Bayanan lissafi

A cikin wannan shafi, zan taƙaita wasu ra'ayoyi game da samar da ƙaddarar hadari daga bayanai marasa gwaji a cikin wani nau'in lissafi. Akwai manyan hanyoyi guda biyu: tsarin zane-zane, wanda ya fi dacewa da Yahudiya Pearl da abokan aiki, da kuma kyakkyawan sakamako, wanda ya fi dacewa da Donald Rubin da abokan aiki. Zan gabatar da matakan sakamako mai yiwuwa domin ya fi dacewa da alaka da ra'ayoyi a cikin bayanan ilmin lissafi a ƙarshen sura ta 3 da 4. Don ƙarin bayani game da tsarin zane-zane, Ina bayar da shawarar Pearl, Glymour, and Jewell (2016) (gabatarwa ) da kuma Pearl (2009) (ci gaba). Domin taƙaitaccen rubutun maganganun ƙaddarar da ke tattare da ma'anar sakamakon da aka samu da kuma tsarin shafuka, ina bayar da shawarar Morgan and Winship (2014) .

Manufar wannan shafi shine don taimaka maka da jin dadi tare da sanarwa da sifa na sakamakon al'amuran da suka dace don ka iya canzawa zuwa wasu daga cikin kayan fasahar da aka rubuta akan wannan batu. Da farko, zan bayyana tsarin da zai yiwu. Bayan haka, zan yi amfani da shi don kara tattauna irin gwaje-gwaje na halitta irin na Angrist (1990) akan sakamakon aikin soja a kan samun kuɗi. Wannan shafukan da ke tattare da shi yana kan gaba ga Imbens and Rubin (2015) .

Tsarin sakamako mai kyau

Tsarin sakamako mai mahimmanci yana da muhimmiyar mahimman abubuwa: raka'a , jiyya , da kuma sakamakon da ya dace . Domin muyi bayanin waɗannan abubuwa, bari muyi la'akari da sakon layi na tambayar da aka yi magana a Angrist (1990) : Mene ne sakamakon aikin soja a kan samun kuɗi? A wannan yanayin, zamu iya ayyana raka'a don zama mutane waɗanda suka cancanci yin rubutun 1970 a Amurka, kuma za mu iya tantance waɗannan mutane ta \(i = 1, \ldots, N\) . Hanyoyin magani a wannan yanayin na iya zama "hidima cikin soja" ko kuma "ba su aiki a cikin soja ba." Zan kira wadannan ka'idoji da kulawa, kuma zan rubuta \(W_i = 1\) idan mutum \(i\) yana cikin yanayin magani kuma \(W_i = 0\) idan mutum \(i\) yana cikin yanayin kulawa. A ƙarshe, sakamakon da ya dace zai zama da wuyar fahimta saboda suna da nasaba da "m" sakamakon; abubuwa da zasu iya faruwa. Ga kowane mutum wanda ya cancanci yin zabin na 1970, zamu iya tunanin adadin da zasu samu a shekarar 1978 idan suna aiki a cikin soja, wanda zan kira \(Y_i(1)\) , da adadin da zasu samu a 1978 idan basu kasance cikin soja ba, wanda zan kira \(Y_i(0)\) . A cikin matakan sakamako, \(Y_i(1)\) da \(Y_i(0)\) suna la'akari da yawaitaccen ƙayyadaddun abu, yayin da \(W_i\) wani lamari ne mai mahimmanci.

Zaɓin raka'a, jiyya, da kuma sakamakon yana da mahimmanci saboda yana bayyana abin da za a iya koya-kuma ba za a iya koya daga binciken ba. Zaɓin raka'a-mutanen da suka cancanta don yin rubutun 1970-ba su haɗa da mata ba, don haka ba tare da ƙarin ra'ayi ba, wannan binciken ba zai gaya mana wani abu game da sakamakon aikin soja a kan mata ba. Sharuɗɗa game da yadda za a bayyana magunguna da kuma sakamakon yana da mahimmanci. Alal misali, ya kamata a mayar da hankali ga kula da sha'awa don yin aiki a cikin soja ko fuskantar gwagwarmaya? Shin sakamako na sha'awa shi ne haɓaka ko samun gamsuwa? Daga ƙarshe, zaɓin raƙuman raka'a, jiyya, da kuma sakamako ya kamata a kaddamar da su ta hanyar kimiyya da manufofin manufofin binciken.

Idan aka ba da zabi na raka'a, jiyya, da sakamakon da ya dace, sakamakon sakamako na jiyya akan mutum \(i\) , \(\tau_i\) , shine

\[ \tau_i = Y_i(1) - Y_i(0) \qquad(2.1)\]

A wasu kalmomi, zamu kwatanta yadda mutum \(i\) zai samu bayan ya yi aiki ga mutum \(i\) zai samu ba tare da yin hidima ba. A gare ni, eq. 2.1 ita ce hanyar da ta fi dacewa don bayyana sakamakon sakamako, kuma koda yake mafi sauki, wannan tsari yana nunawa a cikin hanyoyi masu muhimmanci da ban sha'awa (Imbens and Rubin 2015) .

Lokacin amfani da tsarin sakamako mai mahimmanci, sau da yawa ina taimakawa wajen rubuta wani tebur wanda ya nuna sakamakon da zai iya haifar da maganin jiyya ga dukkan raka'a (tebur 2.5). Idan ba za ku iya kwatanta tebur kamar wannan don nazarinku ba, to, kuna iya zama mafi mahimmanci a cikin ma'anar ku na sassanku, jiyya, da kuma sakamakon da kuka samu.

Lokacin da aka bayyana sakamakon sakamako a wannan hanyar, duk da haka, muna shiga cikin matsala. A kusan dukkanin lokuta, ba zamu iya ganin dukkanin sakamako mai kyau ba. Wato, wani mutum wanda yayi aiki ko ba ya aiki. Saboda haka, muna lura da wani sakamako mai mahimmanci- \(Y_i(1)\) ko \(Y_i(0)\) amma ba duka biyu ba. Rashin iya yin la'akari da sakamako mai kyau shine babbar matsalar da Holland (1986) kira shi Babban Matsala na Causal Inference .

Abin farin cikin, idan muna yin bincike, ba wai muna da mutum ɗaya ba; a maimakon haka, muna da mutane da yawa, kuma wannan yana ba da hanya a kan matsalar Matsala ta Causal Inference. Maimakon ƙoƙari na ƙayyadad da tasirin maganin kowane mutum, zamu iya kimanta tasirin magani na kowane ɗayan:

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

Wannan ƙayyadaddun abu har yanzu an bayyana shi dangane da \(\tau_i\) , wanda ba a sani ba, amma tare da wasu algebra (2.8 na Gerber and Green (2012) ), muna samun

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

Wannan yana nuna cewa idan zamu iya kimanta yawan matsayi na yawan jama'a a karkashin jiyya ( \(N^{-1} \sum_{i=1}^N Y_i(1)\) ) da kuma matsakaicin yawan jama'a a karkashin iko ( \(N^{-1} \sum_{i=1}^N Y_i(1)\) ), to zamu iya kimanta sakamako na jiyya, ko da ba tare da kimanta sakamakon maganin kowane mutum ba.

Yanzu da na yanke shawararmu-abin da muke ƙoƙarin ƙaddamarwa-Zan juya ga yadda za mu iya kwatanta shi da bayanai. Kuma a nan muna tafiya cikin matsala cewa kawai muna ganin daya daga cikin sakamakon da kowa zai iya samu; muna ganin ko dai \(Y_i(0)\) ko \(Y_i(1)\) (tebur 2.6). Zamu iya kimanta yawan maganin magani ta hanyar kwatanta abubuwan da mutane ke bayarwa da ke ba da gudummawa ga mutanen da ba su aiki ba:

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

inda \(N_t\) da \(N_c\) sune lambobin mutane a yanayin kulawa da kulawa. Wannan tsarin zaiyi aiki sosai idan aikin kulawa ya kasance mai zaman kanta daga sakamakon da zai yiwu, wani yanayi wanda ake kira jahilai . Abin takaici, idan babu gwaji, rashin jahilci ba sau da yawa, wanda ke nufin cewa mai ƙididdigewa a cikin eq. 2.4 bazai iya samar da kyakkyawan ƙayyadadden rahoto ba. Wata hanya ta tunani game da shi ita ce, ba tare da samun aikin ba da izini ba, eq. 2.4 ba a kwatanta da kamar; yana kwatanta abubuwan da mutane ke bayarwa. Ko kuma ya bayyana daban-daban daban-daban, ba tare da yin amfani da maganin bazuwar ba, ƙila za a iya rarraba allo don magance sakamakon da zai yiwu.

A cikin sura ta 4, zan bayyana yadda gwajin gwagwarmaya na bazuwar zasu iya taimakawa masu bincike suyi lissafin ƙaddararsu, kuma a nan zan bayyana yadda masu bincike zasu iya amfani da gwaje-gwaje na halitta, kamar su irin caca.

Kwancen gwaji

Ɗaya daga cikin hanyoyin da za a ƙaddamar da ƙaddarawa ba tare da yin gwaji ba ne don neman wani abu da ke faruwa a duniya wanda ya ba da izini a gare ku. An kira wannan hanyar gwaji na halitta . A lokuta da dama, rashin tausayi, yanayi bazai ba da izini ba don neman yawan mutane. Amma wani lokaci, yanayin ba da izinin magani ba. Musamman ma, zan yi la'akari da yanayin idan akwai wani magani na biyu wanda yake karfafa mutane su karbi magani na farko . Alal misali, za a iya yin rubutun a matsayin wani magani na sakandare wanda ba a ba da izini ba wanda ya karfafa wasu mutane su dauki nauyin farko, wanda ke aiki a cikin soja. Wannan zane ana kira wani lokacin karfafa karfafawa . Kuma hanyar bincike da zan bayyana don magance wannan halin da ake ciki a wasu lokutan ana kiran saɓin kayan aiki . A cikin wannan wuri, tare da wasu tsammanin, masu bincike zasu iya amfani da ƙarfafawa don koyo game da sakamakon magani na farko don takamaiman raka'a na raka'a.

Don magance nau'o'in daban-daban - ƙarfafawa da kuma kulawa na farko - muna buƙatar wasu ƙididdiga. Ka yi la'akari da cewa an tsara wasu mutane ( \(Z_i = 1\) ) ko a'a ( \(Z_i = 0\) ); a wannan halin, \(Z_i\) an kira wani kayan aiki wani lokaci.

Daga cikin waɗanda aka tsara, wasu sun yi hidima ( \(Z_i = 1, W_i = 1\) ) kuma wasu basu ( \(Z_i = 1, W_i = 0\) ) ba. Hakazalika, a cikin wadanda ba'a tsara su ba, wasu sun yi hidima ( \(Z_i = 0, W_i = 1\) ) kuma wasu basu ( \(Z_i = 0, W_i = 0\) ) ba. Abubuwan da za a iya samu ga kowane mutum za a iya fadada yanzu don nuna matsayin su don karfafawa da kuma maganin. Alal misali, bari \(Y(1, W_i(1))\) zama haɗin mutum \(i\) idan an kirkiro shi, inda \(W_i(1)\) shine matsayin sabis ne idan aka tsara. Bugu da ari, zamu iya raba jama'a zuwa kungiyoyi hudu: masu ƙira, masu tayar da hankali, masu tayar da hankali, da masu koraushewa (tebur 2.7).

Kafin mu tattauna akan kwatancin sakamako na jiyya (watau soja,) zamu iya bayyana ma'anar ƙarfafawa guda biyu (watau, an tsara). Na farko, zamu iya bayyana ma'anar ƙarfafawa akan maganin farko. Na biyu, zamu iya bayyana sakamakon ƙarfafawa a kan sakamakon. Zai bayyana cewa waɗannan halayen biyu zasu iya haɗuwa don samar da kimantawa game da sakamakon magani a kan wani rukuni na mutane.

Na farko, za a iya bayyana ma'anar karfafawa a kan jiyya ga mutum \(i\) a matsayin

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

Bugu da ari, ana iya bayyana wannan adadi a kan dukan yawan jama'a

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

A ƙarshe, za mu iya ƙayyade \(\text{ITT} _{W}\) ta yin amfani da bayanai:

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

inda \(\bar{W}^{\text{obs}}_1\) shine magani na lura ga wadanda aka karfafa kuma \(\bar{W}^{\text{obs}}_0\) wanda ake kula da magani ga wadanda ba'a karfafa su ba. \(\text{ITT}_W\) an kira wani lokaci sau da yawa .

Bayan haka, za a iya bayyana ma'anar ƙarfafawa a kan sakamakon sakamakon mutum \(i\) kamar:

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

Bugu da ari, ana iya bayyana wannan adadi a kan dukan yawan jama'a

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

A ƙarshe, zamu iya kimanta \(\text{ITT}_{Y}\) ta yin amfani da bayanai:

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

inda \(\bar{Y}^{\text{obs}}_1\) shine sakamako na ƙarshe (misali, samun kuɗi) ga wadanda aka karfafa (misali, \(\bar{W}^{\text{obs}}_0\) ) da \(\bar{W}^{\text{obs}}_0\) shine sakamako na ƙarshe ga waɗanda ba a karfafa su ba.

A ƙarshe, za mu mayar da hankalin mu game da sakamakon sha'awa: sakamakon sakamako na farko (misali, sabis na soja) akan sakamako (misali, albashi). Abin takaici, yana nuna cewa mutum ba zai iya kwatanta wannan tasiri a kan dukkan raka'a ba. Duk da haka, tare da wasu tsammanin, masu bincike zasu iya kwatanta tasirin jiyya akan ƙwararru (watau mutanen da za su bauta wa idan aka tsara da kuma mutanen da ba za su bauta ba idan ba a tsara ba, tebur 2.7). Zan kira wannan ƙididdigar yadda za a iya haifar da sakamako mai yawa (CACE) (wanda ake kira wani lokaci na maganin magani na gida , LATE):

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

inda \(G_i\) ba da rukuni na mutum \(i\) (duba tebur 2.7) da \(N_{\text{co}}\) shi ne adadin masu cikawa. A wasu kalmomi, eq. 2.11 kwatanta albashi na compliers waɗanda aka tsara \(Y_i(1, W_i(1))\) kuma ba a shirya \(Y_i(0, W_i(0))\) . An kiyasta a cikin eq. 2.11 yana da wuya a kiyasta daga lura da bayanai saboda baza'a iya gane masu amfani ba ne kawai ta hanyar yin amfani da bayanan da aka lura (don sanin idan wani ya kwarewa zai buƙaci ko ya yi aiki lokacin da aka tsara kuma ya yi hidima lokacin da ba a tsara shi ba).

Ya juya-wani abu mai ban mamaki-cewa idan akwai wasu ƙwararrun, sa'an nan kuma ya ba da wanda ya sanya ra'ayi guda uku, yana yiwuwa a kwatanta CACE daga bayanan lura. Na farko, dole ne mutum ya ɗauka cewa aikin zuwa magani shi ne bazuwar. A game da batun irin caca wannan ya dace. Duk da haka, a wasu wurare inda gwaje-gwaje na halitta ba su dogara ga bazuwar jiki, wannan zato yana iya zama matsala. Abu na biyu, wanda ya dauka cewa ba su da wani maƙaryata (wannan zato shine wani lokaci ana kiran shi sautin tunani). A cikin mahallin daftarin aiki yana da kyau a ɗauka cewa akwai mutane da yawa waɗanda ba za su bauta wa idan aka tsara su ba kuma za su yi aiki idan ba a rubuta su ba. Na uku, kuma a ƙarshe, shi ne mafi mahimmanci mahimmanci wanda ake kira ƙuntatawa mara izini . A karkashin ƙuntataccen haɓaka, mutum ya ɗauka cewa duk sakamako na aikin aikin magani ya wuce ta hanyar kulawa kanta. A wasu kalmomin, dole ne mutum ya ɗauka cewa babu wani tasiri na ƙarfafawa a kan sakamakon. A misali, game da irin caca, misali, mutum yana buƙatar ɗaukar cewa wannan matsayi ba shi da tasiri a kan samun kuɗi ba tare da aikin soja ba (adadi na 2.11). Za a iya ƙuntata ƙuntatawa ta musamman idan, misali, mutanen da aka tsara su sun fi yawan lokaci a makaranta don kauce wa sabis ko kuma idan ma'aikata ba su iya hayar mutanen da aka tsara.

Figure 2.11: Harkokin ƙuntatawa yana buƙatar ƙarfafawa (rubutun caca) yana da tasirin sakamako (albashi) kawai ta hanyar kulawa (sabis na soja). Za a iya ƙuntata ƙuntatawa idan idan, alal misali, mutanen da aka tsara sun kashe mafi yawan lokuta a makaranta don kauce wa sabis kuma wannan kara yawan lokuta a makaranta ya jagoranci haɓakar haɓaka.

Idan waɗannan yanayi uku (aikin bazuwar zuwa magani, babu masu tsaiko, da ƙuntatawa na ƙuntatawa) an haɗu, to,

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

don haka za mu iya kwatanta CACE:

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

Wata hanya ta tunani game da CACE shine cewa bambanci ne tsakanin sakamakon waɗanda aka karfafa da waɗanda ba'a karfafa su ba, wadanda suka karɓa ta hanyar karbar kudi.

Akwai manyan mahimman abubuwa guda biyu don tunawa. Na farko, ƙuntatawa ta haɓaka shine mai karfi mai karfi, kuma yana buƙatar a sami kuɓuta a kan hanyar da ta shafi shari'ar, wanda akai-akai yana buƙatar ƙwarewar yankin. Ƙuntatawa ta cirewa ba za a iya kubutar da shi ba tare da yin bayani game da ƙarfafawa. Na biyu, ƙalubalen gwagwarmaya tare da kayan aiki na kayan aiki yana zuwa lokacin da ƙarfafawa ba ta da tasiri a kan magance jiyya (lokacin da \(\text{ITT}_W\) ƙananan ne). Wannan ake kira kayan aiki mai rauni , kuma yana haifar da matsaloli masu yawa (Imbens and Rosenbaum 2005; Murray 2006) . Wata hanya ta tunani game da matsala tare da kwarewar kayan aiki shine \(\widehat{\text{CACE}}\) iya zama mai kula da ƙananan ra'ayi a cikin \(\widehat{\text{ITT}_Y}\) - bisa ga Rashin ƙetare ƙuntatawa-saboda waɗannan ƙyamar sunyi girma da karamin \(\widehat{\text{ITT}_W}\) (duba eq 2.13). Da wuya, idan magani da yanayi ya ba shi da tasirin gaske a kan jiyya da kake damu, to, za ka yi wuya a koya game da magani da kake damu.

Dubi babi na 23 da 24 na Imbens and Rubin (2015) don ƙarin fasalin wannan tattaunawa. Hanyar tsarin tattalin arziki ta al'ada ga kayan aiki na kayan aiki an nuna shi a matsayin maƙasudin lissafin ƙididdiga, ba sakamakon sakamako ba. Don gabatarwa daga wannan hangen nesa, duba Angrist and Pischke (2009) , kuma don kwatanta tsakanin hanyoyin biyu, duba sashi 24.6 na Imbens and Rubin (2015) . Wani madadin, dan kadan kaɗan da aka gabatar da kayan aiki mai mahimmanci na kayan aiki an samo su a babi na 6 na Gerber and Green (2012) . Don ƙarin bayani game da ƙuntatawar cirewa, duba D. Jones (2015) . Aronow and Carnegie (2013) bayyana wani ƙarin sashi na zaton da za a iya amfani dasu don kimanta ATE maimakon CACE. Don ƙarin bayani game da irin yadda gwaji na halitta zai iya zama da kyau a fassara, duba Sekhon and Titiunik (2012) . Don ƙarin gabatarwa ga gwaje-gwaje na halitta - wanda ya wuce fiye da ma'anar kayan aiki masu mahimmanci har ma sun haɗa da alamomi irin su tawayar rikici-ga Dunning (2012) .