Bayanan lissafi

Ina tsammanin hanya mafi kyau don fahimtar gwaje-gwajen shine tsarin da zai yiwu (wanda na tattauna a cikin bayanin lissafi a babi na 2). Tsarin sakamako mai mahimmanci yana da dangantaka mai zurfi ga ra'ayoyin daga samfurin samfurin wanda na bayyana a babi na 3 (Aronow and Middleton 2013; Imbens and Rubin 2015, chap. 6) . An rubuta wannan shafi a hanyar da za a jaddada wannan haɗin. Wannan ƙarfafawa ba wani gargajiya bane, amma ina tsammanin haɗin tsakanin samfur da gwaje-gwaje yana da taimako: yana nufin cewa idan kun san wani abu game da samfur sannan ku san wani abu game da gwaje-gwajen da kuma mataimakin. Kamar yadda zan nuna a cikin wadannan bayanan, tsarin da zai yiwu ya nuna ƙarfin nazarin gwaje-gwajen da aka ƙayyade don kimanta sakamakon tasirin, kuma yana nuna iyakokin abin da za a iya yi tare da gwaje-gwajen da aka yi daidai.

A cikin wannan shafukan, zan bayyana tsarin sakamako mai mahimmanci, sake buga wasu daga cikin matakan daga bayanin ilimin lissafi a cikin sura na 2 don yin waɗannan bayanan abubuwan da ke ciki. Bayan haka zan bayyana wasu sakamakon taimako game da ƙayyadadden ƙididdigar ƙididdigar magunguna, ciki har da tattaunawa game da ƙayyadaddun allo da kuma bambancin bambancin bambanci. Wannan shafukan da ke tattare yana jawo hankalin Gerber and Green (2012) .

Tsarin sakamako mai kyau

Don nuna alamun sakamako mai mahimmanci, bari mu koma ga gwajin Restivo da gwagwarmaya na van de Rijt don kiyasta sakamakon samun karbar gine-gine akan gudummawar gaba zuwa Wikipedia. Tsarin sakamako mai mahimmanci yana da muhimmiyar mahimman abubuwa: raka'a , jiyya , da kuma sakamakon da ya dace . A game da Restivo da van de Rijt, wa] annan raka'a sun cancanci masu gyara-wa] anda ke cikin kashi 1% na masu bayar da gudunmawa-wa] anda ba su samu barnstar ba tukuna. Za mu iya rarraba wadannan masu gyara da \(i = 1 \ldots N\) . The jiyya a cikin gwaji kasance "barnstar" ko "ba barnstar," kuma zan rubuta \(W_i = 1\) idan mutum \(i\) ne a lura da yanayin da \(W_i = 0\) in ba haka ba. Hanya na uku na tsarin sakamako mai mahimmanci shine mafi mahimmanci: sakamakon da ya dace . Wadannan suna da wuyar fahimta saboda suna ƙunshe da sakamakon "m" - abubuwan da zasu iya faruwa. Ga kowane editan Wikipedia, wanda zai iya tunanin adadin gyare-gyare da zai yi a yanayin jinin ( \(Y_i(1)\) ) da kuma lambar da zata yi a yanayin kulawa ( \(Y_i(0)\) ).

Ka lura cewa wannan zaɓi na raka'a, jiyya, da kuma sakamakon ya bayyana abin da za a iya koya daga wannan gwaji. Alal misali, ba tare da wani ra'ayi ba, Restivo da van de Rijt ba za su iya yin wani abu game da sakamakon barnstars akan duk masu gyara na Wikipedia ba ko a sakamakon irin su gyara inganci. Gaba ɗaya, zaɓin raka'a, jiyya, da kuma sakamakon dole ne ya dogara akan manufar binciken.

Bada wadannan sakamako mai mahimmanci-wadanda aka taƙaita su a cikin tebur 4.5-wanda zai iya ayyana sakamakon sakamako na jiyya ga mutum \(i\) kamar yadda

\[ \tau_i = Y_i(1) - Y_i(0) \qquad(4.1)\]

A gare ni, wannan daidaituwa ita ce hanyar da ta fi dacewa don bayyana sakamakon sakamako, kuma, kodayake musamman mai sauƙi, wannan tsarin yana nunawa a cikin hanyoyi masu muhimmanci da ban sha'awa (Imbens and Rubin 2015) .

Idan muka ayyana halin da ake ciki 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 editan edita na musamman ya karbi barnstar ko a'a. 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, lokacin da muke yin bincike, ba wai muna da mutum ɗaya ba, muna da mutane da dama, kuma wannan yana ba da hanyar da za a fuskanci matsalar matsala ta Causal Inference. Maimakon ƙoƙarin ƙaddamar da tasirin maganin kowane mutum, zamu iya kimanta sakamako mafi mahimmancin magani:

\[ \text{ATE} = \frac{1}{N} \sum_{i=1}^N \tau_i \qquad(4.2)\]

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

\[ \text{ATE} = \frac{1}{N} \sum_{i=1}^N Y_i(1) - \frac{1}{N} \sum_{i=1}^N Y_i(0) \qquad(4.3)\]

Daidaita 4.3 yana nuna cewa idan zamu iya kimanta yawan matsayi na yawan jama'a a karkashin magani ( \(N^{-1} \sum_{i=1}^N Y_i(1)\) ) da kuma sakamakon yawancin 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. Ina so in yi tunani game da wannan ƙalubalen ƙididdiga a matsayin matsala samfurin (tunani a baya ga bayanin lissafi a babi na 3). Ka yi la'akari da cewa ba mu zaɓi wasu mutane su lura a yanayin yanayin magani ba kuma mun zaɓi wasu mutane su lura a yanayin kulawa, to, zamu iya kiyasta matsakaicin sakamako a kowace yanayin:

\[ \widehat{\text{ATE}} = \underbrace{\frac{1}{N_t} \sum_{i:W_i=1} Y_i(1)}_{\text{average edits, treatment}} - \underbrace{\frac{1}{N_c} \sum_{i:W_i=0} Y_i(0)}_{\text{average edits, control}} \qquad(4.4)\]

inda \(N_t\) da \(N_c\) sune lambobin mutane a yanayin kulawa da kulawa. Daidaita 4.4 shine bambanci-na-ma'ana kiyasta. Saboda samfurin samfurin, mun sani cewa kalma na farko shine mai ƙididdigar ba tare da la'akari ba saboda matsakaicin sakamako a karkashin magani kuma kalma na biyu shine mai ƙididdigar ba tare da la'akari ba a karkashin iko.

Wata hanyar da za ta yi tunani game da abin da ke tattare da shi ba shi ne tabbatar da cewa kwatanta tsakanin kulawa da kungiyoyi masu iko suna da gaskiya saboda ƙididdigar ke tabbatar da cewa ƙungiyoyi biyu za su yi kama da juna. Wannan kamanni yana riƙe da abubuwan da muka auna (faɗi adadin gyare-gyaren a cikin kwanaki 30 kafin gwaji) da abubuwan da ba mu auna ba (ya ce jinsi). Wannan ƙwarewa don tabbatar da daidaituwa a kan abubuwan da aka lura da kuma ba tare da sunaye ba ne. Don ganin ikon daidaitawa ta atomatik a kan abubuwan da ba a sani ba, bari muyi tunanin cewa bincike na gaba ya gano cewa mutane sun fi karfin kyauta fiye da mata. Shin wannan zai ɓace sakamakon sakamakon Restivo da gwagwarmayar van de Rijt? A'a. Ta hanyar ƙaddamarwa, sun tabbatar da cewa duk waɗanda ba a iya ba da su ba za su daidaita, a cikin tsammanin. Wannan kariya akan rashin sani ba abu ne mai iko ba, kuma hanya ce mai mahimmanci cewa gwaje-gwajen daban-daban daga hanyoyin da ba na gwajin da aka bayyana a babi na 2.

Bugu da ƙari ga gano ma'anar magani ga dukan jama'a, yana yiwuwa a ayyana wani sakamako na jiyya don rabon mutane. Wannan ana kiran shi a matsayin magungunan magani (CATE). Alal misali, a cikin binciken da Restivo da van de Rijt suka yi, bari muyi tunanin cewa \(X_i\) shine ko mai yin edita ya kasance a sama ko žasa da adadin lambobi a cikin kwanaki 90 kafin gwaji. Mutum zai iya lissafin bambancin magani ga waɗannan haske da masu gyara.

Ka'idojin sakamako mai mahimmanci shine hanya mai mahimmanci don tunani game da ƙididdigar gwaji da gwaje-gwaje. Duk da haka, akwai karin abubuwan da ke tattare da halayen da za ku iya tuna. Wadannan abubuwa biyu suna da yawa tare da su a karkashin kalmar Stable Unit Treatment Value Assumption (SUTVA). Sashe na farko na SUTVA shine zaton cewa kawai abinda ke damun mutum \(i\) ita ce ko mutumin ya kasance a cikin magani ko yanayin kulawa. A wasu kalmomi, an ɗauka cewa wannan mutumin \(i\) ba shi da tasiri game da maganin da aka ba wasu mutane. Ana kiran wannan a wasu lokutan "babu tsangwama" ko "babu tsaiko", kuma za'a iya rubuta shi kamar:

\[ Y_i(W_i, \mathbf{W_{-i}}) = Y_i(W_i) \quad \forall \quad \mathbf{W_{-i}} \qquad(4.5)\]

inda \(\mathbf{W_{-i}}\) alama ce ta magungunan magani ga kowa da kowa sai mutum \(i\) . Ɗaya hanyar da za a iya keta wannan ita ce idan magani daga mutum daya ya fadi kan wani mutum, ko dai a gaskiya ko mummunan. Komawa zuwa Restivo da gwagwarmayar van de Rijt, zakuyi tunanin aboki biyu \(i\) da \(j\) kuma mutumin nan \(i\) yana karɓar shinge da \(j\) ba. Idan \(i\) karbar barnstar ya sa \(j\) don shirya ƙarin (daga ma'anar gasar) ko gyara kasa (daga rashin damuwa), to, an keta SUTVA. Haka kuma za'a iya karya idan tasiri na jiyya ya dogara da adadin sauran mutanen da ke karbar magani. Alal misali, idan Restivo da van de Rijt sun ba da lita dubu ko 10,000 a maimakon 100, wannan zai iya tasiri tasiri na karbar barnstar.

Batu na biyu da aka rushe zuwa SUTVA shine zaton cewa kawai magani mai dacewa shine wanda mai bincike ya ba da; wannan tsammanin ana kira wani lokaci ba jiyya ko ɓoye ba . Alal misali, a cikin Restivo da van de Rijt, yana iya kasancewa cewa idan ta ba da masu bincike na gwaninta su sa masu gyara su kasance a cikin shafukan masu shahararren mashahuran kuma suna kasancewa a kan shafukan masu shahararren-maimakon karɓar barnstar- wannan ya haifar da canji a gyare-gyare. Idan wannan gaskiya ne, to, sakamakon sakamako na barnstar ba a rarrabe ba daga sakamakon kasancewar a cikin shahararren masu gyara. Tabbas, ba a bayyana ba idan, daga hangen nesa, wannan ya kamata a dauke shi mai kyau ko maras kyau. Wato, zaku iya tunanin wani mai bincike yana cewa cewa sakamakon karbar barnstar ya hada da dukkanin maganin da ake yi wa gine-ginen. Ko kuwa za ku iya tunanin halin da ake ciki a inda bincike zai so ya ware tasirin barnstars daga dukkanin wadannan abubuwa. Wata hanya ta tunani game da ita ita ce a tambayi idan akwai wani abu da zai jagoranci abin da Gerber and Green (2012) (shafi na 41) ya kira "rashin lafiya a alama"? A wasu kalmomi, akwai wani abu banda magani wanda ya sa mutane a yanayin kulawa da kulawa za a bi da su daban? Damuwa game da fasalwar alama shine abin da ke jagorantar marasa lafiya a rukunin kulawa a gwaje-gwajen likita don daukar kwayar cutar. Wannan hanya, masu bincike zasu iya tabbatar da cewa bambanci tsakanin yanayin biyu shine ainihin magani amma ba kwarewar shan kwaya ba.

Don ƙarin bayani game da SUTVA, duba sashi na 2.7 na Gerber and Green (2012) , sashi na 2.5 na Morgan and Winship (2014) , da kuma sashi na 1.6 na Imbens and Rubin (2015) .

Tsaida

A cikin sashe na baya, Na bayyana yadda za a kimanta tasirin magani. A cikin wannan sashe, zan bayar da wasu ra'ayoyi game da bambancin waɗannan ƙididdiga.

Idan kayi tunani game da kimantawa yadda zaku yi la'akari da yadda za a kwatanta bambancin tsakanin samfurori guda biyu, to yana yiwuwa a nuna cewa kuskuren kuskure na magungunan magani shine:

\[ SE(\widehat{\text{ATE}}) = \sqrt{\frac{1}{N-1} \left(\frac{m \text{Var}(Y_i(0))}{N-m} + \frac{(N-m) \text{Var}(Y_i(1))}{m} + 2\text{Cov}(Y_i(0), Y_i(1)) \right)} \qquad(4.6)\]

inda \(m\) mutane suka sanya magani kuma \(Nm\) don sarrafawa (duba Gerber and Green (2012) , eq 3.4). Don haka, lokacin da kake tunani game da mutane da yawa da za a ba su magani da kuma yawancin da za a ba su don sarrafawa, za ka iya ganin cewa idan \(\text{Var}(Y_i(0)) \approx \text{Var}(Y_i(1))\) , to, kana so \(m \approx N / 2\) , muddin farashin magani da iko sun kasance iri ɗaya. Daidaitawa 4.6 ya bayyana dalilin da ya sa zanen Bond da abokan aiki (2012) gwajin game da sakamakon ilimin zamantakewa game da jefa kuri'a (adadi 4.18) ba shi da lissafi. Ka tuna cewa yana da 98% na mahalarta a cikin yanayin magani. Wannan yana nufin cewa rashin halin kirki a yanayin kulawa ba a kiyasta shi daidai ba kamar yadda zai iya kasancewa, wanda hakan yana nufin cewa bambancin bambancin tsakanin magani da kulawa ba a kiyasta shi daidai ba yadda zai iya zama. Don ƙarin bayani a kan mafi kyawun rarraba na mahalarta a yanayi, ciki har da lokacin da farashin ya bambanta tsakanin yanayin, duba List, Sadoff, and Wagner (2011) .

A ƙarshe, a cikin rubutu na ainihi, Na bayyana yadda bambancin bambancin bambance-bambance, wanda aka saba amfani dashi a cikin tsari mai launi, zai iya haifar da ƙaramin bambanci fiye da bambancin-in-wajen kiyasta, wadda aka saba amfani dashi a tsakanin batutuwa zane. Idan \(X_i\) ita ce darajar sakamakon kafin magani, to, yawancin da muke ƙoƙarin kwatanta tare da bambancin-bambanci shine:

\[ \text{ATE}' = \frac{1}{N} \sum_{i=1}^N ((Y_i(1) - X_i) - (Y_i(0) - X_i)) \qquad(4.7)\]

Kuskuren kuskure na wannan yawa shine (duba Gerber and Green (2012) , eq 4.4)

\[ SE(\widehat{\text{ATE}'}) = \sqrt{\frac{1}{N-1} \left( \text{Var}(Y_i(0) - X_i) + \text{Var}(Y_i(1) - X_i) + 2\text{Cov}(Y_i(0) - X_i, Y_i(1) - X_i) \right)} \qquad(4.8)\]

A kwatanta eq. 4.6 da eq. 4.8 ya nuna cewa kuskuren bambancin bambanci zai sami kuskure mafi kuskuren lokacin da (duba Gerber and Green (2012) , eq 4.6)

\[ \frac{\text{Cov}(Y_i(0), X_i)}{\text{Var}(X_i)} + \frac{\text{Cov}(Y_i(1), X_i)}{\text{Var}(X_i)} > 1\qquad(4.9)\]

Da wuya, a lokacin da \(X_i\) ke da tsinkaye na \(Y_i(1)\) da \(Y_i(0)\) , to, za ka iya samun ƙayyadadden ƙididdiga daga bambanci-bambanci fiye da na bambanci- na-ma'ana daya. Wata hanyar yin la'akari da wannan a cikin batun Restivo da gwagwarmaya na van de Rijt shine cewa akwai bambancin yanayi a cikin adadin da mutane ke shirya, don haka wannan ya sa ya dace da yanayin kulawa da kulawa da wuya: yana da wuyar gano dangi Ƙananan sakamako a cikin bayanan bayanan sakamako. Amma idan ka bambanta-fita daga wannan yanayi mai saurin yanayi, to, akwai ƙananan canje-canjen, kuma wannan yana sa ya fi sauƙi don gano ƙananan sakamako.

Dubi Frison and Pocock (1992) don kwatanta kwatancin bambancin-ma'ana, bambancin-bambancin, da kuma hanyoyin ANCOVA a cikin mafi girma inda aka samo ma'aunin wuri kafin a yi masa magani da kuma bayan magani. Musamman ma, suna bada shawara sosai ga ANCOVA, wanda ban rufe a nan ba. Bugu da ari, ga McKenzie (2012) don tattaunawa game da muhimmancin yawan matakan da za a bi bayan magancewa.