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

38e2af7f8bdad4f2087ab3d42b627e33?s=47 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)

38e2af7f8bdad4f2087ab3d42b627e33?s=128

Shuntaro Sato

November 25, 2020
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  1. ,PP !NFEJ@EBUB ࢁຊɹ޸࣍࿠ $IBQ 0VUDPNFSFHSFTTJPOBOEQSPQFOTJUZTDPSFT

  2. ͸͡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
  3. "HFOEB w0VUDPNFSFHSFTTJPO w1SPQFOTJUZTDPSFT w1SPQFOTJUZTUSBUJpDBUJPOBOETUBOEBSEJ[BUJPO w1SPQFOTJUZNBUDIJOH w1SPQFOTJUZNPEFMT TUSVDUVSBMNPEFMT QSFEJDUJWFNPEFMT

  4. 0VUDPNFSFHSFTTJPOͷ֓ཁ ېԎʢ"ʣͱମॏ૿Ճʢ:ʣͷBWFSBHFDBVTBMF⒎FDUΛݟ͖ͯͨɻ ʢ*1XFJHIUɺ4UBOEBSEJ[BUJPOɺHFTUJNBUJPOʣ ্هͷϞσϧͨͪ͸ɺม਺-ͱΞ΢τΧϜ:ͱͷؔ࿈ʹ৮Ε͍ͯͳ͍ɻ 0VUDPNFSFHSFTTJPOͰ͸ɺม਺-ͱΞ΢τΧϜ:ͷؔ࿈Λ໌ࣔతʹදݱ͠ɺҎԼͷΑ͏ʹϞ σϧΛ࡞੒͢Δɻ ɹɹɹɹɹ  ্ه͸DIBQUFSͰɺGBVYNBSHJOBMTUSVDUVSBMNPEFMͱͯ͠঺հ͞Εͨɻ E

    [Ya,c=0 ∣ L] = β0 + β1 a + β2 aL + β3 L ېԎ B ʹΑΔม਺-಺ͷฏۉҼՌޮՌ
  5. 0VUDPNFSFHSFTTJPOͷղऍ  ˢԾʹېԎͨ͠ਓ͕ɺېԎ͠ͳ͔ͬͨΒͲΕ͙Β͍ମॏ͕૿Ճ͢Δ͔ɻ ɹˡɹ0VUDPNFSFHSFTTJPO ˢېԎͨ͠ਓ͸ɺېԎ͠ͳ͔ͬͨਓͱൺ΂ͯɺͲΕ͙Β͍ମॏ͕૿Ճ͔ͨ͠ɻ ม਺-ͷ֤૚಺Ͱ&YDIBOHFBCJMJUZɺ1PTJUJWJUZɺ$POTJTUFOTZ े෼ʹఆٛ͞Εͨհೖʣ͕੒Γཱ ͓ͬͯΓɺม਺-͕ަབྷΛௐ੔͢Δͷʹे෼Ͱ͋Γɺ-ͱ:ͷؔ࿈ੑΛਖ਼͘͠ϞσϧԽͰ͖͍ͯΔ࣌ɺ ͱͳΔɻ E

    [Ya,c=0 ∣ L] = β0 + β1 a + β2 aL + β3 L E[Y ∣ A, C = 0,L] = α0 + α1 A + α2 AL + α3 L α = β
  6. 0VUDPNFSFHSFTTJPOͷղऍ σʔλུ֓ E [Yc=0 ∣ qsmk, L] = β0 +

    β1 qsmk + β2 sex + β2 race . . . β15 qsmk * smokeintensity
  7. 0VUDPNFSFHSFTTJPOͷղऍ  RTNLʹؔ܎ͷͳ͍߲ʹ͸ΛೖΕͯ஋Λਪఆ͢Δͱ ɹ٤Ԏຊ਺̑ຊͩͬͨਓ͕ېԎ͢ΔͱฏۉతʹLHଠΔɻ ɹ٤Ԏຊ਺̑ຊͩͬͨਓ͕ېԎ͢ΔͱฏۉతʹLHଠΔɻ -ͷ֤૚ʹ͓͍ͯېԎ͕ମॏ૿Ճʹ༩͑ΔӨڹͷظ଴஋ͱݴ͑Δɻ ूஂશମ͕ېԎͨ͠৔߹ͷޮՌΛਪఆ͢Δʹ͸ɺ͞Βʹ4UBOEBSEJ[BUJPOΛߦ͏ɻ ˞٤Ԏຊ਺͕૿͑Δͱ ʢҰൠతʹ͸ओޮՌͱݺ͹ΕΔʣ͚ͩɺଠΔͱ͸ղऍͰ͖ͳ͍ɻ E

    [Yc=0 ∣ qsmk, L] = − 1.6 + 2.6qsmk − 1.4sex . . .0.05qsmk * smokeintensity E [Yc=0 ∣ A = 1,L = 5] = 2.8kg E [Yc=0 ∣ A = 1,L = 5] = 0.2kg β3
  8. 0VUDPNFSFHSFTTJPOͷղऍ ͪͳΈʹҰൠతͳQSPEVDUUFSNΛೖΕͳ͍Ϟσϧ৔߹͸ɺ  ͱͳΔɻ ͜ͷͱ͍͏਺஋ͷҙຯ͸ɺɺʁ E [Yc=0 ∣ qsmk, L]

    = − 1.6 + 3.5qsmk − 1.4sex . . .
  9. "HFOEB w0VUDPNFSFHSFTTJPO w1SPQFOTJUZTDPSFT w1SPQFOTJUZTUSBUJpDBUJPOBOETUBOEBSEJ[BUJPO w1SPQFOTJUZNBUDIJOH w1SPQFOTJUZNPEFMT TUSVDUVSBMNPEFMT QSFEJDUJWFNPEFMT

  10. 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]
  11. 1SPQFOTJUZTDPSFTͱަབྷ π(L) = 1 1 + exp−(β0+β1qsmk+β2sex⋯) ฏۉɿ ฏۉɿ ΋͠ɺ෼෍͕౳͍͠৔߹͸ɺ

    -ʹΑΔަབྷ͸ଘࡏ͠ͳ͍ɻ
  12. 1SPQFOTJUZTDPSFTBT#BMBODFEDPWBSJBUFT 1SPQFOTJUZTDPSF͕ಉ͡Ͱ͋ͬͯ΋ɺม਺-͕ಉ͡ͱ͸ݶΒͳ͍ɻ ྫɿ14ͷਓ "ࡀɺঁੑɺ٤Ԏຊ਺ຊɺӡಈश׳ɺɺɺ #ࡀɺஉੑɺ٤Ԏຊ਺ຊɺӡಈश׳ɺɺɺ ݸਓϨϕϧͰݟΔͱɺม਺-͸ҟͳΔ஋͕ͩɺېԎʹׂΓ౰ͯΒΕΔ֬཰ ͸ಉ͡ͳΔɻ ͭ·Γɺ Ͱ৚݅෇͚ͨ৔߹ɺม਺-ͱ"ʢېԎʣ͸ಠཱͱͳΔɻ ˡ#BSBODFEDPWBSJBUFT

    5FDIOJDBM1PJOU#BMBODJOHTDPSFT  ˠಉ͡14ಉ࢜Ͱ͋Ε͹ɺ&YDIBOHFBCJMJUZ͕੒Γཱͭɻ ஫ҙɿͨͩ͠ɺ3$5ͱ͸ҧ͍ɺ6ONFBTVSFEͳަབྷ͸ߟྀ͞Εͳ͍ɻ π(L) π(L) A ⊥ ⊥ L ∣ π(L)
  13. 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)
  14. "HFOEB w0VUDPNFSFHSFTTJPO w1SPQFOTJUZTDPSFT w1SPQFOTJUZTUSBUJpDBUJPOBOETUBOEBSEJ[BUJPO w1SPQFOTJUZNBUDIJOH w1SPQFOTJUZNPEFMT TUSVDUVSBMNPEFMT QSFEJDUJWFNPEFMT

  15. 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)
  16. 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
  17. ࿈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 
  18. "HFOEB w0VUDPNFSFHSFTTJPO w1SPQFOTJUZTDPSFT w1SPQFOTJUZTUSBUJpDBUJPOBOETUBOEBSEJ[BUJPO w1SPQFOTJUZNBUDIJOH w1SPQFOTJUZNPEFMT TUSVDUVSBMNPEFMT QSFEJDUJWFNPEFMT

  19. 14NBUDIJOHͷ֓ཁ ࠶ܝ ಉ͡14ಉ࢜Ͱ͋Ε͹ɺ&YDIBOHFBCJMJUZ͕੒Γཱͭɻ 14ͷ෼෍͕౳͍͠৔߹͸ɺ-ʹΑΔަབྷ͸ଘࡏ͠ͳ͍ɻ 14͕ࣅ͍ͯΔਓಉ࢜ΛϚονϯάͤ͞Ε͹ɺհೖͱඇհೖ ې ԎͱඇېԎ Ͱ෼෍͕౳͘͠ͳΔɻ ͷޡࠩΛڐͯ͠Ϛονϯά͢ΔͳͲڐ༰ൣғΛܾΊΔɻ ڐ༰ൣғେɿ&YDIBOHFBCJMJUZ͕୲อ͞Εͳ͍ɻ

    ڐ༰ൣғখɿϚον͢Δਓ͕গͳ͘ͳΓɺ$*͕޿͘ͳΔɻ
  20. 14NBUDIJOH͕ҙຯ͢Δ͜ͱ 14NBUDIJOH͸ɺλʔήοτͱ͢Δूஂશһ͕Ϛονϯά͢Ε͹ɺλʔήοτूஂͷҼՌޮՌΛਪ ఆ͢Δ͜ͱ͕Ͱ͖Δɻʢৗʹ1PTJUJWJUZ͕੒Γཱͭʣ ېԎͨ͠ਓ͕ɺېԎ͠ͳ͔ͬͨΒͲΕ͙Β͍ମॏ͕มԽ͔ͨ͠ʁ λʔήοτूஂɹɹɹɹɹɹɹɹɹɹɹҼՌޮՌ  ର৅ूஂΛڱΊΔ͜ͱͰɺશͯͷਓΛϚονϯάͤ͞ɺղऍՄೳͳूஂʹ͢Δ͜ͱ͕Ͱ͖Δɻ ʢ೥ྸ࠽ҎԼɺ٤Ԏຊ਺͸ຊҎԼͷूஂͳͲʣ ͔͠͠ɺϚονϯά͠ͳ͍ਓ͕͍Δ৔ ߹ɺಛघͳूஂͷҼՌޮՌʹͳΔՄೳੑ

    ͕͋Δɻ ͜ͷूஂͷਪఆ஋ΛҰൠԽͰ͖Δ͔Ͳ͏ ͔ΛධՁ͢Δ͜ͱ͸ࠔ೉
  21. "HFOEB w0VUDPNFSFHSFTTJPO w1SPQFOTJUZTDPSFT w1SPQFOTJUZTUSBUJpDBUJPOBOETUBOEBSEJ[BUJPO w1SPQFOTJUZNBUDIJOH w1SPQFOTJUZNPEFMT TUSVDUVSBMNPEFMT QSFEJDUJWFNPEFMT

  22. ͭʴͭͷϞσϧ 1SPQFOTJUZNPEFMT 4USVDUVSBMNPEFM 1SFEJDUJWFNPEFMT

  23. ͭʴͭͷϞσϧ 1SPQFOTJUZNPEFMT w Λਪఆˠ  DPOEJUJPOBMFYDIBOHFBCJMJUZ Λ໨ࢦ͢ w ม਺-ͱΞ΢τΧϜ:ͷؔ܎͸ਪఆ͠ͳ͍ɻ 'JOF1PJOU

     w *1XFJHIUJOH HFTUJNBUJPOͳͲ 4USVDUVSBMNPEFM 1SFEJDUJWFNPEFMT P(A = 1|L) Ya ⊥ ⊥ A ∣ L
  24. ͭ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]
  25. ͭʴͭͷϞσϧ 1SPQFOTJUZNPEFMT 4USVDUVSBMNPEFM 1SFEJDUJWFNPEFMT w ΛϞσϧԽ͢ΔɻʢฏۉͰͳͯ͘΋ྑ͍ʣ w $PVUFSGBDUVBM ͸ߟ͑ͣɺ:ͷ༧ଌͷΈʹڵຯ͕͋Δɻ w

    ҼՌͰͳ͘૬ؔͷΈΛߟྀ͢ΔͨΊɺհೖ"ͱม਺-Λ۠ผ͠ͳ͍ɻ E[Y|L] Ya
  26. 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
  27. ͋ΔපӃ9Ͱ͸ɺ٤ԎऀͷΛېԎͤ͞ɺපӃ:Ͱ͸ېԎͤ͞ͳ͍ɻ  ͜ͷ࣌ͷ1SPQFOTJUZTDPSF͸පӃ9ͷूஂͰ͸ɺපӃ:Ͱ͸ͱͳΓɺ*18Λߦ͏ͱແݶେͷॏΈ ʹͳΔɻ ͜ͷΑ͏ͳɺ"Λ׬શʹ༧ଌͰ͖Δ͕ɺ:ʹશ͘د༩͠ͳ͍ม਺-͸ҼՌਪ࿦ʹ͓͍ͯɺແҙຯͰ͋ ΔɻʢΉ͠Ζ༗֐Ͱ͋Δɻʣ Ϟσϧʹجͮ͘͢΂ͯͷҼՌਪ࿦ख๏͸ɺ&YDIBOHFBCJMJUZɺ1PTJUJWJUZɺ$POTJTUFOTZ े෼ʹఆٛ ͞Εͨհೖʣͱ͍͏৚݅Λඞཁͱ͢Δʂ ېԎ(A)

    ମॏ૿Ճ(Y) පӃ(L) ෼ࢄരൃ