Introduction to Bayesian Statistics (Modern epidemiology Chap. 18)

Chapter 18 Introduction to Bayesian Statistics Chuntaro (@Shuntarooo3) 1 Modern
Epidemiology

2 Overview w ϕΠζͷྺ࢙ w Ӹֶσʔλ΁ස౓࿦తํ๏Λ༻͍Δ͜ͱͷ໰୊఺ w ϕΠζਪఆ w ࣄલ෼෍
w ໬౓ w ࣄޙ෼෍ w ۩ମྫ w ࣄલ෼෍Λ3$5Ͱද͢ͱʁ w ֊૚Ϟσϧ w Ϛϧίϑ࿈࠯ϞϯςΧϧϩ๏ w චऀͷࢥ͍

Introduction: History 3 w $IBQʙ͸ɼӸֶͰ༻͍Δස౓࿦తํ๏Λઆ໌ w $IBQ͸ɼϕΠζతํ๏Λ঺հ
5IPNBT#BZFT https://ja.wikipedia.org/wiki/τʔϚεɾϕΠζ, https://ja.wikipedia.org/wiki/ϐΤʔϧʹγϞϯɾϥϓϥε, https://en.wikipedia.org/wiki/Ronald_Fisher, https://en.wikipedia.org/wiki/ Jerzy_Neyman, https://errorstatistics.com/2018/08/11/egon-pearsons-heresy-3/ 1JFSSF4JNPO -BQMBDF 3"'JTIFS +/FZNBO &1FBSTPO ϕΠζ౷ܭͷ։ൃ Ұൠతͳ ౷ܭֶ ϕΠζ౷ܭͷ ࠶ڵ

Introduction: Motivation of Bayesian 4 3BOEPNJ[FE USJBMT 3BOEPNTBNQMF TVSWFZT 'SFRVFOUJTU
NFUIPET ʢස౓࿦తํ๏ʣ 0CTFSWBUJPOBM EBUB w %BUBͷੜ੒ϝΧχζϜ w മ࿐ͷׂ෇͚ w)FBWJMZOPOSBOEPN w1PPSMZVOEFSTUPPE 'SFRVFOUJTU NFUIPET ʢස౓࿦తํ๏ʣ ˡେৎ෉ʁ Ӹֶݚڀͷେ෦෼ ࢖༻ #BZFTJBOBQQSPBDI ͜ΕΒΛ͏·͘ ଊ͑ΒΕΔ͔΋

FREQUENTISM VERSUS SUBJECTIVE BAYESIANISM (p. 329) 5 w ղੳʹ࠾༻ͨ͠Ծఆ΍Ϟσϧ͕ओ؍తͰ͋Δ w
ዞҙੑ͕ੜ͡Δ ස౓࿦೿͔ΒϕΠζ೿΁ͷ൷൑ w ͢΂ͯͷ౷ܭతਪ࿦ʹڞ௨͍ͯ͠Δ w ϕΠζΞϓϩʔν͸ɼ೚ҙͷཁૉΛ໌Β͔ʹ͍ͯ͠Δ͚ͩ

Subjective probabilities should not be arbitrary [1/4] (p. 330) 6
ࣄલʢ֬཰ʣ෼෍ɿQSJPS QSPCBCJMJUZ EJTUSJCVUJPO w ୯ʹɼQSJPSͱ΋ݴ͏ w ɹɹɹɹɹɹɹͱ.PEFSO&QJ͸ද͢ɽҰൠతʹ͸ɼ w σʔλΛ؍࡯͢Δલʹ෼ੳऀ͕ʢओ؍తʹʣܾఆͨ͠฼਺ͷ෼෍ QSJPS TVCKFDUJWF QBSBNFUFS ྫ w །Ұͷύϥϝʔλ͕33 3JTL3BUJP ͱ͢Δ RR > RRmedian RR < RRmedian ಉ͡ ֬৴౓ Pr(RR > RRmedian ) = Pr(RR < RRmedian ) RRlower RRupper ͜͜ʹ33͕͋Δ ֬৴౓͸ Pr(RRlower < RR < RRupper ) = 0.95 Pr(parameters) f(θ)

Subjective probabilities should not be arbitrary [2/4] (p. 330) 7
TVCKFDUJWFQSJPS͸ओ؍త͗͢ͳ͍͔ʁ w ͔֬ʹݸਓؒͰҟͳΔ͔΋͠Εͳ͍ w ͜Ε͸ዞҙతͰ͋Δ͜ͱΛҙຯ͢Δ΋ͷͰ͸ͳ͍ w ଛࣦΛ࠷খݶʹ཈͑Δ͜ͱΛ໨ඪʹϨʔεʹṌ͚Δͱ͖ʹɼ୭΋ ແ࡞ҝʹબ͹ΕͨڝٕऀʹṌ͚Δͷ͕߹ཧతͩͱ͸ࢥΘͳ͍ w աڈͷσʔλʹج͍ͮͯɼṌ͚Δର৅ऀΛબͿ͸ͣ w ϕΠζ౷ܭͰ͸ɼઌߦݚڀͷ݁ՌΛࣄલ֬཰ʹ൓ө͢Δ

8 ໬౓ɿMJLFMJIPPE w ϕΠζ౷ܭʹ͓͚Δೖྗ͸ɼࣄલ֬཰ͱ໬౓ w ɹɹɹɹɹɹɹɹɹͱ.PEFSO&QJ͸ද͢ɽҰൠతʹ͸ɼ w ύϥϝʔλ͕༩͑ΒΕͨͱ͖ʹɼσʔλ͕ಘΒΕΔ֬཰ ֬཰ Pr(data
∣ parameters) f(X ∣ θ) Pr(data ∣ parameters) ύϥϝʔλΛఆ਺ɼ σʔλΛม਺ ύϥϝʔλΛม਺ɼ σʔλΛఆ਺ ໬౓ Subjective probabilities should not be arbitrary [3/4] (p. 330)

9 ໬౓͸ద੾͔ʁ w ద੾ͳ֬཰෼෍Ͱ໬౓Λද͍ͯ͠Δ͔͸ɼ ස౓࿦೿΋ϕΠζ೿΋ಉ͡Α͏ʹٙ೦ʹ࣋ͭ w ͋ΔళฮɹͷΞΠεͷചΓ্͛ɹΛ༧ଌ͢Δ w ചΓ্͛ͷฏۉΛɹͰද͢ w
ചΓ্͛ͷฏۉɹ͸ɼؾԹɹͱ஍Ҭɹʢ੢೔ຊ͔౦೔ຊ͔ʣͷ ӨڹΛड͚ΔͱԾఆ͢Δ w ؾԹ͕౓্͕Δ͝ͱʹɹສԁചΓ্্͕͕͛Δ w ੢೔ຊ͸౦೔ຊʹൺ΂ͯɼɹສԁചΓ্্͕͕͛Δ w ചΓ্͛ͷ෼ࢄ͸ɹͰද͢ ྫ i Yi μi Ti Li βT βL σ2 μi = β0 + βT Ti + βL Li μi Yi ∼ Normal(μi , σ2) (i = 1,...,n) NPEFMJOH ˡద੾͔ͳʁ Subjective probabilities should not be arbitrary [4/4] (p. 330)

The posterior distribution (p. 330) 10 ࣄޙʢ֬཰ʣ෼෍ɿQPTUFSJPS QSPCBCJMJUZ EJTUSJCVUJPO w
୯ʹɼQPTUFSJPSͱ΋ݴ͏ w σʔλ͕༩͑ΒΕͨޙͷ฼਺ͷ৚݅෇͖෼෍ Pr(parameters ∣ data) = Pr(data ∣ parameters) Pr(parameters) Pr(data) Pr(parameters ∣ data) ∝ Pr(data ∣ parameters) Pr(parameters) ࣄޙ֬཰ ໬౓ ࣄલ֬཰ पล໬౓ɹˡύϥϝʔλͷؔ਺Ͱͳ͍ Χʔωϧɿ,BSOFM ࣄલ֬཰ ໬౓ ࣄޙ֬཰

Frequentist-Bayesian parallels (p. 331) 11 w ʮύϥϝʔλ͸ස౓࿦೿ʹ͸ݻఆ͞Εͨ΋ͷͱͯ͠ѻΘΕΔ͕ɼ ϕΠζ೿͸ϥϯμϜʹѻ͏ʯͱ͍͏ͷ͸ؒҧ͍ w ͲͪΒ΋ڵຯͷ͋Δύϥϝʔλ͸ݻఆ͍ͯ͠Δ
ਅ஋͸ݻఆ ϕΠζ೿ ස౓࿦೿ ύϥϝʔλͷෆ࣮֬ੑΛ ࣄޙ෼෍Ͱද͢ ύϥϝʔλ͸఺ਪఆ͠ɼ ͦͷਫ਼౓͸۠ؒਪఆͰද͢ w ස౓࿦తํ๏΋ϕΠζతํ๏΋໬౓ʹ݁Ռ͕ґଘ͢Δ w ద੾ͳϞσϦϯάΛߦ͏ඞཁ͕͋Δ

Empirical Priors (p. 331) 12 w ࣄલ෼෍͕ओ؍తͳ΋ͷͰ͋ͬͯ͸ͳΒͳ͍ͱ͢ΔͳΒ͹ɼ ԿΛ࢖͑͹ྑ͍ͷ͔ʁ w 4UFJOFTUJNBUJPOɿॖখਪఆ
w &NQJSJDBM#BZFTɿϋΠύʔύϥϝʔλ΋σʔλ͔Βਪఆ w 1FOBMJ[FEFTUJNBUJPOɿਖ਼ଇԽʢॖখਪఆʣ w SBOEPNDPF⒏DJFOUɿϋΠύʔύϥϝʔλ΋σʔλ͔Βਪఆ w SJEHFSFHSFTTJPOɿਖ਼ଇԽʢॖখਪఆʣ w ڞ໾ࣄલ෼෍ɿޮ཰ͷྑ͍ܭࢉ w ແ৘ใࣄલ෼෍ɿ੄ͷ޿͍෼෍

Frequentist-Bayesian divergences [1/2] (p. 332) 13 ;ʔΜ

Frequentist-Bayesian divergences [2/2] (p. 332) 14 ֬৴۠ؒʢDSFEJCMFJOUFSWBM$SF*ʣ w ϕΠζ౷ܭͰਪఆ͢Δ৴པ۠ؒ w
৴༻۠ؒͱ΋͍͏ w ͋Δύϥϝʔλͷ֬৴۠ؒͰ͸ɼ ύϥϝʔλࣗ਎͕෼෍ʢࣄޙ෼෍ʣ͢Δ w $SF*͸ɼͦͷ۠ؒ಺ʹύϥϝʔλ͕ଘࡏ͢Δ֬཰͕ ৴པ۠ؒʢDPOpEFODFJOUFSWBM$*ʣ w ස౓࿦Ͱਪఆ͢Δ৴པ۠ؒ w $*ʹਅͷύϥϝʔλ͕ଘࡏ͢ Δ֬཰͸ɼ͔ IUUQTSQTZDIPMPHJTUDPNE$* $*ΛΞχϝʔγϣϯʹͨ͠αΠτ

Frequentist fantasy versus observational reality (p. 332) 15 3BOEPNJ[FE USJBMT
3BOEPNTBNQMF TVSWFZT 'SFRVFOUJTU NFUIPET ʢස౓࿦తํ๏ʣ 0CTFSWBUJPOBM EBUB w %BUBͷੜ੒ϝΧχζϜ w മ࿐ͷׂ෇͚ w)FBWJMZOPOSBOEPN w1PPSMZVOEFSTUPPE 'SFRVFOUJTU NFUIPET ʢස౓࿦తํ๏ʣ ࢖༻ Ӹֶʹ͓͍ͯස౓࿦తํ๏Λ༻͍Δ͜ͱ͸ɼ ؍࡯σʔλΛ͔͋ͨ΋ݫີʹઃܭ͞Εɼ ؅ཧ͞ΕͨϥϯμϜԽ࣮ݧ͔ΒಘΒΕͨͱߟ͑ΔΑ͏ͳ΋ͷ 'BOUBTZ

Summary (p. 333) 16 w ϕΠζతํ๏΁ͷ൷൑͸ɼ w ࣄલ෼෍͕ዞҙత w ࣄલ෼෍͕ɼ༗֐Ͱಛผͳํ๏Ͱओ؍తʢʁʣ
ɹͰͳ͚Ε͹ͳΒͳ͍ w ؍࡯ݚڀͰɼ೔ৗతʹ࢖ΘΕ͍ͯΔղੳϞσϧ΋ዞҙతͰ͋Δ w ϕΠζతํ๏͸ɼස౓࿦తํ๏ͱಉ౳͔ͦΕҎ্ͷՊֶతࠜڌΛ ༩͑Δ w લఏ͕ʮݟ͑ΔԽʯ͞Ε͍ͯΔ͔Β w ൷൑తʹਫ਼ࠪ͞ΕΔඞཁ͸͋Δ

SIMPLE APPROXIMATE BAYESIAN METHODS [1/2] (p. 333) 17 Pr(parameters ∣
data) = Pr(data ∣ parameters) Pr(parameters) Pr(data) ࣄޙ֬཰ ໬౓ ࣄલ֬཰ पล໬౓ ܭࢉେม f(x) = ∫ ∞ −∞ f(x ∣ θ)f(θ)dx ϕΠζ౷ܭͷ࠶ڵ w 1$ͷൃୡ w ϞϯςΧϧϩΞϧΰϦζϜ ΧςΰϦσʔλʹదͨ͠ϕΠζۙࣅ͸ɼස౓࿦ͷۙࣅͱಉఔ౓ʹਖ਼֬ w YදΛྫʹͱΓɼϕΠζతํ๏ͱස౓࿦తํ๏Λൺֱ

SIMPLE APPROXIMATE BAYESIAN METHODS [2/2] (p. 333) 18 w YදΛྫʹͱΓɼϕΠζతํ๏ͱස౓࿦తํ๏Λൺֱ
w PVUDPNF͸ඇৗʹك w SJTLSBUJP SBUFSBUJP PEETSBUJPΛ۠ผ͠ͳ͍ˠ33Ͱදݱ w ɹɹɹ͸ਖ਼ن෼෍ʹै͏ w ਖ਼ن෼෍Ͱ͸࠷ස஋ɼதԝ஋ɼฏۉ͸۠ผ͠ͳ͍ w ɹɹɹɹɹ͸ର਺ਖ਼ن෼෍ʹै͏ w ɹɹɹɹɹɹɹɹɹɹɹ͕੒ΓཱͭͨΊɼҎԼٞ࿦ʹ͸ Λ༻͍Δ ln(RR) RR = eln(RR) medianRR = emedian ln(RR) medianRR

INFORMATION-WEIGHTED AVERAGING (p. 334) 19 w ৘ใʢ·ͨ͸ਫ਼౓ʣ͸ɼ෼ࢄͷٯ਺ w ϕΠζతํ๏ʹ͓͚Δ৘ใʹΑΔॏΈ෇͚͸ɼ
ස౓࿦తํ๏ʹ͓͚Δٯ෼ࢄॏΈ෇͚ʹجͮ͘ਪఆʹରԠ w ࣄલ෼෍ͱ໬౓͸ਖ਼ن෼෍ʹै͏ͱԾఆ͢Δ w ͜ͷԾఆ͸े෼ͳେ͖͞ͷαϯϓϧαΠζΛඞཁͱ͢Δ w DFMMTJ[F͸·ͨ͸͋Ε͹ྑ͍ͩΖ͏ information = precision = 1 variance

A single two-way table [1/5] (p. 334) 20 $BTFDPOUSPMTUVEZGSPN4VBWJUZFUBM
DAVID A. SAVITZ, HOWARD WACHTEL, FRANK A. BARNES, ESTHER M. JOHN, JIRI G. TVRDIK, CASE-CONTROL STUDY OF CHILDHOOD CANCER AND EXPOSURE TO 60-HZ MAGNETIC FIELDS, American Journal of Epidemiology, Volume 128, Issue 1, July 1988, Pages 21–38, https://doi.org/10.1093/oxfordjournals.aje.a114943 w ډॅ஍ͷ࣓քͱখࣇന݂පͱͷਖ਼ͷؔ࿈Λࣔͨ͠ݚڀ w ઌߦݚڀͰ͸Ոఉ಺഑ઢͱന݂පͱͷਖ਼ͷؔ࿈͸ใࠂ͞Ε͍ͯͨ w ډॅ஍ͷ࣓քʢڧ͍ిքޮՌʣͱന݂පͱͷؔ࿈͸௿͍ͱߟ͑ ΒΕ͍ͯͨ தࠃిྗHPΑΓʢhttp://www.energia.co.jp/energy/emf/emfa1.htmlʣ w ࣓ք͸ɼిྲྀ͕ྲྀΕΔ෺ͷपғʹൃੜ w ిք͸ɼిѹ͕͔͔ͬͨ෺ͷपғʹൃੜ

A single two-way table [2/5] (p. 334) 21 w ࣄલ33ͷ֬཰Λ
ͱԾఆ w $POUSPMʹൺ΂ͨ$BTFͷ࣓քമ࿐Φοζͷൺ exp(prior mean − 1.96 prior SD) = 1/4 exp(prior mean + 1.96 prior SD) = 4 Prior mean of ln(RR) = ln(1/4) + ln(4) 2 = 0 Prior SD of ln(RR) = ln(4) − ln(1/4) 2 ⋅ 1.96 = 0.707 ࿨ ࠩ Prior variance of ln(RR) = 0.7072 = 1/2 Prior ln(RR) ∼ N(0, 1/2) ࣄલ෼෍ͷܭࢉ $POUSPMʹൺ΂ͨ$BTFͷ࣓քമ࿐Φοζൺ͸/VMM 33ͷ৴༻۠ؒ

A single two-way table [3/5] (p. 334) 22 DAVID A.
SAVITZ, HOWARD WACHTEL, FRANK A. BARNES, ESTHER M. JOHN, JIRI G. TVRDIK, CASE-CONTROL STUDY OF CHILDHOOD CANCER AND EXPOSURE TO 60-HZ MAGNETIC FIELDS, American Journal of Epidemiology, Volume 128, Issue 1, July 1988, Pages 21–38, https://doi.org/10.1093/oxfordjournals.aje.a114943 X = 1 (>= 3 mG) X = 0 (< 3 mG) Case 3 33 Table odds ratio = RR estimate = 3.51 Controls 5 193 95% CI = 0.80, 15.4 Estimated RR = sample OR = (3 ⋅ 193)/(5 ⋅ 33) = 3.51 ؍࡯σʔλͰͷܭࢉ Estimated variance ln(OR) = 1/3 + 1/33 + 1/5 + 1/193 = 0.569 95 % CI of OR = exp[ln(3.51) ± 1.96 ⋅ 0.5691/2] = 0.80, 15.4

A single two-way table [4/5] (p. 334) 23 DAVID A.
SAVITZ, HOWARD WACHTEL, FRANK A. BARNES, ESTHER M. JOHN, JIRI G. TVRDIK, CASE-CONTROL STUDY OF CHILDHOOD CANCER AND EXPOSURE TO 60-HZ MAGNETIC FIELDS, American Journal of Epidemiology, Volume 128, Issue 1, July 1988, Pages 21–38, https://doi.org/10.1093/oxfordjournals.aje.a114943 Posteriror variance for ln(RR) ≃ 1 1 1/2 + 1 0.569 = 0.266 ࣄޙ෼෍ͷܭࢉ ࣄલ෼෍ͷ৘ใ ؍࡯σʔλͷ৘ใ UPUBMJOGPSNBUJPO Posteriror mean for ln(RR) = expected ln(RR) given data ≃ 0 1/2 + ln(3.51) 0.569 total information = 0 1/2 + ln(3.51) 0.569 1/0.266 = 0.587 ࣄલ෼෍ͷॏΈ෇͚ฏۉ ؍࡯σʔλͷॏΈ෇͚ฏۉ Posteriror median for RR ≃ exp(0.587) = 1.8 95 % posterior limits for RR ≃ exp(0.587 ± 1.96 ⋅ 0.2661/2) = 0.65, 4.94

A single two-way table [5/5] (p. 334) 24
ࣄલ෼෍ σʔλ º ࣄޙ෼෍

Bayesian Interpretation of Frequentist results (p. 335) 25 Posterior mean
for ln(RR) ≃ 0 1/2 + ln(3.51) 0.569 1 1/2 + 1 0.569 = 0.587 Posterior mean for ln(RR) without prior ≃ 0 + ln(3.51) 0.569 0 + 1 0.569 = ln(3.51) Estimated RR = sample OR = (3 ⋅ 193)/(5 ⋅ 33) = 3.51 Posterior mean for RR without prior ≃ exp{ln(3.51)} = 3.51 ࣄલ෼෍ͷ৘ใ͸ͳ͍ ස౓࿦తํ๏ Ұக w ස౓࿦తํ๏͔ΒಘΒΕΔ݁Ռ͸ɼࣄલ৘ใΛશ͘࢖Θͳ͍ϕΠζ࿦తํ๏͔Β ಘΒΕΔ݁Ռͱಉ͡ w ࣄલ৘ใΛ࢖Θͳ͍ͱ͍͏͜ͱ͸ɼ ͱ͍͏33Λ΍ͱಉ͘͡Β͍͋Γ͑Δͱߟ͑Δͱಉ͜͡ͱ w ແ৘ใࣄલ෼෍͸໌ࣔతʹࣄલ෼෍Λ༩͑͸͍ͯ͠Δ఺Ͱස౓࿦ͱҟͳΔ w ͔͠͠ײ֮తʹ͋Γಘͳ͍ࣄલ෼෍΋औΓ͑Δ

Adjustment [1/3] (p. 336) 26 ࣓ք খࣇന݂ප ݚڀʢ$POGPVOEFSʣ ࣄલ৘ใͳ͠ ݚڀ
1SJPS 33 1PTUFSJPS ˠ ˠ ˠ ݚڀ 1SJPS 33 1PTUFSJPS /" ˠ /" ˠ /" ˠ ݚڀ 1SJPS 33 1PTUFSJPS ˠ ˠ ˠ ͜͏͍͏ঢ়ଶ͔ͳʁ ݚڀ͝ͱͷQSJPS͸ͳ͍ ڞ௨ͷQSJPSΛઃఆ͢Δ ૚ผղੳͰަབྷΛௐ੔ Estimated common RR = sample OR = 1.69 ln(OR) = ln(1.69) = 0.525 95 % CI of common OR = 1.28, 2.23 variance of ln(OR) = { ln(2.23) − ln(1.28) 2 ⋅ 1.96 }2 = 0.0201 ؍࡯σʔλͰͷܭࢉ N(0, 1/2)

Adjustment [2/3] (p. 336) 27 Posteriror variance for ln(RR) ≃
1 1 1/2 + 1 0.0201 = 0.0193 ࣄޙ෼෍ͷܭࢉ ڞ௨ࣄલ෼෍ͷ৘ใ ؍࡯σʔλͷ৘ใ UPUBMJOGPSNBUJPO Posteriror mean for ln(RR) = expected ln(RR) given data ≃ 0 1/2 + ln(1.69) 0.0201 total information = 0 1/2 + ln(1.69) 0.0201 1/0.0193 = 0.504 ࣄલ෼෍ͷॏΈ෇͚ฏۉ ؍࡯σʔλͷॏΈ෇͚ฏۉ Posteriror median for RR ≃ exp(0.504) = 1.66 95 % posterior limits for RR ≃ exp(0.504 ± 1.96 ⋅ 0.01931/2) = 1.26, 2.17

Adjustment [3/3] (p. 336) 28
ڞ௨ͷࣄલ෼෍ σʔλ º ࣄޙ෼෍

Varying the prior (p. 336) 29
ࣄલ෼෍ σʔλ ࣄޙ෼෍ σʔλ ࣄޙ෼෍ ࣄલ෼෍ ࣄલ෼෍ σʔλ ࣄޙ෼෍ *OGPSNBUJPO ಉ͙͡Β͍ͩͱ ࣄલ෼෍ʹۙ͘ͳΔ σʔλͷ৘ใ͕େ͖͍ͱ σʔλͷ෼෍ʹۙ͘ͳΔ w ࣄલ෼෍ͷ৘ใ͕େ͖͍ͱ ਫ਼౓ͷߴ͍ࣄલ෼෍ʹۙ͘ͳΔ w /VMMΛӽ͢ʁ

Bayes versus semi-bayes (p. 337) 30 logit(Pr(Y = 1 ∣
X, C)) = β0 + β1 X + β2 C ࣓ք ʢ9ʣ খࣇന݂ප ʢ:ʣ $$POGPVOEFS β1 ∼ N(0, 1/2) ৘ใͳ͠ ηϛ΂Πζ

PRIOR DATA: FREQUENTIST INTERPRETATION OF PRIORS [1/3] (p. 337) 31
ࣄલ෼෍ Prior ln(RR) ∼ N(0, 1/2) ͜ͷࣄલ෼෍ͱಉ͡৘ใΛ༩͑Δස౓࿦తͳ৘ใ͸Կ͔ʁ 3$5ΛԾ૝తʹߟ͑Δ R / / / 9 ˠ" / 9 ˠ" X = 1 X = 0 Case A1 A0 Controls N1-A1 N0-A0 Total N1 N0 &RVBMBMMPDBUJPO/// 3BSFEJTFBTF/" Estimated RR = (A1 /N)/(A0 /N) = A1 /A0 Estimated variance for ln(RR) = 1 A1 + 1 A0 + 1 N − A1 + 1 N − A0 = 1 A1 + 1 A0 N − A1 ≃ N, 1/N ≃ 0 XIFSF

Estimated variance for ln(RR) = 1 A1 + 1 A0 = 1 A + 1 A = 2 A = 1 2 Prior ln(RR) ∼ N(0, 1/2) ʹ߹ΘͤΔ Estimated RR = A1 /A0 = 1 → A1 = A0 = A A1 = A0 = A = 4 X = 1 X = 0 Case 4 4 Controls 99,996 99,996 Total 100,000 100,000 ࣄલ෼෍ Prior ln(RR) ∼ N(0, 1/2) ࣄલ෼෍ͱಉ౳ͷ৘ใΛ༗͢Δঢ়گ͸ۃ୺Ͱ͸ͳ͍͔ʁ ࣄલ෼෍ͷݟ௚͠

X = 1 X = 0 Case 4 4 Controls 99,996 99,996 Total 100,000 100,000 ࣄલ෼෍ Prior ln(RR) ∼ N(0, 1/2) ࣄલ෼෍͔Βɼಉ౳ͷσʔλΛٯࢉ ্هσʔλΛҰͭͷ૚ɼ؍࡯σʔλΛҰͭͷ૚ͱ͠ɼ૚ผղੳ ස౓࿦తํ๏ͷιϑτ΢ΣΞͰϕΠζਪఆΛ͓͜ͳ͏ʹ͸ʁ X = 1 X = 0 Case 3 33 Controls 5 193 ૚ผղੳ ؍࡯σʔλ

Reverse-bayes analysis (p. 338) 34 ࣄલ෼෍ σʔλ ࣄޙ෼෍
σʔλ͕໌ࣔ͞Ε͍ͯͳ͍ͨΊΘ͔Βͳ͍ ࣄޙ෼෍ͷ֬৴۠ؒԼݶΛͱ͢Δͱʜ RRprior = 1 95 % posterior limits for RRprior = 0.85, 1.18 ٯࢉ ඞཁͳࣄલ෼෍ͱͦΕʹରԠ͢Δσʔλ ࣄޙ෼෍ΛԾઆͱͯ͠ઃఆͨ͠৔߹ɼࣄલ෼෍͸Ͳ͏ͳΔ͔ʁ X = 1 X = 0 Case 250 250 Total 7,500,000 7,500,000 ͋·Γʹ΋ۃ୺

Priors with non-null center (p. 339) 35
ࣄલ෼෍ Prior ln(RR) ∼ N(ln(2), 1/2) Posteriror variance for ln(RR) ≃ 1 1 1/2 + 1 0.569 = 0.266 Posteriror mean for ln(RR) ≃ ln(2) 1/2 + ln(3.51) 0.569 total information = 0 1/2 + ln(1.69) 0.0201 1/0.266 = 0.956 Posteriror median for RR ≃ exp(0.956) = 2.60 95 % posterior limits for RR ≃ exp(0.956 ± 1.96 ⋅ 0.2661/2) = 0.95, 7.15 σʔλ ࣄޙ෼෍ X = 1 X = 0 Case 4 4 Total 100,000 200,000

Choosing the sizes of the prior denominators (p. 339) 36
w ɹɹɹɹɹɹɹɹɹɹɹɹɹɹɹͰ͋Ε͹ɼ رগ࣬ױੑ͸ҡ࣋Ͱ͖ͯ໰୊ͳ͍ w SJTLSBUJPSBUFSBUJPPEETSBUJPͷۙࣅ͕ҡ࣋Ͱ͖Δ w ͜Ε·Ͱͷܭࢉʹ࢖͍ͬͯͨɹɹɹɹͷۙࣅʹ͸Өڹ͢Δ͸ͣ N1 > 100 ⋅ A1 and N0 > 100 ⋅ A0 N ≫ A

Non-normal priors (p. 340) 37 Prior ln(RR) ∼ N(μ, σ2)
→ Prior RR ∼ logNormal(μ, σ2) ͜Ε·Ͱͷࣄલ෼෍ ɹɹɹɹɹɹɹɹɹͳΒ͹ɼɹɹɹɹɹɹɹɹɹɹͷํ͕ਖ਼֬ N1 ≫ A1 and N0 ≫ A1 Prior RR ∼ F(2A1 , 2A0 ) A1 ≠ A0 , or A1 , A0 < 3 ͷͱ͖ɼ͜ͷ෼෍ͷ࢖༻͸ෆద log Normal F F RR 95% CreI RR CreI Distribution A1 = A0 = A = 4 (1/4, 4) 93.3% F(8, 8) A1 = A0 = A = 3 (1/5, 5) 92.8% F(6, 6)

Further extensions (p. 340) 38 w߱ѹༀͷޮՌΛݕূ͢Δྟচࢼݧ wಉҰର৅ऀʹରͯ͠౤༩લ͔Β౤༩ޙʹ͔͚ͯෳ਺ճ݂ѹଌఆ w ಉҰର৅ऀͷ݂ѹͷਪҠ͸૬ؔ͋Δ w
ର৅ऀ͝ͱʹϕʔεͷ݂ѹҧ͏ w ର৅ऀ͝ͱʹԼ͕Γํ΋ҧ͏ .VMUJMFWFM )JFSBSDIJDBM NPEFMJOH μij = β0 + βX Xi + βT Tij + βXT Xi Tij + r0i Yij ∼ Normal(μij , σ2) (i = 1,...,n; j = 1,...,k) β0 , βX , βT , βXT , ri ∼ N(0, 1002) ूஂͷฏۉͱͯ͠ճؼ܎਺ ݸਓ͝ͱͷҧ͍ ˞ର৅ऀ͝ͱʹԼ͕ Γํ͕ҧ͏͜ͱ͸ߟ ྀͨ͠ϞσϧͰ͸͋Γ ·ͤΜ

CHECKING THE PRIOR (p. 340) 39 ࣄલ෼෍ΛՃ͑ͯϕΠζਪఆ͢Δલʹɼ ؍࡯σʔλͱͷۉҰੑΛ֬ೝ͢Δ΂͖ Frequentist estimate
− prior estimate ( frequentist variance + prior variance)2 = χscore Ծఆɿࣄલ෼෍͕ਖ਼ن෼෍ʹै͏ɼ؍࡯঱ྫ਺͕े෼େ͖͍ 1஋͕খ͍͞৔߹ɼ ࣄલ෼෍ͱ؍࡯σʔλͰॏΈ෇͚ਪఆ͢Δͷ͸ෆదͰ͋Δ͜ͱΛࣔ͢ ݕఆͯ͠ͳ͍͚Ͳɼ݁ہ1஋ग़͢Μ͔ʔʔʔ͍

DISCUSSION Data alone say nothing at all (p. 341) 40
wස౓࿦తํ๏͸ϕΠζతํ๏͔ΒಘΒΕΔ݁ՌΑΓ΋؍࡯σʔλΛ Α͘൓ө͢Δ wස౓࿦తํ๏͸ϥϯμϜԽൺֱࢼݧɼϥϯμϜαϯϓϦϯάʹجͮ ͍͍ͯΔ w͍ͭ΋ͷ࿩͠ wਪ࿦Ͱ͸ͳ͘σʔλΛهड़͢Δ͚ͩͳΒ͹ɼ ද΍άϥϑͰཁ໿͢΂͖

Data priors as a general diagnostic device (p. 341) 41
wേଇ෇͖໬౓๏͸ɼϕΠζతͳݟํ͔Β΋ղऍͰ͖Δ wଞͷ͍Ζ͍Ζͳ౷ܭख๏΋ϕΠζతͳݟํ͕Ͱ͖Δ͔΋

The role of Markov-Chain Monte Carlo [1/3] (p. 342) 42
Pr(parameters ∣ data) = Pr(data ∣ parameters) Pr(parameters) Pr(data) ࣄޙ෼෍ ฏۉɼதԝ஋ɼύʔηϯλΠϧ఺Λऔಘ͍ͨ͠ˠܭࢉෳࡶ .$.$ʢ.BSLPW$IBJO.POUF$BSMPʣ๏ ʢϚϧίϑ࿈࠯ϞϯςΧϧϩ๏ʣΛ༻͍ͯ౷ܭྔΛऔಘ͠Α͏ w Ϛϧίϑ࿈࠯Λར༻ͯ͠ɼࣄޙ෼෍ʹै͏ཚ਺Λੜ੒͢Δख๏ w ཚ਺ੜ੒ΞϧΰϦζϜͱͯ͠ɼϝτϩϙϦεɾϔΠεςΟϯά ε๏΍ϋϛϧτχΞϯɾϞϯςΧϧϩ๏͕͋Δ

w .$.$๏͸ຊ࣭తʹɼ෼ੳత౷ܭख๏ΑΓ΋ϩόετੑ͕௿͍ ʢ೥࣌఺ͷ8JO#6(4ͷओுʣ w .$.$͔Βಘͨղੳ݁ՌΛνΣοΫ͢Δʹ΋෼ੳత౷ܭख๏͕༗ӹ w ෳࡶͳϞσϦϯάΛߦ͏৔߹Ͳ͏͢Δͷʁ w ׬શʹਖ਼֬ͳϞσϧͰɼे෼௕࣮͘ߦʢੜ੒͢Δཚ਺ͷݸ਺ʣ Ͱ͖Ε͹ɼ෼ੳత౷ܭख๏ΑΓ΋ਖ਼֬ͳ݁ՌΛੜ੒Ͱ͖Δ w Ϟσϧ͸ਖ਼ղ͔෼͔Βͳ͍͔Βɼ .$.$͸ޡΓʹޡΓΛੵΈॏͶΔ͚͔ͩ΋ w શͯͷํ๏͕ͦ͏Ͱ͸ͳ͍͔ʁ w ߟ͑͏Δଥ౰ͳϞσϦϯά͢Ε͹Α͍ͷͰ͸ʁ

w ϕΠζ౷ܭϞσϦϯά͸ɼैདྷͷ౷ܭϞσϦϯάΑΓ΋ॊೈͳઃ ܭ͕Մೳ w ࠷ۙ͸ɼ8JO#6(4Ͱ͸ͳ͘4UBO͕Α͘ར༻͞ΕΔ w 3Ͱ΋3TUBOͱ͍͏΋ͷ͕࢖͑Δ w CSNTͱ͍͏QBDLBHF΋ w NBD04$BUBMJOB͸4UBO͔Βਖ਼ࣜʹΞοϓσʔτ͢Δͳͱܯࠂ ग़͔ͨΒ஫ҙͯ͠ʂʂ w ϕΠζ౷ܭϞσϦϯά໘ന͍Α w ݕఆ͢Δ͜ͱͰͳ͘ɼਪఆʹڵຯ͕͋ΔͳΒੵۃతʹ࢖ͬͯྑ͍ ͷͰ͸ʁ

Connections to sensitivity analysis (p. 342) 45 w ස౓࿦తํ๏ʹΑΓɼݻఆ͞ΕͨύϥϝʔλΛมԽͤͯ͞ɼ ͦΕ͕౷ܭྔʹ༩͑ΔӨڹΛධՁ͢Δ
w ࣄલ෼෍ΛมԽͤͯɼͦΕ͕౷ܭྔʹ༩͑ΔӨڹΛධՁ͢Δ

Some cautions on use of priors (p. 342) 46 w
ࣄલ෼෍ΛऔΓೖΕͯղੳΛ͓͜ͳ͏ͷ͸༰қͰ͋Δ w ࣄલ৘ใΛར༻͢Δ͜ͱ͸ɼ ਪ࿦Λվળ͢Δํ๏ͱͯ͠ड͚ೖΕΒΕ͍ͯΔ w ޡͬͨ৘ใΛઃఆ͢Δͱɼޡͬͨ݁Ռ͕ಘΒΕΔ w ස౓࿦తํ๏Ͱಘͨ݁ՌͱϕΠζతํ๏Ͱಘͨ݁ՌΛൺֱ͢Δ΂͠ w 1஋Λ࢖ͬͯɼซ߹͕ద͍ͯ͠Δ͔νΣοΫ͢Δ

CONCLUSIONS (p. 343) 47 w ϕΠζతํ๏͸ɼස౓࿦తํ๏͔ΒಘΒΕͨਪఆ஋ʹɼࣄલ৘ใ ΛՃ͑Δ͜ͱͰ͓͜ͳ͏͜ͱ͕Ͱ͖Δ w ૚ผղੳͷΑ͏Ͱ͋Δ w
ස౓࿦తํ๏ʹΑΔ݁Ռͷఏ͕ࣔඪ४Ͱ͋ͬͯ΋ɼ ϕΠζΛ౷ܭڭҭʹऔΓೖΕΔͷ͸༗༻Ͱ͋Ζ͏

48 Overview w ϕΠζͷྺ࢙ w Ӹֶσʔλ΁ස౓࿦తํ๏Λ༻͍Δ͜ͱͷ໰୊఺ w ϕΠζਪఆ w ࣄલ෼෍ɼ໬౓ɼࣄޙ෼෍
w ۩ମྫ w ࣄલ෼෍Λ3$5Ͱද͢ͱʁ w ֊૚Ϟσϧ w Ϛϧίϑ࿈࠯ϞϯςΧϧϩ๏ w චऀͷࢥ͍ ࣄલ֬཰ ໬౓ ࣄޙ֬཰

ࢀߟจݙͱҾ༻จݙ അ৔ਅ࠸ 3ͱ4UBOͰ͸͡ΊΔϕΠζ౷ܭϞσϦϯάʹΑΔσʔλ෼ੳೖ໳ ߨஊࣾ w ࠓ࠷΋৽͘͠෼͔Γ΍͍͢ϕΠζϞσϦϯάͷॻ੶ w 4UBO΍CSNTͷઆ໌΋๛෋
দӜ݈ଠ࿠ 4UBOͱ3ͰϕΠζ౷ܭϞσϦϯάڞཱग़൛ w ΑΓ೉ղ͕ͩ෼͔Γ΍͍͢ɽ w 4UBOͷਂΈ͸ͪ͜Βͷํ͕ཧղ͠΍͍͢ େؔਅ೭ ϕΠζਪఆೖ໳ɽΦʔϜࣾ w ෺ޠܗࣜͰອը΋͋Γɼͱ͖ͬͭ΍͍͕͢ɼ಺༰͸͚ͬ͜͏ϔϏʔ w ػցֶशͱͷϦϯΫΛҙ͍ࣝͯ͠Δ͔΋ Ԟଜ੖඙ɼ຀ࢁ޾࢙ɼӝੜਅ໵ 3Ͱָ͠ΉϕΠζ౷ܭೖ໳ɽٕज़ධ࿦ࣾ ਢࢁರࢤ ϕΠζਪ࿦ʹΑΔػցֶशೖ໳ɽߨஊࣾ ๛ాलथ ͸͡Ίͯͷ౷ܭσʔλ෼ੳɽே૔ॻళ ؠ೾σʔλαΠΤϯεץߦҕһձ ؠ೾σʔλαΠΤϯεWPM <ಛू>ϕΠζਪ࿦ͱ.$.$ͷϑϦʔιϑτؠ೾ॻళ 49

Discussion!! 50

Introduction to Bayesian Statistics (Modern epidemiology Chap. 18)

Introduction to Bayesian Statistics (Modern epidemiology Chap. 18)

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