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Outcome regression and propensity scores (Causa...

Avatar for Shuntaro Sato Shuntaro Sato
November 25, 2020

Outcome regression and propensity scores (Causal inference: What if, Chapter 15)

Keywords: 因果推論, Outcome regression(アウトカム回帰),
Propensity score(傾向スコア),層別化,標準化,Propensity matching(傾向スコアマッチング),Predictive model(予測モデル)

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Shuntaro Sato

November 25, 2020
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  1. ͸͡Ίʹ l0VUDPNFSFHSFTTJPOBOEWBSJPVTWFSTJPOTPGQSPQFOTJUZTDPSFBOBMZTFTBSFUIFNPTUDPNNPOMZVTFE QBSBNFUSJDNFUIPETGPSDBVTBMJOGFSFODF:PVNBZSJHIUMZXPOEFSXIZJUUPPLVTTPMPOHUPJODMVEFB DIBQUFSUIBUEJTDVTTFTUIFTFNFUIPET4PGBSXFIBWFEFTDSJCFE*1XFJHIUJOH TUBOEBSEJ[BUJPO BOEH FTUJNBUJPOrUIFHNFUIPET1SFTFOUJOHUIFNPTUDPNNPOMZVTFENFUIPETBGUFSUIFMFBTUDPNNPOMZVTFEPOFT TFFNTBOPEEDIPJDFPOPVSQBSU8IZEJEO`UXFTUBSUXJUIUIFTJNQMFSBOEXJEFMZVTFENFUIPETCBTFEPO PVUDPNFSFHSFTTJPOBOEQSPQFOTJUZTDPSFT 

    #FDBVTFUIFTFNFUIPETEPOPUXPSLJOHFOFSBM .PSFQSFDJTFMZ UIFTJNQMFSPVUDPNFSFHSFTTJPOBOEQSPQFOTJUZTDPSFNFUIPETrBTEFTDSJCFEJOB[JMMJPO QVCMJDBUJPOTUIBUUIJTDIBQUFSDBOOPUQPTTJCMZTVNNBSJ[FrXPSLpOFJOTJNQMFSTFUUJOHT CVUUIFTFNFUIPET BSFOPUEFTJHOFEUPIBOEMFUIFDPNQMFYJUJFTBTTPDJBUFEXJUIDBVTBMJOGFSFODFXJUIUJNFWBSZJOH USFBUNFOUT*O1BSU***XFXJMMBHBJOEJTDVTTHNFUIPETCVUXJMMTBZMFTTBCPVUDPOWFOUJPOBMPVUDPNFSFHSFTTJPO BOEQSPQFOTJUZTDPSFNFUIPET5IJTDIBQUFSJTEFWPUFEUPDBVTBMNFUIPETUIBUBSFDPNNPOMZVTFECVUIBWF MJNJUFEBQQMJDBCJMJUZGPSDPNQMFYMPOHJUVEJOBMEBUBzংจΑΓ ʮҰൠతʹ༻͍ΒΕΔճؼϞσϧ΍܏޲είΞΛͳͥઌʹ঺հ͠ͳ͔ͬͨͷ͔ʁͦͷཧ༝ ͸ɺ໾ʹཱͨͳ͍͔ΒͰ͋ΔɻΑΓਖ਼֬ʹ͸ɺ୯७ͳઃఆͰ͸͏·͘ػೳ͢Δ͕ɺॎஅσʔ λͳͲͷෳࡶͳઃఆͰ͸͏·͘ػೳ͠ͳ͍͔ΒͰ͋ΔɻຊষͰ͸ɺͦΜͳϞσϧͨͪΛ঺հ ͢Δɻʯʢҙ༁ʣ
  2. 0VUDPNFSFHSFTTJPOͷղऍ σʔλུ֓ E [Yc=0 ∣ qsmk, L] = β0 +

    β1 qsmk + β2 sex + β2 race . . . β15 qsmk * smokeintensity
  3. 1SPQFOTJUZTDPSFTུ֓ When *18 $IBQUFS ɺHFTUJNBUJPO $IBQUFS Ͱ͸ɺ ɿېԎʢ"ʣʹׂΓ౰ͯΒΕΔ৚݅෇͖֬཰ΛٻΊͨɻ ͕ʹ͍ۙˠېԎʹׂΓ౰ͯΒΕΔ֬཰͕௿͍ ͕ʹ͍ۙˠېԎʹׂΓ౰ͯΒΕΔ֬཰͕ߴ͍

    Propensity score (PS) 3BOEPNJ[FEUSJBMͰ͸΋ͪΖΜ14͸ʹͳΔɻ 0CTFSWBUJPOBMTUVEJFTͰ͸ɺ"΁ͷׂΓ౰ͯ֬཰͸ݸਓʹΑͬͯҟͳΔɻ ˠσʔλ͔Βਪఆ͢Δඞཁ͕͋Δɻ P[A = 1|L] π(L) : P[A = 1|L] π(L) : P[A = 1|L]
  4. 1SPQFOTJUZTDPSFTͱҼՌ ҼՌΛٻΊΔʹ͸ɺม਺-಺Ͱ ͱ"͕ಠཱ͍ͯ͠Δඞཁ͕͋ͬͨɻ ʢհೖʹׂΓ౰ͯΒΕΔ֬཰ͱɺΞ΢τΧϜͷ஋͸ಠཱʣ  DPOEJUJPOBMFYDIBOHFBCJMJUZ  ͜Ε͸  ͱ΋ݴ͑Δɻ

    Ͱ৚݅෇͚ͨ৔߹Ͱɺ&YDIBOHFBCJMJUZ΍1PTJUJWJUZ͕੒ΓཱͭݶΓɺ w4USBUJpDBUJPO 0VUDPNFSFHSFTTJPO  w4UBOEBSEJ[BUJPO w.BUDIJOH ͳͲͰҼՌؔ܎Λਪఆ͢Δ͜ͱ͕Ͱ͖Δɻ Ya Ya ⊥ ⊥ A ∣ L Ya ⊥ ⊥ A ∣ π(L) π(L)
  5. 1SPQFOTJUZTDPSFTΛ࢖ͬͯ ͋Δ14ͷ஋ʢ ʣͷ΋ͱͰɺฏۉҼՌޮՌʢମॏ૿ՃʣΛٻΊΔɻ  ͔͠͠ɺ ͸dͷ࿈ଓ஋Ͱ͋Γɺಉ͡ Λ࣋ͭਓ͸ɺཧ۶্ଘࡏ͠ͳ͍ɻ ͦ͜Ͱ 14ΛؙΊͯɺ෼Ґʹ૚ผԽ͢Δɻ ֤૚ͰฏۉҼՌޮՌΛٻΊΔɻ

    0VUDPNFSFHSFTTJPOʹ͍ΕͪΌ͏ɻ ࿈ଓྔͷ··ѻ͏ɻ 0VUDPNFSFHSFTTJPOʹ͍ΕͪΌ͏ɻ TUBOEBSEJ[BUJPO π(L) = s E [Y ∣ A = 1,c = 0,π(L) = s] − E [Y ∣ A = 0,c = 0,π(L) = s] π(L) π(L)
  6. 14ΛؙΊͯɺ෼Ґʹ૚ผԽ͢Δɻ  dLHͷฏۉҼՌޮՌ ʢͨͩ͠ɺ৴པ۠ؒ͸͞Βʹ޿͍ɻʣ   RTNLͷ܎਺͸LH $*ɿ  

    ˞F⒎FDUNPEJGZDBUJPO͸ߟ͑ͳ͍ɻ E[Y ∣ A, C = 0,π(L)] = β0 + β1 qsmk + β2 ps2 + . . . ֤૚ͰฏۉҼՌޮՌΛٻΊΔɻ Uݕఆ 0VUDPNFSFHSFTTJPOʹೖΕͪΌ͏ɻ
  7. ࿈ଓྔͷ··ѻ͏   RTNLͷ܎਺͸LH $*ɿ   E[Y ∣ A,

    C = 0,π(L)] = β0 + β1 qsmk + β2 ps  ճͷCPPUTUSBQ .BSHJOBMF⒎FDUʢ฼ूஂશମͷฏۉҼՌޮ Ռʣ͸LH $*ɿ   0VUDPNFSFHSFTTJPOʹೖΕͪΌ͏ɻ 4UBOEBSEJ[BUJPOΛ͢Δɻ ଞʹ΋14ͷྦྷ৐߲΍1SPEVDUUFSNΛೖΕΔ͜ͱ΋Ͱ͖Δɻͨͩ͠ɺղऍ͕ෳࡶʹͳΔɻ 'JOF1PJOU 
  8. ͭʴͭͷϞσϧ 1SPQFOTJUZNPEFMT 4USVDUVSBMNPEFM w ɺ  ΛϞσϧԽ͢Δɻ w "͕:ʹ༩͑Δ௚઀తͳҼՌޮՌΛਪఆ͢Δɻʢ྆൓Ԡؔ܎ͳͲʣ w

    ม਺-ͱհೖ"ͱؔ܎͸ਪఆ͠ͳ͍ɻ 'JOF1PJOU  w TUSVDUVSBMOFTUFENPEFMɺPVUDPNFSFHSFTTJPO GBVYNBSHJOBM TUSVDUVSBMNPEFM  1SFEJDUJWFNPEFMT E[Ya |L] E[Ya=1 |L] E[Ya=0 |L]
  9. ૬ؔWTҼՌ 0VUDPNFSFHSFTTJPO͸ɺҼՌؔ܎ͷਪఆͱɺ༧ଌͷೋͭͷ༻్͕ࠞࡏ͍ͯ͠ΔͨΊɺଟ͘ͷޡղ Λ·Ͷ͘ɻ ࠷΋ଟ͍ޡղ͸ɺม਺બ୒Ͱ͋Δɻ ྫɿGPSXBSETFMFDUJPO CBDLXBSEFMJNJOBUJPO TUFQXJTFTFMFDUJPO  ͜ΕΒ͸ɺߴ࣍ݩͷม਺͔ΒɺΞ΢τΧϜͱ૬͕ؔߴ͍ม਺Λબ୒͢Δʹ͸ྑ͍ख๏Ͱ͋Δɻ ͔͠͠ʂ

    1SPQFOTJUZNPEFMͷม਺-͸ɺ"ʢېԎʣΛ༧ଌ͢Δ͜ͱ͕໨తͰ͸ͳ͘ɺ&YDIBOHFBCJMJUZΛอূ ͢Δ͜ͱ͕໨తͰ͋Δɻ ΋͠ɺ"ͱڧ͍૬͕ؔ͋Δ͕ɺ:ͱશؔ͘܎ͷͳ͍ม਺ΛೖΕΔͱɺਪఆ͞Εͨ:ͷ෼ࢄ͕େ͖͘ͳ Δɻ ˞օ༷ʹ͓ฉ͖͍ͨ͜͠ͱɿ1SPQFOTJUZTDPSFͷਪఆʹػցֶशΛ༻͍Δ͜ͱʹ͍ͭͯ