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

Transcript

1. Shuntaro Sato (Sato@ੜ෺౷ܭՈ) Graphical representation of causal effects Causal Inference: What If ษڧձ
2. Overview 2 • DAGͱ͸Կ͔ཧղ͢Δ • Causal diagramͷߏ੒ͱόΠΞεΛඥ෇͚Δ • ৄࡉ͸ɼChapter 7, 8, 9
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
4. 6.1 Causal diagrams 4 • DAGͷجຊతͳಛ௃Λཧղ͢Δ • Common causeΛཧղ͢Δ
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঱౓
6. 6 DAG Causal DAG
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
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
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
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
11. 6.2 Causal diagrams and marginal independence 11 • DAG্ͷͲͷม਺΋৚݅෇͚ͳ͍࣌ɼ Figure 6.2ʙ6.4͸ͦΕͧΕԿΛҙຯ͢Δͷ͔ཧղ͢Δ • ௚ײʹର͠ɼCausal graphs theory Ͱઆ໌͢Δ
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࿈ͯͦ͠͏
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
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↓
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
16. Causal graphs theory (Observational study Figure 6.3) 16 Common cause LΛ௨ͯ͠ɼ A͔ΒYʢor Y͔ΒAʣ΁ͷؔ࿈ͷྲྀΕ͕Ͱ͖Δ࣌ɼAͱY͸ؔ࿈͢Δ Causal graph theory
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
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
19. Causal graphs theory (Figure 6.4) 19 • Colliders ͸ɼͦΕΒ͕ؒʹ͋Δม਺ؒͷؔ࿈ͷύεΛϒϩοΫ͢Δ • A → L ← Y ͸Collider L ͰϒϩοΫ͞ΕΔͨΊɼA ͱ Y ͸independentͰ͋Δ Causal graph theory
20. ·ͱΊʢ6.2 Causal diagrams and marginal independenceʣ 20 • 2ͭͷม਺͕ɼ(marginally) associatedͱ͸ɼ • Ұํ͕ଞํͷݪҼͰ͋Δ • ͦΕΒ͕ Common cause Ͱ͋Δ • ͦΕҎ֎Ͱ͋Ε͹ɼ(marginally) independent
21. 6.3 Causal diagrams and conditional independence 21 • Figure 6.2ʙ6.4ͷม਺Λ৚͚݅ͮΔͱԿΛҙຯ͢Δͷ͔ཧղ͢Δ • ௚ײʹର͠ɼCausal graphs theory Ͱઆ໌͢Δ No conditioning Conditioning
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
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
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
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?
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
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
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
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?
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
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…
32. 6.4 Positivity and consistency in causal diagrams 32 Causal inferenceʹඞཁͳ৚͕݅ɼDAGͰ͸ͲͷΑ͏ʹѻ͏͔ཧղ͢Δ
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
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
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
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
37. 6.5 Structural classification of bias 37 • Systematic bias ʢܥ౷όΠΞεʣΛཧղ͢Δ • Lack of exchangeability Λཧղ͢Δ • DAG ͔Βൃੜ͠͏Δ systematic bias Λ໌Β͔ʹͰ͖Δ
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
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
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
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
42. DAG → Lack of exchangeability → Bias 42 Common causes Conditioning on common effects Lack of exchangeability Bias Confounding Selection bias Bias
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
44. 6.6 The structure of effect modification 44 • Effect modification Λߟ͑ΔͷʹɼDAG ͸༗ޮ͔ʁ • Effect modifier ͷλΠϓΛ۠ผͰ͖Δ͔ʁ • Causal effect modifier ͱ Surrogate effect modifier Λ۠ผͰ͖Δ͔ʁ
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
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
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
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
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
50. Overview 50 • DAGͱ͸Կ͔ཧղͰ͖·͔ͨ͠ʁ • Causal diagramͷߏ੒ͱόΠΞεΛඥ෇͚ΒΕ·͔͢ʁ • ৄࡉ͸ɼChapter 7, 8, 9 • DAG Λ࢖ͬͯԁ׈ͳίϛϡχέʔγϣϯΛͱΓ·͠ΐ͏ʂ
51. 51 DAG ͬͯ࢖ͬͯΜͷ!?
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.
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
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.
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.
56. 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
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)
58. Technical Point 6.2 Counterfactual models associated with a causal DAG 58 טΈࡅ͚·ͤΜͰͨ͠
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
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
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
62. Fine Point 6.3 Discovery of causal structure 62 Causal diagram Data analysis Causal diagram Data analysis Discovery of causal structure
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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