Shuntaro Sato
June 15, 2020
4.2k

# 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

June 15, 2020

## 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

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

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…

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