## Amanqaku eMathematika

Ndicinga ukuba indlela efanelekileyo yokuqonda iimvavanyo zezikhokelo zeziphumo (endizixubusha kumanqaku eemathematika kwisahluko 2). Isakhelo (Aronow and Middleton 2013; Imbens and Rubin 2015, chap. 6) obusondeleyo kwiimbono ezivela kwisampula esekelwe kwisakhiwo esichazwe kwisahluko 3 (Aronow and Middleton 2013; Imbens and Rubin 2015, chap. 6) . Esi sihlomelo sibhaliwe ngendlela efana nokugxininisa ukuba unxibelelwano. Oku kugxininiswa kukungekona okomveli, kodwa ndicinga ukuba uxhulumaniso phakathi kweempendulo kunye nokulinga luncedo: kuthetha ukuba ukuba uyazi into malunga nesampuli uze wazi okuthile malunga nokulinga kunye nangoononophelo. Njengoko ndiza kubonisa kula manqaku, isakhelo sesiphumo esinokuthi sibonakalisa amandla okuvavanya okulawulwa ngokungapheliyo ekuqikeleleni imiphumo ye-causal, kwaye ibonisa ukulinganiselwa kwezinto ezinokuthi zenziwe ngezilingo ezifezekileyo.

Kule sihlomelo, ndiza kuchaza izikhokelo zeziphumo eziphambili, ukuphinda ezinye zezinto ezivela kumanqaku eemathematika kwisahluko 2 ukwenzela ukuba le ncwadana ibe neyodwa. Emva koko ndiza kuchaza ezinye iziphumo ezifanelekileyo malunga nokuchaneka koqikelelo lwemiphumo yesiganeko yonyango, kubandakanywa ingxoxo yoluhlu olulungileyo kunye noqikelelo olucacisa ukungafani. Esi sihlomelo sisondela kakhulu Gerber and Green (2012) .

Isikhokelo sesiphumo

Ukuze sifanekise izikhokelo zeziphumo, masibuyele kwi-Restivo kunye no-van de Rijt ukuzama ukuqikelela umphumo wokufumana i-barnstar kwiminikelo ezayo kwi-Wikipedia. Isikhokelo sesiphumo esinokubakho sinezinto ezintathu eziphambili: iiyunithi , unyango kunye neziphumo ezinokwenzeka . Kwimeko ye-Restivo kunye ne-van de Rijt, iiyunithi zazifanele abahleli-abo baphezulu 1% yabanikeli-abangazange bafumane i-barnstar. Singaqhotyoshelisa abahleli nge $$i = 1 \ldots N$$ . Unyango lwaloo vavanyo lwalo lwalu "barnstar" okanye "akukho barnstar," kwaye ndiya kubhala $$W_i = 1$$ ukuba umntu $$i$$ $$W_i = 0$$ yonyango kwaye $$W_i = 0$$ ngenye indlela. Inxalenye yesithathu yezikhokelo zeziphumo kubaluleke kakhulu: iziphumo ezinokwenzeka . Ezi zinto zinzima kakhulu ukucinga ngokuba zibandakanya iziphumo "ezinokwenzeka" izinto ezinokwenzeka. Ngomhleli ngamnye we-Wikipedia, umntu unokucinga ngolu hlobo lwezinto azakuzenza kwimeko $$Y_i(1)$$ ( $$Y_i(1)$$ ) kunye nenani ayenzayo kwimeko yokulawula ( $$Y_i(0)$$ ).

Qaphela ukuba olu khetho lweeyunithi, unyango, kunye neziphumo zichaza into enokuyifunda kule mvavanyo. Ngokomzekelo, ngaphandle kweengcamango ezongezelelweyo, i-Restivo kunye ne-van de Rijt abakwazi ukusho nantoni ngemiphumo yamabhengezo kubo bonke abahleli be-Wikipedia okanye kwiziphumo ezinjengekhwalithi yokuhlela. Ngokubanzi, ukhetho lweeyunithi, unyango kunye neziphumo kufuneka zisekelwe kwiinjongo zesifundo.

Ukubonelelwa kwezi ziphumo ezinokuthi zishwankathelwe kwitheyibhile 4.5-enye ingachaza umphumo wonyango lomntu $$i$$ njengoko

$\tau_i = Y_i(1) - Y_i(0) \qquad(4.1)$

Kwimi, eli lingqinisiso yindlela ecacileyo yokuchaza umphumo we-causal, kwaye, nangona ilula kakhulu, esi sikhokelo siphumelele ukuba senzeke kwiindlela ezininzi ezibalulekileyo (Imbens and Rubin 2015) .

Itheyibhile 4.5: Itheyibhile yeziPhumo eziPhambili
Umntu Ukulungiswa kwimeko yonyango Uhlela kwimeko yokulawula Impembelelo yonyango
1 $$Y_1(1)$$ $$Y_1(0)$$ $$\tau_1$$
2 $$Y_2(1)$$ $$Y_2(0)$$ $$\tau_2$$
$$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$
N $$Y_N(1)$$ $$Y_N(0)$$ $$\tau_N$$
kuthetha $$\bar{Y}(1)$$ $$\bar{Y}(0)$$ $$\bar{\tau}$$

Ukuba sichaza indlela ebalulekileyo ngayo, nangona kunjalo, siba yingxaki. Phantse kuzo zonke iimeko, asikwazi ukugcina iziphumo zombini. Okokuthi, umhleli othile we-Wikipedia ufumane i-barnstar okanye ayikho. Ngoko ke, sigcina esinye seziphumo ezinokwenzeka- $$Y_i(1)$$ okanye $$Y_i(0)$$ -kodwa asikho zombini. Ukungakwazi ukugcina iziphumo eziphambili ziyiyona ngxaki enkulu apho i- Holland (1986) ibibiza ngokuba yiNgxaki ebalulekileyo ye-Causal Inference .

Ngethamsanqa, xa senza uphando, asikho nje umntu onye, ​​sinabantu abaninzi, kwaye oku kunika indlela ejikeleze ingxaki ebalulekileyo ye-Causal Inference. Kunokuba uzame ukuqikelela umgangatho wonyango lomntu ngamnye, sinokulinganisela umphumo wonyango ophakathi:

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

Le nto ibonakaliswe ngokwemiqathango ye- $$\tau_i$$ engenakukwazi ukuyenza, kodwa kunye ne-algebra (Eq 2.8 Gerber and Green (2012) ) siyafumana

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

I-Equation 4.3 ibonisa ukuba ukuba sinokuqikelela isiphumo somyinge wabantu ngaphantsi kwonyango ( $$N^{-1} \sum_{i=1}^N Y_i(1)$$ ) kunye nesiphumo somyinge wesigxina phantsi kolawulo ( $$N^{-1} \sum_{i=1}^N Y_i(1)$$ ), ngoko ke sinokuqikelela umgangatho wonyango, ngaphandle kokuqikelela umphumo wonyango kumntu othile.

Ngoku ukuba ndichazile ukuqikelela kwethu-into esizama ukuyiqikelela-ndiya kutshintsha indlela esinokuyiqikelela ngayo ngedatha. Ndiyathanda ukucinga ngolu mngeni wokuqikelela njengengxaki yesampulu (cinga kwakhona kumanqaku eemathematika kwisahluko 3). Khawucinge ukuba sikhetha abantu ngokukhawuleza ukuba bagcine imeko yonyango kwaye sikhethile ngokukhawuleza abanye abantu ukuba bagcine imeko yokulawula, ngoko sinokulinganisela umphumo wesigqibo kwimeko nganye:

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

apho $$N_t$$ kunye $$N_c$$ ngamanani abantu $$N_c$$ kunye nolawulo lweemeko. Ukulingana 4.4 ngumlinganiselo wokulinganisela-of-means estimator. Ngenxa yokucwangciswa kwesampula, siyazi ukuba ikhefu lokuqala lingumlinganiselo wokulinganisela ukungenakulungelelaniswa kwisiphumo esiphakathi kwonyango kunye nekota yesibini ngumqikelelo ongenakulungele phantsi kolawulo.

Enye indlela yokucinga malunga nantoni na eyenziwa yinto eqinisekisa ukuba ukuthelekiswa phakathi kwamayeza kunye namaqela okulawula kulungile kuba uhlahlo lwabiwo luqinisekisa ukuba amaqela amabini aya kufana. Oku kufana nezinto esizilinganisile (zithetha inani lokuhlelwa kwiintsuku ezingama-30 ngaphambi kokuzama) kunye nezinto esingazange sizilinganise (sithetha ngesini). Lo msebenzi ukuqinisekisa imali eseleyo kwi zombini iimeko eziqwalaselwa kuthwethwe lubalulekile. Ukuze sibone amandla okulungelelanisa ngokuzenzekelayo kwizinto ezingabonakaliyo, makhe sicinge ukuba uphando oluzayo lubona ukuba amadoda aphendule kakhulu kumabhaso kunabesifazane. Ngaba loo nto ingavumelekanga iziphumo zeRetivo kunye novavanyo lukaVan de Rijt? Cha. Ngokurhoxisa, baqinisekisa ukuba zonke izinto ezingabonakaliyo ziya kulungelelaniswa, kulindeleke. Olu khuselo oluchasileyo lunamandla kakhulu, kwaye yindlela ebalulekileyo yokuba iimvavanyo zihluke kwiindlela ezingekho zovavanyo ezichazwe kwisiqendu 2.

Ukongezelela ekuchazeni umphumo wonyango kubo bonke abantu, kunokwenzeka ukuchaza umphumo wonyango kwi-subset of people. Oku kubizwa ngokuba yimpembelelo yempatho yomgangatho wonyango (CATE). Ngokomzekelo, kwisifundo se-Restivo kunye ne-van de Rijt, makhe sicinge ukuba $$X_i$$ ukuba ngaba umhleli ungaphezulu okanye ngaphantsi kwenani lamanani ehleliweyo kwiintsuku ezingama-90 ngaphambi kovavanyo. Omnye unako ukubala umphumo wonyango ngokwahlukileyo kulaba bahleli abakhanyayo nabanzima.

Isikhokelo sesiphumo esinamandla sisindlela esinamandla sokucinga malunga ne-causal inference kunye nokulinga. Nangona kunjalo, kukho ezimbini eziyinkimbinkimbi ezongezelelweyo omele uzigcine engqondweni. Ezi nobunzima zombini zidla lumped kunye phantsi Unit Esitalini elithi Treatment Ixabiso kwingcinga (SUTVA). Inxalenye yokuqala ye-SUTVA yindlela yokucinga ukuba into ebalulekileyo kumntu $$i$$ isiphumo kukuba ingaba loo mntu uphantsi kwonyango okanye imeko yolawulo. Ngamanye amazwi, kucingwa ukuba umntu $$i$$ awuchukumiswanga unyango olunikezwa kwabanye abantu. Ngamanye amaxesha kuthiwa "akukho nto yokuphazamiseka" okanye "akukho ukuphazamiseka", kwaye ingabhalwa ngokuthi:

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

apho $$\mathbf{W_{-i}}$$ yimbonakaliso yemimiselo yonyango wonke umntu ngaphandle komntu $$i$$ . Enye indlela oku kuphulwa ngayo ukuba unyango oluvela kumntu omnye luchithela komnye umntu, nokuba luhle okanye lubi. Ukubuyela kwi-Restivo kunye novavanyo lukaVan de Rijt, cinga abahlobo ababili $$i$$ kunye $$j$$ kwaye lowo mntu $$i$$ ufumana i-barnstar kwaye $$j$$ akayi. Ukuba $$i$$ ukufumana izizathu zokubambisa $$j$$ ukuhlela ngaphezulu (ngaphandle komncinci wokhuphiswano) okanye ukuhlela ngaphantsi (ngaphandle kokuphelelwa ithemba), ngoko i-SUTVA iphulwe. Ingaphinde iphulwe ukuba impembelelo yonyango incike kwinani labanye abantu abafumana unyango. Ngokomzekelo, ukuba i-Restivo kunye ne-van de Rijt babanike iibarnstars ezili-1,000 okanye ezili-10 000 esikhundleni se-100, oku kusenokuba nefuthe ekufumaneni i-barnstar.

Umcimbi wesibini ogqitywe kwi-SUTVA yinto yokuthi unyango olulodwa kuphela oluye lwenziwa ngumphandi; le ngxaki ngamanye amaxesha kuthiwa akukho unyango ofihlakeleyo okanye ukungabandakanywa . Ngokomzekelo, kwi-Restivo kunye ne-van de Rijt, mhlawumbi bekuye kwenzeka ukuba ngokunika abagcini barnstar ukuba babonakalise ukuba ngabavakalisi bavezwe kwiphepha labahleli abadumileyo kwaye bebekwe kwiphepha elihleliweyo labahleli-kunokuba bafumane i-barnstar- obangela ukutshintsha kwindlela yokuziphatha. Ukuba oku kuyinyani, umphumo we-barnstar ayikwahluleki kwimpembelelo yokuba kwiphepha elihleliweyo labahleli. Ngokuqinisekileyo, akucaci ukuba, ngokubhekiselele kwenzululwazi, oku kufuneka kuthathelwe ingqwalasela okanye ingathandeki. Oko kukuthi, ungacinga umphandi othi impembelelo yokufumana i-barnstar iquka zonke izibonelelo zonyango ezilandelayo. Okanye unokucinga ngembandela apho uphando lufuna ukuhlukanisa umphumo weebarnstars kuzo zonke ezinye izinto. Enye indlela yokucinga ngayo kukubuza ukuba kukho nayiphi na into ekhokelela Gerber and Green (2012) (umz. 41) ukubiza "ukuchithwa ngokulinganayo"? Ngamanye amazwi, ingaba kukho enye ngaphandle kweyonyango ebangela ukuba abantu kunyango kunye nokulawulwa kweemeko kufuneka baphathwe ngokwahlukileyo? Ukukhathazeka malunga nokuphulwa komlinganiso kukukhokelela izigulane kwiqela lolawulo kwizilingo zonyango ukuze uthathe ipilisi ye placebo. Ngaloo ndlela, abaphandi banokuqiniseka ukuba umlinganiselo ophela phakathi kwezi zimbini iimeko zonyango kwaye akubona amava okuthatha ipilisi.

Ukufumana okungaphezulu kwi-SUTVA, jonga kwicandelo 2.7 Gerber and Green (2012) , icandelo lesi-2.5 Morgan and Winship (2014) , kunye ne-1.6 ye- Imbens and Rubin (2015) .

Ukuchaneka

Kwinqanaba elidlulileyo, ndichazile indlela yokulinganisela umphumo wonyango ophakathi. Kule candelo, ndiza kubonelela ngeengcamango malunga nokuhluka kwala maxabiso.

Ukuba ucinga malunga nokuqikelela umlinganiselo wonyango oqhelekileyo njengoko uqikelele umahluko phakathi kweesampuli zendlela ezimbini, ngoko kunokwenzeka ukuba ubonise ukuba impazamo eqhelekileyo yempembelelo yonyango yile ndlela:

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

apho $$m$$ abantu babelwe unyango kunye $$Nm$$ ukulawula (bona Gerber and Green (2012) , isiqendu 3.4). Ngaloo ndlela, xa ucinga malunga nabangaphi abantu abanokubanika unyango kunye nokuba bangaphi abanikezelayo ukulawula, unokubona ukuba $$\text{Var}(Y_i(0)) \approx \text{Var}(Y_i(1))$$ ke, ufuna $$m \approx N / 2$$ , nje ngokuba iindleko zonyango kunye nokulawula zifanayo. I-Equation 4.6 icacisa ukuba kutheni ukuqulunqwa kweBond kunye nabo basebenzisana nabo (2012) bazama ngemiphumo yenkcazelo yoluntu ngokuvota (umfanekiso 4.18) wawungasebenzi kakuhle. Khumbula ukuba ngaba ne-98% yabathathi-nxaxheba kwimeko yokunyanga. Oku kuthetha ukuba ukuziphatha okubhekiselele kwimeko yokulawula kwakungalinganiswa ngokuchanekileyo njengoko kwakunokwenzeka, oko oko kwakuthetha ukuba ulwahlulo oluqikelelweyo phakathi kwonyango kunye nomqathango wokulawula aluqikelelwa ngokuchanekileyo njengoko kunokwenzeka. Ukufumana okungakumbi malunga nokwabiwa ngokuthe ngqo kwabathathi-nxaxheba kwiimeko, kubandakanywa neendleko ezahlukileyo phakathi kweemeko, bona List, Sadoff, and Wagner (2011) .

Ekugqibeleni, kwimixholo engundoqo, ndacacisa indlela umlinganisi ohluke ngayo-ohlukeneyo, oqhelekileyo owenziwe kwindlela edibeneyo, unokukhokelela ekunciphiseni okuncinci kunomlinganiselo wokulinganisa, ngokuqhelekileyo kusetyenziswa kwizifundo-phakathi uyilo. Ukuba $$X_i$$ lixabiso lempembelelo ngaphambi kokuba unyango, ngoko ubungakanani esilinga ukuqikelela kunye nendlela eyahlukileyo-eyahlukileyo:

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

Iphutha eliqhelekileyo lelo nani (bona Gerber and Green (2012) , eq. 4.4)

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

Ukuthelekiswa kweq. 4.6 kunye neq. 4.8 ibonisa ukuba indlela eyahlukileyo-eyahlukileyo iya kuba nephuso elincinci elincinci xa (khangela Gerber and Green (2012) , eq. 4.6)

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

Ngokukodwa, xa $$X_i$$ kulandelelaniswa kakhulu $$Y_i(1)$$ kunye $$Y_i(0)$$ , ngoko unokufumana uqikelelo oluchanekileyo kwindlela eyahlukileyo-eyahlukileyo kunokuba uvela kummahluko- -yithetha enye. Enye indlela yokucinga ngale nto kumxholo we-Restivo kunye novavanyo lukaVan de Rijt kukuba kukho ukuhlukahluka kwemvelo kwindleko abantu abayihlela ngayo, ngoko oku kukuthelekisa unyango nolawulo lweemeko nzima: kunzima ukubona isihlobo umphumo omncinci kwimpumelelo yesiphumo. Kodwa ukuba ukwahlukana-oku kuhlukahluka ngokwemvelo, ngoko kukho ukuhluka okuncinane, kwaye oko kwenza kube lula ukubona umphumo omncinci.

Jonga Frison and Pocock (1992) ngokuthelekiswa ngokuchanekileyo kwemeko-ye-ithetha, umahluko-weentlukwano, kunye neendlela ze-ANCOVA esisekwe ngokubanzi apho kukho imilinganiselo emininzi ngaphambi kokunyanga kunye nokunyangwa emva kokunyanga. Ngokukodwa, bayancoma kakhulu i-ANCOVA, engayifumananga apha. Ngaphezulu, jonga McKenzie (2012) ngengxoxo ngokubaluleka kwamanyathelo amaninzi emva kokunyangwa kwezonyango.