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Confounding (Causal inference: What if, Chapter 7)

Shuntaro Sato
November 18, 2020

Confounding (Causal inference: What if, Chapter 7)

Keywords: 因果推論, Confounding(交絡), Confounder(交絡因子)Exchangeability(交換可能性), Backdoor criterion(バックドア基準), Single-world intervention graphs(SWIG)

Shuntaro Sato

November 18, 2020
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  1. Confounding
    Causal Inference Chapter7
    @sanokaz368
    2020 ೥ 6 ݄ 21 ೔
    1

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  2. ໨࣍
    7.1 The structure of confounding
    7.2 Confounding and exchangeability
    7.3 Confounding and the backdoor criterion
    7.4 Confounding and confounders
    7.5 Single-world intervention graphs
    7.6 Confounding adjustment
    Technical Points & Fine Points
    2

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  3. ͸͡Ίʹ
    • ؍࡯ݚڀͰ͸ treatment ͱ outcome ͷ૒ํʹӨڹΛ༩͑
    ΔҼࢠ͕͠͹͠͹ଘࡏ͢Δ
    • ͜ͷ࣌ަབྷ (confounding) ͕͋Δͱݴ͍ɺexchangeability
    ͕੒Γཱͨͳ͘ͳΔ
    • ຊষͰ͸ަབྷͷఆٛͱͦΕΛௐ੔͢Δํ๏Λѻ͏
    3

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  4. 7.1 The structure of
    confounding

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  5. ަབྷͷఆٛ
    ަབྷͷఆٛ
    Treatment A ͱ outcome Y ͕ڞ௨ͷݪҼΛ࣋ͭ͜ͱʹΑͬͯ
    ੜ͡ΔόΠΞεΛɺA ͷ Y ΁ͷޮՌʹ͓͚Δަབྷ
    (confounding) ͱݺͿ
    A Y
    L
    Figure 7.1
    ڞ௨ͷݪҼ L ͕ଘࡏ͢Δͱɺ
    association causation
    4

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  6. ަབྷͷྫ: healthy worker bias
    A Y
    L
    Figure 7.1
    A: ফ๷࢜ͱͯ͠ಇ͘
    Y: ࢮ๢
    L: ݈߁Ͱ͋Δ
    5

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  7. ަབྷͷྫ: confounding by indication/channeling
    A Y
    L
    Figure 7.1
    A Y
    L
    U
    Figure 7.2
    A: ༀͷ࢖༻
    Y: ڵຯͷ͋Δ࣬ױ
    L: A ͕࢖༻͞Ε΍͍͢පଶ
    U:Y ͱ L Λͱ΋ʹҾ͖ى͜͢Ҽࢠ
    ಉछͷༀͷதͰɺױऀݻ༗ͷϦεΫҼࢠ L ʹΑͬͯҩࢣ͕ༀ
    A Λ࢖͏܏޲͕͋Δ৔߹ɺಛʹ channeling ͱݺͿ͜ͱ͕ଟ͍ɻ
    6

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  8. ަབྷͷྫ: reverse causation
    A Y
    L
    U
    Figure 7.3
    A: ӡಈ
    Y: ڵຯͷ͋Δ࣬ױ
    U: ແ঱ީੑͷ࣬ױ
    L: U ʹΑͬͯੜ͡ΔҼࢠ
    L ͕ະ஌ͩͱɺY ΛҾ͖ى͜͢Α͏ͳ U ʹΑͬͯӡಈෆ଍͕
    ੜ͍ͯ͡Δͷʹ΋ؔΘΒͣɺӡಈෆ଍ʹΑͬͯ Y ͕Ҿ͖ى͜
    ͞Ε͍ͯΔ͔ͷΑ͏ʹ൑அ͞Ε΍͍͢ɻ
    7

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  9. ަབྷͷྫ: linkage disequilibrium
    A Y
    L
    U
    Figure 7.3
    A: DNA ഑ྻ
    Y: ԿΒ͔ͷੑ࣭
    L: A ͱϦϯΫ͍ͯ͠Δ DNA ഑ྻ
    U: ຽ଒ͳͲɺA ͱ L ΛϦϯΫͤ͞
    ΔཁҼ
    ෳ਺ͷຽ଒ΛؚΉूஂͰݚڀΛߦ͏৔߹ʹɺಛʹ population
    stratification ͱݺͿ͜ͱ͕ଟ͍ɻ
    8

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  10. ຊষʹ͓͚ΔԾఆ
    ຊষʹ͓͍ͯ͸ҎԼΛԾఆ͢Δɻ
    • σʔλ͸ର৅ूஂ͔Βແ࡞ҝʹநग़͞Ε͍ͯΔ (selection
    node ͕ͳ͍)
    • ҼՌ DAG ͷ͢΂ͯͷ node ͸׬શʹଌఆ͞Ε͍ͯΔ
    • Random variability ͕ଘࡏ͠ͳ͍
    ্ه 3 ఺͸ͦΕͧΕɺ8 ষɺ9 ষɺ10 ষͰѻ͏ɻ
    9

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  11. 7.2 Confounding and
    exchangeability

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  12. ҼՌޮՌͷࣝผ
    ؍࡯ݚڀ͔ΒҼՌޮՌΛࣝผ͢ΔͨΊͷ 3 ৚͕݅
    • Consistency
    • Postivity
    • Exchangeablity
    Ͱ͋ͬͨɻ͜͜Ͱ͸ consistency ͱ positivity ͸੒Γཱͭ΋ͷ
    ͱͯ͠Ծఆ͢Δɻަབྷ͕͋ͬͯ΋
    Conditional exchangeability: Ya A | L
    ͕੒Γཱͯ͹ɺҼՌޮՌΛࣝผͰ͖Δɻ
    10

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  13. Conditional exchangeability ͸੒Γཱͭͷ͔ʁ
    Conditional exchangeablitiy ͕੒ΓཱͭΑ͏ͳڞมྔ L ͷू߹
    ͕͋Δ͔Ͳ͏͔͸൑ఆͰ͖Δͷ͔ʁ

    • σʔλΛੜΈग़ͨ͠ਅͷҼՌ DAG ͕Θ͔Ε͹ɺ ͜ͷ໰
    ͍ʹ౴͑ΒΕΔ
    • ౴͑Δํ๏͸ backdoor ج४ͱ SWIG ͷ 2 ͕ͭ͋Δ
    11

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  14. Backdoor path
    Path
    ಉ͡ node Λ௨Βͳ͍Α͏ʹ edge Λͭͳ͍ͰͰ͖Δϧʔτ
    Causal path
    Causal path: arrow ͷ޲͖͕͢΂ͯಉ͡Ͱ͋Δ path
    Noncausal path: arrow ͷ޲͖͕Ұக͍ͯ͠ͳ͍ path
    Backdoor path
    Treatment ͱ outcome Λ݁Ϳ non causal path ͷ͏ͪɺ
    treatment ʹྲྀΕࠐΉ޲͖ͷ΋ͷ
    A Y
    12

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  15. Open backdoor path
    Selection node ͕ͳ͍ͱ͍͏ԾఆͷԼɺopen backdoor path ͸
    • Treatment ͱ outcome Λ݁Ϳ
    • Treatment ʹྲྀΕࠐΉ (← Ͱελʔτ)
    • Arrow ͷ޲͖͕Ұக͍ͯ͠ͳ͍ (޲͖͕ 1 ճҎ্มΘΔ)
    • Collider ͕ͳ͍ (→ ͔Β͸޲͖͕มΘΒͳ͍)
    ͱͳΔ path Ͱ͋Γɺڞ௨ͷݪҼͷଘࡏΛҙຯ͢Δɻ
    A Y
    A ͱ Y Λ݁Ϳ open path ͸ causal path ͔ open backdoor path
    ͷ͍ͣΕ͔Ͱ͋Δɻ
    13

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  16. Backdoor ج४
    Backdoor ج४
    ڞมྔͷू߹ L ͸ҎԼͷ৚݅Λຬͨ͢ͱ͖ɺbackdoor ج
    ४Λຬͨ͢ɺͱݴ͏ɻ
    • L Ͱ৚݅෇͚Δ͜ͱͰ͢΂ͯͷ backdoor path ͕ด͡Δ
    • L ͷதʹɺA ͷ descendant ؚ͕·Εͳ͍
    Conditional exchangeablity ⇐⇒ backdoor ج४
    Failthfulness ͱ Technical Pont 7.1 Ͱٞ࿦͢Δ৚݅ͷԼͰ
    Ya A | L ⇐⇒ L ͕ backdoor ج४Λຬͨ͢
    14

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  17. Backdoor ج४Λຬͨ͢ 2 ͭͷঢ়گ
    ަབྷ͕ͳ͍ (no confounding)
    ަབྷ͕ͳ͍৔߹ɺ΋ͱ΋ͱ open backdoor path ͕ଘࡏ͠ͳ
    ͍ɻMarginal exchangeability Ya A ͕੒Γཱͭɻ
    ະଌఆͷަབྷ͕ͳ͍ (no unmeasured confounding)
    ަབྷ͕͋Δ͕ɺL(ଌఆ͞Ε͍ͯͯɺA ͷ descendant Λؚ·
    ͳ͍) Ͱ৚݅෇͚Δ͜ͱͰ͢΂ͯͷ backdoor path Λด͡Δ
    ͜ͱ͕Ͱ͖Δ৔߹ɺconditional exchangeability Ya A | L ͕
    ੒ΓཱͭɻL ͷ͜ͱΛ sufficient set for confounding
    adjustment ͱݺͿɻ
    15

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  18. ަབྷͷ޲͖ͱڧ͞
    • Backdoor ج४͸ަབྷͷ޲͖΍ڧ͞ʹؔͯ͠͸Կ΋ޠΒ
    ͳ͍
    • ະଌఆͷަབྷͷ໰୊͸”all or nothing” Ͱ͸ͳ͍ͷͰɺ༧૝
    ͞ΕΔ޲͖΍ڧ͞Λߟ͑Δ͜ͱ͕ॏཁͰ͋Δ (Fine Point
    7.1)
    16

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  19. 7.3 Confounding and the
    backdoor criterion

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  20. ަབྷ͕͋Δ͕ɺYa A | L ͕੒Γཱͭ৔߹
    A Y
    L
    Figure 7.1
    A Y
    L
    U
    Figure 7.2
    A Y
    L
    U
    Figure 7.3
    ͍ͣΕͷ৔߹΋
    • ަབྷ͕ଘࡏ͢Δ
    • L Ͱ৚݅෇͚Δ͜ͱͰ͢΂ͯ
    ͷ backdoor path ͕ด͡Δ
    ͨΊ Ya A | L ͕੒Γཱͪɺະଌ
    ఆͷަབྷ͸ͳ͍ɻ
    17

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  21. ަབྷ͸ͳ͍͕ɺYa A | L ͕੒Γཱͨͳ͍৔߹
    A Y
    L
    U2
    U1
    Figure 7.4
    A Y
    L
    U2
    U1
    Figure 7.4’
    ͜ͷ৔߹
    • ަབྷ͸ͳ͍
    • Backdoor path ͸͋Δ͕
    collider L ʹΑΓด͍ͯ͡Δ
    Ͱ͋ΓɺYa A
    ٯʹ L Ͱ৚͚݅ͮΔͱ
    • Backdoor path ͕։͘
    ͷͰ Ya A | L ͸੒Γཱͨͳ͍
    (selection bias)ɻ
    18

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  22. ަབྷ͕͋ΓɺYa A | L ΋੒Γཱͨͳ͍৔߹
    A Y
    L
    U2
    U1
    Figure 7.5
    A Y
    L
    U2
    U1
    L1
    L2
    Figure 7.6
    Figure 7.5 ͸
    • ަབྷ͋Γ (U1
    ͕ڞ௨ͷݪҼ)
    • L Ͱ৚͚݅ͮΔͱ৽ͨͳ
    backdoor path ͕։͘
    ͱͳΔͷͰɺYa A ΋ Ya A | L
    ΋੒ཱ͠ͳ͍ɻ
    Figure 7.6 ͷΑ͏ʹ L1
    ΍ L2
    ͕ଌ
    ఆͰ͖Ε͹ɺL1
    ΋͘͠͸ L2
    ɾL Ͱ
    ৚͚݅ͮΔ͜ͱͰ͢΂ͯͷ
    backdoor path ͕ด͡Δɻ
    19

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  23. ަབྷͷ 2 ͭͷఆٛ
    ަབྷͷఆٛ 1 (structural definition)
    A ͱ Y ͕ڞ௨ͷݪҼΛ࣋ͭ͜ͱʹΑͬͯੜ͡ΔόΠΞε
    ަབྷͷఆٛ 2
    A ͱ Y ͷؒͷ open backdoor path ʹΑͬͯੜ͡ΔόΠΞε
    =A Λແ࡞ҝʹׂΓ෇͚Δͱফࣦ͢Δܥ౷తόΠΞε
    {
    Selection node ͕ͳ͍৔߹ɿ2 ͭͷఆ͕ٛҰக͢Δ
    Selection node ͕͋Δ৔߹ɿඞͣ͠΋Ұக͠ͳ͍ (Figure 7.4’)
    ఆٛ 1 ͸ղੳͷํ๏ʹґΒͳ͍͕ɺఆٛ 2 Ͱ͸ղੳͷํ๏ʹ
    Αͬͯަབྷͷ༗ແ͕มΘΔɻ
    20

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  24. ަབྷҼࢠ (confounder)
    ަབྷҼࢠ (confounder)
    Treatment A ͱ outcome Y ʹؔ͢Δσʔλ͚ͩͰ͸ҼՌޮՌ
    ΛࣝผͰ͖ͣɺL Λ༻͍Δ͜ͱͰࣝผͰ͖ΔΑ͏ʹͳΔ࣌ɺ
    L ΛަབྷҼࢠ (confounder) ͱݺͿ
    = Ya A ͕੒ཱ͠ͳ͍͕ Ya A | L ͕੒ཱ͢Δ࣌
    21

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  25. ަབྷҼࢠͷྫ
    A Y
    L
    U
    Figure 7.2
    A Y
    L
    U
    Figure 7.3
    • 7.1 Ͱ͸ L ͸ڞ௨ͷݪҼͰ͋Γ
    ަབྷҼࢠ
    • 7.2-7.3 Ͱ͸ L ͸ڞ௨ͷݪҼͰ͸
    ͳ͍͕ަབྷҼࢠ
    • 7.4 Ͱ͸ L ͸ަབྷҼࢠͰ͸ͳ͍
    U ͕ଌఆ͞ΕΕ͹ɺU Ͱ open backdoor path Λด͡Δ͜ͱ͕
    Ͱ͖ΔͷͰɺU ͷԼͰ͸ L ͸ަབྷҼࢠʹͳΒͳ͍ɻ͜ͷΑ͏
    ʹަབྷ͸ઈରతͳ֓೦͕ͩɺަབྷҼࢠ͸૬ରతͳ֓೦Ͱ͋Δɻ
    22

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  26. 7.4 Confounding and
    confounders

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  27. ަབྷͷߟ͑ํͷൺֱ
    ຊষͰͷަབྷͷߟ͑ํ (structural approach)
    1. ҼՌؔ܎ʹ͍ͭͯͷΞϓϦΦϦͳ஌ࣝʹج͍ͮͯɺA ͱ
    Y ͷ͢΂ͯͷڞ௨ͷݪҼΛؚΉΑ͏ʹɺҼՌ DAG Λॻ͘
    2. ڞ௨ͷݪҼ͕͋Ε͹ަབྷ͕ଘࡏ͢Δ
    3. Backdoor ج४ʹΑΓௐ੔͢Δม਺ (ަབྷҼࢠ) ΛܾΊΔ
    ఻౷తͳަབྷͷߟ͑ํ
    1. ౷ܭతͳؔ࿈ʹج͍ͮͯɺ” ަབྷҼࢠ” Λݟ͚ͭΔ
    2. "ަབྷҼࢠ"Λௐ੔͢Δ
    3. ௐ੔ͨ͠ਪఆ஋ͱௐ੔͍ͯ͠ͳ͍ਪఆ஋͕ҟͳΕ͹” ަ
    བྷ” ͕ଘࡏ͢Δ
    23

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  28. ఻౷తͳߟ͑ํʹ͓͚Δ” ަབྷҼࢠ” ͷఆٛ
    ఻౷తͳߟ͑ํʹ͓͚Δ” ަབྷҼࢠ” ͷఆٛ
    ҎԼͷ 3 ৚݅Λຬͨ͢ม਺Λ” ަབྷҼࢠ” ͱݺͿɻ
    • Treatment ͱؔ࿈͍ͯ͠Δ
    • Treatment Ͱ৚͚݅ͮͨԼͰ outcome ͱؔ࿈͍ͯ͠Δ
    • Treatment ͱ outcome ͷؒͷҼՌܦ࿏্ʹଘࡏ͠ͳ͍
    Figure 7.1-7.3 Ͱ͸͜ͷఆٛͷԼɺL ͕ަབྷҼࢠͱͳΓɺզʑ
    ͷఆٛͱҰக͢Δɻ
    24

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  29. ఻౷తͳߟ͑ํͰෆ౎߹͕ੜ͡Δ৔߹
    A Y
    L
    U2
    U1
    Figure 7.4
    A L Y
    U
    Figure 7.7
    Figure 7.4 ΋ 7.7 ΋
    • ަབྷ͸ͳ͘ Ya A
    • ͔͠͠ L ͸఻౷తͳ” ަབྷҼ
    ࢠ” ͷ 3 ৚݅Λຬͨ͢
    • L Ͱௐ੔͢Δͱٯʹ selection
    bias ͕ൃੜ
    • Selection bias ͷ͍ͤͰௐ੔ͷ
    ༗ແʹΑͬͯਪఆ஋͕ҟͳΔ
    ͷͰ” ަབྷ” ͱ൑அ͞ΕΔ
    25

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  30. मਖ਼൛” ަབྷҼࢠ” ͷఆٛ
    Figure 7.4 ΍ 7.7 ʹରԠ͢ΔͨΊɺ3 ৚݅ͷ͏ͪ 2 ൪໨Λ
    • Treatment Ͱ৚͚݅ͮͨԼͰ outcome ͱؔ࿈͍ͯ͠Δ

    • Outcome ͷݪҼͰ͋Δ
    ͱमਖ਼͠Α͏ͱ͢Δͱࠓ౓͸ Figure 7.2 Ͱ L ΛަབྷҼࢠͱΈ
    ͳ͞ͳ͘ͳͬͯ͠·͏ɻ
    26

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  31. Structural approach ͷར఺
    ަབྷʹର͢Δ structural approach Ͱ͸ɺҼՌؔ܎ʹؔ͢Δݚڀ
    ऀͷߟ͑΍ԾఆΛҼՌ DAG Ͱද͠ɺͦΕʹج͍ͮͯௐ੔͢΂
    ͖ม਺ΛܾΊΔ͜ͱͰɺ
    • ԾఆʹԊͬͨద੾ͳղੳ͕ߦ͑Δ
    • Ծఆ͕໌֬ʹͳΔ͜ͱͰɺͦͷԾఆʹ͍ͭͯଞͷݚڀऀ
    ͱٞ࿦Ͱ͖Δ
    27

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  32. 7.5 Single-world intervention
    graphs

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  33. Counterafactural Λάϥϑʹ૊ΈࠐΉ
    ͜͜·ͰͷҼՌ DAG ʹ͓͍ͯ͸ɺάϥϑ্ʹ Ya ͕ग़ݱ͠ͳ
    ͍ͷʹؔΘΒͣɺҎԼͷ 2 ͭ
    • άϥϑ͔Βಋ͔ΕΔ backdoor ج४ͱ
    • Ya A | L
    ͕݁ͼ͍͍ͭͯͨɻ
    ຊηΫγϣϯͰ͸ Single-world intervention graphs (SWIGs) ʹ
    ͓͍ͯɺYa Λάϥϑʹ૊ΈࠐΉ͜ͱͰ exchangeability ΛධՁ
    ͢Δɻ
    28

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  34. Figure7.2 ʹ͓͚Δ SWIG
    A Y
    L
    U
    Figure 7.2
    A | a Ya
    L
    U
    Figure 7.9
    A: ΞεϐϦϯɺY: ೴ଔத
    L: ৺࣬ױɺU:ಈ຺ߗԽ
    ΞεϐϦϯ͕ॲํ͞ΕΔॠؒʹɺॲ
    ํᝦΛճऩ͢Δͱ͍͏Ծ૝తͳੈք
    Λߟ͑Δɻ
    • શһ a = 0 ͱ͍͏ single
    intervention Λड͚Δ
    • Ծ૝తͳੈքʹ͓͚Δ Y ͸ Ya=0
    • A ͸ natural value of treatment ͳ
    ͷͰྲྀΕࠐΉ໼ҹ͸ͦͷ··
    • a ͸ A ͔Βग़Δ໼ҹΛҾ͖ܧ͙
    • A ͔Β a ΁ͷ໼ҹ͸ͳ͍
    29

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  35. SWIG ʹ͓͚Δ exchangeability ͷධՁ
    A | a Ya
    L
    U
    Figure 7.9
    A | a Ya
    L
    U2
    U1
    Figure 7.10
    L Ͱ৚͚݅ͮΔͱ A ͱ Ya ͸
    d-separate ͞Ε͍ͯΔͷͰ
    Ya A | L
    Figure 7.4 Λ SWIG ʹม׵ͨ͠
    Figure7.10 Ͱ͸ɺL Ͱ৚͚݅ͮͣʹ
    d-separate ͞Ε͍ͯΔͷͰ
    Ya A
    SWIG Ͱ͸ɺBackdoor ج४ͱ
    Ya A | L ͱͷؔ܎͕Θ͔Γ΍͍͢
    30

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  36. 7.6 Confounding adjustment

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  37. ަབྷͷௐ੔
    Conditional exchangeabilityYa A | L ͷԼͰަབྷҼࢠΛௐ੔
    ͢Δํ๏ʹ͸ҎԼͷ 2 छྨ͕͋Δɻ
    G-method
    L ΛؚΉ backdoor path ͕ଘࡏ͠ͳ͍৔߹ͷ A ͱ Y ͷؔ࿈Λ
    γϛϡϨʔγϣϯ͠ɺpopulation શମ͋Δ͍͸ͦͷҰ෦ʹ͓
    ͚ΔҼՌޮՌΛਪఆ͢ΔɻBackdoor path Λফ͢ʂ
    ۩ମྫɿstandardizatiosn, IP weighting, g-estimation
    Stratification-based method
    L ͷ஋ʹΑܾͬͯ·Δ subpopulation ʹ͓͚Δ conditional
    effect ΛٻΊΔɻL Ͱ৚͚݅ͮΔʂ
    ۩ମྫɿstratification, restriction, outcome regression,
    matching
    31

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  38. Time-varying treatment
    Time-varying treatment ͕͋Δ৔߹
    • G-method: ަབྷͷௐ੔ʹ༻͍Δ͜ͱ͕Ͱ͖Δɹ (Part III)
    • Stratification-based method: ৚͚݅ͮΔ͜ͱʹΑͬͯ
    selection bias ͕ੜ͡ΔڪΕ͕͋Δ (Chapter 20)
    • Conditional exchangeability Ҏ֎ΛԾఆ͢Δख๏ɿҰൠʹ
    ༻͍Δ͜ͱ͕Ͱ͖ͳ͍
    ͱͳ͍ͬͯΔͨΊɺg-method Λ༻͍Δඞཁ͕͋Δɻ
    32

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  39. Conditional exchangeablity Λ༻͍ͣʹަབྷΛௐ੔͢Δ
    ํ๏
    ަབྷ͸ conditional exchangeablity Λ༻͍ͳͯ͘΋
    • Difference-in-Difference (Technical Point 7.3)
    • Instrumental variable estimation (Chapter 16)
    • The front door criterion (Technical Point 7.4)
    ͳͲʹΑͬͯରॲ͢Δ͜ͱ͕Ͱ͖Δ͕ɺ͜ΕΒͷख๏΋
    conditional exchangeability ͱಉ༷ʹɺݕূෆՄೳͳԾఆΛඞཁ
    ͱ͢ΔͨΊɺख๏ΛબͿࡍʹ͸ɺͲͷԾఆ͕ΑΓ͔֬Β͍͠
    ͔Λߟ͑Δඞཁ͕͋Δɻ
    33

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  40. ؍࡯σʔλʹΑΔҼՌਪ࿦ʹ͓͚Δෆ࣮֬ੑ
    • Expert knowledge Λ༻͍Δ
    • ม਺ (treatment ͷ non-descendant) ΛͳΔ΂͘ଟ͘ଌఆ
    ͜ͱͰ conditional exchangebility ʹۙͮ͜͏ͱͯ͠΋ɺ੒Γཱ
    ͭอূ͸ͳ͍ɻ
    • ෳ਺ͷ causal structure ʹ͍ͭͯͦΕͧΕ෼ੳΛߦ͏
    ৔߹Ͱ΋ͦ͜ʹਅͷ causal structure ؚ͕·ΕΔอূ͸ͳ͍ɻ
    34

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  41. ؍࡯ݚڀΛͲͷΑ͏ʹ൷൑͢΂͖͔
    ͲͷΑ͏ͳ؍࡯ݚڀͰ͋Εަབྷ͕ੜ͡ΔՄೳ͕͋ΔͨΊ
    ໾ʹཱͨͳ͍൷൑
    ʮަབྷ͕ଘࡏ͢ΔՄೳੑ͕͋ΔͨΊɺ͜ͷ؍࡯ݚڀ͔Βಋ͔
    ΕΔਪ࿦͸ޡ͍ͬͯΔ͔΋͠Εͳ͍ʯ
    Ͱ͸ͳ͘ɺ
    ద੾ͳ൷൑
    • ͦͷ݁Ռͱ૬൓͢ΔݚڀΛҾ༻͢Δ
    • ަབྷΛੜͤ͡͞Δ۩ମతͳݪҼΛࢦఠ͢Δ
    ͜ͱͰ൷൑͢΂͖Ͱ͋Δɻ
    35

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  42. Technical Points & Fine Points

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  43. Technical Point 7.1
    Conditional exchangeability ← backdoor ج४
    Faithfulness ͕ͳͯ͘΋੒ཱ
    Conditional exchangeability → backdoor ج४
    FFRCISTG ϞσϧͷԼͰɺfaithfulness ͕͋Δ৔߹ʹ੒ཱ
    Cross-world independence ΛԾఆ͢Δ NPSEM-IE ϞσϧͰ͸
    backdoor ج४͕੒Γཱͨͳͯ͘΋ conditional exchangeablity
    ͕੒ΓཱͪಘΔɻ
    36

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  44. Fine Point 7.1: ަབྷͷํ޲
    Signed causal diagram Λ࢖͏ͱɺަབྷ͕Ͳͷํ޲ʹ bias Λ΋
    ͨΒ͔͕͢Θ͔Δɻ
    A Y
    L
    Figure 7.1
    L → A : +, L → Y : + ͷͱ͖ɺ+ ʹ bias
    L → A : −, L → Y : − ͷͱ͖ɺ+ ʹ bias
    L → A : +, L → Y : − ͷͱ͖ɺ− ʹ bias
    L → A : −, L → Y : + ͷͱ͖ɺ− ʹ bias
    37

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  45. Fine Point 7.1: ަབྷͷڧ͞
    • ڧ͍ަབྷʹ͸ڧ͍ confounder-treatment ؔ܎ͱڧ͍
    confounder-outcome ؔ܎͕ඞཁ
    • Confounder ͕཭ࢄม਺ͷ৔߹ɺconfounder ͷ prevalence
    ʹΑͬͯަབྷͷڧ͕͞มΘΔ
    • Confounder ͕ະ஌ͷ৔߹ɺbias ͷڧ͞Λਪଌ͢Δͷʹ
    sensitivity analysis Λߦ͏͜ͱ͕༗༻
    38

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  46. Fine Point 7.2
    A Y
    L
    U2
    U1
    L1
    L2
    Figure 7.6
    Figure 7.6 ʹ͓͍ͯ͸
    • Ya A | L, L2
    • Ya A | L1
    ͕੒Γཱ͕ͭɺ
    • Ya A | L
    • Ya A | L2
    ͸੒Γཱͨͳ͍ɻैͬͯɺ
    ʮL ͷΈʯ
    ɺ
    ͋Δ͍͸ʮL2
    ͷΈʯͰ͸ҼՌޮՌ͸
    ࣝผͰ͖ͣɺ
    ʮL ͱ L2
    ʯ͋Δ͍͸
    ʮL1
    ʯ͕ଌఆ͞ΕΔඞཁ͕͋Δɻ
    39

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  47. Fine Point 7.3: Surrogate confounder
    L A Y
    U
    Figure 7.8
    • ະଌఆͷ U ʹΑͬͯަབྷ͕ൃੜ
    • L ͸ backdoor path ্ʹͳ͍
    • L ͸ U ͱؔ࿈͍ͯ͠Δ
    L ͕ೋ஋Ͱ U ͷ nondifferential ͳޡ෼ྨͰ͋Ε͹ɺL Ͱ৚݅ͮ
    ͚Δ͜ͱͰ backdoor path ͕෦෼తʹϒϩοΫ͞ΕΔɻ
    Surrogate confounder
    Backdoor path ্ʹ͸ଘࡏ͠ͳ͍͕ɺަབྷʹΑΔ bias ΛݮΒ
    ͢ͷʹ࢖͑Δม਺Λ surrogate confounder ͱݺͿ
    ަབྷʹରॲ͢Δํ๏ͱͯ͠Ͱ͖Δ͚ͩଟ͘ͷ surrogate
    confounder Λଌఆ͢Δɺͱ͍͏ઓུ͕ՄೳͰ͋Δɻ
    40

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  48. Technical Point 7.2
    ม਺ͷू߹ L ͱ U ͕ Ya A | L, U Λຬͨ͢ͱ͖ɺҎԼͷ৚
    ݅Λຬͨͤ͹ɺL ͷΈͰे෼ (Ya A | L) Ͱ͋Δɻ
    U ͕ޓ͍ʹૉͳू߹ U1
    ͱ U2
    ʹ෼͚ΒΕɺ
    • L ʹΑΓܾ·Δ૚ʹ͓͍ͯ U1
    ͱ A ͕ؔ࿈͍ͯ͠ͳ͍
    • A, L, U1
    ʹΑΓܾ·Δ૚ʹ͓͍ͯ U2
    ͱ Y ͕ؔ࿈͍ͯ͠
    ͳ͍
    41

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  49. Fine Point 7.4
    A L Y
    U
    Figure 7.11
    A | a La Ya
    U
    Figure 7.12
    L ͸
    • Backdoor path Λด͡Δ͕
    • A ͷ descendant
    ͳͷͰɺYa A | L ͸੒Γཱͨ
    ͳ͍ɻ
    SWIG Λඳ͘ͱɺ੒Γཱͭͷ͸
    Ya A | La Ͱ͋Δ͜ͱ͕Θ͔Δɻ
    ॏཁͳͷ͸ʮL ͕ A ΑΓ࣌ؒతʹ
    લ͔ޙ͔ʯͰ͸ͳ͘ɺʮL ͕ A ͷ
    descendant ͔൱͔ʯ
    Ͱ͋Δɻ
    42

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  50. Technical Point 7.3
    C A Y
    U
    Figure 7.13
    ະଌఆͷ U ʹΑΔަབྷ͕͋Δͱ͖ɺ
    • A ͱͷؔ܎ʹ͓͍ͯ U ʹΑΔަ
    བྷ͕ੜ͡Δ͕
    • A ʹΑΔӨڹΛड͚ͳ͍
    ม਺Λ negative outcome control ͱͯ͠༻͍Δ͜ͱ͕͋Δɻ
    43

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  51. Technical Point 7.3
    Treatment ͷલͷ outcome ͷ஋Λ C ͱ͢Δͱɺnegative
    outcome control ͱͯ͠࢖༻Ͱ͖ɺ
    Additive equi-confounding
    E[Y0 | A = 1] − E[Y0 | A = 0] = E[C | A = 1] − E[C | A = 0]
    ͷԼͰɺॲஔ܈ʹ͓͚Δ A ͷ Y ΁ͷޮՌΛܭࢉͰ͖Δɻ͜ͷ
    ํ๏͸ difference-in-differences ͱͯ͠஌ΒΕ͍ͯΔɻΑΓऑ
    ͍Ծఆʹجͮ͘ख๏΋ఏҊ͞Ε͍ͯΔɻ
    44

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  52. Technical Point 7.4
    A M Y
    U
    Figure 7.14
    7.14 Ͱ͸ backdoor path Λด͡Δͷ
    ʹ U Ͱ৚͚݅ͮΔඞཁ͕͋Δ͕ɺU
    ͕ະଌఆͰ΋ front door formula Λ༻
    ͍ͯ Pr[Ya = 1] ΛࣝผͰ͖Δɻ
    A ʹΑΔӨڹ͸ M ͔ΒͷΈड͚ΔͷͰɺPr[Ya = 1] ͸

    m
    Pr[Ma = m]Pr[Ym = 1 | Ma = m]
    =

    m
    Pr[Ma = m]
    A MaΑΓࣝผ
    Pr[Ym = 1]
    Ym M|A ΑΓࣝผ
    (∵ Ma Ym)
    =

    m
    Pr[M = m|A = a]

    a′
    Pr[Y = 1|M = m, A = a′]Pr[A = a′]
    45

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  53. ·ͱΊ
    • ަབྷͱ͸ڞ௨ͷݪҼʹΑͬͯੜ͡Δ bias Ͱ͋Δ
    • ҼՌ DAG ʹج͖ͮɺbackdoor ج४ΛదԠͯ͠ௐ੔͢Δ
    ม਺ΛܾΊΔ
    • Ͳͷ؍࡯ݚڀʹ΋ަབྷ͕ੜ͡ΔՄೳੑ͕͋Γɺͦͷରॲ
    ʹ͸ݕূෆՄೳͳԾఆ͕ඞཁͰ͋Δ
    46

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