pogojna verjetnost in bayesov izrek¡f...false positive fraction = 1.0 specificity true positive...
TRANSCRIPT
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Pogojna verjetnost in Bayesov izrek
Iztok Grabnar
Univerza v Ljubljani, Fakulteta za farmacijo
Maj, 2012
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Pogojna verjetnost
Verjetnost dogodka “a”
pri pogoju, da se je zgodil dogodek “b”
Zapis:
| pomeni “če je”, če imamo”, “če velja”
Uporaba:
Diagnostični testi
(občutljivost/specifičnost)
Analiza podatkov, primerjava modelov
Markovski
procesi
bap
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Pogojna verjetnost
Verjetnost, da bo otrok deček (p=0,51)
Verjetnost krvne skupine A (p=0,43)
Kakšna je verjetnost, da bo otrok deček s krvno skupino A?
Verjetnost barvne slepote (p=0,04)
Kakšna je verjetnost, da bo otrok deklica z barvno slepoto?
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Odločitveni model
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Igralne karte
♠ ♣ ♥ ♦13
13
13
13
13
13
7 13
P(S)P(F)P(S|F)P(F|S)
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Primer pogojne verjetnosti
Diagnostični klinični test
Dogodka
Dogodek
A: Pacient ima bolezen
Dogodek
B: Test je pozitiven
Zapis:
Verjetnost, da ima pacient bolezen in pozitivni izid testa
Verjetnost, da ima pacient bolezen, a je izid testa
negativen (lažno
negativen)
Verjetnost, da
pacient
nima bolezni, a je izid
testa pozitiven
(lažno
pozitiven)
Verjetnost, da ima pacient bolezen, če je izid testa pozitiven
Verjetnost, da ima pacient bolezen, če je izid testa negativen
'
''
BApBApBApBApBAp
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30% pacientov, ki so bili v letu 2008 sprejeti v enoti nujne medicinske pomoči UKC Ljubljana zaradi bolečine v trebušni votlini, je imelo akutno vnetje slepiča. 70% bolnikov z vnetjem slepiča je imelo telesno temperaturo nad 37,5°C. Pri pacientih, ki niso imeli vnetja slepiča, pa je bila telesna temperatura višja od 37,5°C pri 40% bolnikov.
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Kaj lahko rečemo o dogodku A, če poznamo izid B?
Pozitiven test
= bolezen, negativen test
= zdrav?
Diagnostični testi niso idealni!
Odvisnost dogodkov A in B
Ali je
p(A) odvisna od
p(B)?
Ali je
p(B) odvisna od
p(A)?
'' ApABpApABp
ApABpBp
ApABpBAp
BpBAp
BpBApBAp
ApABpBpBApBAp
1'
'''
![Page 10: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/10.jpg)
Bayesov
teorem
Dogodki
Ai
kjer
je i
= 1 do
k so:
Nezdružljivi: za vse
i,j
Za vsak dogodek B
Bayesov
teorem
10
1
k
ji
AAPAA
k
iii
kk
ApABpBp
ApABpApABpBp
1
11
)()()(
)()()()()(
k
iii
jjj
ApABp
ApABpBp
BApBAp
1)()(
)()()(
)()(
Thomas Bayes (1702Thomas Bayes (1702--1761)1761)
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Podatek, informacija in znanje
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Učenje
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Občutljivost in specifičnost
Občutljivost TP=a/(a+b)Specifičnost TN=d/(c+d)Pozitivna napovedna vrednost PNV=a/(a+c)Negativna napovedna vrednost NNV=d/(b+d)
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Understanding sensitivity and specificity with the right
side
of the brain
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Občutljivost
24/30=80%
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Specifičnost
56/70=80%
![Page 21: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/21.jpg)
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Pozitivna napovedna vrednost
24/38=63%
![Page 23: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/23.jpg)
Negativna napovedna vrednost
56/62=90%
![Page 24: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/24.jpg)
Zdravi Bolni
Vrednost diagnostičnega parametraali
Subjektivna ocena verjetja bolezni
Threshold
![Page 25: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/25.jpg)
Zdravi Bolni
Vrednost diagnostičnega parametraali
Subjektivna ocena verjetja bolezni
Threshold
![Page 26: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/26.jpg)
Zdravi Bolni
Vrednost
diagnostičnega
parametraali
Subjektivna
ocena
verjetja
bolezni
Običajneje:
![Page 27: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/27.jpg)
Mejna vrednost
TPF,
sen
sitiv
ityFPF, 1-specificity
less aggressivemindset
Zdravi
Bolni
![Page 28: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/28.jpg)
Mejna vrednost
TPF,
sen
sitiv
ityFPF, 1-specificity
moderatemindset
Zdravi
Bolni
![Page 29: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/29.jpg)
Mejna vrednost
TPF,
sen
sitiv
ityFPF, 1-specificity
more aggressive
mindset
Zdravi
Bolni
![Page 30: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/30.jpg)
Mejna vrednost
Zdravi
Bolni TPF,
sen
sitiv
ityFPF, 1-specificity
Entire ROC curve
![Page 31: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/31.jpg)
TPF,
sen
sitiv
ityFPF, 1-specificity
Entire ROC curve
Reader Skilland/or
Level of Technology
chanc
e line
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ROC
Receiver Operating Characteristic(historično ime iz raziskav z radarjem)
Relative Operating Characteristic
Operating Characteristic
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ROC (Relative
operating
characteristics)
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Nekaj
ROC krivulj
. . .
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Kateri diagnostični test je boljši?
False Positive Fraction= 1.0
Specificity
True
Pos
itive
Fra
ctio
n=
Sens
itivi
ty
1.0
1.00.0
0.0
Modality A
Modality B
![Page 36: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/36.jpg)
False Positive Fraction= 1.0
Specificity
True
Pos
itive
Fra
ctio
n=
Sens
itivi
ty
1.0
1.00.0
0.0
Modality A
Modality B
Za odgovor moramo poznati ROC krivulji(prva
možnost):
B je boljši:
večji
TPF pri istem
FPF, ali
manjši
FPF
priistem
TPF
![Page 37: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/37.jpg)
Druga
možnost: Ista
ROC
False Positive Fraction= 1.0
Specificity
True
Pos
itive
Fra
ctio
n=
Sens
itivi
ty
1.0
1.00.0
0.0
Modality A
Modality B
![Page 38: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/38.jpg)
False Positive Fraction= 1.0
Specificity
True
Pos
itive
Fra
ctio
n=
Sens
itivi
ty
1.0
1.00.0
0.0
Modality A
Modality B
. . . In še tretja:
A
je boljši:
večji
TPF priistem
FPF, ali
manjši
FPF priistem
TPF
![Page 39: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/39.jpg)
ROC
1-specifičnost
obču
tljiv
ost
![Page 41: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/41.jpg)
AUROC
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Youdenov indeks
J=občutljivost+specifičnost-1
False Positive Fraction= 1.0
Specificity
True
Pos
itive
Fra
ctio
n=
Sens
itivi
ty1.0
1.00.0
0.0
A
B
![Page 43: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/43.jpg)
Tiroksin in hipotiroidizem
V preglednici so podani rezultati klinične diagnostike in laboratorijske analize tiroksina pri 125 bolnikih s sumom na hipotiroidizem.
Koncentracija
T4(ng/mL)
Hipotiroidizem
DA NE
≤
5 18 1
5,1 do 7,0 7 17
7,1 do 9,0 4 36
> 9 3 39
Kakšni sta specifičnost in občutljivost laboratorijskega testa za hipotiroidizem pri mejnih koncentracijah T4 5.0, 7.0 in 9.0 ng/mL?
Skiciraj ROC diagram!
![Page 44: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/44.jpg)
ROC
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1-specifičnost
obču
tljiv
ost
![Page 45: Pogojna verjetnost in Bayesov izrek¡f...False Positive Fraction = 1.0 Specificity True Positive Fraction = Sensitivity 1.0 1.0 0.0 0.0 Modality A Modality B Za odgovor moramo poznati](https://reader033.vdocuments.mx/reader033/viewer/2022041705/5e43f906f89cff11b7014fbf/html5/thumbnails/45.jpg)
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Gonick, Smith. Cartoon
guide
to statistics
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Tiroksin in hipotiroidizem
Z uporabo Bayesove formule izračunaj kakšna je pri naključno izbranem pacientu verjetnost hipotiroidizma, če je izid testa pozitiven in kot mejno vrednost izberemo 5.0, 7.0 in 9.0 ng/mL. Pri tem upoštevaj epidemiološke podatke, ki kažejo, da ima hipotiroidizem 0.25% prebivalcev?
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PNV se s povečevanjem prevalence povečuje.
Odvisna od občutljivosti in specifičnosti diagnostičnega testa.
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Bayesovo
adaptivno vodenje
Identifikacija
Opazovanje
Individualni režim odmerjanja
Terapevtskicilj
Predhodno vedenje p(D)
Thomas Bayes
(1702-1761)
Dnevni odmerek varfarina (mg/dan)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Šte
vilo
bol
niko
v
0
10
20
30
40
50
60
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Primer Kri v blatu pri bolnikih z rakom debelega črevesja
+ -
+ 20 180 200
- 10 1820 1830
30 2000 2030
Rezultat endoskopije
Kri v
bla
tu
Občutljivost, specifičnost, PNV, NNV