A decomposition of Fisher's information to inform sample size for developing fair and precise clinical prediction models -- part 1: binary outcomes

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'!<2(,!'%%9( '@,!($4F!A(,)$%K23!<2),+!T&,3!@&,( ,!'(,!Q, T!,P,3 @4I!ZJHI!U["_]^_"M&^`\`I! ikI ! j '3!G29T,+$3B, 3!H1I!0&($3F'B,!'3)!A,3'+$a ,)!+$F ,+$&22)!'4!<,@&2)4!@2!$.!H( I.!8,,!;8.!J'(F!*ZI!J9+KP'($'5+,!A( 2B324K%!<2),+4M!$449,4!$3!),P ,+2A$3B! <2),+4.!,P'+9' K3B!'449H I!W&,!:3'+-4$4!2Q!Z$3'(-!*'@ '!X4,%23)!,)$K23YI!823)23M!1&'A<'3!'3)! G'++]!"k`kI! _\I ! o'B,+F ,(F ,!oI!:!32@,!23!'!B,3,( '+!),V3$K23!2Q!@&,!%2,g%$,3 @!2Q!),@,(<$3' K23I! Z$2<,@($F 'I!"kk"]p`M\k"R UI ! ! ^_ ! _pI ! ;'A+'3!H I!*,%$4$23!W&,2(-!'3)!@&,!E'%•$3)$3B!?( 2%,44I!0@'3Q2( )!8'T!#,P$,T I! "k\`]U[X\YM"[\_ R kUI ! ! ! ! ! ! ^\ ! SUPPLEMENT AR Y MA TERIAL S 1: Further detail s of exis 3ng sample siz e calcula3on Criterion (i): sample siz e to tar get a pr ecise es4mate of the ov e r all outcome risk ?(,%$4$23!$3!@&,!2P ,( '++!($4F!$4!@&,!V(4 @!4@ '5$+$@-!+,P,+!),V3,)!5 -!#$+,-!'3)!12++$34I 20 !=Q!@&,! 4'C ^_ % g # z { $> ' z { "7~ ! !!!!! >dI ! X "[ Y! 6 ,!B,3,('++-!( ,%2<<,3)!'$<$3B!Q2(! % x 7lC lr .!'3)!@&94!%23V),3%,!$3 @,(P'+!T$)@&! x 7lC> ! 59@!'!4<'++,(!<'(B$3!2Q!,(( 2(!<' -!5,!4,34$5+,!Q2(!+2T!X2(!&$B&Y!, P,3@!($4F 4I!! ! Criterion (ii): sample siz e to tar get small o verfi;ng of pr edictor eff ects 0&($3F'B ,!X'+42!F32T3!'4!A,3'+$4'K23!2(!( ,B9+'($4'K23Y!<,@&2)4!'$l• Y.!'3)!@&,!'3K%$A'@,)!<2),+! A,(Q 2(<'3%,!'4!),V3,)!5-!@&,!12 D R 03,++!# R4d9'(,)!4@ 'K4K%! $! !" # " I 15, 50, 55 ! W&,!<$3$<9"7 RY f> ' ! !" # € g ƒ ! !!!!! >dI ! X "" Y! ! W&,!39<5,(!2Q!A'('<, @,( 4!X • Y!4&2 9+).!'@!+,'4@. !%2((,4A23)!@ 2!'!%2( ,!4,@!2Q!A(,)$%@ 2(4!F32 T3! @2!5,!$r ` j„& $ ! !" # " 0 …† :c: 7‡b; $ ! !" # " . > ' Dz > $ > ' z " +/> E # "" .!49%&!@&'@!"_n! 2Q!@&,!@2@ '+!P'($'3%,!$4!, DA+'$3,)I 15 !W&,!P'+9,!2Q! ! !" # !%'3!'+42!5,!),($P,)!Q( 2dI ! X "U Y! ! 6 ,!49B B,4@! % !$4!'!4<'++!P'+9,.!49%&!'4! x ![I[_I!W&,!25@'$3,)!P '+9,!2Q! € !%'3!@&,3!5,!A+'%,)! $3@2!@&,!A( ,P$294!,d9'K23!Q(2'5'9$%98%<7"67".%9.%$96%6 9%."?2"56%#% ;'9:5=%'$%#$%'$)'-')2#/%67#6%!#=%9.%!#=%$96%7#-"%:.956#6"%>#$>". % ! The individual’s chosen utilities also define their risk threshold at which they would be willing to choose a biopsy. It corresponds to where the expected uti lity from biopsy ( N ,H &!'()* . ) exceeds the expected utility from n o biopsy (e.g., N ,H +'&!'()* . ): " N ,H &!'()* . O N ,H +' , &!'()* . This can be expressed in terms of the individual’s probability of prostate cancer ( - ! ) weighted by the utility values: , - ! P H0 ! . 5 Q , 0 1 - ! . P HG ! R O , - ! P HL ! . 5 Q , 0 1 - ! . P HM ! R Rearranging identifies the ris k threshold at which the indi vidual prefers a biopsy. 57 ! ^k ! - -./01.234 ! " O / 0 5 H0 ! 1 HL ! HM ! 1 HG ! 2 5" For example, let us return to the individual who expresses their utility of each outcome state as H0 ! 3 0II J H G ! 3 KJ HL ! 3 I" and HM ! 3 0I . Then: - -./01.234 ! " O / 0 5 H0 ! 1 HL ! HM ! 1 HG ! 2 5" 3 / 0 5 0II 1 I 0I 1 K 2 5" 3 I: IMS If a model estimates this individual’s risk of prostate cance r to be - T ! 3 I: IK0 , then this suggests their preference is a biopsy, as their point estimate of risk exceeds their personal risk threshold, and so their expected utility of a biopsy is larger than their expected utility of no biopsy: N ,H &!'()* . 3 , - T ! P H0 ! . 5 Q , 0 1 - T ! . P HG ! R 3 , I: IK0 P 0II . 5 Q , 0 1 I: IK0 . P KR 3 U: SK N QH +' , &!'()* R 3 , - T ! P HL ! . 5 Q , 0 1 - T ! . P HM ! R 3 ,I: IK0 P I. 5 ,,0 1 I: IK0 . P 0I . 3 U: MU ! > P,3!@&29B&!@&$4!49BB,4@4!@&,!% 2((,%@!),%$4$23!$4!@ 2!5$2A4- .!@&,( ,!<'-!4 K++!5,!93%,(@'$3@-! '529@!@&$4!),%$4$23!),A,3)$3B!23!@&,!A(,%$4$23!2Q! W X , I!=3),,).!93),(4 @'3)$3B!@&,!A( ,)$%K23! 93%,(@'$3@- !<'-!B$P ,!)2%@2( 4.!&,'+@&!A(2Q ,44$23'+4!'3)!(,B9+'@ 2(4!'449( '3%,!@2!94,!2(! ,3)2(4,!@&,!<2),+!$3!@&,!V( 4@!A+'%,.!2(!$),3KQ-!T&,3!Q9(@&,(!(,4,'( %&!$4!3,,),)I!j$%F ,(4!,@! '+I!49AA2(@!@&$4!'(B9<,3@. 33 !32K3B!@&'@!u ),%$4$23!'3'+-4$4!@,++4!94!T&$%&!),%$4$23!@ 2!<'F ,!Q2(! 32T .!59@!T ,!<'-!'+42!T '3@!@2!F32 T!&2T!<9%&!%23V),3%,!T ,!4&29+)!&'P ,!$3!@&'@!),%$4$23I! =Q!T,!'( ,!$349g%$,3@+-!%23V),3 @!@&'@!T,!'( ,!($B&@.!Q9(@&,(!(,4,'( %&!$4!T'(( '3@,)I v!=3!@&$4! %23@ ,D@.!@&,!'$ '%&!Q 2(,4 @!T'4!'++2T,)!"[[! @(,,4!T$@&!@( ,,!),A@&!2Q!^.!'3)!T,!'AA+$,)!,'%&!2Q!@&,!"[[[!<2),+4!@ 2!,'%&!A'(K%$A'3@!$3! @&,!Q9++!4-3 @&,K%!)'@ '4,@!@2!25@ '$3!"[[[!,4K<'@,)!($4F 4!Q2(!, P,(-!$3)$P$)9'+I!W&,!UI_n!'3)! kpI_n!A,(%,3 K+,!P'+9,4!T ,(,!@&,3!94,)!@2!),($P ,!'!k_n!93%,(@'$3@-!$3@ ,(P'+!Q 2(!,'%&! $3)$P$)9'+c 4!($4F.!'4!4&2T3!$3! 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")'>G9$5%8.9!%67"%4DDD%!9)"/5I%09.%>9!:#.'59$(%:#$"/%&)+% 579<5%67"%2$>".6 #'$6=%'$6 ".-#/5%)".'-")%8.9!%92.%5#!:/"%5'P"%>#/>2/#G9$%;#5") % 9$%67"%5#!"%2$'6%'$89.!#G9$%#5% 67"%QR")%!9)" /5(%#$)%6725%1'-"%')"$G>#/%."52/65%6 9%:#$"/%&;+I % ) ) ) ) ) ) ) ) ) ) E..G V'6%,+3.'+B).'+ %,O3;-)I ",)+9%)>.5'%B)% S3#$;%)I ", )O3,."0-)#"5%;;.'()3$$, "369%-) 0 .2 .4 .6 .8 1 95% uncertainty interval for true risk 0 .2 .4 .6 .8 1 true risk 0 .2 .4 .6 .8 1 95% uncertainty interval for true risk 0 .2 .4 .6 .8 1 true risk 0 .2 .4 .6 .8 1 95% uncertainty interval for true risk 0 .2 .4 .6 .8 1 true risk 0 .2 .4 .6 .8 1 95% uncertainty interval for true risk 0 .2 .4 .6 .8 1 true risk ! ii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uncertainty interval for true risk 0 .2 .4 .6 .8 1 true risk 0 .2 .4 .6 .8 1 95% uncertainty interval for true risk 0 .2 .4 .6 .8 1 true risk 0 .2 .4 .6 .8 1 95% uncertainty interval for true risk 0 .2 .4 .6 .8 1 true risk 0 .2 .4 .6 .8 1 95% uncertainty interval for true risk 0 .2 .4 .6 .8 1 true risk