t rs r Prts - - PowerPoint PPT Presentation

t r s r pr t s r
SMART_READER_LITE
LIVE PREVIEW

t rs r Prts - - PowerPoint PPT Presentation

r s t r t t s t t s t s t t s


slide-1
SLIDE 1

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❋❛❝✉❧t② ♦❢ ❍❡❛❧t❤ ❙❝✐❡♥❝❡s

❘✷✲t②♣❡ ❈✉r✈❡s ❢♦r ❉②♥❛♠✐❝ Pr❡❞✐❝t✐♦♥s ❢r♦♠ ❏♦✐♥t ▲♦♥❣✐t✉❞✐♥❛❧✲❙✉r✈✐✈❛❧ ▼♦❞❡❧s

■♥❢❡r❡♥❝❡ ✫ ❛♣♣❧✐❝❛t✐♦♥ t♦ ♣r❡❞✐❝t✐♦♥ ♦❢ ❦✐❞♥❡② ❣r❛❢t ❢❛✐❧✉r❡

P❛✉❧ ❇❧❛♥❝❤❡

❥♦✐♥t ✇♦r❦ ✇✐t❤

▼✲❈✳ ❋♦✉r♥✐❡r ✫ ❊✳ ❉❛♥t❛♥ ✭◆❛♥t❡s✱ ❋r❛♥❝❡✮

❏✉❧② ✷✵✶✺ ❙❧✐❞❡ ✶✴✷✾

slide-2
SLIDE 2

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♥t❡①t ✫ ▼♦t✐✈❛t✐♦♥

  • ▼❡❞✐❝❛❧ r❡s❡❛r❝❤❡rs ❤♦♣❡ t♦ ✐♠♣r♦✈❡ ♣❛t✐❡♥t ♠❛♥❛❣❡♠❡♥t

✉s✐♥❣ ❡❛r❧✐❡r ❞✐❛❣♥♦s❡s

  • ❙t❛t✐st✐❝✐❛♥s ❝❛♥ ❤❡❧♣ ❜② ✜tt✐♥❣ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧s
  • ❚❤❡ ♠❛❦✐♥❣ ♦❢ s♦✲❝❛❧❧❡❞ ✧❞②♥❛♠✐❝✧ ♣r❡❞✐❝t✐♦♥s ❤❛s r❡❝❡♥t❧②

r❡❝❡✐✈❡❞ ❛ ❧♦t ♦❢ ❛tt❡♥t✐♦♥

  • ■♥ ♦r❞❡r t♦ ❜❡ ✉s❡❢✉❧ ❢♦r ♠❡❞✐❝❛❧ ♣r❛❝t✐❝❡✱ ♣r❡❞✐❝t✐♦♥s s❤♦✉❧❞

❜❡ ✧❛❝❝✉r❛t❡✧ ❍♦✇ s❤♦✉❧❞ ✇❡ ❡✈❛❧✉❛t❡ ❞②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥ ❛❝❝✉r❛❝②❄

❙❧✐❞❡ ✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-3
SLIDE 3

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉❛t❛ ✫ ❈❧✐♥✐❝❛❧ ❣♦❛❧

◮ ❉❛t❛✿ ❉■❱❆❚ ❝♦❤♦rt ❞❛t❛ ♦❢ ❦✐❞♥❡② tr❛♥s♣❧❛♥t r❡❝✐♣✐❡♥ts

✭s✉❜s❛♠♣❧❡✱ n = ✹, ✶✶✾✮

◮ ❈❧✐♥✐❝❛❧ ❣♦❛❧✿

  • ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥ ♦❢ r✐s❦ ♦❢ ❦✐❞♥❡②

❣r❛❢t ❢❛✐❧✉r❡ ✭❞❡❛t❤ ♦r r❡t✉r♥ t♦ ❞✐❛❧②s✐s✮

  • ❯s✐♥❣ r❡♣❡❛t❡❞ ♠❡❛s✉r❡♠❡♥ts ♦❢

s❡r✉♠ ❝r❡❛t✐♥✐♥❡

❙❧✐❞❡ ✷✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-4
SLIDE 4

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉■❱❆❚ ❞❛t❛ s❛♠♣❧❡ ✭♥❂✹✱✶✶✾✮

  • ❋r❡♥❝❤ ❝♦❤♦rt
  • ❆❞✉❧t r❡❝✐♣✐❡♥ts
  • ❚r❛♥s♣❧❛♥t❡❞ ❛❢t❡r ✷✵✵✵
  • ❈r❡❛t✐♥✐♥❡ ♠❡❛s✉r❡❞ ❡✈❡r② ②❡❛r

✻ ❝❡♥t❡rs ✭✇✇✇✳❞✐✈❛t✳❢r✮

❙❧✐❞❡ ✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-5
SLIDE 5

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉■❱❆❚ ❞❛t❛ s❛♠♣❧❡ ✭♥❂✹✱✶✶✾✮

  • ❋r❡♥❝❤ ❝♦❤♦rt
  • ❆❞✉❧t r❡❝✐♣✐❡♥ts
  • ❚r❛♥s♣❧❛♥t❡❞ ❛❢t❡r ✷✵✵✵
  • ❈r❡❛t✐♥✐♥❡ ♠❡❛s✉r❡❞ ❡✈❡r② ②❡❛r
  • ✻ ❝❡♥t❡rs

✭✇✇✇✳❞✐✈❛t✳❢r✮

❙❧✐❞❡ ✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-6
SLIDE 6

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙t❛t✐st✐❝❛❧ ❝❤❛❧❧❡♥❣❡s ❞✐s❝✉ss❡❞

❍♦✇ t♦ ❡✈❛❧✉❛t❡ ❛♥❞✴♦r ❝♦♠♣❛r❡ ❞②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s❄ ◮ ❯s✐♥❣ ❝♦♥❝❡♣ts ♦❢✿

  • ❉✐s❝r✐♠✐♥❛t✐♦♥
  • ❈❛❧✐❜r❛t✐♦♥

◮ ❆❝❝♦✉♥t✐♥❣ ❢♦r✿

  • ❉②♥❛♠✐❝ s❡tt✐♥❣
  • ❈❡♥s♦r✐♥❣ ✐ss✉❡

❙❧✐❞❡ ✹✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-7
SLIDE 7

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❇❛s✐❝ ✐❞❡❛ ✫ ❝♦♥❝❡♣ts ❢♦r ❡✈❛❧✉❛t✐♥❣ ♣r❡❞✐❝t✐♦♥s

❇❛s✐❝ ✐❞❡❛✿ ❝♦♠♣❛r✐♥❣ ♣r❡❞✐❝t✐♦♥s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s ❈♦♥❝❡♣ts✿

❉✐s❝r✐♠✐♥❛t✐♦♥✿ ❆ ♠♦❞❡❧ ❤❛s ❤✐❣❤ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♣♦✇❡r ✐❢ t❤❡ r❛♥❣❡ ♦❢ ♣r❡❞✐❝t❡❞ r✐s❦s ✐s ✇✐❞❡ ❛♥❞ s✉❜❥❡❝ts ✇✐t❤ ❧♦✇ ✭❤✐❣❤✮ ♣r❡❞✐❝t❡❞ r✐s❦ ❛r❡ ♠♦r❡ ✭❧❡ss✮ ❧✐❦❡❧② t♦ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t✳ ❈❛❧✐❜r❛t✐♦♥✿ ❆ ♠♦❞❡❧ ✐s ❝❛❧✐❜r❛t❡❞ ✐❢ ✇❡ ❝❛♥ ❡①♣❡❝t t❤❛t ① s✉❜❥❡❝ts ♦✉t ♦❢ ✶✵✵ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t ❛♠♦♥❣ ❛❧❧ s✉❜❥❡❝ts t❤❛t r❡❝❡✐✈❡ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ①✪ ✭✏✇❡❛❦✑ ❞❡✜♥✐t✐♦♥✮✳

❙❧✐❞❡ ✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-8
SLIDE 8

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❇❛s✐❝ ✐❞❡❛ ✫ ❝♦♥❝❡♣ts ❢♦r ❡✈❛❧✉❛t✐♥❣ ♣r❡❞✐❝t✐♦♥s

❇❛s✐❝ ✐❞❡❛✿ ❝♦♠♣❛r✐♥❣ ♣r❡❞✐❝t✐♦♥s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s (simple!) ❈♦♥❝❡♣ts✿

❉✐s❝r✐♠✐♥❛t✐♦♥✿ ❆ ♠♦❞❡❧ ❤❛s ❤✐❣❤ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♣♦✇❡r ✐❢ t❤❡ r❛♥❣❡ ♦❢ ♣r❡❞✐❝t❡❞ r✐s❦s ✐s ✇✐❞❡ ❛♥❞ s✉❜❥❡❝ts ✇✐t❤ ❧♦✇ ✭❤✐❣❤✮ ♣r❡❞✐❝t❡❞ r✐s❦ ❛r❡ ♠♦r❡ ✭❧❡ss✮ ❧✐❦❡❧② t♦ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t✳ ❈❛❧✐❜r❛t✐♦♥✿ ❆ ♠♦❞❡❧ ✐s ❝❛❧✐❜r❛t❡❞ ✐❢ ✇❡ ❝❛♥ ❡①♣❡❝t t❤❛t ① s✉❜❥❡❝ts ♦✉t ♦❢ ✶✵✵ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t ❛♠♦♥❣ ❛❧❧ s✉❜❥❡❝ts t❤❛t r❡❝❡✐✈❡ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ①✪ ✭✏✇❡❛❦✑ ❞❡✜♥✐t✐♦♥✮✳

❙❧✐❞❡ ✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-9
SLIDE 9

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❇❛s✐❝ ✐❞❡❛ ✫ ❝♦♥❝❡♣ts ❢♦r ❡✈❛❧✉❛t✐♥❣ ♣r❡❞✐❝t✐♦♥s

❇❛s✐❝ ✐❞❡❛✿ ❝♦♠♣❛r✐♥❣ ♣r❡❞✐❝t✐♦♥s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s (simple!) ❈♦♥❝❡♣ts✿

◮ ❉✐s❝r✐♠✐♥❛t✐♦♥✿ ❆ ♠♦❞❡❧ ❤❛s ❤✐❣❤ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♣♦✇❡r ✐❢ t❤❡ r❛♥❣❡ ♦❢ ♣r❡❞✐❝t❡❞ r✐s❦s ✐s ✇✐❞❡ ❛♥❞ s✉❜❥❡❝ts ✇✐t❤ ❧♦✇ ✭❤✐❣❤✮ ♣r❡❞✐❝t❡❞ r✐s❦ ❛r❡ ♠♦r❡ ✭❧❡ss✮ ❧✐❦❡❧② t♦ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t✳ ◮ ❈❛❧✐❜r❛t✐♦♥✿ ❆ ♠♦❞❡❧ ✐s ❝❛❧✐❜r❛t❡❞ ✐❢ ✇❡ ❝❛♥ ❡①♣❡❝t t❤❛t ① s✉❜❥❡❝ts ♦✉t ♦❢ ✶✵✵ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t ❛♠♦♥❣ ❛❧❧ s✉❜❥❡❝ts t❤❛t r❡❝❡✐✈❡ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ①✪ ✭✏✇❡❛❦✑ ❞❡✜♥✐t✐♦♥✮✳

❙❧✐❞❡ ✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-10
SLIDE 10

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❇❛s✐❝ ✐❞❡❛ ✫ ❝♦♥❝❡♣ts ❢♦r ❡✈❛❧✉❛t✐♥❣ ♣r❡❞✐❝t✐♦♥s

❇❛s✐❝ ✐❞❡❛✿ ❝♦♠♣❛r✐♥❣ ♣r❡❞✐❝t✐♦♥s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s (simple!) ❈♦♥❝❡♣ts✿

◮ ❉✐s❝r✐♠✐♥❛t✐♦♥✿ ❆ ♠♦❞❡❧ ❤❛s ❤✐❣❤ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♣♦✇❡r ✐❢ t❤❡ r❛♥❣❡ ♦❢ ♣r❡❞✐❝t❡❞ r✐s❦s ✐s ✇✐❞❡ ❛♥❞ s✉❜❥❡❝ts ✇✐t❤ ❧♦✇ ✭❤✐❣❤✮ ♣r❡❞✐❝t❡❞ r✐s❦ ❛r❡ ♠♦r❡ ✭❧❡ss✮ ❧✐❦❡❧② t♦ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t✳ ◮ ❈❛❧✐❜r❛t✐♦♥✿ ❆ ♠♦❞❡❧ ✐s ❝❛❧✐❜r❛t❡❞ ✐❢ ✇❡ ❝❛♥ ❡①♣❡❝t t❤❛t ① s✉❜❥❡❝ts ♦✉t ♦❢ ✶✵✵ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t ❛♠♦♥❣ ❛❧❧ s✉❜❥❡❝ts t❤❛t r❡❝❡✐✈❡ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ①✪ ✭✏✇❡❛❦✑ ❞❡✜♥✐t✐♦♥✮✳

❙❧✐❞❡ ✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-11
SLIDE 11

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❇❛s✐❝ ✐❞❡❛ ✫ ❝♦♥❝❡♣ts ❢♦r ❡✈❛❧✉❛t✐♥❣ ♣r❡❞✐❝t✐♦♥s

❇❛s✐❝ ✐❞❡❛✿ ❝♦♠♣❛r✐♥❣ ♣r❡❞✐❝t✐♦♥s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s (simple!) ❈♦♥❝❡♣ts✿

◮ ❉✐s❝r✐♠✐♥❛t✐♦♥✿ ❆ ♠♦❞❡❧ ❤❛s ❤✐❣❤ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♣♦✇❡r ✐❢ t❤❡ r❛♥❣❡ ♦❢ ♣r❡❞✐❝t❡❞ r✐s❦s ✐s ✇✐❞❡ ❛♥❞ s✉❜❥❡❝ts ✇✐t❤ ❧♦✇ ✭❤✐❣❤✮ ♣r❡❞✐❝t❡❞ r✐s❦ ❛r❡ ♠♦r❡ ✭❧❡ss✮ ❧✐❦❡❧② t♦ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t✳ ◮ ❈❛❧✐❜r❛t✐♦♥✿ ❆ ♠♦❞❡❧ ✐s ❝❛❧✐❜r❛t❡❞ ✐❢ ✇❡ ❝❛♥ ❡①♣❡❝t t❤❛t ① s✉❜❥❡❝ts ♦✉t ♦❢ ✶✵✵ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t ❛♠♦♥❣ ❛❧❧ s✉❜❥❡❝ts t❤❛t r❡❝❡✐✈❡ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ①✪ ✭✏✇❡❛❦✑ ❞❡✜♥✐t✐♦♥✮✳

❙❧✐❞❡ ✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-12
SLIDE 12

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮ ✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-13
SLIDE 13

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮ ✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-14
SLIDE 14

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮ ✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-15
SLIDE 15

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮ ✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-16
SLIDE 16

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

  • s✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮

✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300 s = 4 years s = 4 years Landmark time ''s''

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-17
SLIDE 17

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

  • s✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮

✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300 s = 4 years

  • s = 4 years

Landmark time ''s''

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-18
SLIDE 18

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

  • s✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮
  • t✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300 s = 4 years

  • s = 4 years

s+t = 9 years Landmark time ''s'' Horizon ''t'' Horizon ''t''

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-19
SLIDE 19

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

  • s✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮
  • t✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300 s = 4 years s+t = 9 years

  • 0 %

100 % 1−Pr(Graft Failure in (s,s+t]) Landmark time ''s'' Horizon ''t'' 1 − πi(s, t) = 64%

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-20
SLIDE 20

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❘✐❣❤t ❝❡♥s♦r✐♥❣ ✐ss✉❡

▲❛♥❞♠❛r❦ t✐♠❡ s

✭✇❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡✮

❚✐♠❡ s + t

✭❡♥❞ ♦❢ ♣r❡❞✐❝t✐♦♥ ✇✐♥❞♦✇✮

t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❋♦r s✉❜❥❡❝t ❝❡♥s♦r❡❞ ✇✐t❤✐♥ t❤❡ st❛t✉s Di(s, t) = ✶ ✶{❡✈❡♥t ♦❝❝✉rs ✐♥ (s, s + t]} ✐s ✉♥❦♥♦✇♥✳

❙❧✐❞❡ ✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-21
SLIDE 21

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❘✐❣❤t ❝❡♥s♦r✐♥❣ ✐ss✉❡

▲❛♥❞♠❛r❦ t✐♠❡ s

✭✇❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡✮

❚✐♠❡ s + t

✭❡♥❞ ♦❢ ♣r❡❞✐❝t✐♦♥ ✇✐♥❞♦✇✮

t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❋♦r s✉❜❥❡❝t ❝❡♥s♦r❡❞ ✇✐t❤✐♥ t❤❡ st❛t✉s Di(s, t) = ✶ ✶{❡✈❡♥t ♦❝❝✉rs ✐♥ (s, s + t]} ✐s ✉♥❦♥♦✇♥✳

❙❧✐❞❡ ✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-22
SLIDE 22

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❘✐❣❤t ❝❡♥s♦r✐♥❣ ✐ss✉❡

▲❛♥❞♠❛r❦ t✐♠❡ s

✭✇❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡✮

❚✐♠❡ s + t

✭❡♥❞ ♦❢ ♣r❡❞✐❝t✐♦♥ ✇✐♥❞♦✇✮

t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❋♦r s✉❜❥❡❝t ❝❡♥s♦r❡❞ ✇✐t❤✐♥ t❤❡ st❛t✉s Di(s, t) = ✶ ✶{❡✈❡♥t ♦❝❝✉rs ✐♥ (s, s + t]} ✐s ✉♥❦♥♦✇♥✳

❙❧✐❞❡ ✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-23
SLIDE 23

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❘✐❣❤t ❝❡♥s♦r✐♥❣ ✐ss✉❡

▲❛♥❞♠❛r❦ t✐♠❡ s

✭✇❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡✮

❚✐♠❡ s + t

✭❡♥❞ ♦❢ ♣r❡❞✐❝t✐♦♥ ✇✐♥❞♦✇✮

t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❋♦r s✉❜❥❡❝t i ❝❡♥s♦r❡❞ ✇✐t❤✐♥ (s, s + t] t❤❡ st❛t✉s Di(s, t) = ✶ ✶{❡✈❡♥t ♦❝❝✉rs ✐♥ (s, s + t]} ✐s ✉♥❦♥♦✇♥✳

❙❧✐❞❡ ✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-24
SLIDE 24

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ ✶ ❳

✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮ ✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ ❳ ✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-25
SLIDE 25

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

◮ ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ πi(s, t) ✶ ❳

✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮ ✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ ❳ ✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-26
SLIDE 26

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

◮ ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ πi(s, t) = P

ξ

ξ✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮ ✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ ❳ ✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-27
SLIDE 27

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

◮ ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ πi(s, t) = P

ξ

  • Di(s, t) = ✶

ξ✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮ ✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ ❳ ✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-28
SLIDE 28

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

◮ ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ πi(s, t) = P

ξ

  • Di(s, t) = ✶
  • Ti > s, Yi(s),

ξ✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮

  • Yi(s)✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ s

❳ ✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-29
SLIDE 29

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

◮ ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ πi(s, t) = P

ξ

  • Di(s, t) = ✶
  • Ti > s, Yi(s), ❳i

ξ✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮

  • Yi(s)✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ s
  • ❳i✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-30
SLIDE 30

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

Pr❡❞✐❝t✐✈❡ ❛❝❝✉r❛❝②

❍♦✇ ❝❧♦s❡ ❛r❡ t❤❡ ♣r❡❞✐❝t❡❞ r✐s❦s πi(s, t) t♦ t❤❡ ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦ P

  • ❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪
  • ✐♥❢♦r♠❛t✐♦♥ ❛t s

Pr❡❞✐❝t✐♦♥ ❊rr♦r✿ P❊

t❤❡ ❧♦✇❡r t❤❡ ❜❡tt❡r P❊ ❇✐❛s✷ ✰ ❱❛r✐❛♥❝❡ ❡✈❛❧✉❛t❡s ❜♦t❤ ❈❛❧✐❜r❛t✐♦♥ ❛♥❞ ❉✐s❝r✐♠✐♥❛t✐♦♥ ❞❡♣❡♥❞s ♦♥

❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪ ❛t r✐s❦ ❛t s

♦❢t❡♥ ❝❛❧❧❡❞ ✧❊①♣❡❝t❡❞ ❇r✐❡r ❙❝♦r❡✧

❙❧✐❞❡ ✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-31
SLIDE 31

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

Pr❡❞✐❝t✐✈❡ ❛❝❝✉r❛❝②

❍♦✇ ❝❧♦s❡ ❛r❡ t❤❡ ♣r❡❞✐❝t❡❞ r✐s❦s πi(s, t) t♦ t❤❡ ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦ P

  • ❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪
  • ✐♥❢♦r♠❛t✐♦♥ ❛t s

◮ Pr❡❞✐❝t✐♦♥ ❊rr♦r✿ P❊π(s, t) = E

  • D(s, t) − π(s, t)

  • T > s
  • t❤❡ ❧♦✇❡r t❤❡ ❜❡tt❡r

P❊ ❇✐❛s✷ ✰ ❱❛r✐❛♥❝❡ ❡✈❛❧✉❛t❡s ❜♦t❤ ❈❛❧✐❜r❛t✐♦♥ ❛♥❞ ❉✐s❝r✐♠✐♥❛t✐♦♥ ❞❡♣❡♥❞s ♦♥

❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪ ❛t r✐s❦ ❛t s

♦❢t❡♥ ❝❛❧❧❡❞ ✧❊①♣❡❝t❡❞ ❇r✐❡r ❙❝♦r❡✧

❙❧✐❞❡ ✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-32
SLIDE 32

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

Pr❡❞✐❝t✐✈❡ ❛❝❝✉r❛❝②

❍♦✇ ❝❧♦s❡ ❛r❡ t❤❡ ♣r❡❞✐❝t❡❞ r✐s❦s πi(s, t) t♦ t❤❡ ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦ P

  • ❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪
  • ✐♥❢♦r♠❛t✐♦♥ ❛t s

◮ Pr❡❞✐❝t✐♦♥ ❊rr♦r✿ P❊π(s, t) = E

  • D(s, t) − π(s, t)

  • T > s
  • t❤❡ ❧♦✇❡r t❤❡ ❜❡tt❡r
  • P❊ ≈ ❇✐❛s✷ ✰ ❱❛r✐❛♥❝❡
  • ❡✈❛❧✉❛t❡s ❜♦t❤ ❈❛❧✐❜r❛t✐♦♥ ❛♥❞ ❉✐s❝r✐♠✐♥❛t✐♦♥
  • ❞❡♣❡♥❞s ♦♥ P
  • ❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪
  • ❛t r✐s❦ ❛t s
  • ♦❢t❡♥ ❝❛❧❧❡❞ ✧❊①♣❡❝t❡❞ ❇r✐❡r ❙❝♦r❡✧

❙❧✐❞❡ ✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-33
SLIDE 33

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) =E

  • D(s, t) − π(s, t)

  • T > s
  • E
  • D(s, t)+ E
  • D(s, t)
  • Hπ(s)
  • ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦

−π(s, t)

  • T > s
  • ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t

t✐♠❡ ✳ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s♠❛❧❧❡r ❱❛r ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-34
SLIDE 34

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) =E

  • D(s, t) − E
  • D(s, t)
  • Hπ(s)

✷ E

  • D(s, t) + E
  • D(s, t)
  • Hπ(s)
  • ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦

− π(s, t) ✷

  • T > s
  • ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t

t✐♠❡ ✳ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s♠❛❧❧❡r ❱❛r ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-35
SLIDE 35

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) =E

  • D(s, t) − E
  • D(s, t)
  • Hπ(s)

  • T > s
  • + E
  • E
  • D(s, t)
  • Hπ(s)
  • ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦

− π(s, t) ✷

  • T > s
  • Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t

t✐♠❡ s✳ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s♠❛❧❧❡r ❱❛r ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-36
SLIDE 36

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) = E

  • D(s, t) − E
  • D(s, t)
  • Hπ(s)

  • T > s
  • ■♥s❡♣❛r❛❜✐❧✐t②

+ E

  • E
  • D(s, t)
  • Hπ(s)
  • − π(s, t)

  • T > s
  • ❇✐❛s✴❈❛❧✐❜r❛t✐♦♥

Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t t✐♠❡ s✳ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s♠❛❧❧❡r ❱❛r ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-37
SLIDE 37

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) = E

  • ❱❛r
  • D(s, t)
  • Hπ(s)
  • T > s
  • ❉✐s❝r✐♠✐♥❛t✐♦♥

+ E

  • E
  • D(s, t)
  • Hπ(s)
  • − π(s, t)

  • T > s
  • ❈❛❧✐❜r❛t✐♦♥

Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t t✐♠❡ s✳ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s♠❛❧❧❡r ❱❛r ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-38
SLIDE 38

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) = E

  • ❱❛r
  • D(s, t)
  • Hπ(s)
  • T > s
  • ❉✐s❝r✐♠✐♥❛t✐♦♥

+ E

  • E
  • D(s, t)
  • Hπ(s)
  • − π(s, t)

  • T > s
  • ❈❛❧✐❜r❛t✐♦♥

Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t t✐♠❡ s✳ ◮ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ Hπ(s) t❤❡ s♠❛❧❧❡r ❱❛r

  • D(s, t)
  • Hπ(s)
  • ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-39
SLIDE 39

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) = E

  • ❱❛r
  • D(s, t)
  • Hπ(s)
  • T > s
  • ❉✐s❝r✐♠✐♥❛t✐♦♥

+ E

  • E
  • D(s, t)
  • Hπ(s)
  • − π(s, t)

  • T > s
  • ❈❛❧✐❜r❛t✐♦♥

Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t t✐♠❡ s✳ ◮ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ Hπ(s) t❤❡ s♠❛❧❧❡r ❱❛r

  • D(s, t)
  • Hπ(s)
  • ◮ E
  • D(s, t)
  • Hπ(s)
  • − π(s, t) ≡ ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-40
SLIDE 40

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) = E

  • ❱❛r
  • D(s, t)
  • Hπ(s)
  • T > s
  • ❉♦❡s ◆❖❚ ❞❡♣❡♥❞ ♦♥ π(s,t)

+ E

  • E
  • D(s, t)
  • Hπ(s)
  • − π(s, t)

  • T > s
  • ❉❡♣❡♥❞s ♦♥ π(s,t)

Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t t✐♠❡ s✳ ◮ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ Hπ(s) t❤❡ s♠❛❧❧❡r ❱❛r

  • D(s, t)
  • Hπ(s)
  • ◮ E
  • D(s, t)
  • Hπ(s)
  • − π(s, t) ≡ ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-41
SLIDE 41

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π✲t②♣❡ ❝r✐t❡r✐♦♥ ◮ ❇❡♥❝❤♠❛r❦ PE✵

❚❤❡ ❜❡st ✧♥✉❧❧✧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ✭♠❛r❣✐♥❛❧✮ ♣r❡❞✐❝t❡❞ r✐s❦ S(s + t|s) = E

  • D(s, t)
  • H✵(s)
  • ,

H✵(s) = {T > s} t♦ ❛❧❧ s✉❜❥❡❝ts ❧❡❛❞s t♦

PE✵(s, t) = ❱❛r{D(s, t)|T > s} = S(s + t|s)

  • ✶ − S(s + t|s)
  • .

❙✐♠♣❧❡ ✐❞❡❛

✶ P❊ P❊✵

❙❧✐❞❡ ✶✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-42
SLIDE 42

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π✲t②♣❡ ❝r✐t❡r✐♦♥ ◮ ❇❡♥❝❤♠❛r❦ PE✵

❚❤❡ ❜❡st ✧♥✉❧❧✧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ✭♠❛r❣✐♥❛❧✮ ♣r❡❞✐❝t❡❞ r✐s❦ S(s + t|s) = E

  • D(s, t)
  • H✵(s)
  • ,

H✵(s) = {T > s} t♦ ❛❧❧ s✉❜❥❡❝ts ❧❡❛❞s t♦

PE✵(s, t) = ❱❛r{D(s, t)|T > s} = S(s + t|s)

  • ✶ − S(s + t|s)
  • .

◮ ❙✐♠♣❧❡ ✐❞❡❛ R✷

π(s, t) = ✶ − P❊π(s, t)

P❊✵(s, t)

❙❧✐❞❡ ✶✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-43
SLIDE 43

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❲❤② ❜♦t❤❡r❄

◮ R✷

π(s, t) ❛✐♠s t♦ ❝✐r❝✉♠✈❡♥t t❤❡ ❞✐✣❝✉❧t ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢✿

  • t❤❡ s❝❛❧❡ ♦♥ ✇❤✐❝❤ PE(s, t) ✐s ♠❡❛s✉r❡❞
  • ✐♥t❡r♣r❡t❛t✐♦♥ ❢♦r tr❡♥❞ ♦❢ PE(s, t) ✈s s

❇❡❝❛✉s❡ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ s❝❛❧❡ ♦♥ ✇❤✐❝❤ P❊✭s✱t✮ ✐s ♠❡❛s✉r❡❞ ❝❤❛♥❣❡s ✇✐t❤ s✱ ❛♥ ✐♥❝r❡❛s✐♥❣✴❞❡❝r❡❛s✐♥❣ tr❡♥❞ ❝❛♥ ❜❡ ❞✉❡ t♦ ❝❤❛♥❣❡s ✐♥✿ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥s ❛♥❞✴♦r t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥

❙❧✐❞❡ ✶✷✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-44
SLIDE 44

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❲❤② ❜♦t❤❡r❄

◮ R✷

π(s, t) ❛✐♠s t♦ ❝✐r❝✉♠✈❡♥t t❤❡ ❞✐✣❝✉❧t ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢✿

  • t❤❡ s❝❛❧❡ ♦♥ ✇❤✐❝❤ PE(s, t) ✐s ♠❡❛s✉r❡❞
  • ✐♥t❡r♣r❡t❛t✐♦♥ ❢♦r tr❡♥❞ ♦❢ PE(s, t) ✈s s

◮ ❇❡❝❛✉s❡ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ s❝❛❧❡ ♦♥ ✇❤✐❝❤ P❊✭s✱t✮ ✐s ♠❡❛s✉r❡❞ ❝❤❛♥❣❡s ✇✐t❤ s✱ ❛♥ ✐♥❝r❡❛s✐♥❣✴❞❡❝r❡❛s✐♥❣ tr❡♥❞ ❝❛♥ ❜❡ ❞✉❡ t♦ ❝❤❛♥❣❡s ✐♥✿

  • t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥s

❛♥❞✴♦r • t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥

❙❧✐❞❡ ✶✷✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-45
SLIDE 45

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈❤❛♥❣❡s ✐♥ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥s

✧❊ss❡♥t✐❛❧❧②✱ ❛❧❧ ♠♦❞❡❧s ❛r❡ ✇r♦♥❣✱ ❜✉t s♦♠❡ ❛r❡ ✉s❡❢✉❧✳✧✱ ●✳ ❇♦① ❚❤❡ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧ ❢r♦♠ ✇❤✐❝❤ ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ t❤❡ ♣r❡❞✐❝t✐♦♥s ❝❛♥ ❜❡ ✏♠♦r❡ ✇r♦♥❣✑ ❢♦r s♦♠❡ t❤❛♥ ❢♦r s♦♠❡ ♦t❤❡rs✳ ❈❛❧✐❜r❛t✐♦♥ t❡r♠ ♦❢ ❝❤❛♥❣❡s ✇✐t❤ ❲❡ ❝❛♥ ✇♦r❦ ♦♥ ✐t✦

❙❧✐❞❡ ✶✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-46
SLIDE 46

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈❤❛♥❣❡s ✐♥ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥s

✧❊ss❡♥t✐❛❧❧②✱ ❛❧❧ ♠♦❞❡❧s ❛r❡ ✇r♦♥❣✱ ❜✉t s♦♠❡ ❛r❡ ✉s❡❢✉❧✳✧✱ ●✳ ❇♦① ◮ ❚❤❡ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧ ❢r♦♠ ✇❤✐❝❤ ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ t❤❡ ♣r❡❞✐❝t✐♦♥s ❝❛♥ ❜❡ ✏♠♦r❡ ✇r♦♥❣✑ ❢♦r s♦♠❡ s t❤❛♥ ❢♦r s♦♠❡ ♦t❤❡rs✳

  • ❈❛❧✐❜r❛t✐♦♥ t❡r♠ ♦❢ PE(s, t) ❝❤❛♥❣❡s ✇✐t❤ s
  • ❲❡ ❝❛♥ ✇♦r❦ ♦♥ ✐t✦

❙❧✐❞❡ ✶✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-47
SLIDE 47

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈❤❛♥❣❡s ✐♥ t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥

❆♥ ❡①❛♠♣❧❡✿

  • P❛t✐❡♥ts ✇✐t❤ ❝❛r❞✐♦✈❛s❝✉❧❛r ❤✐st♦r② ✭❈❱✮ ❛❧❧ ❞✐❡ ❡❛r❧②✳
  • ❖♥❧② t❤♦s❡ ✇✐t❤♦✉t ❈❱ r❡♠❛✐♥ ❛t r✐s❦ ❢♦r ❧❛t❡ s✳
  • ❚❤❡♥✿
  • t❤❡ ❡❛r❧✐❡r s t❤❡ ♠♦r❡ ❤♦♠♦❣❡♥❡♦✉s t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥
  • ❈❱ ✐s ✉s❡❢✉❧ ❢♦r ♣r❡❞✐❝t✐♦♥ ❢♦r ❡❛r❧② s ❜✉t ✉s❡❧❡ss ❢♦r ❧❛t❡ s✳

❚❤❡ ❛✈❛✐❧❛❜❧❡ ✐♥❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ ♠♦r❡ ✐♥❢♦r♠❛t✐✈❡ ❢♦r s♦♠❡ s t❤❛♥ ❢♦r s♦♠❡ ♦t❤❡rs✳ ❉✐s❝r✐♠✐♥❛t✐♥❣ t❡r♠ ♦❢ ❝❤❛♥❣❡s ✇✐t❤ ❚❤✐s ✐s ❥✉st ❤♦✇ ✐t ✐s✱ t❤❡r❡ ✐s ♥♦t❤✐♥❣ ✇❡ ❝❛♥ ❞♦✦

✭✇❡ ❝❛♥ ♦♥❧② ✇♦r❦ ✇✐t❤ t❤❡ ❞❛t❛ ✇❡ ❤❛✈❡✮

❙❧✐❞❡ ✶✹✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-48
SLIDE 48

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈❤❛♥❣❡s ✐♥ t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥

❆♥ ❡①❛♠♣❧❡✿

  • P❛t✐❡♥ts ✇✐t❤ ❝❛r❞✐♦✈❛s❝✉❧❛r ❤✐st♦r② ✭❈❱✮ ❛❧❧ ❞✐❡ ❡❛r❧②✳
  • ❖♥❧② t❤♦s❡ ✇✐t❤♦✉t ❈❱ r❡♠❛✐♥ ❛t r✐s❦ ❢♦r ❧❛t❡ s✳
  • ❚❤❡♥✿
  • t❤❡ ❡❛r❧✐❡r s t❤❡ ♠♦r❡ ❤♦♠♦❣❡♥❡♦✉s t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥
  • ❈❱ ✐s ✉s❡❢✉❧ ❢♦r ♣r❡❞✐❝t✐♦♥ ❢♦r ❡❛r❧② s ❜✉t ✉s❡❧❡ss ❢♦r ❧❛t❡ s✳

◮ ❚❤❡ ❛✈❛✐❧❛❜❧❡ ✐♥❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ ♠♦r❡ ✐♥❢♦r♠❛t✐✈❡ ❢♦r s♦♠❡ s t❤❛♥ ❢♦r s♦♠❡ ♦t❤❡rs✳

  • ❉✐s❝r✐♠✐♥❛t✐♥❣ t❡r♠ ♦❢ PE(s, t) ❝❤❛♥❣❡s ✇✐t❤ s
  • ❚❤✐s ✐s ❥✉st ❤♦✇ ✐t ✐s✱ t❤❡r❡ ✐s ♥♦t❤✐♥❣ ✇❡ ❝❛♥ ❞♦✦

✭✇❡ ❝❛♥ ♦♥❧② ✇♦r❦ ✇✐t❤ t❤❡ ❞❛t❛ ✇❡ ❤❛✈❡✮

❙❧✐❞❡ ✶✹✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-49
SLIDE 49

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π(s, t) ✐♥t❡r♣r❡t❛t✐♦♥ ◮ ❆❧✇❛②s tr✉❡✿

▼❡❛s✉r❡ ♦❢ ❤♦✇ t❤❡ ♣r❡❞✐❝t✐♦♥ t♦♦❧ π(s, t) ♣❡r❢♦r♠s ❝♦♠♣❛r❡❞ t♦ t❤❡ ❜❡♥❝❤♠❛r❦ ♥✉❧❧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ♣r❡❞✐❝t❡❞ r✐s❦ t♦ ❛❧❧ s✉❜❥❡❝ts ✭♠❛r❣✐♥❛❧ r✐s❦✮✳

❲❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ❝❛❧✐❜r❛t❡❞ ✭str♦♥❣❧②✮✿

❱❛r ❱❛r ❡①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥ ❈♦rr✷ ❝♦rr❡❧❛t✐♦♥ ✶ ♠❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡ ✵

✭❛❢t❡r ❧✐tt❧❡ ❛❧❣❡❜r❛✮

❙❧✐❞❡ ✶✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-50
SLIDE 50

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π(s, t) ✐♥t❡r♣r❡t❛t✐♦♥ ◮ ❆❧✇❛②s tr✉❡✿

▼❡❛s✉r❡ ♦❢ ❤♦✇ t❤❡ ♣r❡❞✐❝t✐♦♥ t♦♦❧ π(s, t) ♣❡r❢♦r♠s ❝♦♠♣❛r❡❞ t♦ t❤❡ ❜❡♥❝❤♠❛r❦ ♥✉❧❧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ♣r❡❞✐❝t❡❞ r✐s❦ t♦ ❛❧❧ s✉❜❥❡❝ts ✭♠❛r❣✐♥❛❧ r✐s❦✮✳

◮ ❲❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ❝❛❧✐❜r❛t❡❞ ✭str♦♥❣❧②✮✿ R✷

π(s, t) = ❱❛r{π(s, t)|T > s}

❱❛r{D(s, t)|T > s} ❡①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥ ❈♦rr✷ ❝♦rr❡❧❛t✐♦♥ ✶ ♠❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡ ✵

✭❛❢t❡r ❧✐tt❧❡ ❛❧❣❡❜r❛✮

❙❧✐❞❡ ✶✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-51
SLIDE 51

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π(s, t) ✐♥t❡r♣r❡t❛t✐♦♥ ◮ ❆❧✇❛②s tr✉❡✿

▼❡❛s✉r❡ ♦❢ ❤♦✇ t❤❡ ♣r❡❞✐❝t✐♦♥ t♦♦❧ π(s, t) ♣❡r❢♦r♠s ❝♦♠♣❛r❡❞ t♦ t❤❡ ❜❡♥❝❤♠❛r❦ ♥✉❧❧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ♣r❡❞✐❝t❡❞ r✐s❦ t♦ ❛❧❧ s✉❜❥❡❝ts ✭♠❛r❣✐♥❛❧ r✐s❦✮✳

◮ ❲❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ❝❛❧✐❜r❛t❡❞ ✭str♦♥❣❧②✮✿ R✷

π(s, t) = ❱❛r{π(s, t)|T > s}

❱❛r{D(s, t)|T > s} ❡①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥ = ❈♦rr✷ D(s, t), π(s, t)

  • T > s
  • ❝♦rr❡❧❛t✐♦♥

✶ ♠❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡ ✵

✭❛❢t❡r ❧✐tt❧❡ ❛❧❣❡❜r❛✮

❙❧✐❞❡ ✶✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-52
SLIDE 52

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π(s, t) ✐♥t❡r♣r❡t❛t✐♦♥ ◮ ❆❧✇❛②s tr✉❡✿

▼❡❛s✉r❡ ♦❢ ❤♦✇ t❤❡ ♣r❡❞✐❝t✐♦♥ t♦♦❧ π(s, t) ♣❡r❢♦r♠s ❝♦♠♣❛r❡❞ t♦ t❤❡ ❜❡♥❝❤♠❛r❦ ♥✉❧❧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ♣r❡❞✐❝t❡❞ r✐s❦ t♦ ❛❧❧ s✉❜❥❡❝ts ✭♠❛r❣✐♥❛❧ r✐s❦✮✳

◮ ❲❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ❝❛❧✐❜r❛t❡❞ ✭str♦♥❣❧②✮✿ R✷

π(s, t) = ❱❛r{π(s, t)|T > s}

❱❛r{D(s, t)|T > s} ❡①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥ = ❈♦rr✷ D(s, t), π(s, t)

  • T > s
  • ❝♦rr❡❧❛t✐♦♥

= E

  • π(s, t)
  • D(s, t) = ✶, T > s
  • ♠❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡

− E

  • π(s, t)
  • D(s, t) = ✵, T > s
  • .

✭❛❢t❡r ❧✐tt❧❡ ❛❧❣❡❜r❛✮

❙❧✐❞❡ ✶✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-53
SLIDE 53

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❖❜s❡r✈❛t✐♦♥s ✫ ■P❈❲ P❊ ❡st✐♠❛t♦r

◮ ❖❜s❡r✈❛t✐♦♥s ✭✐✳✐✳❞✳✮

Ti, ∆i, πi(·, ·)

  • , i = ✶, . . . , n
  • ✇❤❡r❡

Ti = Ti ∧ Ci, ∆i = ✶ ✶{Ti ≤ Ci}

■♥❞✐❝❛t♦r ♦❢ ✏♦❜s❡r✈❡❞ ❡✈❡♥t ♦❝❝✉rr❡♥❝❡✑ ✐♥ ✿

✶ ✶ ✶ ✶

✿ ❡✈❡♥t ♦❝❝✉rr❡❞

✿ ❡✈❡♥t ❞✐❞ ♥♦t ♦❝❝✉r ♦r ❝❡♥s♦r❡❞ ♦❜s✳

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤t✐♥❣ ✭■P❈❲✮ ❡st✐♠❛t♦r✿

✶ ✷

❛♥❞

❙❧✐❞❡ ✶✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-54
SLIDE 54

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❖❜s❡r✈❛t✐♦♥s ✫ ■P❈❲ P❊ ❡st✐♠❛t♦r

◮ ❖❜s❡r✈❛t✐♦♥s ✭✐✳✐✳❞✳✮

Ti, ∆i, πi(·, ·)

  • , i = ✶, . . . , n
  • ✇❤❡r❡

Ti = Ti ∧ Ci, ∆i = ✶ ✶{Ti ≤ Ci}

◮ ■♥❞✐❝❛t♦r ♦❢ ✏♦❜s❡r✈❡❞ ❡✈❡♥t ♦❝❝✉rr❡♥❝❡✑ ✐♥ (s, s + t]✿

  • Di(s, t) = ✶

✶{s < Ti ≤ s + t, ∆i = ✶} =    ✶

✿ ❡✈❡♥t ♦❝❝✉rr❡❞

✿ ❡✈❡♥t ❞✐❞ ♥♦t ♦❝❝✉r ♦r ❝❡♥s♦r❡❞ ♦❜s✳

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤t✐♥❣ ✭■P❈❲✮ ❡st✐♠❛t♦r✿

✶ ✷

❛♥❞

❙❧✐❞❡ ✶✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-55
SLIDE 55

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❖❜s❡r✈❛t✐♦♥s ✫ ■P❈❲ P❊ ❡st✐♠❛t♦r

◮ ❖❜s❡r✈❛t✐♦♥s ✭✐✳✐✳❞✳✮

Ti, ∆i, πi(·, ·)

  • , i = ✶, . . . , n
  • ✇❤❡r❡

Ti = Ti ∧ Ci, ∆i = ✶ ✶{Ti ≤ Ci}

◮ ■♥❞✐❝❛t♦r ♦❢ ✏♦❜s❡r✈❡❞ ❡✈❡♥t ♦❝❝✉rr❡♥❝❡✑ ✐♥ (s, s + t]✿

  • Di(s, t) = ✶

✶{s < Ti ≤ s + t, ∆i = ✶} =    ✶

✿ ❡✈❡♥t ♦❝❝✉rr❡❞

✿ ❡✈❡♥t ❞✐❞ ♥♦t ♦❝❝✉r ♦r ❝❡♥s♦r❡❞ ♦❜s✳

◮ ■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤t✐♥❣ ✭■P❈❲✮ ❡st✐♠❛t♦r✿

  • PE π(s, t) = ✶

n

n

  • i=✶
  • Di(s, t) − πi(s, t)

✷ ❛♥❞

❙❧✐❞❡ ✶✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-56
SLIDE 56

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❖❜s❡r✈❛t✐♦♥s ✫ ■P❈❲ P❊ ❡st✐♠❛t♦r

◮ ❖❜s❡r✈❛t✐♦♥s ✭✐✳✐✳❞✳✮

Ti, ∆i, πi(·, ·)

  • , i = ✶, . . . , n
  • ✇❤❡r❡

Ti = Ti ∧ Ci, ∆i = ✶ ✶{Ti ≤ Ci}

◮ ■♥❞✐❝❛t♦r ♦❢ ✏♦❜s❡r✈❡❞ ❡✈❡♥t ♦❝❝✉rr❡♥❝❡✑ ✐♥ (s, s + t]✿

  • Di(s, t) = ✶

✶{s < Ti ≤ s + t, ∆i = ✶} =    ✶

✿ ❡✈❡♥t ♦❝❝✉rr❡❞

✿ ❡✈❡♥t ❞✐❞ ♥♦t ♦❝❝✉r ♦r ❝❡♥s♦r❡❞ ♦❜s✳

◮ ■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤t✐♥❣ ✭■P❈❲✮ ❡st✐♠❛t♦r✿

  • PE π(s, t) = ✶

n

n

  • i=✶
  • Wi(s, t)
  • Di(s, t) − πi(s, t)

✷ ❛♥❞

❙❧✐❞❡ ✶✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-57
SLIDE 57

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❖❜s❡r✈❛t✐♦♥s ✫ ■P❈❲ P❊ ❡st✐♠❛t♦r

◮ ❖❜s❡r✈❛t✐♦♥s ✭✐✳✐✳❞✳✮

Ti, ∆i, πi(·, ·)

  • , i = ✶, . . . , n
  • ✇❤❡r❡

Ti = Ti ∧ Ci, ∆i = ✶ ✶{Ti ≤ Ci}

◮ ■♥❞✐❝❛t♦r ♦❢ ✏♦❜s❡r✈❡❞ ❡✈❡♥t ♦❝❝✉rr❡♥❝❡✑ ✐♥ (s, s + t]✿

  • Di(s, t) = ✶

✶{s < Ti ≤ s + t, ∆i = ✶} =    ✶

✿ ❡✈❡♥t ♦❝❝✉rr❡❞

✿ ❡✈❡♥t ❞✐❞ ♥♦t ♦❝❝✉r ♦r ❝❡♥s♦r❡❞ ♦❜s✳

◮ ■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤t✐♥❣ ✭■P❈❲✮ ❡st✐♠❛t♦r✿

  • PE π(s, t) = ✶

n

n

  • i=✶
  • Wi(s, t)
  • Di(s, t) − πi(s, t)

✷ ❛♥❞

  • R✷

π(s, t) = ✶ −

  • PE π(s, t)
  • PE ✵(s, t)

❙❧✐❞❡ ✶✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-58
SLIDE 58

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤ts

  • Wi(s, t) =

✶ ✶ + ✶ ✶ + ✵

✇✐t❤ G(u|s) t❤❡ ❑❛♣❧❛♥✲▼❡✐❡r ❡st✐♠❛t♦r ♦❢ P(C > u|C > s)✳ ▲❛♥❞♠❛r❦ t✐♠❡ s ❚✐♠❡ s + t t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❙❧✐❞❡ ✶✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-59
SLIDE 59

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤ts

  • Wi(s, t) = ✶

✶{s < Ti ≤ s + t}∆i

  • G(

Ti|s) + ✶ ✶ + ✵

✇✐t❤ G(u|s) t❤❡ ❑❛♣❧❛♥✲▼❡✐❡r ❡st✐♠❛t♦r ♦❢ P(C > u|C > s)✳ ▲❛♥❞♠❛r❦ t✐♠❡ s ❚✐♠❡ s + t t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❙❧✐❞❡ ✶✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-60
SLIDE 60

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤ts

  • Wi(s, t) = ✶

✶{s < Ti ≤ s + t}∆i

  • G(

Ti|s) + ✶ ✶{ Ti > s + t}

  • G(s + t|s)

+ ✵

✇✐t❤ G(u|s) t❤❡ ❑❛♣❧❛♥✲▼❡✐❡r ❡st✐♠❛t♦r ♦❢ P(C > u|C > s)✳ ▲❛♥❞♠❛r❦ t✐♠❡ s ❚✐♠❡ s + t t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❙❧✐❞❡ ✶✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-61
SLIDE 61

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤ts

  • Wi(s, t) = ✶

✶{s < Ti ≤ s + t}∆i

  • G(

Ti|s) + ✶ ✶{ Ti > s + t}

  • G(s + t|s)

+ ✵

✇✐t❤ G(u|s) t❤❡ ❑❛♣❧❛♥✲▼❡✐❡r ❡st✐♠❛t♦r ♦❢ P(C > u|C > s)✳ ▲❛♥❞♠❛r❦ t✐♠❡ s ❚✐♠❡ s + t t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❙❧✐❞❡ ✶✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-62
SLIDE 62

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❆s②♠♣t♦t✐❝ ✐✳✐✳❞✳ r❡♣r❡s❡♥t❛t✐♦♥

▲❡♠♠❛✿ ❆ss✉♠❡ t❤❛t t❤❡ ❝❡♥s♦r✐♥❣ t✐♠❡ C ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ (T, η, π(·, ·)) ❛♥❞ ❧❡t θ ❞❡♥♦t❡ ❡✐t❤❡r PEπ✱ R✷

π ♦r ❛ ❞✐✛❡r❡♥❝❡ ✐♥ P❊

♦r R✷

π✱ t❤❡♥

√n

  • θ(s, t) − θ(s, t)
  • =

✶ √n

n

  • i=✶

■❋θ( Ti, ∆i, πi(s, t), s, t) + op (✶) ✇❤❡r❡ ■❋θ( Ti, ∆i, πi(s, t), s, t) ❜❡✐♥❣ ✿ ◮ ③❡r♦✲♠❡❛♥ ✐✳✐✳❞✳ t❡r♠s ◮ ❡❛s② t♦ ❡st✐♠❛t❡ ✭✉s✐♥❣ ◆❡❧s♦♥✲❆❛❧❡♥ ✫ ❑❛♣❧❛♥✲▼❡✐❡r✮

❙❧✐❞❡ ✶✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-63
SLIDE 63

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

P♦✐♥t✇✐s❡ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧ ✭✜①❡❞ s✮

  • ❆s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t②✿

√n

  • θ(s, t) − θ(s, t)
  • D

− → N

  • ✵, σ✷

s,t

  • ✾✺✪ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧✿
  • θ(s, t) ± z✶−α/✷
  • σs,t

√n

  • ✇❤❡r❡ z✶−α/✷ ✐s t❤❡ ✶ − α/✷ q✉❛♥t✐❧❡ ♦❢ N(✵, ✶)✳
  • ❱❛r✐❛♥❝❡ ❡st✐♠❛t♦r✿
  • σ✷

s,t = ✶

n

n

  • i=✶
  • ■❋θ(

Ti, ∆i, πi(s, t), s, t) ✷

❙❧✐❞❡ ✶✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-64
SLIDE 64

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

  • θ(s, t) ±

q(S,t)

✶−α

  • σs,t

√n

  • ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢

❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r

✶ ✱ s❛② ✹✵✵✵✱ ❞♦✿

✶ ●❡♥❡r❛t❡

❢r♦♠ ✐✳✐✳❞✳ ✵ ✶ ✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r ■❋

✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

  • ✷ ❈♦♠♣✉t❡

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-65
SLIDE 65

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

  • θ(s, t) ±

q(S,t)

✶−α

  • σs,t

√n

  • ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠

✭ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡

❢r♦♠ ✐✳✐✳❞✳ ✵ ✶ ✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r ■❋

✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

  • ✷ ❈♦♠♣✉t❡

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-66
SLIDE 66

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

  • θ(s, t) ±

q(S,t)

✶−α

  • σs,t

√n

  • ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡

❢r♦♠ ✐✳✐✳❞✳ ✵ ✶ ✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r ■❋

✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

  • ✷ ❈♦♠♣✉t❡

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-67
SLIDE 67

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

  • θ(s, t) ±

q(S,t)

✶−α

  • σs,t

√n

  • ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡ {ωb

✶, . . . , ωb n} ❢r♦♠ n ✐✳✐✳❞✳ N(✵, ✶)✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r ■❋

✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

  • ✷ ❈♦♠♣✉t❡

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-68
SLIDE 68

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

  • θ(s, t) ±

q(S,t)

✶−α

  • σs,t

√n

  • ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡ {ωb

✶, . . . , ωb n} ❢r♦♠ n ✐✳✐✳❞✳ N(✵, ✶)✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r

■❋θ(·)✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

√n

n

  • i=✶

ωb

i

  • ■❋θ(

Ti, ∆i, πi(s, t), s, t)

  • σs,t
  • ✷ ❈♦♠♣✉t❡

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-69
SLIDE 69

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

  • θ(s, t) ±

q(S,t)

✶−α

  • σs,t

√n

  • ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡ {ωb

✶, . . . , ωb n} ❢r♦♠ n ✐✳✐✳❞✳ N(✵, ✶)✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r

■❋θ(·)✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

√n

n

  • i=✶

ωb

i

  • ■❋θ(

Ti, ∆i, πi(s, t), s, t)

  • σs,t
  • ✷ ❈♦♠♣✉t❡

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-70
SLIDE 70

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

  • θ(s, t) ±

q(S,t)

✶−α

  • σs,t

√n

  • ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡ {ωb

✶, . . . , ωb n} ❢r♦♠ n ✐✳✐✳❞✳ N(✵, ✶)✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r

■❋θ(·)✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

√n

n

  • i=✶

ωb

i

  • ■❋θ(

Ti, ∆i, πi(s, t), s, t)

  • σs,t
  • ✷ ❈♦♠♣✉t❡

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-71
SLIDE 71

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

  • θ(s, t) ±

q(S,t)

✶−α

  • σs,t

√n

  • ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡ {ωb

✶, . . . , ωb n} ❢r♦♠ n ✐✳✐✳❞✳ N(✵, ✶)✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r

■❋θ(·)✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

√n

n

  • i=✶

ωb

i

  • ■❋θ(

Ti, ∆i, πi(s, t), s, t)

  • σs,t
  • ✷ ❈♦♠♣✉t❡

q(S,t)

✶−α ❛s t❤❡ ✶✵✵(✶ − α)t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

  • Υ✶, . . . , ΥB

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-72
SLIDE 72

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉■❱❆❚ s❛♠♣❧❡

  • P♦♣✉❧❛t✐♦♥ ❜❛s❡❞ st✉❞② ♦❢ ❦✐❞♥❡② r❡❝✐♣✐❡♥ts ✭♥❂✹✱✶✶✾✮
  • ❙♣❧✐t t❤❡ ❞❛t❛ ✐♥t♦ tr❛✐♥✐♥❣ ✭✷✴✸✮ ❛♥❞ ✈❛❧✐❞❛t✐♦♥ ✭✶✴✸✮ s❛♠♣❧❡s

✿ t✐♠❡ ❢r♦♠ ✶✲②❡❛r ❛❢t❡r tr❛♥s♣❧❛♥t❛t✐♦♥ t♦ ❣r❛❢t ❢❛✐❧✉r❡ ✇❤✐❝❤ ✐s✿ ❉❡❛t❤ ❘❡t✉r♥ t♦ ❞✐❛❧②s✐s ❖❘ ❈❡♥s♦r✐♥❣ ❞✉❡ t♦✿ ❞❡❧❛②❡❞ ❡♥tr✐❡s✿ ✷✵✵✵✲✷✵✶✸ ❡♥❞ ♦❢ ❢♦❧❧♦✇✲✉♣✿ ✷✵✶✹ ❇❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s✿ ❛❣❡✱ s❡①✱ ❝❛r❞✐♦✈❛s❝✉❧❛r ❤✐st♦r② ▲♦♥❣✐t✉❞✐♥❛❧ ❜✐♦♠❛r❦❡r ✭②❡❛r❧②✮✿ s❡r✉♠ ❝r❡❛t✐♥✐♥❡

❙❧✐❞❡ ✷✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-73
SLIDE 73

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉■❱❆❚ s❛♠♣❧❡

  • P♦♣✉❧❛t✐♦♥ ❜❛s❡❞ st✉❞② ♦❢ ❦✐❞♥❡② r❡❝✐♣✐❡♥ts ✭♥❂✹✱✶✶✾✮
  • ❙♣❧✐t t❤❡ ❞❛t❛ ✐♥t♦ tr❛✐♥✐♥❣ ✭✷✴✸✮ ❛♥❞ ✈❛❧✐❞❛t✐♦♥ ✭✶✴✸✮ s❛♠♣❧❡s
  • T✿ t✐♠❡ ❢r♦♠ ✶✲②❡❛r ❛❢t❡r tr❛♥s♣❧❛♥t❛t✐♦♥ t♦ ❣r❛❢t ❢❛✐❧✉r❡ ✇❤✐❝❤ ✐s✿

❉❡❛t❤ ❘❡t✉r♥ t♦ ❞✐❛❧②s✐s ❖❘ ❈❡♥s♦r✐♥❣ ❞✉❡ t♦✿ ❞❡❧❛②❡❞ ❡♥tr✐❡s✿ ✷✵✵✵✲✷✵✶✸ ❡♥❞ ♦❢ ❢♦❧❧♦✇✲✉♣✿ ✷✵✶✹ ❇❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s✿ ❛❣❡✱ s❡①✱ ❝❛r❞✐♦✈❛s❝✉❧❛r ❤✐st♦r② ▲♦♥❣✐t✉❞✐♥❛❧ ❜✐♦♠❛r❦❡r ✭②❡❛r❧②✮✿ s❡r✉♠ ❝r❡❛t✐♥✐♥❡

❙❧✐❞❡ ✷✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-74
SLIDE 74

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉■❱❆❚ s❛♠♣❧❡

  • P♦♣✉❧❛t✐♦♥ ❜❛s❡❞ st✉❞② ♦❢ ❦✐❞♥❡② r❡❝✐♣✐❡♥ts ✭♥❂✹✱✶✶✾✮
  • ❙♣❧✐t t❤❡ ❞❛t❛ ✐♥t♦ tr❛✐♥✐♥❣ ✭✷✴✸✮ ❛♥❞ ✈❛❧✐❞❛t✐♦♥ ✭✶✴✸✮ s❛♠♣❧❡s
  • T✿ t✐♠❡ ❢r♦♠ ✶✲②❡❛r ❛❢t❡r tr❛♥s♣❧❛♥t❛t✐♦♥ t♦ ❣r❛❢t ❢❛✐❧✉r❡ ✇❤✐❝❤ ✐s✿

❉❡❛t❤ ❘❡t✉r♥ t♦ ❞✐❛❧②s✐s ❖❘

  • ❈❡♥s♦r✐♥❣ ❞✉❡ t♦✿
  • ❞❡❧❛②❡❞ ❡♥tr✐❡s✿ ✷✵✵✵✲✷✵✶✸
  • ❡♥❞ ♦❢ ❢♦❧❧♦✇✲✉♣✿ ✷✵✶✹
  • ❇❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s✿ ❛❣❡✱ s❡①✱ ❝❛r❞✐♦✈❛s❝✉❧❛r ❤✐st♦r②
  • ▲♦♥❣✐t✉❞✐♥❛❧ ❜✐♦♠❛r❦❡r ✭②❡❛r❧②✮✿ s❡r✉♠ ❝r❡❛t✐♥✐♥❡

❙❧✐❞❡ ✷✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-75
SLIDE 75

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉❡s❝r✐♣t✐✈❡ st❛t✐st✐❝s ✫ ❝❡♥s♦r✐♥❣ ✐ss✉❡

  • s ∈ S = {✵, ✵.✺, . . . , ✺}
  • t = ✺ ②❡❛rs

s=0 s=1 s=2 s=3 s=4 s=5 Censored in (s,s+5] Known as event−free at s+5 Observed failure in (s,s+5] landmark time (year)

  • No. of subjects

200 600 1000 1400

❙❧✐❞❡ ✷✷✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-76
SLIDE 76

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❏♦✐♥t ♠♦❞❡❧

◮ ▲♦♥❣✐t✉❞✐♥❛❧ log

  • Yi(tij)
  • = (β✵ + b✵i) + β✵,❛❣❡❆●❊i + β✵,s❡①❙❊❳i

+

  • β✶ + b✶i + β✶,❛❣❡❆●❊i
  • × tij + ǫij

= ♠i(t) + εij ❙✉r✈✐✈❛❧ ✭❤❛③❛r❞✮

❡①♣

❛❣❡❆●❊ ❈❱❈❱ ✶♠ ✷

✭✜tt❡❞ ✉s✐♥❣ ♣❛❝❦❛❣❡ ❏▼✮

❙❧✐❞❡ ✷✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-77
SLIDE 77

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❏♦✐♥t ♠♦❞❡❧

◮ ▲♦♥❣✐t✉❞✐♥❛❧ log

  • Yi(tij)
  • = (β✵ + b✵i) + β✵,❛❣❡❆●❊i + β✵,s❡①❙❊❳i

+

  • β✶ + b✶i + β✶,❛❣❡❆●❊i
  • × tij + ǫij

= ♠i(t) + εij ◮ ❙✉r✈✐✈❛❧ ✭❤❛③❛r❞✮ hi(t) = h✵(t) ❡①♣

  • γ❛❣❡❆●❊i + γ❈❱❈❱i

+ α✶♠i(t) + α✷ d♠i(t) dt

  • ✭✜tt❡❞ ✉s✐♥❣

♣❛❝❦❛❣❡ ❏▼✮

❙❧✐❞❡ ✷✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-78
SLIDE 78

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π(s, t) ✈s s ✭t❂✺ ②❡❛rs✮

Landmark time s (years) Rπ

2(s, t)

1 2 3 4 5 0 % 10 % 20 % 30 %

  • 95% pointwise CI

95% simultaneous CB

❙❧✐❞❡ ✷✹✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-79
SLIDE 79

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s (years) Rπ

2(s, t)

1 2 3 4 5 0 % 10 % 20 % 30 %

  • T ~ Age + CV + m(t) + m'(t) (JM)

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-80
SLIDE 80

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s (years) Rπ

2(s, t)

1 2 3 4 5 0 % 10 % 20 % 30 %

  • T ~ Age + CV + m(t) + m'(t) (JM)

T ~ Age + CV

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-81
SLIDE 81

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s (years) Rπ

2(s, t)

1 2 3 4 5 0 % 10 % 20 % 30 %

  • T ~ Age + CV + m(t) + m'(t) (JM)

T ~ Age + CV + Y(t=0) T ~ Age + CV

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-82
SLIDE 82

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s (years) Rπ

2(s, t)

1 2 3 4 5 0 % 10 % 20 % 30 %

  • T ~ Age + CV + m(t) + m'(t) (JM)

T ~ Age + CV + Y(t=0)

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-83
SLIDE 83

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s Difference in Rπ

2(s,t)

  • −5 %

5 % 10 % 20 % 0 % 1 2 3 4 5

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-84
SLIDE 84

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s Difference in Rπ

2(s,t)

  • −5 %

5 % 10 % 20 % 0 % 1 2 3 4 5

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-85
SLIDE 85

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈❛❧✐❜r❛t✐♦♥ ♣❧♦t ✭❡①❛♠♣❧❡ ❢♦r s = ✸ ②❡❛rs✮

(0,10) (0;3.5] (10,20) (3.5;5.3] (20,30) (5.3;8] (30,40) (8;10.4] (40,50) (10.4;13.1] (50,60) (13.1;16.5] (60,70) (16.5;22.3] (70,80) (22.3;33] (80,90) (33;55.1] (90,100) (55.1;100] Predicted Observed

Risk groups (quantile groups, in %) 5−year risk (%)

20 40 60 80 100 2% 8(4)% 4% 15(6)% 7% 15(5)% 9% 13(5)% 12% 16(6)% 15% 21(7)% 19% 26(6)% 27% 29(7)% 42% 36(9)% 77% 74(8)%

❙❧✐❞❡ ✷✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-86
SLIDE 86

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❆r❡❛ ✉♥❞❡r t❤❡ ❘❖❈(s, t) ❝✉r✈❡ ✈s s

Landmark time s (years) AUCπ(s, t) 1 2 3 4 5 50% 60% 70% 80% 100%

  • T ~ Age + CV + m(t) + m'(t) (JM)

T ~ Age + CV + Y(t=0) T ~ Age + CV

❙❧✐❞❡ ✷✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-87
SLIDE 87

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✉♠♠✐♥❣ ✉♣

◮ ❘✷✲t②♣❡ ❝✉r✈❡

  • s✉♠♠❛r✐③❡s ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♥❣ s✐♠✉❧t❛♥❡♦✉s❧②
  • ❤❛s ❛♥ ✉♥❞❡rst❛♥❞❛❜❧❡ tr❡♥❞

◮ ❙✐♠♣❧❡ ♠♦❞❡❧ ❢r❡❡ ✐♥❢❡r❡♥❝❡

  • ♣r❡❞✐❝t✐♦♥s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛♥② ♠♦❞❡❧
  • ✇❡ ❞♦ ♥♦t ❛ss✉♠❡ ❛♥② ♠♦❞❡❧ t♦ ❤♦❧❞
  • ❛❧❧♦✇s ❢❛✐r ❝♦♠♣❛r✐s♦♥s ♦❢ ❞✐✛❡r❡♥t ♣r❡❞✐❝t✐♦♥s

◮ ❚❤❡ ♠❡t❤♦❞ ❛❝❝♦✉♥ts ❢♦r✿

  • ❈❡♥s♦r✐♥❣
  • ❉②♥❛♠✐❝ s❡tt✐♥❣ ✭t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥ ❝❤❛♥❣❡s✮

❙❧✐❞❡ ✷✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-88
SLIDE 88

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉✐s❝✉ss✐♦♥

◮ ❚❤❡ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❛❧❧♦✇s ❞✐✛❡r❡♥t ✐♥t❡r❡st✐♥❣ ✐♥t❡r♣r❡t❛t✐♦♥s✿

  • ❊①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥
  • ❈♦rr❡❧❛t✐♦♥
  • ▼❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡

◮ ❯♥❢♦rt✉♥❛t❡❧②

  • t❤❡ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥♥♦t ❜❡ ❝❤❡❝❦❡❞

✭❝✉rs❡ ♦❢ ❞✐♠❡♥s✐♦♥❛❧✐t②✮

◮ ❍♦✇❡✈❡r

  • ✇❡❛❦ ❛♥❞ str♦♥❣ ❞❡✜♥✐t✐♦♥s ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞✿
  • str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ✐♠♣❧✐❡s ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥
  • ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥ ✏♦❢t❡♥✑ ❜❡ s❡❡♥ ❛s ❛ r❡❛s♦♥❛❜❧❡

❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ✐♥ ♣r❛❝t✐❝❡

  • ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥ ❜❡ ❛ss❡ss❡❞ ✭♣❧♦ts✮

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❙❧✐❞❡ ✷✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

slide-89
SLIDE 89

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉✐s❝✉ss✐♦♥

◮ ❚❤❡ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❛❧❧♦✇s ❞✐✛❡r❡♥t ✐♥t❡r❡st✐♥❣ ✐♥t❡r♣r❡t❛t✐♦♥s✿

  • ❊①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥
  • ❈♦rr❡❧❛t✐♦♥
  • ▼❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡

◮ ❯♥❢♦rt✉♥❛t❡❧②

  • t❤❡ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥♥♦t ❜❡ ❝❤❡❝❦❡❞

✭❝✉rs❡ ♦❢ ❞✐♠❡♥s✐♦♥❛❧✐t②✮

◮ ❍♦✇❡✈❡r

  • ✇❡❛❦ ❛♥❞ str♦♥❣ ❞❡✜♥✐t✐♦♥s ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞✿
  • str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ✐♠♣❧✐❡s ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥
  • ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥ ✏♦❢t❡♥✑ ❜❡ s❡❡♥ ❛s ❛ r❡❛s♦♥❛❜❧❡

❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ✐♥ ♣r❛❝t✐❝❡

  • ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥ ❜❡ ❛ss❡ss❡❞ ✭♣❧♦ts✮

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❙❧✐❞❡ ✷✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳