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# Theory to consider an inaccurate testing and how to determine the prior probability

I presented a mathematical theory on a medical testing method. This fundamental theory can be taken account of both cases when the resource of the testing is limited or not. One implication is that "negative proof" may not function well, and another implication is that excessively high specificity and accuracy are required for meaningful diagnosis unless the careful usage of the diagnosis is considered.

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### Theory to consider an inaccurate testing and how to determine the prior probability

1. 1. 2020 5 30 ( )
2. 2. : https://www.amazon.co.jp/gp/product/B07QYZ3CXH/ 2500
3. 3. : https://www.springernature.com/gp/librarians/news- events/all-news-articles/industry-news-initiatives/free- access-to-textbooks-for-institutions-affected-by- coronaviru/17855960
4. 4. :http://www.chugaiigaku.jp/upfile/browse/browse2906.pdf https://en.wikipedia.org/wiki/Pre-_and_post- test_probability Mikael Häggström
5. 5. Prob( hᵢ | e ) ∝ Prob( hᵢ ) Prob( e | hᵢ ) (∝ ) Prob( hᵢ | e ) = Prob( hᵢ ) Prob( e | hᵢ ) / Prob ( e ) P( e ∩ hᵢ ) / P ( e ) { P ( hᵢ ) P ( e ∩ hᵢ ) } / { P ( hᵢ ) P ( e ) } P ( hᵢ )
6. 6. (1702-1761) Prob( hᵢ | e ) ∝ Prob( hᵢ ) Prob( e | hᵢ ) • • ( 0 ) • • 100% 100% • • • Wikipedia
7. 7. 1. (hypothesis) : h₀ h₁ 2. (evidence) Prob ( e | h₀ ) : Prob ( e | h₁ ) • : (100% - ) • (100% - ) : 3. : 99% 70% • ( ) 99% : 30% = 3.3 : 1 ( ) 1% : 70% = 1 : 70 • : 3.3 70 • 100% ( ) a : b 3.3 a : b a : 70 b
8. 8. • • • 3 • : • : • , ROC • R ROCR
9. 9. 1. 1/5000 5000 2. 1, 2, 5 10 3. 1, 2, 5, 10, 20, 50, 100, 20, 500, 1000, 2000, 5000 (1, 1/2, 1/5, 1/10, ..) 4. 5. 70 ( ) 1/3.33 ( ) 6. 70 5% 79% Y = L X / ( L X + ( 1 – X ) )
10. 10. Y = L X / ( L X + ( 1 – X ) ) 0 ≦ X ≦ 1 , 0 ≦ Y ≦ 1 L=M 10 ⁻ᴱ ; E ∈ { } (1) : M=1: -3 ≦ E ≦ 3 (2) : M=2,5: -4 ≦ E ≦ 3 (3) : M=3,4,6,7,8,9: -3≦E≦ 2 (4) : M=1.2, 1.4, 1.6, 1.8, 2.5, 3.5, 4.5 : -2≦ E ≦ 2 L X Y
11. 11. p log ( p / (1-p) ) [0, 1] ( 0% 100%) [-∞ , +∞] e ˣ / ( 1 + e ˣ )
12. 12. • p (p ) log(p/(1-p)) • • 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, .. 尤度比ごとのベイズ更新の様子 (黒太線は尤度比が1000のべき乗) 事前確率 (ロジット表示) 事後確率 0% 25% 50% 75% 100% 1/100万 1/10万 0.01% 0.1% 1% 10% 50% 90% 99% 99.9%
13. 13. • 70 1/3.3 • or
14. 14. • ( ) : 1. ( ; ) 2. ( ; ) 3. (DAG ) • 3 • : • 3 • ( ) • ? (1)
15. 15. ? (3) 1. Prior Probability = ∨ (1 – e ⁻ ᴱˣᵖᵒˢᵘʳᵉ ᴺᵘᵐᵇᵉʳ , “Caused Propagation”) 2. Exposure Number = Σ { ( ) A} + Σ { B} + C • • • ( ) • ( ) , , , ; ( ) , .. 3. Caused Propagation = ∨ ( “symptoms”, “infections to others”) 4. Prior Probability 1 ( : / / 2 )
16. 16. 1. : (100% - ) 2. ( ) 3. 30% 99% i.e. 3.3 4. ↓ 5. 1: 75% 5:1( 80% ) 0.5 (75%→37.5% ) 6. 2: 99% 99% 1 99.99% i.e. Y = 1 / ( X + ( 1 – X ) / L )
17. 17. PCR : 1. • 95% 3 99.9875% ( = 1 - 0.05³) ∵( ) • 70% 20% 3 48.8% ( = 1 - 0. 8³ ) 2. : • 100% ( ) • ( : ) • PCR 3. : • PCR 1 • RNA (10⁻⁴ / )
18. 18. : • • 1 • • • • ( ) ( ) ( log₁₀it) •
19. 19. R (1) library(matlab) par(family= "HiraKakuProN-W3",mai=c(1,1.2,1.2,1)) plot(NA,NA,yaxt="n",xaxt="n", xaxs="i", yaxs="i", xlab=" ",ylab=" ",cex=2,xlim=0:1,ylim=0:1,cex.lab=1.4,main=" ¥n( 10 )",cex.main=1.6) points(meshgrid(0:100/100,0:100/100),pch=3,cex=0.1,col="gray80") points(meshgrid(0:20/20,0:20/20),pch=3,cex=0.4,col="gray50") points(meshgrid(0:4/4,0:4/4),pch=3,cex=2.0) axis(1,0:4/4,c("0%","25%","50%","75%","100%"),las=1,cex=3) axis(2,0:4/4,c("0%","25%","50%","75%","100%"),las=1,cex=3) x=0:200/200 ; for(a in c(2,5) %x%10^(-4:3)) { y = x*a/(x*a+(1-x)) ; points(x,y,type="l") } for(a in 10^(-3:3)) { y = x*a/(x*a+(1-x)) ; points(x,y,type="l",lwd=2) } for(a in c(1/3.3, 70) ) {y = x*a/(x*a+(1-x)) ; points(x,y,type="l",col=rgb(0,0,1,0.4),lwd=3)} 5*70/(5*70+95)*100 # 78.651
20. 20. R (2) library(matlab) par(family= "HiraKakuProN-W3",mai=c(0.2,0.2,0.2,0.2)) plot(NA,NA,yaxt="n",xaxt="n", xaxs="i", yaxs="i", xlab="",ylab="",cex=2,xlim=0:1,ylim=0:1,,main="") points(meshgrid(0:100/100,0:100/100),pch=3,cex=0.1,col="gray80") points(meshgrid(0:20/20,0:20/20),pch=3,cex=0.4,col="gray50") points(meshgrid(0:4/4,0:4/4),pch=3,cex=2.0) x=0:400/400 ; for(a in 10^(-3:3)) { y = x*a/(x*a+(1-x)) ; points(x,y,type="l",lwd=2.5) } for(a in c(2,5)%x%10^(-4:3)) { y = x*a/(x*a+(1-x)) ; points(x,y,type="l",lwd=1.6) } for(a in c(3,4,6,7,8,9) %x%10^(-3:2)) { y = x*a/(x*a+(1-x)) ; points(x,y,type="l",lwd=0.8,col=rgb(0,0,0,0.8)) } for(a in c(6:9/5,2.5,3.5,4.5) %x%10^(-2:2)) { y = x*a/(x*a+(1-x)) ; points(x,y,type="l",lwd=0.5,col=rgb(0,0,0,0.6)) }
21. 21. R (3) library(matlab) LG <- function(x) log (x/(1-x)) iLG <- function(x) exp(x)/(1+exp(x)) par(family= "HiraKakuProN-W3",mai=c(1,1.2,1.2,1)) plot(NA,NA,yaxt="n",xaxt="n", xaxs="i", yaxs="i", xlab=" ( )",ylab=" ",cex=2,xlim=LG(c(1e-6,1-1e-3)),ylim=0:1,cex.lab=1.4,main=" ¥n( 1000 )",cex.main=1.6) points(meshgrid(LG(c(10^(-6:-1)%x%c(1:9),1-10^(-3:-2)%x%c(1:9))),0:20/20),pch=3,cex=0.4,col="gray50") points(meshgrid(LG(c(10^(-6:-1),1/2,1-10^(-3:-1))),0:4/4),pch=3,cex=2.0) axis(2,0:4/4,c("0%","25%","50%","75%","100%"),las=1,cex=3) X<-c(10^(-6:-1),0.5,1-10^(-1:-3)); axis(1,LG(X),c('1/100 ','1/10 ’, '0.01%','0.1%','1%','10%','50%','90%','99%','99.9%'),las=1,cex=3) x= iLG(-300:200/20) ; for(a in c(2,5)%x%10^(-5:8)) { y = x*a/(x*a+(1-x)) ; points(LG(x),y,type="l") } for(a in 10^(-5:8)) { y = x*a/(x*a+(1-x)) ; points(LG(x),y,type="l",lwd=3.2) } for(a in 10^c(-9,-6,-3,0,3,6) ) {y = x*a/(x*a+(1-x)) ; points(LG(x),y,type="l",col=rgb(0,0,0,0.4),lwd=7)} for(a in c(1/(3.3^(1:8) )) ) {y = x*a/(x*a+(1-x)) ; points(LG(x),y,type="l",col=rgb(0,0,1,0.5),lwd=3)} for(a in c(70^(1:4)) ) {y = x*a/(x*a+(1-x)) ; points(LG(x),y,type="l",col=rgb(1,.5,0,1),lwd=3)}
22. 22. R (4) par(family= "HiraKakuProN-W3",mai=c(1,1.2,1.2,1)) plot(NA,NA,yaxt="n",xaxt="n", xaxs="i", yaxs="i", xlab=" ",ylab=" ",cex=2,xlim=0:1,ylim=c(0,3),cex.lab=1.4,main=" ?¥n( 70% 99% )",cex.main=1.6) points(meshgrid(0:20/20,0:30/10),pch=3,cex=0.4,col="gray50") points(meshgrid(0:4/4,0:6/2),pch=3,cex=2.0) axis(1,0:4/4,c("0%","25%","50%","75%","100%"),las=1,cex=3) axis(2,0:3,c('0 ','1 ','2 ','3 '),las=1,cex=3) x=0:200/200 ; for(a in c(2,5)%x%10^(-4:3)) { y = a/(x*a+(1-x)) ; points(x,y,type="l") } for(a in 10^(-3:3)) { y = a/(x*a+(1-x)) ; points(x,y,type="l",lwd=2.5, col="gray30") } for(a in c(1/3.3) ) {y = a/(x*a+(1-x)) ; points(x,y,type="l",col=c(rgb(0,0,1,0.8)),lwd=3)} for(a in c(70) ) {y = a/(x*a+(1-x)) ; points(x,y,type="l",col=c(rgb(1,.5,0,1)),lwd=3)}