࣍ 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
ަ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
ަབྷͷྫ: confounding by indication/channeling A Y L Figure 7.1 A Y L U Figure 7.2 A: ༀͷ༻ Y: ڵຯͷ͋Δ࣬ױ L: A ͕༻͞Ε͍͢පଶ U:Y ͱ L ΛͱʹҾ͖ى͜͢Ҽࢠ ಉछͷༀͷதͰɺױऀݻ༗ͷϦεΫҼࢠ L ʹΑͬͯҩࢣ͕ༀ A Λ͏͕͋Δ߹ɺಛʹ channeling ͱݺͿ͜ͱ͕ଟ͍ɻ 6
ަབྷͷྫ: reverse causation A Y L U Figure 7.3 A: ӡಈ Y: ڵຯͷ͋Δ࣬ױ U: ແީੑͷ࣬ױ L: U ʹΑͬͯੜ͡ΔҼࢠ L ͕ະͩͱɺY ΛҾ͖ى͜͢Α͏ͳ U ʹΑͬͯӡಈෆ͕ ੜ͍ͯ͡ΔͷʹؔΘΒͣɺӡಈෆʹΑͬͯ Y ͕Ҿ͖ى͜ ͞Ε͍ͯΔ͔ͷΑ͏ʹஅ͞Ε͍͢ɻ 7
ަབྷͷྫ: linkage disequilibrium A Y L U Figure 7.3 A: DNA ྻ Y: ԿΒ͔ͷੑ࣭ L: A ͱϦϯΫ͍ͯ͠Δ DNA ྻ U: ຽͳͲɺA ͱ L ΛϦϯΫͤ͞ ΔཁҼ ෳͷຽΛؚΉूஂͰݚڀΛߦ͏߹ʹɺಛʹ population stratification ͱݺͿ͜ͱ͕ଟ͍ɻ 8
Backdoor ج४ Backdoor ج४ ڞมྔͷू߹ L ҎԼͷ݅Λຬͨ͢ͱ͖ɺbackdoor ج ४Λຬͨ͢ɺͱݴ͏ɻ • L Ͱ͚݅Δ͜ͱͰͯ͢ͷ backdoor path ͕ด͡Δ • L ͷதʹɺA ͷ descendant ؚ͕·Εͳ͍ Conditional exchangeablity ⇐⇒ backdoor ج४ Failthfulness ͱ Technical Pont 7.1 Ͱٞ͢Δ݅ͷԼͰ Ya A | L ⇐⇒ L ͕ backdoor ج४Λຬͨ͢ 14
Backdoor ج४Λຬͨ͢ 2 ͭͷঢ়گ ަབྷ͕ͳ͍ (no confounding) ަབྷ͕ͳ͍߹ɺͱͱ open backdoor path ͕ଘࡏ͠ͳ ͍ɻMarginal exchangeability Ya A ͕Γཱͭɻ ະଌఆͷަབྷ͕ͳ͍ (no unmeasured confounding) ަབྷ͕͋Δ͕ɺL(ଌఆ͞Ε͍ͯͯɺA ͷ descendant Λؚ· ͳ͍) Ͱ͚݅Δ͜ͱͰͯ͢ͷ backdoor path Λด͡Δ ͜ͱ͕Ͱ͖Δ߹ɺconditional exchangeability Ya A | L ͕ ΓཱͭɻL ͷ͜ͱΛ sufficient set for confounding adjustment ͱݺͿɻ 15
ަ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 Ͱ͚݅Δ͜ͱͰͯ͢ ͷ backdoor path ͕ด͡Δ ͨΊ Ya A | L ͕Γཱͪɺະଌ ఆͷަབྷͳ͍ɻ 17
ަབྷͳ͍͕ɺYa A | L ͕Γཱͨͳ͍߹ A Y L U2 U1 Figure 7.4 A Y L U2 U1 Figure 7.4’ ͜ͷ߹ • ަབྷͳ͍ • Backdoor path ͋Δ͕ collider L ʹΑΓด͍ͯ͡Δ Ͱ͋ΓɺYa A ٯʹ L Ͱ͚݅ͮΔͱ • Backdoor path ͕։͘ ͷͰ Ya A | L Γཱͨͳ͍ (selection bias)ɻ 18
ަ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 Ͱ͚݅ͮΔͱ৽ͨͳ backdoor path ͕։͘ ͱͳΔͷͰɺYa A Ya A | L ཱ͠ͳ͍ɻ Figure 7.6 ͷΑ͏ʹ L1 L2 ͕ଌ ఆͰ͖ΕɺL1 ͘͠ L2 ɾL Ͱ ͚݅ͮΔ͜ͱͰͯ͢ͷ backdoor path ͕ด͡Δɻ 19
ަབྷҼࢠ (confounder) ަབྷҼࢠ (confounder) Treatment A ͱ outcome Y ʹؔ͢Δσʔλ͚ͩͰҼՌޮՌ ΛࣝผͰ͖ͣɺL Λ༻͍Δ͜ͱͰࣝผͰ͖ΔΑ͏ʹͳΔ࣌ɺ L ΛަབྷҼࢠ (confounder) ͱݺͿ = Ya A ཱ͕͠ͳ͍͕ Ya A | L ཱ͕͢Δ࣌ 21
ަ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 ަབྷҼࢠͰͳ͍ U ͕ଌఆ͞ΕΕɺU Ͱ open backdoor path Λด͡Δ͜ͱ͕ Ͱ͖ΔͷͰɺU ͷԼͰ L ަབྷҼࢠʹͳΒͳ͍ɻ͜ͷΑ͏ ʹަབྷઈରతͳ֓೦͕ͩɺަབྷҼࢠ૬ରతͳ֓೦Ͱ͋Δɻ 22
౷తͳߟ͑ํͰෆ߹͕ੜ͡Δ߹ A Y L U2 U1 Figure 7.4 A L Y U Figure 7.7 Figure 7.4 7.7 • ަབྷͳ͘ Ya A • ͔͠͠ L ౷తͳ” ަབྷҼ ࢠ” ͷ 3 ݅Λຬͨ͢ • L Ͱௐ͢Δͱٯʹ selection bias ͕ൃੜ • Selection bias ͷ͍ͤͰௐͷ ༗ແʹΑͬͯਪఆ͕ҟͳΔ ͷͰ” ަབྷ” ͱஅ͞ΕΔ 25
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 ํᝦΛճऩ͢Δͱ͍͏Ծతͳੈք Λߟ͑Δɻ • શһ a = 0 ͱ͍͏ single intervention Λड͚Δ • Ծతͳੈքʹ͓͚Δ Y Ya=0 • A natural value of treatment ͳ ͷͰྲྀΕࠐΉҹͦͷ·· • a A ͔Βग़ΔҹΛҾ͖ܧ͙ • A ͔Β a ͷҹͳ͍ 29
SWIG ʹ͓͚Δ exchangeability ͷධՁ A | a Ya L U Figure 7.9 A | a Ya L U2 U1 Figure 7.10 L Ͱ͚݅ͮΔͱ A ͱ Ya d-separate ͞Ε͍ͯΔͷͰ Ya A | L Figure 7.4 Λ SWIG ʹมͨ͠ Figure7.10 ͰɺL Ͱ͚݅ͮͣʹ d-separate ͞Ε͍ͯΔͷͰ Ya A SWIG ͰɺBackdoor ج४ͱ Ya A | L ͱͷ͕ؔΘ͔Γ͍͢ 30
Fine Point 7.1: ަབྷͷํ Signed causal diagram Λ͏ͱɺަབྷ͕Ͳͷํʹ 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
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 ͋Δ͍ʮL2 ͷΈʯͰҼՌޮՌ ࣝผͰ͖ͣɺ ʮL ͱ L2 ʯ͋Δ͍ ʮL1 ʯ͕ଌఆ͞ΕΔඞཁ͕͋Δɻ 39
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 ͳޡྨͰ͋ΕɺL Ͱ݅ͮ ͚Δ͜ͱͰ backdoor path ͕෦తʹϒϩοΫ͞ΕΔɻ Surrogate confounder Backdoor path ্ʹଘࡏ͠ͳ͍͕ɺަབྷʹΑΔ bias ΛݮΒ ͢ͷʹ͑ΔมΛ surrogate confounder ͱݺͿ ަབྷʹରॲ͢Δํ๏ͱͯ͠Ͱ͖Δ͚ͩଟ͘ͷ surrogate confounder Λଌఆ͢Δɺͱ͍͏ઓུ͕ՄೳͰ͋Δɻ 40
Technical Point 7.2 มͷू߹ L ͱ U ͕ Ya A | L, U Λຬͨ͢ͱ͖ɺҎԼͷ ݅ΛຬͨͤɺL ͷΈͰे (Ya A | L) Ͱ͋Δɻ U ͕ޓ͍ʹૉͳू߹ U1 ͱ U2 ʹ͚ΒΕɺ • L ʹΑΓܾ·Δʹ͓͍ͯ U1 ͱ A ͕ؔ࿈͍ͯ͠ͳ͍ • A, L, U1 ʹΑΓܾ·Δʹ͓͍ͯ U2 ͱ Y ͕ؔ࿈͍ͯ͠ ͳ͍ 41
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 Λඳ͘ͱɺΓཱͭͷ Ya A | La Ͱ͋Δ͜ͱ͕Θ͔Δɻ ॏཁͳͷʮL ͕ A ΑΓ࣌ؒతʹ લ͔ޙ͔ʯͰͳ͘ɺʮL ͕ A ͷ descendant ͔൱͔ʯ Ͱ͋Δɻ 42
Technical Point 7.4 A M Y U Figure 7.14 7.14 Ͱ backdoor path Λด͡Δͷ ʹ U Ͱ͚݅ͮΔඞཁ͕͋Δ͕ɺU ͕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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