Graphical representation of causal effects (Causal inference: What if, Chapter 6)

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Shuntaro Sato (Sato@ੜ෺౷ܭՈ) Graphical representation of causal effects Causal Inference: What If ษڧձ

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Overview 2 • DAGͱ͸Կ͔ཧղ͢Δ • Causal diagramͷߏ੒ͱόΠΞεΛඥ෇͚Δ • ৄࡉ͸ɼChapter 7, 8, 9

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Introduction 3 A (treatment) Y (outcome) L (covariates) • ؔ܎ • Ծఆ Causal inference ݱ࣮͸ෳࡶ ϕʔεϥΠϯͷڞมྔ͸ॳճമ࿐ʹӨڹ͠ɼ ͜ͷമ࿐͸࣍ͷ࣌఺ͰͷڞมྔʹӨڹ͠ɼ ͜ͷڞมྔ͸࣍ͷമ࿐ʹӨڹ͠ɼ ͦΕ͸Ξ΢τΧϜʹӨڹ͢Δɽ ͦΕͧΕͷڞมྔʹ͸ະ஌ͷڞมྔ͕Өڹ͢Δ… DAG

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6.1 Causal diagrams 4 • DAGͷجຊతͳಛ௃Λཧղ͢Δ • Common causeΛཧղ͢Δ

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DAG 5 - " : Node Edge DAG: Directed acyclic graph • Node͸֬཰ม਺ • ໼ҹʢEdgeʣ͸Node͔ΒNode΁ͷӨڹͷํ޲Λҙຯ͢Δ • ໼ҹ͕͋Δ͜ͱ͸ɼूஂ಺ͷগͳ͘ͱ΋Ұਓʹ͓͍ͯɼ direct causal effect͕͋Δ͜ͱΛࣔ͢ • ໼ҹ͕ͳ͍͜ͱ͸ɼूஂ಺Ͱ୭Ұਓɼdirected causal effect͕ͳ͍͜ͱΛࣔ͢ • ໼ҹ͸ɼͲͷΑ͏ͳӨڹ͕͋Δ͔͸ࣔ͞ͳ͍ • ໼ҹ͸ɼ८ճ͠ͳ͍ʢAcyclicʣ • Y: ࢮ๢ • A: ৺Ҡ২ • L: ࣬පͷॏ঱౓

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6 DAG Causal DAG

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Causal DAG (Technical Point 6.1 Causal directed acyclic graphs) 7 Causal DAGͷ৚݅ • Vj ͔ΒVm ΁ͷ໼ҹ͕ແ͍ͱ͸ɼ Vj ͔ΒVm ΁ͷdirect causal effect͸ͳ͍ɼͱղऍ͢Δ 7K 7N • άϥϑ্ͷม਺ͷϖΞͷ͢΂ͯͷCommon causeʢڞ௨ݪҼʣ͸ɼ ଌఆ͞Ε͍ͯͳͯ͘΋άϥϑʹ͋Δ΂͖ 7K 7N 6 Unmeasured Common cause • ม਺͸ɼͦͷࢠଙͷݪҼͰ͋Δ

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Conditionally or Marginally Randomized experiment? 8 Conditionally randomized experiment • ॏ঱౓L͸৺Ҡ২AʹӨڹΛ༩͑Δ • ݸਓ͸ɼॏ঱౓LʹΑΓɼ ৺Ҡ২AͷϥϯμϜׂ෇͚֬཰͕ҟͳΔ • ৺Ҡ২A͸ࢮ๢YʹӨڹΛ༩͑Δ • ॏ঱౓L͸ࢮ๢YʹӨڹΛ༩͑Δ Common cause Marginally randomized experiment • ॏ঱౓L͸৺Ҡ২AʹӨڹΛ༩͑ͳ͍ • ݸਓ͸ɼॏ঱౓LʹؔΘΒͣɼ ৺Ҡ২AΛϥϯμϜʹׂΓ෇͚ΒΕΔ • ৺Ҡ২A͸ࢮ๢YʹӨڹΛ༩͑Δ

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Observational studiesͰͷԾఆ 9 Figure 6.1ʹ͓͍ͯɼObservational studiesͰ͸࣍ͷԾఆΛ͓͘ • ৺Ҡ২ A ͸ࢮ๢ Y ʹӨڹΛ༩͑Δ • ॏ঱౓ L ͸৺Ҡ২ A ʹӨڹΛ༩͑Δ • ॏ঱౓ L ͸ࢮ๢ Y ʹӨڹΛ༩͑Δ • ৺Ҡ২ A ͱࢮ๢ Y ͷ Common cause ͸ L ͷΈͰ͋Δ ະଌఆͰ͋ͬͨͱͯ͠΋ɼcommon cause ͸ DAG ʹؚΊΔඞཁ͕͋Δ Ya ⊥ ⊥ A|L for all a The assumption of conditional exchangeability

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Graphical and counterfactual approach 10 • Graphical approachͷํ͕ɼCounterfactual approachΑΓ௚ײతʹѻ͑Δ • ͜ΕΒ͸ີ઀ʹϦϯΫ͍ͯ͠Δ ैདྷͷDAG͸ɼCounterfactual approachΛදͤͳ͍ SWIG (Single World Intervention Graph) ֦ு DAG + Counterfactual approach Chapter 7

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6.2 Causal diagrams and marginal independence 11 • DAG্ͷͲͷม਺΋৚݅෇͚ͳ͍࣌ɼ Figure 6.2ʙ6.4͸ͦΕͧΕԿΛҙຯ͢Δͷ͔ཧղ͢Δ • ௚ײʹର͠ɼCausal graphs theory Ͱઆ໌͢Δ

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௚ײతʹؔ࿈ͦ͠͏ 12 • ΞεϐϦϯ࢖༻ A ͸ ৺ଁපͷϦεΫ Y ʹରͯ͠༧๷ͷcausal effectΛ࣋ͭ • ৚݅෇͚ͳ͍ͰΞεϐϦϯ࢖༻ A ΛϥϯμϜׂ෇ͨ͠ Pr(Ya=1 = 1) ≠ Pr(Ya=0 = 1) • ϥΠλʔॴ࣋ A ͸ɼ ୭ʹରͯ͠΋ഏ͕ΜͷϦεΫ Y ͷ ʢ༠Ҿ or ༧๷ʣcausal effect ͸ͳ͍ • ٤Ԏ L ͸ɼA ͱ Y ͷ྆ํʹ causal effect Λ࣋ͭ Pr(Ya=1 = 1) = Pr(Ya=0 = 1) Figure 6.2΋6.3Ͱ΋ɼAͱY͸௚ײతʹؔ࿈ͯͦ͠͏

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Causal graphs theory (Randomized experiment) 13 A ͕ Y ΁ causal effect Λ࣋ͭ࣌ɼҰൠతʹɼA ͱ Y ͸ؔ࿈͢Δ͜ͱ͕ظ଴͞ΕΔ Causal graph theory Unconditional exchangeability Ͱ͋Δ Ideal randomized experiment Ͱ͸ɼ Causation ͸ Association Λࣔ͢ʢٯ΋ಉ༷ʣ Pr(Ya=1 = 1) ≠ Pr(Ya=0 = 1) Pr(Y = 1|A = 1) ≠ Pr(Y = 1|A = 0) Pr(Y = 1|A = 1) ≠ Pr(Y = 1|A = 0) Pr(Ya=1 = 1) ≠ Pr(Ya=0 = 1) Causation Association Association Causation

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Figure 6.3Ͱ͸ɼͳͥ௚ײతʹؔ࿈͋Δͱࢥͬͨͷ͔ʁ 14 1. ͋Δݚڀऀ͸ɼ ʮϥΠλʔॴ࣋ A ͸ഏ͕ΜͷϦεΫ Y ΁ͷ effect ͕͋Δ͔ʯݚڀ͢Δ͜ͱʹͨ͠ 2. ൴͸ଟ͘ͷਓʹɼϥΠλʔΛॴ͍࣋ͯ͠Δ͔ฉ͖ɼ޲͜͏5೥ؒͰ൴Β͕ഏ͕Μͱ ਍அ͞Ε͔ͨͲ͏͔ه࿥ͨ͠ 3. Hera͸ɼϥΠλʔΛॴ͍࣋ͯͨ͠ 4. ΋͠Hera͕ϥΠλʔΛॴ͍࣋ͯ͠ΔͳΒ͹ɼ൴ঁ͸Smoker (L)Ͱ͋ΔՄೳੑ͕ߴ͍ 5. ͞ΒʹɼHera͸SmokerͰ͋ΔͳΒ͹ഏ͕ΜϦεΫ΋ߴ͍Ͱ͋Ζ͏ 6. ௚ײతʹɼϥΠλʔॴ࣋ͱഏ͕ΜͷϦεΫYʹ͸ؔ࿈͕͋ΔͩΖ͏ ϥΠλʔॴ࣋ ٤Ԏ↑ ഏ͕ΜϦεΫ↑ ϥΠλʔະॴ࣋ ٤Ԏ↓ ഏ͕ΜϦεΫ↓

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Prediction = Association 15 • AͱYͷؒʹؔ࿈͕ظ଴͞ΕΔ • A = 1ͷूஂͰͷഏ͕ΜϦεΫͱA = 0ͷूஂͰͷഏ͕ΜϦεΫ͸ҟͳΔ Pr(Y = 1|A = 1) ≠ Pr(Y = 1|A = 0) Association Prediction • A ʹΑΓɼY ͷ༧ଌೳྗ্͕͕Δ • ྫ͑ɼA ͸ Y΁ͷcausal effectΛ࣋ͨͳͯ͘΋ Association = Prediction ϥΠλʔॴ࣋ ٤Ԏ↑ ഏ͕ΜϦεΫ↑ ϥΠλʔະॴ࣋ ٤Ԏ↓ ഏ͕ΜϦεΫ↓ ഏ͕Μ

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Causal graphs theory (Observational study Figure 6.3) 16 Common cause LΛ௨ͯ͠ɼ A͔ΒYʢor Y͔ΒAʣ΁ͷؔ࿈ͷྲྀΕ͕Ͱ͖Δ࣌ɼAͱY͸ؔ࿈͢Δ Causal graph theory

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Figure 6.4ͷಛ௃ 17 • A͸ϋϓϩλΠϓ • ϋϓϩλΠϓ A ͸ɼ୭Ұਓͱͯ͠٤Ԏ Y ΁ͷ causal effect Λ࣋ͨͳ͍ • ϋϓϩλΠϓ A ΋ɼ٤Ԏ Y ΋৺ଁප L ΁ͷcausal effectΛ࣋ͭ • L͸ɼA ͱ Y ͷCommon effectʢڞ௨ޮՌʣͰ͋Δ • Common effect Ͱ͋Δ L ΛɼColliderʢ߹ྲྀ఺ʣͱ͍͏ʢA → L ← Yʣ Collider Common effect

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Figure 6.4ʹ͓͍ͯAͱY͸ؔ࿈͠ͳ͍ 18 1. ͋Δݚڀऀ͸ɼ ʮ٤Ԏ Y ΁ϋϓϩλΠϓ A ͷ effect ͕͋Δ͔ʯݚڀ͢Δ͜ͱʹͨ͠ 2. ൴ঁ͸ɼଟ͘ͷࢠڙୡͷϋϓϩλΠϓΛௐ΂ɼ ͦͷࢠͲ΋͕ͨͪ٤ԎऀʹͳΔ͔Ͳ͏͔Λه࿥ͨ͠ 3. Apollo͸ɼϋϓϩλΠϓ A Λ࣋ͨͳ͍ʢA = 0ʣ 4. ൴͸٤Ԏ͢ΔʢY = 1ʣՄೳੑ͕ߴ͍͔ʁɹ௿͍͔ʁ 5. ϋϓϩλΠϓ A ͷ༗ແʹΑΓɼ٤ԎऀʹͳΔϦεΫ Y ͸ಉ͡ͳͷͰɼ ϋϓϩλΠϓ A ͸٤Ԏ Y Λ༧ଌ͢ΔೳྗΛ޲্ͤ͞ͳ͍ 6. ௚ײతʹɼϋϓϩλΠϓ A ͱ٤Ԏ Y ͸ؔ࿈͠ͳ͍ͩΖ͏ Pr(Y = 1|A = 1) = Pr(Y = 1|A = 0) A⊥ ⊥ Y

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Causal graphs theory (Figure 6.4) 19 • Colliders ͸ɼͦΕΒ͕ؒʹ͋Δม਺ؒͷؔ࿈ͷύεΛϒϩοΫ͢Δ • A → L ← Y ͸Collider L ͰϒϩοΫ͞ΕΔͨΊɼA ͱ Y ͸independentͰ͋Δ Causal graph theory

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·ͱΊʢ6.2 Causal diagrams and marginal independenceʣ 20 • 2ͭͷม਺͕ɼ(marginally) associatedͱ͸ɼ • Ұํ͕ଞํͷݪҼͰ͋Δ • ͦΕΒ͕ Common cause Ͱ͋Δ • ͦΕҎ֎Ͱ͋Ε͹ɼ(marginally) independent

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6.3 Causal diagrams and conditional independence 21 • Figure 6.2ʙ6.4ͷม਺Λ৚͚݅ͮΔͱԿΛҙຯ͢Δͷ͔ཧղ͢Δ • ௚ײʹର͠ɼCausal graphs theory Ͱઆ໌͢Δ No conditioning Conditioning

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Mediator (Figure 6.5) 22 • ΞεϐϦϯ࢖༻ A ͱ৺ଁප Y ͸ؔ࿈͢Δ • ΞεϐϦϯͷ࢖༻͸ɼ ৺ଁපϦεΫ΁ͷcausal effectΛ͔࣋ͭΒ • ΞεϐϦϯ࢖༻ A ͸ɼ݂খ൘ͷڽݻ B ΁causal effectΛ࣋ͭ • ݂খ൘ͷڽݻΛݮΒ͢ • ݂খ൘ͷڽݻ͸ɼ৺ଁපϦεΫ΁ͷcausal effectΛ࣋ͭ • ม਺ؒͷύεͷؒʹ͋Δม਺ΛMediatorʢഔհҼࢠʣͱ͍͏ Mediator ৘ใ௥Ճ

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MediatorΛ৚͚݅ͮΔͱʁ 23 • Mediator BΛ৚͚݅ͮΔͱɼA ͱ Y͸ؔ࿈͢Δ͔ʁ • Bͷ৘ใ͕͋Δ࣌ɼA͸Yͷ༧ଌೳྗΛ޲্ͤ͞Δ͔ʁ ม਺Λ࢛֯ͰғΉ͜ͱ͸ɼ BͰ৚͚݅ͮΔ͜ͱΛҙຯ͢Δ BͰϑΟϧλʔΛ͔͚ͨதͰͷ AͱYͷؔ࿈Λߟ͑Δ • ݂খ൘ͷڽݻ͕গͳ͍ਓʢB = 0ʣ͸ɼ৺ଁපͷฏۉతͳϦεΫ͸௿͍ • B = 0 ͷதͰ͸ɼ ΞεϐϦϯͷ࢖༻ A ͷ༗ແʹؔΘΒͣɼ৺ଁපͷϦεΫ͸௿͍ • ΞεϐϦϯ࢖༻͸ B Λ௨ͯ͠ͷΈ৺ଁපʹӨڹ͢ΔͷͰɼ ΞεϐϦϯ࢖༻ͷ৘ใ͸ɼ৺ଁපͷϦεΫΛ༧ଌ͢Δ͜ͱʹߩݙ͠ͳ͍ Conditioning

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Causal graphs theory (Figure 6.5) 24 • A ͱ Y ͕ marginally associated Ͱ΋ɼ Mediator B Ͱ৚͚݅ͮΔͱɼA ͱ Y ͸ conditionally independent Ͱ͋Δ • ؔ࿈͠ͳ͍ Causal graph theory Pr(Y = 1|A = 1, B = b) = Pr(Y = 1|A = 0, B = b) for all b A⊥ ⊥ Y|B Block

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Common causeΛ৚͚݅ͮΔͱʁ 25 • Common cause LΛ৚͚݅ͮΔͱɼA ͱ Y͸ؔ࿈͢Δ͔ʁ • Lͷ৘ใ͕͋Δ࣌ɼA͸Yͷ༧ଌೳྗΛ޲্ͤ͞Δ͔ʁ • Nonsmoker ʹݶఆ͢Δ • ϥΠλʔॴ࣋ A ͷ༗ແʹؔΘΒͣɼഏ͕ΜͷϦεΫ͸௿͍ • ݸਓ͕ϥΠλʔΛॴ࣋͢Δ͜ͱΛ஌͍ͬͯͯ΋ɼ ٤Ԏͷ৘ใ͕͋Ε͹ɼഏ͕ΜϦεΫͷ༧ଌͷೳྗ͸޲্͠ͳ͍ͩΖ͏ A: ϥΠλʔॴ࣋ Y: ഏ͕Μ L: ٤Ԏ Association Association?

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Causal graphs theory (Figure 6.6) 26 • A ͱ Y ͕ marginally associated Ͱ΋ɼ Common cause L Ͱ৚͚݅ͮΔͱɼA ͱ Y ͸ conditionally independent Ͱ͋Δ • ؔ࿈͠ͳ͍ Causal graph theory Pr(Y = 1|A = 1,L = l) = Pr(Y = 1|A = 0,L = l) for all l A⊥ ⊥ Y|L Block

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Common effect (Collider) Λ৚͚݅ͮΔͱʁ 27 Common effect (Collider) LΛ৚͚݅ͮΔͱɼA ͱ Y͸ؔ࿈͢Δ͔ʁ A: ϋϓϩλΠϓ Y: ٤Ԏ L: ৺ଁප Unassociation Association? • ৺ଁපΛ࣋ͭݸਓʹݶఆ͠ɼϋϓϩλΠϓ A ͷ٤Ԏ Y ͷؔ࿈ΛධՁ͢Δ • A ͱ Y ͷΈ͕ L ͷݪҼͩͱ͢Δ • ৺ଁපΛ࣋ͭूஂͰɼ Ծʹશһ͕ϋϓϩλΠϓͳͩ͠ͱ͢Δͱɼશһ͕٤ԎऀͰͳ͍ͱ͍͚ͳ͍ Ծʹશһ͕ඇ٤Ԏऀͩͱ͢Δͱɼશһ͕ϋϓϩλΠϓ͋ΓͰͳ͍ͱ͍͚ͳ͍ Common effect (Collider) LΛ৚͚݅ͮΔͱɼA ͱ Y ͸ؔ࿈͢Δ

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Causal graphs theory (Figure 6.7) 28 • Common effect (Collider)Ͱ৚͚݅ͮΔͱɼ ৚݅෇͚Λ͍ͯ͠ͳ͍࣌ʹ͸ Block ͞Ε͍ͯͨ A → L ← Y ͷܦ࿏͕։͔ΕΔ • ؔ࿈͢Δ Causal graph theory Open

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Common effect ͷࢠଙΛ৚͚݅ͮΔͱʁ 29 Common effect (Collider) ͷࢠଙ C Λ৚͚݅ͮΔͱɼA ͱ Y͸ؔ࿈͢Δ͔ʁ Unassociation Association • Figure 6.7 ʹར೘࣏ྍ C Λ௥Ճ • ͜ͷ࣏ྍ͸৺ଁපͷ਍அͷ݁Ռ࢖ΘΕΔ • C ͸ Common effect L ͷӨڹΛड͚Δ Common effect (Collider) Lͷࢠଙ C Λ৚͚݅ͮΔͱɼA ͱ Y ͸ؔ࿈͢Δ Association?

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Causal graphs theory (Figure 6.8) 30 • Common effect (Collider)ͷࢠଙ C Λ৚͚݅ͮΔͱɼ ৚݅෇͚Λ͍ͯ͠ͳ͍࣌ʹ͸ Block ͞Ε͍ͯͨ A → L ← Y ͷܦ࿏͕։͔ΕΔ • ؔ࿈͢Δ Causal graph theory Open A → L ← Y ͷύεΛϒϩοΫ͢Δʹ͸ɼL ͱ C ͷ྆ํΛ৚݅෇͚ͳ͍

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·ͱΊʢ6.3 Causal diagrams and conditional independenceʣ 31 No conditioning Conditioning (Mediator) (Marginally) Association (Conditionally) Independent Common cause (Marginally) Association (Conditionally) Independent Common effect (Marginally) Independent (Conditionally) Association Association between A and Y is…

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6.4 Positivity and consistency in causal diagrams 32 Causal inferenceʹඞཁͳ৚͕݅ɼDAGͰ͸ͲͷΑ͏ʹѻ͏͔ཧղ͢Δ

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Causal inference ʹඞཁͳ৚݅ 33 දݱʢcounterfactuals or graphsʣʹؔΘΒͣɼ Standardization or IP weightingΛ༻͍ͨCausal inferenceʹ͸ɼ3ͭͷ৚͕݅ඞཁ • Exchangeability • Positivity • Consistency Chapter 7 and 8 Ͱѻ͏ ͜ͷChapterͰѻ͏ ͜ΕΒͷ৚͕݅੒ཱ͠ͳ͍ͱɼղੳ͔ΒಘΒΕΔ਺஋Λద੾ʹղऍͰ͖ͳ͍

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Positivity 34 Standardized risk for treatment level a - " : ∑ l Pr(Y = 1|A = a, L = l) Pr(L = l) Pr(A = a|L = l) > 0 for all l with Pr(L = l) ≠ 0 Positivity Well-define͢Δ৚݅ L ͔Β A ΁ͷ໼ҹ͕ܾఆతͰ͸ͳ͍͔ʁ Treatment A Covariate L n 1 1 20 20/20 = 1 1 0 50 50 /100 = 0.5 0 1 0 0/20 = 0 0 0 50 50/100 = 0.5 Pr(A = a|L = l) n Covariate Treatment L = 1 L = 0 A = 1 20 50 A = 0 0 50 ͜ͷॻ੶Ͱ͸ɼಛʹஅΓͷແ͍ݶΓɼ Positivity͸੒ཱ͢Δ

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Consistency 35 Pr(A = a|L = l) Treatment A Well-deﬁned Well-definedͳॲஔ͕ ଌఆͰ͖Δ͔? Outcome Y Consistency 1 Consistency 2 Consistency 1 and 2 ͕੒ཱ͢Ε͹ Ya = Y - " : • ͜ͷॻ੶Ͱ͸ɼಛʹஅΓͷແ͍ݶΓɼTreatment ͸ well-diﬁne ͞Ε͍ͯΔ • Compound / multiple treatment ͷ৔߹ɼ஫ҙ͕ඞཁ

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Compound / multiple treatment (Figure 6.10) 36 • R = 1 ͸ɼຖ೔গͳ͘ͱ΋30෼ӡಈ͢Δ͜ͱ • R = 0 ͸ɼຖ೔30෼ະຬͷӡಈΛ͢Δ͜ͱ • ӡಈ͕࣌ؒ30, 31, 32…෼Ͱ΋ɼR = 1 • ӡಈ͕࣌ؒ29, 28, 27…෼Ͱ΋ɼR = 0 • A (r = 1) ͸ɼR = 1 ͷશͯͷӡಈ࣌ؒΛද͢ • A (r = 0) ͸ɼR = 0 ͷશͯͷӡಈ࣌ؒΛද͢ • R ͸ compound treatment • A͸े෼ʹఆٛ͞Ε͍ͯΔͷͰɼR ͔Β Y ΁ͷ௚઀໼ҹ͸ͳ͍ Compound treatment ͱ ͦͷ version ͱͳΔ treatment Λ well-difine ͢Δ͜ͱͰɼ Causal inference Λద੾ʹ͢ΔͨΊͷม਺ɾؔ܎Λݕ౼Ͱ͖Δ ॏཁͳ step

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6.5 Structural classification of bias 37 • Systematic bias ʢܥ౷όΠΞεʣΛཧղ͢Δ • Lack of exchangeability Λཧղ͢Δ • DAG ͔Βൃੜ͠͏Δ systematic bias Λ໌Β͔ʹͰ͖Δ

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Systematic bias 38 Sample size Error Systematic bias Random error • Sample size ͕ແݶͰ͋ͬͯ΋ɼࣝผՄೳੑ͕ෆे෼ͳͱ͖ʹൃੜ͢ΔΤϥʔ • Systematic bias ͸ sample size ͷӨڹΛड͚ͳ͍ • Random error ͸ sample size ͕ແݶʹͳΔͱ0ʹͳΔ ؔ৺ͷ͋Δ฼ूஂʹ͓͚Δɼ treatmentͱoutcomeͷҼՌؔ܎ʹ༝དྷ͠ͳ͍ɼ treatment ͱ outcome ͱͷؔ࿈Λ systematic bias ͱ͍͏ʢඇެࣜʣ

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(Unconditional) bias 39 (Unconditional) bias ͕ແ͍ͱ͍͏͜ͱ͸ɼ Pr(Ya=1 = 1) − Pr(Ya=0 = 1) ≠ Pr(Y = 1|A = 1) − Pr(Y = 1|A = 0) Ya ⊥ ⊥ A (Unconditional) exchangeability ͸੒ཱ͠ͳ͍ Association measure Effect measure Consistent estimator Pr(Ya = 1) = ̂ Pr(Ya = 1) = ̂ Pr(Y = 1|A = a) (Unconditional) bias Lack of exchangeability between the treated and the untreated

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Bias under the null 40 Lack of exchangeability Bias under the null Treatment ͕ outcome ʹରͯ͠ɼcausal effect Λ࣋ͨͳ͍৔߹Ͱ΋ Treatment ͱ outcome ͸ؔ࿈͢Δ Causal risk ratio = 1 (Null) Associational risk ratio = 1.26 Bias Null Ͱͳͯ͘΋ ಉ͡ཧ༝Ͱ bias͞ΕΔ

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Conditional bias 41 Bias under the null Pr(Ya=1 = 1|L = l) − Pr(Ya=0 = 1|L = l) ≠ Pr(Y = 1|L = l, A = 1) − Pr(Y = 1|L = l, A = 0) গͳ͘ͱ΋̍ͭͷɹͰ Conditional bias Ya ⊥ ⊥ A|L = l for all a and l Conditional exchangeability ͸੒ཱ͠ͳ͍ Conditional bias ͕ແ͍ͱ͍͏͜ͱ͸ɼ Association measure Effect measure Consistent estimator Pr(Ya = 1) = ̂ Pr(Ya = 1) = ∑ l ̂ Pr(Ya = 1|L = l) ̂ Pr(L = l) = ∑ l ̂ Pr(Y|L = l, A = a) ̂ Pr(L = l) l

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DAG → Lack of exchangeability → Bias 42 Common causes Conditioning on common effects Lack of exchangeability Bias Confounding Selection bias Bias

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Next three chapters 43 • Chapter 7 Confounding • Chapter 8 Selection bias • Chapter 9 Measurement bias • ม਺ʹଌఆΤϥʔ͕͋Δ৔߹ɼͲͷΑ͏ͳBias͕ൃੜ͢Δ͔ • ͜ΕΒͷ Systematic bias ͸ Randomized experimentsͰ΋ى͜ΓಘΔ • ͜Ε·Ͱͷ Observational studies ͸ɼ ideal randomized experimentsͷෆ׬શͳܗͱͯ͠આ໌͖ͯͨ͠ • Loss-to-follow-up ͕ͳ͍ • ࢀՃऀׂ͕Γ෇͚ͨ Treatment Λ׬ᘳʹ९क • Treatment ΛࢀՃऀ΋ݚڀऀ΋஌Βͳ͍

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6.6 The structure of effect modification 44 • Effect modification Λߟ͑ΔͷʹɼDAG ͸༗ޮ͔ʁ • Effect modifier ͷλΠϓΛ۠ผͰ͖Δ͔ʁ • Causal effect modifier ͱ Surrogate effect modifier Λ۠ผͰ͖Δ͔ʁ

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Effect modifier Λؚ·ͳͯ͘΋ DAG ͸ଥ౰ (Figure 6.11 and 6.12) 45 • ݚڀऀ͸ɼݸਓ͕͔͔ͬͨපӃͷҩྍͷ࣭ V ʹΑͬͯɼ ৺ଁҠ২ͷҼՌؔ܎͕ҟͳΔͷͰ͸ͳ͍͔ͱߟ͑ͨ • V ͔Β A ʹ͸໼ҹ͕ͳ͍ • A ͸ϥϯμϜʹׂΓ෇͚ΒΕ͍ͯΔ͔Β • V ͸ A ͱ Y ͷڞ௨ݪҼͰͳ͍͔Β • Question Λ໌֬ʹ͢ΔͨΊʹɼDAGʹؚΜ͚ͩͩ Figure 6.11 ͸ɼV Λؚ·ͳͯ͘΋ଥ౰ͳDAGͰ͋Δ V ͔Β Y ͷܦ࿏ʹ͋Δม਺΋ effet modiﬁer ͱͯ͠ద౰ • ߹ซ঱ N

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DAG Ͱ͸ effect modification ͷλΠϓΛ۠ผͰ͖ͳ͍ (Figure 6.11) 46 Y A V = 0 Y A V = 1 Y A V = 0 Y A V = 1 Y A V = 0 Y A V = 1 + + + - + ಉ͡ํ޲ ٯํ޲ ยํ͚ͩ

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Surrogate effect modifier (Figure 6.13) 47 • S ͸࣏ྍίετ • S ͸ V ͷӨڹ͸ड͚Δ͕ɼY ΁ͷޮՌ͸ͳ͍ • S Ͱ૚ผͯ͠΋ɼeffect modification ͸ݟΒΕΔ Surrogate effect modiﬁer Causal effect modiﬁer ͜Εࣗ਎͸ɼ effect Λ modify ͠ͳ͍ Heterogeneity of causal effect

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Surrogate effect modifier (Figure 6.14) 48 Common cause Surrogate effect modifier ͸ɼ୯ʹ Causal effect modifier ͱؔ࿈͕͋Δ͚ͩ Causal effect modifier Surrogate effect modifier • Y: ࢮ๢ • A: ৺Ҡ২ • V: ҩྍͷ࣭ • U: ډॅ஍ • P: ύεϙʔτ্ͷࠃ੶ Common cause U Λ௨ͯ͠ɼP ͸ Surrogate effect modifierʹͳΔ

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Surrogate effect modifier (Figure 6.15) 49 Common effect Causal effect modiﬁer Surrogate effect modiﬁer • Y: ࢮ๢ • A: ৺Ҡ২ • V: ҩྍͷ࣭ • S: ࣏ྍίετ • W: Ӄ಺ਫͱͯ͠ਫಓਫͰ͸ ͳ͘ɼϛωϥϧ΢Υʔλʔ Λ࢖͍ͬͯΔ͔ʁ Common effect S Λ৚͚݅ͮΔͱɼW ͸ Surrogate effect modifierʹͳΔ • ઌਐࠃͰ͸ɼW ͸ S ʹӨڹ͢Δ͕ɼYʹ͸Өڹ͠ͳ͍ • ௿ίετೖӃͰϛωϥϧ΢ΥʔλʔΛ࢖͏ʹ͸ɼҩྍͷ࣭Λ௿͘͢Δ • ௿ίετೖӃͰҩྍͷ࣭Λߴ͘͢ΔͨΊʹ͸ɼϛωϥϧ΢ΥʔλʔΛ࢖Θͳ͍

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Overview 50 • DAGͱ͸Կ͔ཧղͰ͖·͔ͨ͠ʁ • Causal diagramͷߏ੒ͱόΠΞεΛඥ෇͚ΒΕ·͔͢ʁ • ৄࡉ͸ɼChapter 7, 8, 9 • DAG Λ࢖ͬͯԁ׈ͳίϛϡχέʔγϣϯΛͱΓ·͠ΐ͏ʂ

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51 DAG ͬͯ࢖ͬͯΜͷ!?

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Review (1) 52 Tennant, P. W. G. et al. (2019) “Use of directed acyclic graphs (DAGs) in applied health research: review and recommendations,” Epidemiology. medRxiv. doi: 10.1101/2019.12.20.19015511.

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Review (2) 53 Tennant, P. W. G. et al. (2019) “Use of directed acyclic graphs (DAGs) in applied health research: review and recommendations,” Epidemiology. medRxiv. doi: 10.1101/2019.12.20.19015511. ظؒ 1999 - 2017 ݕࡧ Word “directed acyclic graphs” or similar or citing DAGitty ഔମ Scopus, Web of Science, Medline, and Embase

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Review (3) 54 Tennant, P. W. G. et al. (2019) “Use of directed acyclic graphs (DAGs) in applied health research: review and recommendations,” Epidemiology. medRxiv. doi: 10.1101/2019.12.20.19015511.

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Review (4) 55 Tennant, P. W. G. et al. (2019) “Use of directed acyclic graphs (DAGs) in applied health research: review and recommendations,” Epidemiology. medRxiv. doi: 10.1101/2019.12.20.19015511.

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Markov (Technical Point 6.1 Causal directed acyclic graphs) 56 7 7 7 ਌ʢPA2 ʣ ઌ૆ɾඇࢠଙ ࢠ ࢠଙ 7 7 7 ઌ૆ɾඇࢠଙ 7 7 7 ࢠ ਌ʢPA3 ʣ ࢠଙ DAG಺ͷ೚ҙͷ֬཰ม਺ Vj ͕ɼ ͦͷ਌Ͱ৚͚݅ͮͨ࣌ɼͦͷඇࢠଙʢ਌Λআ͍ͨʣͱ৚͖݅ͭಠཱͰ͋Δ 7 7 7 Vj ⊥ ⊥ nd(Vj ) \pa(Vj ) | pa(Vj ) ඇࢠଙʢ਌Λআ͍ͨʣ V3 ⊥ ⊥ V1 | V2 Local directed Markov propertyʢہॴత༗޲Ϛϧίϑੑʣ ͜ΕΛہॴత༗޲Ϛϧίϑੑͱ͍͏

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Technical Point 6.2 Counterfactual models associated with a causal DAG 58 טΈࡅ͚·ͤΜͰͨ͠

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Fine Point 6.1 D-separation 59 1. non-collider ͕৚͚݅ͮ͞ΕΔ 2. ৚͚͍݅ͮͯͳ͍ collider ΛؚΉ 3. Collider ͷ ࢠଙ΋৚͚͍݅ͮͯͳ͍ Path ͸࣍ͷ৚݅ͷͱ͖ʹݶΓɼblock ͞ΕΔ 2ม਺ؒͷ͢΂ͯͷ path ͕ block ͞ΕΔ D-separation $ # " A is d-separation from B conditional on C. A is statistically independent of B given C. D-separation and statistically independent

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Fine Point 6.2 Faithfulness (1) 60 : " The sharp null hypothesis of no causal effects of A on any individual’s Y holds : " A has a causal effect on the outcome Y of at least one individual in the population Null Pr(Ya=1 = 1) ≠ Pr(Ya=0 = 1) Pr(Y = 1|A = 1) ≠ Pr(Y = 1|A = 0) Null ʹͳΔ͕࣌͋Δ Y A V = 0 Y A V = 1 + - Cancel out d-separation Ͱͳ͚Ε͹ɼnon-zero association Faithfulness d-connect Ͱ͋Ε͹ɼcancel out Ͱ zero associationʹ͸ͳΒΜΑ

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Fine Point 6.2 Faithfulness (2) 61 Faithfulness ͸ɼݚڀσβΠϯʹΑΓ੒ཱ͠ͳ͘ͳΔ Prospective study Ͱ Matching ͢Δ L ͱ A ͸ؔ࿈͠ͳ͍ Lͷ෼෍ΛA = 1ͱA = 0Ͱἧ͑Δ S ͸ Matching ʹ selection Λද͢ ղੳ͞ΕΔͷ͸ɼS = 1 ͷΈ S ͸ collider L → A L → S ← A L ͱ A ͷؒʹ͸2ͭͷؔ࿈ͷύε ໃ६ L → A L → S ← A Cancel out L ͱ A ͸ؔ࿈͠ͳ͍

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Fine Point 6.3 Discovery of causal structure 62 Causal diagram Data analysis Causal diagram Data analysis Discovery of causal structure

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·ͱΊ 63 No conditioning Conditioning (Mediator) (Marginally) Association (Conditionally) Independent Common cause (Marginally) Association (Conditionally) Independent Common effect (Marginally) Independent (Conditionally) Association Association between A and Y is…

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• Hernán MA, Robins JM. (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC. • Pearl J, (མւߒ, ༁). (2019). ೖ໳ ౷ܭతҼՌਪ࿦. ே૔ॻళ. • ٶ઒խາ. (2004). ౷ܭతҼՌਪ࿦ −ճؼ෼ੳͷ৽͍͠࿮૊Έ−. ே૔ॻళ. • ࠇ໦ֶ. (2017). ߏ଄తҼՌϞσϧͷجૅ. ڞཱग़൛. • Pearl J, (ࠇ໦ֶ, ༁). (2009). ౷ܭతҼՌਪ࿦ Ϟσϧɾਪ࿦ɾਪଌ. ڞཱग़൛. ࢀߟจݙ 64