IP Weighting and Marginal Structural Models(Causal inference: What if, Chapter 12)

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Causal Inference: What If ಡॻձ Chapter 12: IP Weighting and Marginal Structural Models Azusa Matsumoto Reardon August 2, 2020 1 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 1 12.1 The Causal Question 2 12.2 Estimating IP wights via modeling 3 12.3 Stabilized IP weights 4 12.4 Marginal structural models 5 12.5 Effect modiﬁcation and marginal structural models 6 12.6 Censoring and missing data 7 Fine point 12.1 Setting a bad example 12.2 Checking positivity 8 Technical point 12.1 Horvitz-Thompson estimators 12.2 More on stabilized wights 1 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 12.1 The causal question ېԎˠମॏ૿ྔͷҼՌޮՌ ؍ଌσʔλʹର͠ IP Weighting Λ༻͍Δ 2 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point σʔλ ถࠃࠃຽ݈߁ӫཆௐࠪ NHEFS/National Health and Nutrition Examination Survey Data I Epidemiologic Follow-up Study ΞϝϦΧࠃཱӴੜ౷ܭηϯλʔ/National Center for Health Statistics ΞϝϦΧࠃཱ࿝Խݚڀॴ/National Institute on Aging ΞϝϦΧެऺӴੜہ/United States Public Health Service 1971-75 ೥ͷ Baseline ௐࠪˍ 1982 ೥ͷ Follow-up ௐࠪ ؚ·ΕΔαϯϓϧ 25-74 ࡀͷݸਓ Baseline Ͱ٤Ԏ͍ͯ͠Δͱճ౴ ෼ੳʹؚΉม਺ͷ஋ʹܽམ͕ͳ͍ʢˠ 12.6ʣ N = 1566 3 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ୯७ൺֱ Treatment (A) ېԎ A = 1: Baseline ͱ Follw-up ͷؒʹېԎΛͨ͠ݸਓ A = 0: ্هҎ֎ Outcome (Y) ମॏ૿Ճ Y =(Follow-up ͷମॏ)-(Baseline ͷମॏ) ېԎͨ͠άϧʔϓͷฏۉମॏ૿Ճྔ E[Y|A = 1] = 4.5kg(n = 403) ېԎ͠ͳ͔ͬͨάϧʔϓͷฏۉମॏ૿Ճྔ E[Y|A = 0] = 2.0kg(n = 1163) ୯७ൺֱ͢Δͱʜ E[Y|A = 1]−E[Y|A = 0] = 2.5kg(1.7,3.4) 4 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ஌Γ͍ͨҼՌޮՌ ͳͥ୯७ൺֱ͡ΌͩΊͳͷ͔? ˠશ͘ҟͳͬͨूஂͷൺֱʹͳͬͯ͠·͏ɻ 5 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ஌Γ͍ͨҼՌޮՌ ΋͠શһ͕ېԎ͍ͯͨ͠ΒΈΒΕͨͰ͋Ζ͏ฏۉମॏ૿ Ճྔ (൓ࣄ࣮) E[Ya=1] ΋͠શһ͕ېԎ͍ͯ͠ͳ͔ͬͨΒΈΒΕͨͰ͋Ζ͏ฏۉ ମॏ૿Ճྔ (൓ࣄ࣮) E[Ya=0] ஌Γ͍ͨҼՌޮՌ E[Ya=1]−E[Ya=0] ̸= E[Y|A = 1]−E[Y|A = 0] 6 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Surrogate Confounder ؍ଌͰ͖Δ L ͱ؍ଌͰ͖ͳ͍ U ͕૬ؔ͢ΔɻL Λௐ੔͢΂ ͖͔ʁ L Λௐ੔͢Δ͜ͱͰ A ← U → Y ͷόοΫυΞύεΛ෦෼తʹ ϒϩοΫͰ͖Δ 7 ষͷ Technical Point 7.3 ࢀর ʢA ʹӡಈश׳ɺY=৺ଁ࣬ױɺU ʹࣾձܦࡁతഎܠɺL=ऩೖʣ 7 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Surrogate Confounder ͜ͷষͰௐ੔͢Δ Confounder 1 ੑผ (0: male, 1: female) 2 ೥ྸ (years) 3 ਓछ (0: white, 1: otherwise) 4 ڭҭྺ (5 ΧςΰϦʔ) 5 ٤Ԏྔ (Ұ೔ʹٵ͏ຊ਺) 6 ٤Ԏྺ (years) 7 ೔ৗతͳӡಈྔ (3 ΧςΰϦʔ) 8 εϙʔπ׆ಈ (3 ΧςΰϦʔ) 9 ମॏ (kg) Ͳͷม਺Λௐ੔͢΂͖͔ʁ ˠʢChapter 18) 8 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 12.2 Estimating IP wights via modeling IPW Ͱٙࣅूஂ (pseudo-population) Λ࡞Γͩͦ͏ͱͯ͠ ͍Δɻ ΋͠શһ͕ېԎ͍ͯͨ͠Βʁ ΋͠શһ͕ېԎ͍ͯ͠ͳ͔ͬͨΒʁ ٙࣅूஂͷ΋ͭ̎ͭͷੑ࣭ A ͱ L ͕ಠཱ Eps[Y|A = a] ٙࣅूஂͷ mean = ∑ l E[Y|A = a,L = l]Pr[L = l] ࣮ࡍͷूஂͷ standardized mean 9 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Conditional Exchangeablity Ya ⊥ ⊥ A|L ɹ 1 Conditional Exchangeability ͕੒ΓཱͭͷͰ͋Ε͹ɺ E[Ya] ͸ٙࣅूஂͱ࣮ࡍͷूஂͱ΋ʹಉ͡ͱͳΔɻ 2 ٙࣅूஂͰ Unconditional exchangeability ͕੒Γཱͭɻ ަབྷͳ͠ɻ 3 E[Ya] ൓ࣄ࣮ੈքͷฏۉ஋ = Eps[Y|A = a] ٙࣅूஂͷฏۉ஋ 4 Αͬͯɺٙࣅूஂʹ͓͚Δ૬ؔ͸ҼՌؔ܎ͱղऍͰ͖Δɻ 10 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ٙࣅूஂͷͭ͘Γ͔ͨ Ͳ͏΍ͬͯͦΜͳٙࣅूஂΛͭ͘Δͷʁ ˠ Treatment level Λݸਓ͕ड͚Δ֬཰ (conditional on L) ͷ ٯ਺Λ weight ͱͯ͠࢖͏ L Ͱ৚݅෇͚ͨېԎ͢Δ֬཰ Pr[A = 1|L] L Ͱ৚݅෇͚ͨېԎ͠ͳ͍֬཰ Pr[A = 0|L] = 1−Pr[A = 1|L] Treatment A ʹର͢Δݸਓͷ IP weight WA = 1 f(A|L) L Ͱ৚݅෇͚ͨېԎ͢Δ֬཰ 11 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ٙࣅूஂͷͭ͘Γ͔ͨ Figure 2.1 2 ষͰ͸ Causally interpreted tree Ͱ non-parametrically ʹܭࢉͨ͠ɻ ࠓճͷΑ͏ͳߴ࣍ݩͳσʔλͩͱɺେมͳ͜ͱʹͳͬͯ ͠·͏ʂ 9 confounders, up to 6 levels, 2 million branches! ͦ͏ͩɺϞσϧʹཔΖ͏ɻ 12 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point εςοϓ 1 ͸͡ΊʹɺPr[A = 1|L] Λ Parametric ʹਪܭ ˆ Pr[A = 1|L] Λ 1566 ਓͷ͢΂ͯͷ L ͷ஋ͷίϯϏωʔγϣϯ ͰٻΊΔ ੍໿ ࿈ଓม਺͸ Linearʢ௚ઢؔ܎ʣͱ quadratic termʢೋ࣍ ؔ਺తʣ No product term ʢަޓ࡞༻Λߟྀ͠ͳ͍ɺ Unsaturatedʣ IP weights (WA) ͷฏۉ: 2.00(min : 1.05,max : 16.7) 13 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point εςοϓ 2 IP weights ʹΑͬͯ࡞ΒΕͨٙࣅूஂʹ͓͚Δࠩ ˆ Eps[Y|A = 1]− ˆ Eps[Y|A = 0] ஌Γ͍ͨҼՌޮՌ E[Ya=1]−E[Ya=0] IP weights (WA) ʹΑͬͯ࡞ΒΕͨٙࣅूஂʹ͓͚Δࠩ͸஌ Γ͍ͨҼՌޮՌͱͳΔɻ ٙࣅूஂͷ A ʹަབྷ͕ͳ͍ Pr[A = 1|L] ͷϞσϧ͕ਖ਼͍͠ 14 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ·ͱΊ Weighted Least Squares E[Y|A] = θ0 +θ1A ېԎͨ͠ਓͷ Estimated IP weights ˆ W 1 ˆ Pr[A = 1|L] ېԎ͠ͳ͔ͬͨਓͷ Estimated IP weights ˆ W 1 1− ˆ Pr[A = 1|L] ˆ θ1 = 3.4kg(2.4,4.5) ... ېԎ͢Δ͜ͱͰ૿͑Δମॏ 15 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Conﬁdence Interval Weights Λߟྀͨ͠ CI ͸Ͳ͏΍ͬͯٻΊΔͷʁ 1 ౷ܭཧ࿦ʹج͖ͮରԠ͢Δ Variance Λͭ͘Δ ଟ͘ͷ౷ܭιϑτ͕ѻ͍ͬͯͳ͍ɻ 2 Nonparametric boostrap Ͱۙࣅˠ Technical Point 3.1 Ϛγϯύϫʔ͕ඞཁ 3 Robust variance อकతͰଟ͘ͷ౷ܭιϑτͰ͸σϑΥϧτͰઃఆ͞Εͯ ͍Δ ͜ͷষͷ CI ͸͢΂ͯ (3) Pr[A = 1|L] ͷϞσϧ͕ਖ਼͘͠ͳ͍৔߹ɺθ0 ͱ θ1 ʹόΠΞεɺ CI ͕ਅ࣮ͷ஋ΛؚΉ֬཰͸ 95 ˋͱͳΒͳ͍ 16 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 12.3 Stabilized IP weights IP weights WA = 1 f(A|L) ͸ɺStudy population શһͷίϐʔ Λ̎ͭ࡞Δɻ Expected mean of the weights WA = 2 17 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ͸ΜͿΜ͜ ΄͔ͷ΍Γ͔ͨ΋͋Δ YOɻͨͱ͑͹͜Μͳٙࣅूஂ Pseudo-population with P = 0.5 (Probability of receiving A = 1) = 0.5 (Probability of receiving A = 0) = 0.5 IP weights: 0.5 f(A|L) ਺ֶతʹ͸ɺ͜Ε·Ͱͷ Pseudo-population(̎ͭίϐʔ ͨ͠৔߹) ͷ weight Λ̎Ͱׂͬͨ΋ͷͱಉ͡ Expected mean of the weights WA= 1 Effect estimate ΋ಉ͡ ଞͷ֬཰ p f(A|L) 0 < p ≤ 1 Ͱ͋Ε͹ɺͲΕ΋ಉ͡ ˠ Tehnical Point 12.2 18 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ٙࣅूஂʹඞཁͳͷ͸ɺTreatment A ͱ Confounder L ͕ಠ ཱͰ͋Δ͜ͱɻTreatment A Λड͚Δ֬཰ p ͷ஋͕ L ʹґଘ ͠ͳ͍ͷͰ͋Ε͹ɺͲΜͳ஋Ͱ΋໰୊ͳ͍ɻ Α͘࢖ΘΕΔͷ͸ݩͷूஂͷ A ͷׂ߹ɿ f(A) f(A|L) Treated: Pr[A = 1] in the original population = 0.257(403/1566) Pr[A = 1] f(A|L) Untreated: Pr[A = 0] = in the original population = 0.743(1163/1566) Pr[A = 0] f(A|L) 19 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Ծ૝ͷϥϯμϜԽ࣮ݧ ୈ 2 ষ Figure 2.1 ͷσʔλͷ৔߹ Stabilized IP weights: f(A) f(A|L) Pr[A = 1] = 13 20 = 0.65 Pr[A = 0] = 7 20 = 0.35 ͜ͷ̓ٙࣅूஂ͕໛฿͢Δ ɹ 65% ͕ A = 1 ɹ 35% ͕ A = 0 ͷԾ૝ͷϥϯμϜԽ࣮ݧ 20 / 48 Figure 12.1

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Weight ʹΑΔҧ͍ Type Weight Min. Max. Mean Nonstabilized weights WA = 1 f(A|L) 1.05 16.7 2 Stabilized weights SWA = f(A) f(A|L) 0.33 4.30 1 Stabilized weights ͷ΄͏͕Ϩϯδ͕ڱ͍ 21 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ېԎͱମॏ૿ՃͷέʔεͰ SWA (stabilized weights) Λ࢖ͬ ͯܭࢉ͠ͳ͓ͯ͠ΈΑ͏ɻ 1 ෼฼ Pr[A = 1|L] ϩδεςΟοΫճؼͰ Study population 1566 ਓͷ conditional probability ΛٻΊΔʢSection 12.2 ͱಉ͡ʣ 2 ෼ࢠ Pr[A = 1] 403 1566 Saturated logistic model 3 ҼՌޮՌ E[Ya=1]−E[Ya=0] E[Y|A] = θ0 +θ1A ېԎऀͷ SWA: ˆ Pr[A=1] ˆ Pr[A=1|L] ඇېԎऀͷ SWA: (1− ˆ Pr[A=1]) (1− ˆ Pr[A=1|L]) SWA Λ࢖ͬͨͱ͖ͷ݁Ռɿ ˆ θ1 = 3.5kg(2.4,4.5) WA Λ࢖ͬͨͱ͖ͷ݁Ռɿ ˆ θ1 = 3.5kg(2.4,4.5) 22 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ಉ݁͡ՌʹͳΔͳΒɺͳΜͰ Stabilized weights Λ͔ͭ ͏ͷʁ ˠ Unsaturated ͷ৔߹ɺConﬁdence Interval ͕ Narrow ʹ ͳΔɻ Time-varying treatment ΍ continuous treatment ͳͲͰ͸ɺ શͯͷόϦΤʔγϣϯΛϞσϧʹՃ͑Δ͜ͱ͸ݱ࣮తͰ͸ ͳ͍ɻ ͜Ε·ͰͷྫͰ͸ɺTreatment A ͸ 2 ஋͔͠ͱΕͳ͔ͬͨ (Saturated) E[Y|A] = θ0 +θ1A Continuous treatment ͷ৔߹ (ˠ TP12.2) 23 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 12.4 Marginal structural models Marginal Structural Mean Model: Outcome ͕ unobservable ݱ࣮ͷσʔλͰϑΟοτ͢Δ͜ͱ͸Ͱ͖ͳ͍ɻ E[Ya] ൓ࣄ࣮ = β0 +β1a a = 0ʢېԎ͠ͳ͍ʣ ͱ͖... E[Ya] = β0 a = 1ʢېԎ͢Δʣ ɹͱ͖... E[Ya] = β0 +β1 β1 = E[Ya=1]−E[Ya=0] 24 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Marginal structural models IP weight ͰٙࣅूஂΛ࡞ΓɺWLS ͰϑΟοτ E[Y|A] = θ0 +θ1A ੍໿ʢAssumptionʣͷ΋ͱɺٙࣅूஂͰͷ૬ؔ͸ҼՌؔ܎ͱ ղऍͰ͖Δɻ ٙࣅूஂͷ૬ؔ ˆ θ1 ˣ ҼՌޮՌ β1 = E[Ya=1]−E[Ya=0] 25 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Continuous treatment ͜Ε·Ͱͷ Treatment (A) ͸ೋ஋ (1: ېԎͨ͠ɺ0: ͠ͳ͔ͬͨʣ E[Ya] = β0 +β1a Ϟσϧ͔ΒશͯͷόϦΤʔγϣϯ͕ਪఆͰ͖Δɻ(Saturated) ࠨลͷ unknownɿE[Ya=1],E[Ya=1] ӈลͷ unknownɿβ0,β1 Treatment A ͕࿈ଓม਺ͷ৔߹͸ʁ ৽͍͠ Treatment: ٤ԎྔͷมԽ A = (1 ೔ͷλόί٤Ԏຊ਺ɿfollow-up) ɹ − (1 ೔ͷλόί٤Ԏຊ਺ɿbaseline) ର৅αϯϓϧɿϕʔεϥΠϯͰͷ 1 ೔٤Ԏຊ਺͕ 25 ຊ ҎԼͷݸਓʢN=1162ʣ 26 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Continuous Treatment ஌Γ͍ͨҼՌޮՌɿ E[Ya]−E[Ya′ ] Marginal structural model: E[Ya] = β0 +β1a+β2a2 a2 = a×a β0 = E[Ya=0] a = 0ʢ٤Ԏྔ͕มΘΒͳ͔ͬͨ৔߹ʣͷฏۉମॏ૿Ճྔɻ 27 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ஌Γ͍ͨҼՌޮՌɿ E[Ya]−E[Ya′ ] Marginal structural model: E[Ya] = β0 +β1a+β2a2 ྫɿ1 ೔ 20 ຊ૿͑ͨਓͱมΘΒͳ͔ͬͨਓͷൺֱ: E[Ya=20]−E[Ya=0] β0 Marginal structural model: E[Ya=20] = β0 +20β1 +400β2 E[Ya=20]−E[Ya=0] = 20β1 +400β2 ඞཁͳͷ͸ β1 ͱ β2 28 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ࣍ͷεςοϓɿIP weights ͰٙࣅूஂΛ࡞ͬͯϞσϧʹ ϑΟοτ͠Α͏ɻ E[Y|A] = θ0 +θ1A+θ2A2 Stabilized weight (SWA) = f(A) f(A|L) ΛٻΊΔʹ͸ʜ ೋ஋ͷ A ͷ৔߹͸ logistic model Ͱ Pr[A = 1|L] Λٻ Ίͨ ࿈ଓͷ A ͷ৔߹͸ Probability density function (PDF) PDF ʹର͢ΔԾఆ f(A|L): Normal (Gaussian) with mean µL = E[A|L] f(A|L): Constant variance σ2 f(A): Normal (Gaussian) Estimated SWA: min=0.19, max=5.10, mean=1.00 29 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Marginal structural model: E[Ya] = β0 +β1a+β2a2 ݁Ռɻ ˆ β0 = 2.005 ˆ β1 = −0.109 ˆ β2 = 0.003 ΋͠٤ԎྔΛม͑ͳ͔ͬͨΒˠ 2.0kg(1.4,3.5) ΋͠٤ԎྔΛ 1 ೔ 20 ຊ૿΍ͨ͠Βˠ 0.9kg(−1.7,3.5) 30 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ೋ஋ͷΞ΢τΧϜͷ৔߹ ৽͍͠ೋ஋ͷΞ΢τΧϜɿېԎͱࢮ๢ ېԎ (A)1992 ೥·Ͱʹ A = 1: ېԎͨ͠, A = 0: ېԎ͠ͳ͔ͬͨ ࢮ๢ (D)1992 ೥·Ͱʹ D = 1: ࢮ๢ͨ͠, D = 0: ࢮ๢͠ͳ͔ͬͨ Marginal structural logistic model logitPr[Da = 1] = α0 +α1a IP weight Ͱ࡞ͬͨٙࣅूஂͰਪܭ logitPr[D = 1|A] = θ0 +θ1A exp( ˆ θ1) = 1.0(0.8,1.4) Causal odds ratio 31 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 12.5 Effect modiﬁcation and marginal structural models Covariates ʹ͍ͭͯɿ ݚڀͷλʔήοτͱ͢Δ parameter ͕ average causal effect Ͱ͋Δ৔߹͸جຊՃ͑ͳ͍ɻ Effect modiﬁcation ʹؔ৺͕͋Δ৔߹͸Ճ͑Δɻ ݟ͍ͨ΋ͷɿV ͷϨϕϧؒͰͷ Treatment ͷޮՌͷҧ͍ɻ ྫɿېԎͷޮՌ͸உঁͰҧ͏͔ʁ sex V (1: woman, 0: man) E[Ya|V] = β0 +β1a+β2Va+β3V V ͰίϯσΟγϣϯͯ͠͠·͍ͬͯΔͷͰɺݫີʹ ͸”marginal model” Ͱ͸ͳ͍ɻ IPW Ͱௐ੔ͯ͠ϑΟοτɿ E[Y|A,V] = θ0 +θ1A+θ2VA+θ3V 32 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ஫ҙ WA ·ͨ͸ SWA ΛٻΊΔࡍɺcovariates L ʹ V Λ௥Ճ͢Δඞ ཁ͕͋Δɻ V ͕ confounder Ͱ͸ͳ͍৔߹΋௥Ճ͸ඞཁɻ SWA ͷ෼ࢠɺ f[A] Λ࢖͏΂͖͔ f[A|V] Λ࢖͏΂͖͔ʁ SWA(V) = f[A|V] f[A|L] V ΛՃ͑Δͱ CI ͕ڱ͘ͳΔɻ෼ࢠͱ෼฼Ͱ V ͕ variation Λ ٵऩ͢Δˠ SWA ͷϨϯδڱ͘ͳΔˠ narrow CI 33 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Covarite L ʹՃ͑Δαϒηοτ V ʹؔͯ͠͸ɺݚڀऀ͕ 1 Effect modiﬁer Ͱ͋Δͱڧ͘ߟ͑Δ 2 ର৅ʹڧ͍ؔ৺͕͋Δʢશମͷूஂͦͷ΋ͷΑΓ΋ʣ ৔߹ʹݶΔ΂͖ɻ Τηपลߏ଄Ϟσϧ (faux marginal structural model) ΋͢͠΂ͯͷม਺ L Λ marginal structural model ʹՃ͑ͨ ͱͨ͠ΒɺSWA(L) = 1 ͱͳΓɺͦ΋ͦ΋ IPW ͢Δඞཁ΋ͳ ͍ɻී௨ʹ L ͍ΕͯճؼΛճ͚ͩ͢Ͱ OK Confouder ͷௐ੔ͱ Effect modiﬁcation ͸ผ෺ 2 ͭͷ Treatment A ͱ B ͷަޓ࡞༻ʹؔ৺͕͋Δ৔߹ A ͱ B ͷύϥϝʔλʔͲͪΒ΋ marginal structural model ʹೖΕΔ IP weights ͷ෼฼͸ Treatment A ͱ B ͷ joint probabilityPr(A∩B) Exchangeability, positivity, consistency ͷԾఆ͕ඞཁ 34 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 12.6 Censoring and missing data ܽམม਺ N = 1566: 1982 ೥ͷϑΥϩʔΞοϓௐࠪ࣌ͷମॏͷσʔλ͕ ܽམ͍ͯ͠Δ 63 ਓ͕আ֎ (censoring) ͞Ε͍ͯΔɻ ˠ Selection bias ͷՄೳੑ͕͋Δ Censoring Censoring (C): 1982 ೥ͷମॏଌఆ C = 1: ମॏଌఆ͞Εͳ͔ͬͨʢআ֎͞Εͨʣ C = 0: ମॏଌఆ͞Εͨʢআ֎͞Εͳ͔ͬͨʣ ͜Ε·Ͱͷ෼ੳʢ12.2, 12.4ʣ ɺ࣮͸... E[Y|A] = θ0 +θ1A E[Y|A,C = 0] = θ0 +θ1A 35 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 36 / 48 আ֎͞Εͳ͔ͬͨݸਓ (c = 0) ͚ͩΛର৅ͱ͢ΔͱɺC ͕ A ͱ Y ͷ Collider ͋Δ͍͸ Collider ͷࢠଙͰ͋Δ৔߹ʹɺόΠ ΞεͷڪΕ͕͋Δɻ

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Example data Ͱ͸໰୊͋Γͦ͏ɻ Treatment A ͱ C ʹ૬ؔ: ېԎऀ (A = 1) ͷ 5.8 ˋ͕ Censored ൱ېԎऀ (A = 0) ͷ 3.2 ˋ͕ Censored Y ͷ predictor ͱ C ʹ૬ؔɿϕʔεϥΠϯ࣌ͷମॏ আ֎ऀ (C = 1) 76.6kg ൱আ֎ऀ (C = 0) 70.8kg ஌Γ͍ͨҼՌޮՌɿA ͱ C ͷ joint effect E[Ya=1,c=0] શһېԎˍআ֎ͳ͠ − E[Ya=0,c=0] શһ൱ېԎˍআ֎ͳ͠ 37 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ͲΜͳ IP weight Λ࢖͏ͷ͔ʁ WA,C = WA ×WC আ֎͞Εͳ͔ͬͨਓͷ WC = 1 Pr[C=0|L,A] আ֎͞Εͨਓͷɹɹɹ WC = 0 ԾఆɿIdentiﬁability conditions 1 Exchangeability Ya,c=0 ⊥ ⊥ (A,C)|L 2 Positivity for (A = a,C = 0) 3 Consistency 38 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Weighted modelɿ E[Y|A,C = 0] = θ0 +θ1A Marginal structural model: E[Ya,c=0] = β0 +β1a WC ʹΑͬͯ࡞ΒΕΔٙࣅूஂͷ N ͸ Censor લͷ population ͱಉ͡਺ (N = 1566+63 = 1629) L ˠ C ·ͨ͸ A ˠ C ͸ͳ͍ Selection ͦͷ΋ͷ͕ͳ͍ 39 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Stabilized IP weights for Censoring ܽམʹରͯ͠ Stabilized IP weights ͸࢖͑Δͷʁ SWA,C = SWA ×SWC SWC = Pr[C = 0|A] Pr[C = 0|L,A] ٙࣅूஂͷ N ͸ Censor ޙͷ population ͱಉ͡਺ (N = 1566) L ˠ C ͸ͳ͍ Censoring ͸ L ʹґଘ͢ΔܗͰͳ͞ΕΔɻSelection ͸ ͋Δ͕ Selection bias ͸ͳ͍ɻ SWA,C Λ࢖ͬͨͱ͖ͷ݁Ռɿ ˆ θ1 = 3.5kg(2.4,4.5) SWA Λ࢖ͬͨͱ͖ͷ݁Ռɿ ˆ θ1 = 3.5kg(2.4,4.5) WA Λ࢖ͬͨͱ͖ͷ݁Ռɿ ˆ θ1 = 3.5kg(2.4,4.5) 40 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 12.1 Setting a bad example ʮېԎͱମॏ૿Ճʯ͸ສਓʹΘ͔Γ΍ͯ͘͢ศར͕ͩɺ Selection bias ͷՄೳੑ͕͋Δѱ͍ྫɻ ʮېԎʯͱ͸ʜ 1 1971-75 ೥ͷϕʔεϥΠϯௐࠪ࣌ʹ٤ԎऀͰ͋Δɻ 2 1982 ೥ʹېԎ͍ͯ͠Δɻ αόΠόϧόΠΞε Time-varying treatment (Part 3) ηϨΫγϣϯόΠΞεɿTreatment ޙͷΠϕϯτ (survey ࢀՃ) Λ΋ͱʹ Censor ͢Δݚڀऀ 41 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 12.2 Checking positivity ͱ͋Δ L ͷίϯϏωʔγϣϯͷશһ͕ېԎ͠ͳ͔ͬͨɻ ྫɿ66 ࡀɺനਓɺঁੑͷ 4 ਓͱ΋٤ԎܧଓʢPr(A = 1|L) = 0) Structural violation ͋Δ L ͷ஋Λ΋ͭਓʑ͕ Treatʢ·ͨ͸ Untreat) ͞ΕΔ ͜ͱ͕ߏ଄తʹෆՄೳͳ৔߹ɻ ྫɿ৬৔Ͱͷༀ෺๫࿐ˠࢮ๢ʹؔ৺͕͋Δέʔεɻ࢓ࣄ Λ͍ͯ͠ͳ͍ਓ͸ Treat ͞ΕΔ͜ͱ͕ͳ͍ɻ IP Weighting ΍ Standardization ͰҼՌਪ࿦Λߦ͏͜ͱ ͸Ͱ͖ͳ͍ɻPositivity ͕୲อ͞Ε؍ଌͰ͖Δ Strata ͷ ΈΛݚڀͷର৅ͱ͢Δɻ Random violation ༗ݶͷαϯϓϧԼʹ͓͍ͯɺ” ͨ·ͨ·” ͍͔ͭ͘ͷ L ͷ ίϯϏωʔγϣϯͰ 0 ͕؍ଌ͞ΕΔ৔߹ɻ ྫɿ66 ࡀനਓঁੑͱ 67 ࡀനਓঁੑ͸શһ A=0 ͕ͩɺ 65 ࡀനਓঁੑͱ 69 ࡀനਓঁੑ͸ A=1 ΛؚΉɻ Parametric model Λ࢖͏͜ͱͰεϜʔζΞ΢τ͞ΕΔɻ Random nonpositivity ͷ Assumption ͕ඞཁɻ 42 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 12.1 Horvitz-Thompson estimators Positivity ͱ Exchangeability ͕୲อ͞ΕΔ৔߹ɺ E I(A = a)Y f(A|L) IP weighted mean (ch.3) = E[Ya] Counterfactual mean Horvitz-Thompson (1952) estimator: ˆ E I(A = a)Y f(A|L) Modiﬁed Horvitz-Thompson estimator (Robins 1998): ˆ E I(A=a)Y f(A|L) ˆ E I(A=a) f(A|L) 43 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point Positivity ͕୲อ͞ΕΔ৔߹, E I(A = a) f(A|L) = 1 ͳͷͰɺ E I(A=a)Y f(A|L) E I(A=a) f(A|L) = E I(A = a)Y f(A|L) ೋ஋ͷ Y ͷ৔߹ɺ͔ͳΒͣ 0 ͔Β̍ͷ஋ΛͱΔɻ 44 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point E I(A=a)Y f(A|L) E I(A=a) f(A|L) Positivity ͕୲อ͞Εͳ͍৔߹ɺ = ∑ l E[Y|A = a,L = l,L ∈ Q(a)]Pr[L = l|L ∈ Q(a)] Exchangeability ͕୲อ͞Εͳ͍৔߹ɺ E[Ya|L ∈ Q(a)] where Q(a) = l;Pr(A−a|L = l) > 0 is the set of values l for which A = a may be observed with positive probability. Positivity ͕୲อ͞Εͳ͚Ε͹ɺHorvitz-Thompson estimator Ͱ a = 1 ͱ a = 0 ΛൺֱΛߦͬͯ΋ɺҼՌతղऍ͸ Ͱ͖ͳ͍ɻ 45 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point 12.2 More on stabilized wights SWA = f[A] f[A|L] ⇒ g[A] f[A|L] g[A]: A ͷؔ਺͔ͭ L ʹґଘ͠ͳ͍ؔ਺ɻ جຊతʹɺ 1 f[A|L] Ͱ͸ͳ͘ g[A] f[A|L] Λ࢖͏ͱྑ͍ɻ Nonsaturated marginal structural model ʹద༻͢ΔࡍɺΑ Γ Efﬁcient ͱͳΔɻ 46 / 48

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12.1 12.2 12.3 12.4 12.5 12.6 Fine point Technical point ͋Γ͕ͱ͏͍͟͝·ͨ͠ʂ 48 / 48