Confounding (Causal inference: What if, Chapter 7)

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Confounding Causal Inference Chapter7 @sanokaz368 2020 ೥ 6 ݄ 21 ೔ 1

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

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7.1 The structure of confounding

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

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ަ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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ަ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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ަ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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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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7.2 Confounding and exchangeability

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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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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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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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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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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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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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ަ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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7.3 Confounding and the backdoor criterion

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ަ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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ަ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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ަ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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ަ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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ަ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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ަ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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7.4 Confounding and confounders

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ަ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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఻౷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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఻౷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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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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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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7.5 Single-world intervention graphs

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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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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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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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7.6 Confounding adjustment

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ަ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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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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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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؍࡯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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؍࡯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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Technical Points & Fine Points

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