist and Pischke (2009) , ndipo poyerekeza pakati pa njira ziwiri, onani gawo 24.6 la Imbens and Rubin (2015) . Njira ina, njira yochepetsera zochepa zowonjezereka mwa njirayi imaperekedwa mu chaputala 6 cha Gerber and Green (2012) . Kuti mudziwe zambiri pazitsulo zotsalira, onani D. Jones (2015) . Aronow and Carnegie (2013) akulongosola ziganizo zina zomwe zingagwiritsidwe ntchito kulingalira ATE m'malo mwa CACE. Kuti mudziwe zambiri momwe zowonetsera zachilengedwe zingakhale zovuta kwambiri kutanthauzira, onani Sekhon and Titiunik (2012) . Kuti mudziwe zambiri zowonongeka kwachilengedwe-zomwe zimangopitirira njira zothandizira zowonjezereka kuphatikizapo mapangidwe monga regression discontinuity-onani Dunning (2012) .