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）

Transcript

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

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2. ͸͡Ίʹ
   l0VUDPNFSFHSFTTJPOBOEWBSJPVTWFSTJPOTPGQSPQFOTJUZTDPSFBOBMZTFTBSFUIFNPTUDPNNPOMZVTFE
   QBSBNFUSJDNFUIPETGPSDBVTBMJOGFSFODF:PVNBZSJHIUMZXPOEFSXIZJUUPPLVTTPMPOHUPJODMVEFB
   DIBQUFSUIBUEJTDVTTFTUIFTFNFUIPET4PGBSXFIBWFEFTDSJCFE*1XFJHIUJOH TUBOEBSEJ[BUJPO BOEH
   FTUJNBUJPOrUIFHNFUIPET1SFTFOUJOHUIFNPTUDPNNPOMZVTFENFUIPETBGUFSUIFMFBTUDPNNPOMZVTFEPOFT
   TFFNTBOPEEDIPJDFPOPVSQBSU8IZEJEO`UXFTUBSUXJUIUIFTJNQMFSBOEXJEFMZVTFENFUIPETCBTFEPO
   PVUDPNFSFHSFTTJPOBOEQSPQFOTJUZTDPSFT
   #FDBVTFUIFTFNFUIPETEPOPUXPSLJOHFOFSBM
   .PSFQSFDJTFMZ UIFTJNQMFSPVUDPNFSFHSFTTJPOBOEQSPQFOTJUZTDPSFNFUIPETrBTEFTDSJCFEJOB[JMMJPO
   QVCMJDBUJPOTUIBUUIJTDIBQUFSDBOOPUQPTTJCMZTVNNBSJ[FrXPSLpOFJOTJNQMFSTFUUJOHT CVUUIFTFNFUIPET
   BSFOPUEFTJHOFEUPIBOEMFUIFDPNQMFYJUJFTBTTPDJBUFEXJUIDBVTBMJOGFSFODFXJUIUJNFWBSZJOH
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   BOEQSPQFOTJUZTDPSFNFUIPET5IJTDIBQUFSJTEFWPUFEUPDBVTBMNFUIPETUIBUBSFDPNNPOMZVTFECVUIBWF
   MJNJUFEBQQMJDBCJMJUZGPSDPNQMFYMPOHJUVEJOBMEBUBzংจΑΓ
   ʮҰൠతʹ༻͍ΒΕΔճؼϞσϧ΍܏޲είΞΛͳͥઌʹ঺հ͠ͳ͔ͬͨͷ͔ʁͦͷཧ༝
   ͸ɺ໾ʹཱͨͳ͍͔ΒͰ͋ΔɻΑΓਖ਼֬ʹ͸ɺ୯७ͳઃఆͰ͸͏·͘ػೳ͢Δ͕ɺॎஅσʔ
   λͳͲͷෳࡶͳઃఆͰ͸͏·͘ػೳ͠ͳ͍͔ΒͰ͋ΔɻຊষͰ͸ɺͦΜͳϞσϧͨͪΛ঺հ
   ͢Δɻʯʢҙ༁ʣ
   

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

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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
   

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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
   

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6. 0VUDPNFSFHSFTTJPOͷղऍ
   σʔλུ֓
   E [Yc=0 ∣ qsmk, L] = β0
   + β1
   qsmk + β2
   sex + β2
   race . . . β15
   qsmk * smokeintensity
   

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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
   

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8. 0VUDPNFSFHSFTTJPOͷղऍ
   ͪͳΈʹҰൠతͳQSPEVDUUFSNΛೖΕͳ͍Ϟσϧ৔߹͸ɺ
   
   ͱͳΔɻ
   ͜ͷͱ͍͏਺஋ͷҙຯ͸ɺɺʁ
   E [Yc=0 ∣ qsmk, L] = − 1.6 + 3.5qsmk − 1.4sex . . .
   

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9. "HFOEB
   w0VUDPNFSFHSFTTJPO
   w1SPQFOTJUZTDPSFT
   w1SPQFOTJUZTUSBUJpDBUJPOBOETUBOEBSEJ[BUJPO
   w1SPQFOTJUZNBUDIJOH
   w1SPQFOTJUZNPEFMT TUSVDUVSBMNPEFMT QSFEJDUJWFNPEFMT
   

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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]
    

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

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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)
    

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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)
    

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14. "HFOEB
    w0VUDPNFSFHSFTTJPO
    w1SPQFOTJUZTDPSFT
    w1SPQFOTJUZTUSBUJpDBUJPOBOETUBOEBSEJ[BUJPO
    w1SPQFOTJUZNBUDIJOH
    w1SPQFOTJUZNPEFMT TUSVDUVSBMNPEFMT QSFEJDUJWFNPEFMT
    

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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)
    

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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
    

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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
    
    

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18. "HFOEB
    w0VUDPNFSFHSFTTJPO
    w1SPQFOTJUZTDPSFT
    w1SPQFOTJUZTUSBUJpDBUJPOBOETUBOEBSEJ[BUJPO
    w1SPQFOTJUZNBUDIJOH
    w1SPQFOTJUZNPEFMT TUSVDUVSBMNPEFMT QSFEJDUJWFNPEFMT
    

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

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20. 14NBUDIJOH͕ҙຯ͢Δ͜ͱ
    14NBUDIJOH͸ɺλʔήοτͱ͢Δूஂશһ͕Ϛονϯά͢Ε͹ɺλʔήοτूஂͷҼՌޮՌΛਪ
    ఆ͢Δ͜ͱ͕Ͱ͖Δɻʢৗʹ1PTJUJWJUZ͕੒Γཱͭʣ
    ېԎͨ͠ਓ͕ɺېԎ͠ͳ͔ͬͨΒͲΕ͙Β͍ମॏ͕มԽ͔ͨ͠ʁ
    λʔήοτूஂɹɹɹɹɹɹɹɹɹɹɹҼՌޮՌ
    
    ର৅ूஂΛڱΊΔ͜ͱͰɺશͯͷਓΛϚονϯάͤ͞ɺղऍՄೳͳूஂʹ͢Δ͜ͱ͕Ͱ͖Δɻ
    ʢ೥ྸ࠽ҎԼɺ٤Ԏຊ਺͸ຊҎԼͷूஂͳͲʣ
    ͔͠͠ɺϚονϯά͠ͳ͍ਓ͕͍Δ৔
    ߹ɺಛघͳूஂͷҼՌޮՌʹͳΔՄೳੑ
    ͕͋Δɻ
    ͜ͷूஂͷਪఆ஋ΛҰൠԽͰ͖Δ͔Ͳ͏
    ͔ΛධՁ͢Δ͜ͱ͸ࠔ೉
    

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21. "HFOEB
    w0VUDPNFSFHSFTTJPO
    w1SPQFOTJUZTDPSFT
    w1SPQFOTJUZTUSBUJpDBUJPOBOETUBOEBSEJ[BUJPO
    w1SPQFOTJUZNBUDIJOH
    w1SPQFOTJUZNPEFMT TUSVDUVSBMNPEFMT QSFEJDUJWFNPEFMT
    

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22. ͭʴͭͷϞσϧ
    1SPQFOTJUZNPEFMT
    4USVDUVSBMNPEFM
    1SFEJDUJWFNPEFMT
    

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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
    

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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]
    

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25. ͭʴͭͷϞσϧ
    1SPQFOTJUZNPEFMT
    4USVDUVSBMNPEFM
    1SFEJDUJWFNPEFMT
    w ΛϞσϧԽ͢ΔɻʢฏۉͰͳͯ͘΋ྑ͍ʣ
    w $PVUFSGBDUVBM ͸ߟ͑ͣɺ:ͷ༧ଌͷΈʹڵຯ͕͋Δɻ
    w ҼՌͰͳ͘૬ؔͷΈΛߟྀ͢ΔͨΊɺհೖ"ͱม਺-Λ۠ผ͠ͳ͍ɻ
    E[Y|L]
    Ya
    

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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
    

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27. ͋ΔපӃ9Ͱ͸ɺ٤ԎऀͷΛېԎͤ͞ɺපӃ:Ͱ͸ېԎͤ͞ͳ͍ɻ
    
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