Mathematical notes

Ndinofunga kuti nzira yakanakisisa yekuedza kuongororwa ndiyo inogona kugadzirisa zvirongwa (izvo zvandakakurukura mubhuku remasvomhu muchitsauko 2). Izvo zvinogona kugadziriswa zvirongwa zvine ukama hwepedyo nemanzwiro kubva pane zvakagadzirirwa-zvidzidzo zvandakarondedzera muchitsauko 3 (Aronow and Middleton 2013; Imbens and Rubin 2015, chap. 6) . Izvi zvinyorwa zvakanyorwa nenzira yakadai yekusimbisa kuwirirana ikoko. Izvi zvinosimbiswa zvisiri zvisiri zvepanyama, asi ndinofunga kuti kuwirirana pakati pekuenzanisa uye kuedza kunobatsira: zvinoreva kuti kana iwe uchiziva chimwe chinhu pamusoro pekuenzanisa iwe unoziva chimwe chinhu pamusoro pekuedza uye pamwe chete. Sezvandicharatidza mune izvi zvinyorwa, zvigadziriro zvingagadziriswa zvinoratidzira simba rekuongororwa kwakaitwa nemaitiro akaenzana ekufungidzira zvinokonzerwa nemagumisiro, uye rinoratidza kukwana kwezvingaitwa nekutongerwa zvakaenzana kuongororwa.

Muchidimbu ichi, ndicharondedzera sarudzo dzinogona kuitika, kudzokorora zvimwe zvemashoko kubva mune zvinyorwa zvemasvomhu muchitsauko 2 kuitira kuti zvinyorwa izvi zvive zvakanyanya. Ipapo ini ndicharondedzera mimwe inobatsira pamusoro pechakarurama chekufungidzirwa kwemareji ehutano hwekurapa, kusanganisira kukurukurirana kwehuwandu hwekugoverwa uye kusiyana-siyana-kwe-kusiyana kwekufungidzira. Izvi zvinyorwa zvinowedzera zvikuru kuna Gerber and Green (2012) .

Mikana inogona kuitika

Kuti tienzanise mamiriro ekugadzirisa zvinogona kuitika, ngatidzokerei kuRetivo uye kuedza kwaVan de Rijt kuenzana nemigumisiro yekuwana barnstar pamipiro inotevera ye Wikipedia. Izvo zvinogona kugadzirisa zvirongwa zvine zvinhu zvitatu zvikuru: zvikamu , marapirwo , nemigumisiro inogona kuitika . Mune mhaka yeRestivo naVan de Rijt, mauniti aiva akakodzera vaparidzi-avo vari pamusoro 1% yevatsigiri-vakanga vasati vagamuchira barnstar. Tinogona kunyora vatongi ava ne \(i = 1 \ldots N\) . Mishonga mukuedza kwavo yaive "barnstar" kana "kwete barnstar," uye ini ndichanyora \(W_i = 1\) kana munhu \(i\) ari mumamiriro ekurapa uye \(W_i = 0\) zvimwe. Chimwe chechitatu chekugadziriswa kwekugadzirisa zvinokonzera ndicho chakakosha zvikuru: zvinogona kuitika . Izvi zvinyanyisa zvinonzwisisika zvakaoma nokuti zvinosanganisira "zvingaitika" zviitiko-zvinhu zvinogona kuitika. Pamusoro pemumwe mutori we Wikipedia, munhu anogona kufungidzira kuwanda kwezvinhu zvaaizoita mumamiriro ekurapa ( \(Y_i(1)\) ) uye nhamba yaaizoita mumamiriro ekudzora ( \(Y_i(0)\) ).

Cherechedza kuti kusarudzwa kwemauniti, marapirwo, nemigumisiro kunotsanangura izvo zvingadzidza kubva pane izvi. Semuenzaniso, pasina mamwe mafungiro, Restivo navan de Rijt havagoni kutaura pamusoro pemigumisiro yemabarnstars kune vese vanyori ve Wikipedia kana pamigumisiro yakadai sekugadzirisa kunaka. Nenzira yakawanda, kusarudzwa kwemasuniti, marapirwo, nemigumisiro inofanira kunge yakavakirwa pazvinangwa zvekudzidza.

Zvichigoverwa izvi zvingagoneka-izvo zvakapfupikiswa mufurafura 4.5-imwe inogona kutsanangura kukanganisa kwekurapa kwemunhu \(i\) se

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

Kwandiri, kuenzanisa iyi inzira yakanakisisa yekutsanangura chinokonzera kukanganisa, uye, kunyange zvazvo iri nyore kwazvo, urongwa uhwu (Imbens and Rubin 2015) munzira dzakawanda dzinokosha uye dzinofadza (Imbens and Rubin 2015) .

Kana tikatsanangura maitiro nenzira iyi, zvakadaro, tinomhanya mune dambudziko. Munenge mamiriro ezvinhu ose, hatiiti kuti tione zvose zvinogona kuitika. Izvi zvinoreva kuti rimwe diki re Wikipedia rakagadziriswa kana rakagamuchira barnstar kana kwete. Saka, tinocherechedza imwe yemigumisiro inokwanisa- \(Y_i(1)\) kana \(Y_i(0)\) asi kwete zvose. Kukundikana kwekuona zvose zvinogona kuitika ndeye dambudziko guru iyo Holland (1986) yakadana iyo Inokosha Dambudziko re Causal Inference .

Nenzira yakanaka, patinenge tichitsvakurudza, hatisi munhu mumwe chete, tine vanhu vazhinji, uye izvi zvinopa nzira yakapoteredza Chinetso Chaicho cheCausal Inference. Panzvimbo pokuedza kuenzanisa munhu-wehutano hwehutano hwehutano, tinogona kuenzana nehuwandu hwehutano hwehutano:

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

Izvi zvichiri kuratidzwa maererano ne \(\tau_i\) izvo zvisingabviriki, asi neimwe algebra (Eq 2.8 Gerber and Green (2012) ) tinowana

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

Mukana wokuti vanhu vanga 4.3 inoratidza kuti kana tinogona kufungidzira vanhu avhareji mugumisiro pasi kurapwa ( \(N^{-1} \sum_{i=1}^N Y_i(1)\) ) uye vanhu avhareji mugumisiro kudzora ( \(N^{-1} \sum_{i=1}^N Y_i(1)\) ), saka tinogona kuenzana nehuwandu hwehutano hwehutano, kunyange pasina kuongororwa kwehutano hwemunhu chero upi zvake.

Iye zvino zvandakatsanangura maonero edu-chinhu chatinenge tichiedza kufungidzira-ndichazochinja kuti tingakwanisa sei kuenzana ne data. Ndinoda kufunga pamusoro pekutsvaga kwekufungidzira sechinetso chemuenzaniso (funga kumashure kumatematical notes muchitsauko 3). Fungidzira kuti isu tinosarudza dzimwe nguva kuti tifunge mumamiriro ekurapa uye isu tinosarudza vamwe vanhu kuti vatarise mumamiriro ekudzivirira, saka tinogona kuenzanisa chiyero chemugumo mumamiriro ezvinhu ose:

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

apo \(N_t\) uye \(N_c\) ndiwo nhamba dzevanhu vari mukurapa uye kutonga mamiriro. Kuenzanisa 4.4 ichitsinhanisi-ye--i-estimator. Pamusana pekugadzirisa shanduro, tinoziva kuti izwi rekutanga nderekufungidzira kusingasaruriki kwevheji yepamusoro pakugadziriswa pasi pekurapwa uye izwi rechipiri nderekufungidzira kusina kukodzera pasi pekutonga.

Imwe nzira yekufunga pamusoro pekushandiswa kwemaitiro kunobatsira kuti kuenzanisa pakati pemishonga pamwe nekutonga mapoka kwakarurama nokuti randomization inovimbisa kuti mapoka acho maviri achafanana. Izvi zvakafanana nezvezvinhu zvatakazviyera (taura nhamba yezvigadziriswa mumazuva makumi matatu isati yasvika) uye zvinhu zvatisina kuenzanisa (taura zvepabonde). Iko kukwanisa kugadzirisa hutano pazvinhu zviviri zvakacherechedza uye zvinhu zvisingaverengeki zvakakosha. Kuti uone simba rekuenzanisa zvakananga pane zvisiri izvo, ngatimbofungidzira kuti kutsvakurudza kweramangwana kunoona kuti varume vari vazhinji vanoteerera zvipo kupfuura vakadzi. Izvozvo zvingavhiringidza zvigaro zveRetivo uye kuedza kwa van de Rijt here? Nha. Nokuronga, vakavimbisa kuti zvose zvisingabatsiri zvingave zvakanyatsogadziriswa, zvichitarisira. Izvi zvinodzivirira kune zvisingazivikanwi zvakasimba kwazvo, uye inzira inokosha iyo miedzo yakasiyana nedzimwe dzisiri dzekuedza dzinotsanangurwa muchitsauko 2.

Mukuwedzera pakutsanangura mamiriro ekurapa kwehuwandu hwevanhu, zvinokwanisika kurondedzera mishonga yekurapa kwechikamu chevanhu. Izvi yemanyorero kunzi zvimiso avhareji kurapwa maturo (CATE). Somuenzaniso, mukudzidza naRetivo uye van de Rijt, ngatimbofungidzira kuti \(X_i\) ndeyekuti mhariri yakanga iri pamusoro kana pasi pehuwandu hwemapakati emazuva makumi mapfumbamwe asati asvika. Mumwe anogona kuverenga mutezo wekurapa zvakasiyana nekuda kwevatori vechiedza uye vane simba.

Izvo zvinogona kugadzirisa zvirongwa ndezve nzira yakasimba yekufunga pamusoro pezvinyorwa zvitsva uye kuedza. Zvisinei, pane zvimwe zviviri zvakaoma zvaunofanira kuchengeta mupfungwa. Izvo zvinetso zviviri zvinowanzobatanidzwa pamwechete pasi pezwi rinonzi Stable Unit Treatment Value Assumption (SUTVA). Chikamu chekutanga cheSUTVA ndechokufungidzira kuti chinhu chimwe chete chakakosha kumunhu \(i\) 's's outcome) kana munhu uyu ari mukurapa kana kuti kutonga mamiriro. Mune mamwe mazwi, zvinofungidzirwa kuti munhu \(i\) haapindirwi nemishonga yakapiwa kune vamwe vanhu. Izvi dzimwe nguva zvinonzi "hapana kupindira" kana "kwete spillovers", uye inogona kunyorwa se:

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

apo \(\mathbf{W_{-i}}\) isvector yemitemo yekurapa kune munhu wese kunze kwomunhu \(i\) . Imwe nzira iyo iyi inogona kutyorwa ndeyokuti kurapwa kubva kune mumwe munhu kunopararira kune mumwe munhu, zvingave zvakanaka kana zvisina kunaka. Kudzokera kuRetivo uye kuedza kwavan van Rijt, fungidzira shamwari mbiri \(i\) uye \(j\) uye munhu uyo \(i\) anogamuchira barnstar uye \(j\) haiti. Kana \(i\) kugamuchira zvikonzero zvekuchengetedza \(j\) kugadzira zvakawanda (kunze kwechikonzero chekumhanyirira) kana kuronga zvishoma (kunze kwekuora mwoyo), ipapo SUTVA yakaputsika. Inogonawo kutyorwa kana kukanganisa kwehutano kunobva pane nhamba yevamwe vanhu vanogamuchira kurapwa. Semuenzaniso, kana Restivo nevan de Rijt vakapa mabarnstar ane chiuru kana gumi panzvimbo pe 100, izvi zvinogona kunge zvakakonzera mhinduro yekugamuchira barnstar.

Nyaya yechipiri yakabatanidzwa muSUTVA ndeyekufungidzira kuti kurapwa chete kwakakodzera ndiyo iyo muongorori anopa; iyi kufungidzira dzimwe nguva kunonzi maitiro asina kuvanzika kana kusabatwa . Somuenzaniso, muRetivo uye van de Rijt, zvingava zvakaitika kuti kuburikidza nekupa barnstar vatsvakurudzi vakaita kuti vaparidzi vauiswe pane peji rakakurumbira revaridzi uye kuti raive riri pane vanozivikanwa peji peji-pane kugamuchira barnstar- izvo zvakakonzera kuchinja kwehutano hwekuita. Kana iri ichokwadi, zvinozoitika barnstar hazvisi kusiyanisa kubva pamigumisiro yekuve pane peji rakakurumbira vatori. Zvechokwadi, hazvisi pachena kana, kubva mune zvesayenzi, izvi zvinofanirwa kuonekwa sezvakanaka kana zvisingadi. Izvozvo zvinoreva kuti iwe unogona kufungidzira mumwe muongorori achitaura kuti kukwanisa kugamuchira barnstar kunosanganisira zvose zvinotevera zvinorapwa izvo barnstar inotanga. Kana iwe unogona kufungidzira mamiriro ezvinhu apo tsvakurudzo ingada kuvhara mhinduro yezvinyorwa kubva kune zvimwe zvinhu zvimwe. Imwe nzira yekufunga nayo ndeyokubvunza kana pane chimwe chinhu chinotungamirira kune izvo Gerber and Green (2012) (p. 41) vanodana "kuparara mukuenzanisa"? Mune mamwe mazwi, pane chimwe chinhu kunze kwekurapa kunokonzera vanhu mukurapa uye kudzora mamiriro ekuti vashandiswe zvakasiyana here? Kunetseka pamusoro pekuparadzanisa kwemasimirwo ndiko kunotungamirira vatungamiri muboka rekutonga mumayera ezvokurapa kutora piritsi ye placebo. Nenzira iyo, vatsvakurudzi vanogona kuva nechokwadi chokuti misiyano chete pakati pemamiriro maviri aya ndiye mushonga chaiye uye kwete chiitiko chekutora mapiritsi.

Kune zvakawanda pamusoro peSUTVA, ona chikamu 2.7 Gerber and Green (2012) , chikamu 2.5 Morgan and Winship (2014) , uye chikamu 1.6 Imbens and Rubin (2015) .

Precision

Muchikamu chekare, ndakatsanangura maonero ekufungidzira kushandiswa kwehuwandu hwehutano. Muchikamu chino, ini ndichapa dzimwe pfungwa pamusoro pekusiyana kweizvikwanganiso.

Kana iwe uchifunga nezvekufungidzira kushandiswa kwehuwandu hwehutano sekuenzanisa kukosha pakati pemiti miviri yekuenzanisira, zvino zvinokwanisika kuratidza kuti kukanganisa kwakakwana kwehuwandu hwehutano hwekuita ndewekuti:

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

apo \(m\) vanhu vakagoverwa kurapwa uye \(Nm\) kudzora (ona Gerber and Green (2012) , peji 3.4). Saka, paunofunga nezvehuwandu hwevanhu vanogovera kurapwa uye kuti vangani vanozopa kuti vatungamirire, unogona kuona kuti kana \(\text{Var}(Y_i(0)) \approx \text{Var}(Y_i(1))\) ipapo iwe unoda \(m \approx N / 2\) , chero bedzi mari yekurapa nekutonga yakafanana. Kuenzanisa 4.6 kunotsanangura kuti nei kuumbwa kweBond uye pamwe chete nevamwe (2012) kuedza pamusoro pemigumisiro yehupenyu hwevanhu pamusoro pekuvhota (chirevo 4.18) chakanga chisina kukwana nhamba. Yeuka kuti yakanga ine 98% yevatori vechikamu mumamiriro ekurapa. Izvi zvaireva kuti chiito chekutaura mumamiriro ekudzivirira chakanga chisingafungidzirwi sechinyatsokwanisa sezvazvaigona kunge chaive, izvo zvaireva kuti kusiyana kwakatarwa pakati pekurapa uye chimiro chekutonga kwakanga kusingatariswi sezviri nani sezvazvaigona kuva. Kuti uwane zvakawanda pamusoro pekugoverwa kwakakwana kwevatori vechikamu kune mamiriro ezvinhu, kusanganisira kana zvikasiyana zvakasiyana pakati pemamiriro ezvinhu, ona List, Sadoff, and Wagner (2011) .

Pakupedzisira, mumutsara mukuru, ndakatsanangura kuti kuenzanisa-mu-kusiyana-kuenzanisa, izvo zvinowanzoshandiswa mukugadzirwa kwakasanganiswa, kunogona kutungamirira kumusiyano muduku kudarika kuenzanisa-mu-kusaridzira, izvo zvinowanzoshandiswa pane-zvidzidzo design. Kana \(X_i\) inokosha yemugumisiro usati watapwa, ipapo huwandu hwatinoedza kuenzanisa nemusiyano-mu-kusiyana-siyana ndewekuti:

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

Iko kukanganisa kwakawanda kwehuwandu ihwo (ona Gerber and Green (2012) , kureva 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 comparative of eq. 4.6 uye eq. 4.8 inoratidza kuti kusiyana-kwe-kusiyana-siyana kuchaita kuti zviduku zvikanganiso zvakakwana apo (ona Gerber and Green (2012) , kureva 4.6)

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

Zvichida, apo \(X_i\) inonyanya kufungidzirwa \(Y_i(1)\) uye \(Y_i(0)\) , iwe unogona kuwana kuongororwa kwakajeka kubva musiyano-we-kusiyana kusiyana nekubva mutsauko- ye-inoreva imwe. Imwe nzira yekufunga pamusoro peizvi mumamiriro ezvinhu eRetivo uye kuedza kwaVan de Rijt ndeyekuti pane zvakawanda zvakasiyana-siyana zvepanyama muchiyero icho vanhu vanogadzira, saka izvi zvinotanidza kurapwa uye kutonga mamiriro ezvinhu akaoma: zvakaoma kuona mukoma zvishoma pamigumisiro inofadza yekugadzirisa data. Asi kana iwe ukasiyana-kunze kwekusiyana kwechiitiko ichi, ipapo pane kuchinja kwakanyanya kudarika, uye izvozvo zvinoita kuti zvive nyore kuona chiduku chiduku.

Ona Frison and Pocock (1992) kuti uenzanise zvakakwana kusiyana-kwe-zvinoreva, kusiyana-kwe-kusiyana-siyana, uye nzira dzeAnCOVA-based inonyanya kugadziriswa apo pane zviyero zvakasiyana-siyana kusati kwarapa uye pashure pokurapa. Kunyanya, vanonyatsokurudzira ANCOVA, iyo yandisina kumbovhara pano. Uyezve, ona McKenzie (2012) kuti uwane hurukuro pamusoro pekukosha kwemaitiro akawanda ekugadzirisa matanho emigumisiro.