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

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
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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
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models
7 Fine point
8 Technical point
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12.1 The causal question
ېԎˠମॏ૿ྔͷҼՌޮՌ
؍ଌσʔλʹର͠ IP Weighting Λ༻͍Δ
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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
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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)
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஌Γ͍ͨҼՌޮՌ
ͳͥ୯७ൺֱ͡ΌͩΊͳͷ͔?
ˠશ͘ҟͳͬͨूஂͷൺֱʹͳͬͯ͠·͏ɻ
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஌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]
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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
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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)
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models
7 Fine point
8 Technical point
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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
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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
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ٙ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཰
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ٙ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
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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)
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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
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·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
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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
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models
7 Fine point
8 Technical point
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12.3 Stabilized IP weights
IP weights WA = 1
f(A|L)
͸ɺStudy population શһͷίϐʔ
Λ̎ͭ࡞Δɻ
Expected mean of the weights WA = 2
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͸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
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ٙ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)
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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
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Figure 12.1

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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
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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)
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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)
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models
7 Fine point
8 Technical point
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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]
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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]
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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
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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
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஌Γ͍ͨҼՌޮՌɿ
E[Ya]−E[Ya′
]
ྫɿ1 ೔ 20 ຊ૿͑ͨਓͱมΘΒͳ͔ͬͨਓͷൺֱ:
E[Ya=20]−E[Ya=0]
β0
E[Ya=20] = β0 +20β1 +400β2
E[Ya=20]−E[Ya=0] = 20β1 +400β2
ඞཁͳͷ͸ β1
ͱ β2
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࣍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
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݁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)
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ೋ஋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
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models
7 Fine point
8 Technical point
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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
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஫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
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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
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models
7 Fine point
8 Technical point
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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
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আ֎͞Εͳ͔ͬͨݸਓ (c = 0) ͚ͩΛର৅ͱ͢ΔͱɺC ͕ A ͱ
Y ͷ Collider ͋Δ͍͸ Collider ͷࢠଙͰ͋Δ৔߹ʹɺόΠ
ΞεͷڪΕ͕͋Δɻ

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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
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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
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Weighted modelɿ
E[Y|A,C = 0] = θ0 +θ1A
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
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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)
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models
7 Fine point
8 Technical point
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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
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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
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models
7 Fine point
8 Technical point
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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)
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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
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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
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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
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͋Γ͕ͱ͏͍͟͝·ͨ͠ʂ
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IP Weighting and Marginal Structural Models(Causal inference: What if, Chapter 12)

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

Shuntaro Sato

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