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

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

Keywords: 因果推論, DAG, Directed Acyclic Graph, Confounding, Selection bias, Collider, Common cause, Common effect

Shuntaro Sato

June 15, 2020
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  1. Shuntaro Sato
    (Sato@ੜ෺౷ܭՈ)
    Graphical representation of
    causal effects
    Causal Inference: What If ษڧձ

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

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  3. 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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  4. 6.1 Causal diagrams
    4
    • DAGͷجຊతͳಛ௃Λཧղ͢Δ
    • Common causeΛཧղ͢Δ

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  5. 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. 6
    DAG Causal DAG

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  7. 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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  8. 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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  9. 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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  10. 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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  11. 6.2 Causal diagrams and marginal independence
    11
    • DAG্ͷͲͷม਺΋৚݅෇͚ͳ͍࣌ɼ
    Figure 6.2ʙ6.4͸ͦΕͧΕԿΛҙຯ͢Δͷ͔ཧղ͢Δ
    • ௚ײʹର͠ɼCausal graphs theory Ͱઆ໌͢Δ

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  12. ௚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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  13. 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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  14. 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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  15. 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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  16. Causal graphs theory (Observational study Figure 6.3)
    16
    Common cause LΛ௨ͯ͠ɼ
    A͔ΒYʢor Y͔ΒAʣ΁ͷؔ࿈ͷྲྀΕ͕Ͱ͖Δ࣌ɼAͱY͸ؔ࿈͢Δ
    Causal graph theory

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  17. 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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  18. 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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  19. 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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  20. ·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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  21. 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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  22. 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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  23. 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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  24. 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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  25. 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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  26. 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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  27. 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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  28. 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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  29. 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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  30. 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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  31. ·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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  32. 6.4 Positivity and consistency in causal diagrams
    32
    Causal inferenceʹඞཁͳ৚͕݅ɼDAGͰ͸ͲͷΑ͏ʹѻ͏͔ཧղ͢Δ

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  33. 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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  34. 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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  35. 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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  36. 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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  37. 6.5 Structural classification of bias
    37
    • Systematic bias ʢܥ౷όΠΞεʣΛཧղ͢Δ
    • Lack of exchangeability Λཧղ͢Δ
    • DAG ͔Βൃੜ͠͏Δ systematic bias Λ໌Β͔ʹͰ͖Δ

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  38. 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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  39. (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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  40. 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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  41. 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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  42. DAG → Lack of exchangeability → Bias
    42
    Common causes Conditioning on common effects
    Lack of exchangeability
    Bias
    Confounding Selection bias
    Bias

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  43. 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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  44. 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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  45. 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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  46. 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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  47. 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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  48. 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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  49. 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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  50. 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. 51
    DAG ͬͯ࢖ͬͯΜͷ!?

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  52. 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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  53. 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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  54. 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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  55. 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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  56. Markov (Technical Point 6.1 Causal directed acyclic graphs)
    56
    7 7 7
    ਌ʢPA2
    ʣ
    ઌ૆ɾඇࢠଙ

    ࢠଙ
    7 7 7
    ઌ૆ɾඇࢠଙ
    7 7 7

    ਌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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  57. ஞ࣍తҼ਺෼ղͷ๏ଇ (Technical Point 6.1 Causal directed acyclic graphs)
    57
    f(v) = ΠM
    j=1
    f(vj
    |paj
    )
    DAG಺ͷ֬཰ม਺ V ͷಉ࣌෼෍͸ɼ࣍ͷΑ͏ʹද͢͜ͱ͕Ͱ͖Δ
    Y
    A L
    Pr(Y, A, L) = Pr(Y|L) Pr(L|A) Pr(A)
    Pr(Y, A, L) = Pr(Y|A, L) Pr(L, A) Pr(A)
    = Pr(Y|A, L) Pr(L|A) Pr(A)
    Chain rule ʹҼՌਪ࿦ͷ৭Λ͚ͭΔ
    Chain rule
    MarkovੑΑΓ ਌ͷӨڹ͔͠ड͚ͳ͍
    Pr(Y, A, L) = Pr(Y|A, L) Pr(A|L) Pr(L)
    Pr(Y, A, L) = Pr(Y|L) Pr(A|L) Pr(L)

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

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  59. 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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  60. 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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  61. 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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  62. 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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  63. ·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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  64. • 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

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