Korte beschrijving: Dit proefschrift gaat over theoretische en practische aspecten van levensduuranalyse waarbij een gedeelte van de waarnemingen gecensureerd is. Dit soort analyses komen in allerlei situaties voor. In het proefschrift worden echter enkel toepassingen op medisch gebied beschouwd. Een waarneming heet gecensureerd wanneer de werkelijke levensduur (nog) niet kan worden waargenomen, maar men wel heeft kunnen achterhalen dat deze duur groter is dan een bepaalde ondergrens. ...

Zie: Summary

Kardaun, Otto Jan Willem Frederik,

Korte beschrijving: Dit proefschrift gaat over theoretische en practische aspecten van levensduuranalyse waarbij een gedeelte van de waarnemingen gecensureerd is. Dit soort analyses komen in allerlei situaties voor. In het proefschrift worden echter enkel toepassingen op medisch gebied beschouwd. Een waarneming heet gecensureerd wanneer de werkelijke levensduur (nog) niet kan worden waargenomen, maar men wel heeft kunnen achterhalen dat deze duur groter is dan een bepaalde ondergrens. ... Zie: Summar

Kardaun, Otto Jan Willem Frederik

NARCIS 

On statistical Survival Analysis and its Applications in Medicine

University of Groningen

   University of GroningenOn statistical Survival Analysis and its Applications in MedicineKardaun, Otto Jan Willem FrederikIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1986Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Kardaun, O. J. W. F. (1986). On statistical Survival Analysis and its Applications in Medicine. s.n.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019SurnmaryI r r  th is  thcs is  sonre  theore t icz r l  anc l  p rac t ica l  aspec ts  o f  surv iva l - t i tne  ar ta -l1 's is  n ' i th  censo led  da ta  a re  cor ts ic le red .  Surv iva l - t i r t te  ana lys is  does  ap-pcar  in  var ious  contex ts .  App l ica t ions  in  the  thes is ,  ho lve l 'e r ,  a re  re -s t r i c tcd  to  the  med ica l  f i c ld .  An observa t ion  is  ca l led  censored i f  oneknorvs that thc true survival  t ime, r ,vhich cantrot (yet)  be observcd, islarger than a certain lor,ver bound. This happens for example in -su1f i1,3l-t i r r re  ana lys is  o l  pa t ien ts :  I f  a  pa t ien t  i s  s t i l l  a l i ve  a t  t l re  c los ing  t la te  o fthe  s tud ; . ,  then t l rc  t rue  surv i r ' ; r l  t i rnc  (mea, . r r r r :c l  ,  fo r  exa t r rp lc ' .  f r 'o t t t  hedate of diagnosis onlvards) is not observable, l . rut  one krtou's that thist i rnc is lnrger than the t i rne betr 'veen <l iagnosis and the closing dzrte ofthc  s tu i l y .  Orn i t t ing  the  censorcd  observa t ions  f rom thc  ana lys is  i s  no t  tobe adviscd, as this wi l l  sclect inc' l iv iduals rvi th a relat ively short  l i fet imefrom thc avai lable sample.Aftcr the Second World War, var ious stat ist ical  techniques have beendeve loped fo r  cor rcc t l y  incorpora t ing  censored obscrva t io r rs  in  t l re  usua ls ta t i s t i ca l  s i tL ra t , ions .  The cx is t ing  l i t c ra tu rc  i s  norv  ex tens ive .  Bcnch-mark  ar t i c les  z r rc  I (A I ,LAN & \ l l l tnn  (1958)  on  the  proc luc t - l im i t  cs t ina-to r ,  a  d is t r ibu t io r r - f ree  s t i rna tor  fo r  the  surv iva l  f t rnc t ion  S( l )  :  P{ ' I '  >l )  ( rv l i i ch  c lenotcs  the  probab i l i t y  t l r : r t  the  t rL re  - "u lv iva l  t i rne  i - .  la lgcrthan t ) ,  and Cox (1972) ,  u 'h ich  in l roduccs  t l re  p ropor t io r ra I  haz ; r rc1smodc lS( t  I  e1 . .  .  . ,  z k )  -  , c (1  10 , , 0 ) " " P ( l l t z t *  l 0 r z r )to express the i rr f lucnce of r isk factors (also cal led covarial- i les) orr  thesurv iva l  t in re .  Here ,  z r , .  .  . )  zk  a re  the  scores  o f  the  co" 'a r iab les ,  and{3r , . . . ,B6  regress ion  coef f i c ien ts ,  wh ich  are  to  be  es t imated  (as  is  a lsot h e  b a s e - l i n e  s u r v i v a l  f u n c t i o n  S ( ,  1 0 , . . . , 0 ) ) .In the f i rst  part  of  the thesis (chapters I  and I I) ,  survival- t ime analysisis used for the i rr terpretat ion and evaluat ion of t rvo medical  data sets.In  the  Ias t  par t  (chapter  I I I ) ,  sonre  theore t ica l  aspec ts  a re  cons idered,rvhich are rnot ivated by the previous chapters.The f i rst  c lrapter is concerned with the (remaining) survival  t ime(measured frorn diagnosis onwards) of male larynx-cancer pat ients.  ' Ihestage of the cancer (1 to 4) and the age of the pat ient are considcredas covariables. The data are strat i f ied with respect to stage, rvhi le forage Cox 's  p ropor t iona l  hazards  mode i  i s  used.  For  a  pa t ien t  rv i th  s tagex ii ( i : 1 , . . . , 4 )  o f  t h e  c a n c e r  a n d  o f  a g e  z ,  a  p r o g n o s i s  ( w h i c h  i n c l u d e s: ta t i s t i ca l  uncer ta in t i cs )  i s  made o f  t l re  remain ing  l i fe t ime.  FurLhermore ,a conrparison is made with the l i fct ime of the normal populat ion, takinginto account the effects of age. Final ly,  the adequacy of the proport ionalhazards assumption to model the inf luence of age is invest igated.In chapter I I  a more compl icated case-study is considered. The sur-vival  t ime (measured frorn the date of diagnosis unt i l  terminal renalfai lure) of pat ients with a certain kidney disease (IgA nephropathy)is invest igated. Several  correlated r isk factors, such as the glomerularf i l t ra t ion  ra te  o f  thc  k idneys ,  h1 'per tens ion ,  p ro te inur ia  and hematur ia(macroscop ic  as  lve i l  as  mic roscop ic )  seem to  in f luence the  surv iva l  t ime.Bes ides ,  there  are  k idney  b iopsy  var iab les  (w i th  some miss ing  obscrva-t ions) ,  rvh ich  are  poss ib ly  re la tec l  to  the  ser iousness  o f  the  d isease.  Thetype of hypertension treatrnent changed rather drast ical ly dur ing t l ieperiod of invest igat ion. The problem of select ing a sui table set of  covari-ables is addressed expl ic i t ly.  A pract ical  result  of  this chapter consists inthe construcbion of two nomogralns) from which one can reacl,  for an ar-b i t ra ry  pa t ien t  ( rv i th  r i sk - fac to rscores  z r t . . . , z r ) ,  the  es t in ra ted  surv iva lf u n c t i o n  S 1 t 1 r r , . . . , z r )  a n < 1  ( f o r  a  f i x e d  t i m e  t )  t h e  i n v o l v e d  s t a t i s t i c a luncer ta in ty .  N{orcovcr ,  a  res idua l  ana lys is  i s  deve loped as  a  g loba l  moc le lcheck of the proport ional hazarcls assumption. ' Ih is,  hol l 'e l 'cr ,  lcads toeas i l y  in te rprc tab le  resu l ts  on ly  in  the  s imp le  casc  tha t  there  are  l lo  cen-sored obscrvat ions. A general  interact ive cornputer prograrn has beendeveloped to support  the analysis of this chapter.In chapter I I I  t rvo theoret ical  developments, motivated by pract ice,are  d iscussed.  These are  (1 )  the  cons t ruc t ion  o f  s imu l taneous conf i -d e n c e  b a n d s  f o r  t h e  s u r v i v a l  f u n c t i o n  S ( t l z ) :  , 9 ( l  1 " t , . . . , 2 p )  a n d  ( 2 )the i rnplementat ion of the results of mult ivar iable survival  analysis,  asdescribcd in chapter I I ,  in decision analysis.  This implementat ion isperforrncd from the medical as lvel l  as from the mathcmatical  s ide.A (trvo-sided) simultaneous conf idence band is a general isat ion of aconf idence interval  for a f ixed t ime t .  I t  consists of a lorver boundaryS( t  l , z )  and an  upper  boundary  S1t111 such tha t ,  fo r  a  g iven in te rva lI t , , ;n , t ^o , l  and a  g iven a  €  [0 ,1 ] ,  Lhe probab i l i t y  tha t  the  unknorvn  sur -v i ia l  f r rnc t ion  i ies  be tween S( t l z )  and S( t  l z )  fo r  every  I  € l t ,n , , , , t ,no , fis equal to 1 -  a .  The calculat ion of the rvidth of a band rvi th a real ist icshape presents a non-tr iv ial  mathematical  problem, rvhich lvas solvedxiiby using an integral  equat ion approach. Such a conf idence band can beu s e d  ( a m o n g  o t h e r s )  t o  t e s t  t h e  n u l l  h y p o t h e s i s  1 y ' 6 :  S ( t l z ) :  ^ 9 0 ( l ) ,where 5s( l)  is a refercnce survival  funct ion ( for example of the normalpopu la t ion  a t  a  cer ta in  age) .In the decision analysis sect ion of chapter I I I ,  the fol lowing problemis considered. Suppose one is interested in choosing betrveen trvo treat-ments ,  rvh ich  we sha l l  ca l l  A  and B.  (A  may denote  ' to  t rea t '  and  B 'no tto treat '  according to some method.) We further assume that data areavai lable frorn pat ients wiro have been treated according to A and from(o ther )  pa t ien ts  who have been t rea ted  accord ing  to  B .  The ques t ion  iswhether  fo r  a  ne lv  pa t ien t  r ,v i th  covar iab les  z t t .  . .  )  zk )  on  the  bas is  o f  theava i lab le  da ta ,  (1 )  t rea tmcnt  A  has  to  be  recornmended,  (2 )  t rea tmcntB has  to  be  recommenc led ,  o r  (3 )  no  recomnendat ion  shou ld  be  made.Our analysis of this problem is based on the est imated survival  funct ionsof the'effect ive l i fet ime'under each treatment.  ' Ihe effect ive l i fet ime isdetermined by subject ive rat ios, establ ished in a discussion with the pa-t ient,  of  the qual i t ies of l i fe under var ious condit ions. Sometimes, thedif ference between two 'ef fect ive'survival  funct ions is wel l  expressed by aone-dimensional parameter (such as the rat io of the mean, or rnedian, ef-fect ive survival  t imes),  the est imator of which is asymptot ical ly normal lydistr ibuted. For such a si tuat ion, the ( trvo-sided) Neyman-Pearson test-ing theory is replaced by a loss-funct ion formulat ion. This formulat ionIeads to the same part i t ion of the outcome space IR as does some two-sided test.  I t  adequately ref lects,  however,  the three-decision characterof the above mentioncd problem, and al lows a motivated choice of thelevel a of the corresponding (two-sided) test.  In a tradi t ional approach,th is  leve l  i s  o f ten  se t  to  5% (or  1%) .

University of Groningen Digital Archive

ARTS repository - University of Groningen

   University of GroningenOn statistical Survival Analysis and its Applications in MedicineKardaun, Otto Jan Willem FrederikIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1986Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Kardaun, O. J. W. F. (1986). On statistical Survival Analysis and its Applications in Medicine. s.n.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 20-06-2022SurnmaryI r r  th is  thcs is  sonre  theore t icz r l  anc l  p rac t ica l  aspec ts  o f  surv iva l - t i tne  ar ta -l1 's is  n ' i th  censo led  da ta  a re  cor ts ic le red .  Surv iva l - t i r t te  ana lys is  does  ap-pcar  in  var ious  contex ts .  App l ica t ions  in  the  thes is ,  ho lve l 'e r ,  a re  re -s t r i c tcd  to  the  med ica l  f i c ld .  An observa t ion  is  ca l led  censored i f  oneknorvs that thc true survival  t ime, r ,vhich cantrot (yet)  be observcd, islarger than a certain lor,ver bound. This happens for example in -su1f i1,3l-t i r r re  ana lys is  o l  pa t ien ts :  I f  a  pa t ien t  i s  s t i l l  a l i ve  a t  t l re  c los ing  t la te  o fthe  s tud ; . ,  then t l rc  t rue  surv i r ' ; r l  t i rnc  (mea, . r r r r :c l  ,  fo r  exa t r rp lc ' .  f r 'o t t t  hedate of diagnosis onlvards) is not observable, l . rut  one krtou's that thist i rnc is lnrger than the t i rne betr 'veen <l iagnosis and the closing dzrte ofthc  s tu i l y .  Orn i t t ing  the  censorcd  observa t ions  f rom thc  ana lys is  i s  no t  tobe adviscd, as this wi l l  sclect inc' l iv iduals rvi th a relat ively short  l i fet imefrom thc avai lable sample.Aftcr the Second World War, var ious stat ist ical  techniques have beendeve loped fo r  cor rcc t l y  incorpora t ing  censored obscrva t io r rs  in  t l re  usua ls ta t i s t i ca l  s i tL ra t , ions .  The cx is t ing  l i t c ra tu rc  i s  norv  ex tens ive .  Bcnch-mark  ar t i c les  z r rc  I (A I ,LAN & \ l l l tnn  (1958)  on  the  proc luc t - l im i t  cs t ina-to r ,  a  d is t r ibu t io r r - f ree  s t i rna tor  fo r  the  surv iva l  f t rnc t ion  S( l )  :  P{ ' I '  >l )  ( rv l i i ch  c lenotcs  the  probab i l i t y  t l r : r t  the  t rL re  - "u lv iva l  t i rne  i - .  la lgcrthan t ) ,  and Cox (1972) ,  u 'h ich  in l roduccs  t l re  p ropor t io r ra I  haz ; r rc1smodc lS( t  I  e1 . .  .  . ,  z k )  -  , c (1  10 , , 0 ) " " P ( l l t z t *  l 0 r z r )to express the i rr f lucnce of r isk factors (also cal led covarial- i les) orr  thesurv iva l  t in re .  Here ,  z r , .  .  . )  zk  a re  the  scores  o f  the  co" 'a r iab les ,  and{3r , . . . ,B6  regress ion  coef f i c ien ts ,  wh ich  are  to  be  es t imated  (as  is  a lsot h e  b a s e - l i n e  s u r v i v a l  f u n c t i o n  S ( ,  1 0 , . . . , 0 ) ) .In the f i rst  part  of  the thesis (chapters I  and I I) ,  survival- t ime analysisis used for the i rr terpretat ion and evaluat ion of t rvo medical  data sets.In  the  Ias t  par t  (chapter  I I I ) ,  sonre  theore t ica l  aspec ts  a re  cons idered,rvhich are rnot ivated by the previous chapters.The f i rst  c lrapter is concerned with the (remaining) survival  t ime(measured frorn diagnosis onwards) of male larynx-cancer pat ients.  ' Ihestage of the cancer (1 to 4) and the age of the pat ient are considcredas covariables. The data are strat i f ied with respect to stage, rvhi le forage Cox 's  p ropor t iona l  hazards  mode i  i s  used.  For  a  pa t ien t  rv i th  s tagex ii ( i : 1 , . . . , 4 )  o f  t h e  c a n c e r  a n d  o f  a g e  z ,  a  p r o g n o s i s  ( w h i c h  i n c l u d e s: ta t i s t i ca l  uncer ta in t i cs )  i s  made o f  t l re  remain ing  l i fe t ime.  FurLhermore ,a conrparison is made with the l i fct ime of the normal populat ion, takinginto account the effects of age. Final ly,  the adequacy of the proport ionalhazards assumption to model the inf luence of age is invest igated.In chapter I I  a more compl icated case-study is considered. The sur-vival  t ime (measured frorn the date of diagnosis unt i l  terminal renalfai lure) of pat ients with a certain kidney disease (IgA nephropathy)is invest igated. Several  correlated r isk factors, such as the glomerularf i l t ra t ion  ra te  o f  thc  k idneys ,  h1 'per tens ion ,  p ro te inur ia  and hematur ia(macroscop ic  as  lve i l  as  mic roscop ic )  seem to  in f luence the  surv iva l  t ime.Bes ides ,  there  are  k idney  b iopsy  var iab les  (w i th  some miss ing  obscrva-t ions) ,  rvh ich  are  poss ib ly  re la tec l  to  the  ser iousness  o f  the  d isease.  Thetype of hypertension treatrnent changed rather drast ical ly dur ing t l ieperiod of invest igat ion. The problem of select ing a sui table set of  covari-ables is addressed expl ic i t ly.  A pract ical  result  of  this chapter consists inthe construcbion of two nomogralns) from which one can reacl,  for an ar-b i t ra ry  pa t ien t  ( rv i th  r i sk - fac to rscores  z r t . . . , z r ) ,  the  es t in ra ted  surv iva lf u n c t i o n  S 1 t 1 r r , . . . , z r )  a n < 1  ( f o r  a  f i x e d  t i m e  t )  t h e  i n v o l v e d  s t a t i s t i c a luncer ta in ty .  N{orcovcr ,  a  res idua l  ana lys is  i s  deve loped as  a  g loba l  moc le lcheck of the proport ional hazarcls assumption. ' Ih is,  hol l 'e l 'cr ,  lcads toeas i l y  in te rprc tab le  resu l ts  on ly  in  the  s imp le  casc  tha t  there  are  l lo  cen-sored obscrvat ions. A general  interact ive cornputer prograrn has beendeveloped to support  the analysis of this chapter.In chapter I I I  t rvo theoret ical  developments, motivated by pract ice,are  d iscussed.  These are  (1 )  the  cons t ruc t ion  o f  s imu l taneous conf i -d e n c e  b a n d s  f o r  t h e  s u r v i v a l  f u n c t i o n  S ( t l z ) :  , 9 ( l  1 " t , . . . , 2 p )  a n d  ( 2 )the i rnplementat ion of the results of mult ivar iable survival  analysis,  asdescribcd in chapter I I ,  in decision analysis.  This implementat ion isperforrncd from the medical as lvel l  as from the mathcmatical  s ide.A (trvo-sided) simultaneous conf idence band is a general isat ion of aconf idence interval  for a f ixed t ime t .  I t  consists of a lorver boundaryS( t  l , z )  and an  upper  boundary  S1t111 such tha t ,  fo r  a  g iven in te rva lI t , , ;n , t ^o , l  and a  g iven a  €  [0 ,1 ] ,  Lhe probab i l i t y  tha t  the  unknorvn  sur -v i ia l  f r rnc t ion  i ies  be tween S( t l z )  and S( t  l z )  fo r  every  I  € l t ,n , , , , t ,no , fis equal to 1 -  a .  The calculat ion of the rvidth of a band rvi th a real ist icshape presents a non-tr iv ial  mathematical  problem, rvhich lvas solvedxiiby using an integral  equat ion approach. Such a conf idence band can beu s e d  ( a m o n g  o t h e r s )  t o  t e s t  t h e  n u l l  h y p o t h e s i s  1 y ' 6 :  S ( t l z ) :  ^ 9 0 ( l ) ,where 5s( l)  is a refercnce survival  funct ion ( for example of the normalpopu la t ion  a t  a  cer ta in  age) .In the decision analysis sect ion of chapter I I I ,  the fol lowing problemis considered. Suppose one is interested in choosing betrveen trvo treat-ments ,  rvh ich  we sha l l  ca l l  A  and B.  (A  may denote  ' to  t rea t '  and  B 'no tto treat '  according to some method.) We further assume that data areavai lable frorn pat ients wiro have been treated according to A and from(o ther )  pa t ien ts  who have been t rea ted  accord ing  to  B .  The ques t ion  iswhether  fo r  a  ne lv  pa t ien t  r ,v i th  covar iab les  z t t .  . .  )  zk )  on  the  bas is  o f  theava i lab le  da ta ,  (1 )  t rea tmcnt  A  has  to  be  recornmended,  (2 )  t rea tmcntB has  to  be  recommenc led ,  o r  (3 )  no  recomnendat ion  shou ld  be  made.Our analysis of this problem is based on the est imated survival  funct ionsof the'effect ive l i fet ime'under each treatment.  ' Ihe effect ive l i fet ime isdetermined by subject ive rat ios, establ ished in a discussion with the pa-t ient,  of  the qual i t ies of l i fe under var ious condit ions. Sometimes, thedif ference between two 'ef fect ive'survival  funct ions is wel l  expressed by aone-dimensional parameter (such as the rat io of the mean, or rnedian, ef-fect ive survival  t imes),  the est imator of which is asymptot ical ly normal lydistr ibuted. For such a si tuat ion, the ( trvo-sided) Neyman-Pearson test-ing theory is replaced by a loss-funct ion formulat ion. This formulat ionIeads to the same part i t ion of the outcome space IR as does some two-sided test.  I t  adequately ref lects,  however,  the three-decision characterof the above mentioncd problem, and al lows a motivated choice of thelevel a of the corresponding (two-sided) test.  In a tradi t ional approach,th is  leve l  i s  o f ten  se t  to  5% (or  1%) .

English

   University of GroningenOn statistical Survival Analysis and its Applications in MedicineKardaun, Otto Jan Willem FrederikIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1986Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Kardaun, O. J. W. F. (1986). On statistical Survival Analysis and its Applications in Medicine. s.n.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 16-07-2023SurnmaryI r r  th is  thcs is  sonre  theore t icz r l  anc l  p rac t ica l  aspec ts  o f  surv iva l - t i tne  ar ta -l1 's is  n ' i th  censo led  da ta  a re  cor ts ic le red .  Surv iva l - t i r t te  ana lys is  does  ap-pcar  in  var ious  contex ts .  App l ica t ions  in  the  thes is ,  ho lve l 'e r ,  a re  re -s t r i c tcd  to  the  med ica l  f i c ld .  An observa t ion  is  ca l led  censored i f  oneknorvs that thc true survival  t ime, r ,vhich cantrot (yet)  be observcd, islarger than a certain lor,ver bound. This happens for example in -su1f i1,3l-t i r r re  ana lys is  o l  pa t ien ts :  I f  a  pa t ien t  i s  s t i l l  a l i ve  a t  t l re  c los ing  t la te  o fthe  s tud ; . ,  then t l rc  t rue  surv i r ' ; r l  t i rnc  (mea, . r r r r :c l  ,  fo r  exa t r rp lc ' .  f r 'o t t t  hedate of diagnosis onlvards) is not observable, l . rut  one krtou's that thist i rnc is lnrger than the t i rne betr 'veen <l iagnosis and the closing dzrte ofthc  s tu i l y .  Orn i t t ing  the  censorcd  observa t ions  f rom thc  ana lys is  i s  no t  tobe adviscd, as this wi l l  sclect inc' l iv iduals rvi th a relat ively short  l i fet imefrom thc avai lable sample.Aftcr the Second World War, var ious stat ist ical  techniques have beendeve loped fo r  cor rcc t l y  incorpora t ing  censored obscrva t io r rs  in  t l re  usua ls ta t i s t i ca l  s i tL ra t , ions .  The cx is t ing  l i t c ra tu rc  i s  norv  ex tens ive .  Bcnch-mark  ar t i c les  z r rc  I (A I ,LAN & \ l l l tnn  (1958)  on  the  proc luc t - l im i t  cs t ina-to r ,  a  d is t r ibu t io r r - f ree  s t i rna tor  fo r  the  surv iva l  f t rnc t ion  S( l )  :  P{ ' I '  >l )  ( rv l i i ch  c lenotcs  the  probab i l i t y  t l r : r t  the  t rL re  - "u lv iva l  t i rne  i - .  la lgcrthan t ) ,  and Cox (1972) ,  u 'h ich  in l roduccs  t l re  p ropor t io r ra I  haz ; r rc1smodc lS( t  I  e1 . .  .  . ,  z k )  -  , c (1  10 , , 0 ) " " P ( l l t z t *  l 0 r z r )to express the i rr f lucnce of r isk factors (also cal led covarial- i les) orr  thesurv iva l  t in re .  Here ,  z r , .  .  . )  zk  a re  the  scores  o f  the  co" 'a r iab les ,  and{3r , . . . ,B6  regress ion  coef f i c ien ts ,  wh ich  are  to  be  es t imated  (as  is  a lsot h e  b a s e - l i n e  s u r v i v a l  f u n c t i o n  S ( ,  1 0 , . . . , 0 ) ) .In the f i rst  part  of  the thesis (chapters I  and I I) ,  survival- t ime analysisis used for the i rr terpretat ion and evaluat ion of t rvo medical  data sets.In  the  Ias t  par t  (chapter  I I I ) ,  sonre  theore t ica l  aspec ts  a re  cons idered,rvhich are rnot ivated by the previous chapters.The f i rst  c lrapter is concerned with the (remaining) survival  t ime(measured frorn diagnosis onwards) of male larynx-cancer pat ients.  ' Ihestage of the cancer (1 to 4) and the age of the pat ient are considcredas covariables. The data are strat i f ied with respect to stage, rvhi le forage Cox 's  p ropor t iona l  hazards  mode i  i s  used.  For  a  pa t ien t  rv i th  s tagex ii ( i : 1 , . . . , 4 )  o f  t h e  c a n c e r  a n d  o f  a g e  z ,  a  p r o g n o s i s  ( w h i c h  i n c l u d e s: ta t i s t i ca l  uncer ta in t i cs )  i s  made o f  t l re  remain ing  l i fe t ime.  FurLhermore ,a conrparison is made with the l i fct ime of the normal populat ion, takinginto account the effects of age. Final ly,  the adequacy of the proport ionalhazards assumption to model the inf luence of age is invest igated.In chapter I I  a more compl icated case-study is considered. The sur-vival  t ime (measured frorn the date of diagnosis unt i l  terminal renalfai lure) of pat ients with a certain kidney disease (IgA nephropathy)is invest igated. Several  correlated r isk factors, such as the glomerularf i l t ra t ion  ra te  o f  thc  k idneys ,  h1 'per tens ion ,  p ro te inur ia  and hematur ia(macroscop ic  as  lve i l  as  mic roscop ic )  seem to  in f luence the  surv iva l  t ime.Bes ides ,  there  are  k idney  b iopsy  var iab les  (w i th  some miss ing  obscrva-t ions) ,  rvh ich  are  poss ib ly  re la tec l  to  the  ser iousness  o f  the  d isease.  Thetype of hypertension treatrnent changed rather drast ical ly dur ing t l ieperiod of invest igat ion. The problem of select ing a sui table set of  covari-ables is addressed expl ic i t ly.  A pract ical  result  of  this chapter consists inthe construcbion of two nomogralns) from which one can reacl,  for an ar-b i t ra ry  pa t ien t  ( rv i th  r i sk - fac to rscores  z r t . . . , z r ) ,  the  es t in ra ted  surv iva lf u n c t i o n  S 1 t 1 r r , . . . , z r )  a n < 1  ( f o r  a  f i x e d  t i m e  t )  t h e  i n v o l v e d  s t a t i s t i c a luncer ta in ty .  N{orcovcr ,  a  res idua l  ana lys is  i s  deve loped as  a  g loba l  moc le lcheck of the proport ional hazarcls assumption. ' Ih is,  hol l 'e l 'cr ,  lcads toeas i l y  in te rprc tab le  resu l ts  on ly  in  the  s imp le  casc  tha t  there  are  l lo  cen-sored obscrvat ions. A general  interact ive cornputer prograrn has beendeveloped to support  the analysis of this chapter.In chapter I I I  t rvo theoret ical  developments, motivated by pract ice,are  d iscussed.  These are  (1 )  the  cons t ruc t ion  o f  s imu l taneous conf i -d e n c e  b a n d s  f o r  t h e  s u r v i v a l  f u n c t i o n  S ( t l z ) :  , 9 ( l  1 " t , . . . , 2 p )  a n d  ( 2 )the i rnplementat ion of the results of mult ivar iable survival  analysis,  asdescribcd in chapter I I ,  in decision analysis.  This implementat ion isperforrncd from the medical as lvel l  as from the mathcmatical  s ide.A (trvo-sided) simultaneous conf idence band is a general isat ion of aconf idence interval  for a f ixed t ime t .  I t  consists of a lorver boundaryS( t  l , z )  and an  upper  boundary  S1t111 such tha t ,  fo r  a  g iven in te rva lI t , , ;n , t ^o , l  and a  g iven a  €  [0 ,1 ] ,  Lhe probab i l i t y  tha t  the  unknorvn  sur -v i ia l  f r rnc t ion  i ies  be tween S( t l z )  and S( t  l z )  fo r  every  I  € l t ,n , , , , t ,no , fis equal to 1 -  a .  The calculat ion of the rvidth of a band rvi th a real ist icshape presents a non-tr iv ial  mathematical  problem, rvhich lvas solvedxiiby using an integral  equat ion approach. Such a conf idence band can beu s e d  ( a m o n g  o t h e r s )  t o  t e s t  t h e  n u l l  h y p o t h e s i s  1 y ' 6 :  S ( t l z ) :  ^ 9 0 ( l ) ,where 5s( l)  is a refercnce survival  funct ion ( for example of the normalpopu la t ion  a t  a  cer ta in  age) .In the decision analysis sect ion of chapter I I I ,  the fol lowing problemis considered. Suppose one is interested in choosing betrveen trvo treat-ments ,  rvh ich  we sha l l  ca l l  A  and B.  (A  may denote  ' to  t rea t '  and  B 'no tto treat '  according to some method.) We further assume that data areavai lable frorn pat ients wiro have been treated according to A and from(o ther )  pa t ien ts  who have been t rea ted  accord ing  to  B .  The ques t ion  iswhether  fo r  a  ne lv  pa t ien t  r ,v i th  covar iab les  z t t .  . .  )  zk )  on  the  bas is  o f  theava i lab le  da ta ,  (1 )  t rea tmcnt  A  has  to  be  recornmended,  (2 )  t rea tmcntB has  to  be  recommenc led ,  o r  (3 )  no  recomnendat ion  shou ld  be  made.Our analysis of this problem is based on the est imated survival  funct ionsof the'effect ive l i fet ime'under each treatment.  ' Ihe effect ive l i fet ime isdetermined by subject ive rat ios, establ ished in a discussion with the pa-t ient,  of  the qual i t ies of l i fe under var ious condit ions. Sometimes, thedif ference between two 'ef fect ive'survival  funct ions is wel l  expressed by aone-dimensional parameter (such as the rat io of the mean, or rnedian, ef-fect ive survival  t imes),  the est imator of which is asymptot ical ly normal lydistr ibuted. For such a si tuat ion, the ( trvo-sided) Neyman-Pearson test-ing theory is replaced by a loss-funct ion formulat ion. This formulat ionIeads to the same part i t ion of the outcome space IR as does some two-sided test.  I t  adequately ref lects,  however,  the three-decision characterof the above mentioncd problem, and al lows a motivated choice of thelevel a of the corresponding (two-sided) test.  In a tradi t ional approach,th is  leve l  i s  o f ten  se t  to  5% (or  1%) .

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On statistical Survival Analysis and its Applications in Medicine

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