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❈❛❧✐❜r❛t✐♦♥ ♣❧♦ts ❢♦r r✐s❦ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧s ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝♦♠♣❡t✐♥❣ r✐s❦s

❚❤♦♠❛s ❆ ●❡r❞s✱ ❚❤♦♠❛s ❍ ❙❝❤❡✐❦❡✱ P❡r ❑ ❆♥❞❡rs❡♥ ❛♥❞ ▼✐❝❤❛❡❧ ❲ ❑❛tt❛♥ ❏✉♥❡ ✷✻✱ ✷✵✶✹

✶ ✴ ✷✽

▼♦t✐✈❛t✐♦♥✿ ♣❛t✐❡♥t ❝♦✉♥s❡❧✐♥❣

❯s✐♥❣ ❛ st❛t✐st✐❝❛❧ ♠♦❞❡❧✱ ❛ ❞❛t❛❜❛s❡ ❝❛♥ ❜❡ q✉❡r✐❡❞ t♦ ♦❜t❛✐♥ ❛ t❛✐❧♦r❡❞ ♣r❡❞✐❝t✐♦♥ ❢♦r t❤❡ ♣r❡s❡♥t ♣❛t✐❡♥t✳ ❆ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ✶✼✪ ✐s ❝❛❧❧❡❞ r❡❧✐❛❜❧❡✱ ✐❢ ✐t ❝❛♥ ❜❡ ❡①♣❡❝t❡❞ t❤❛t t❤❡ ❡✈❡♥t ✇✐❧❧ ♦❝❝✉r t♦ ❛❜♦✉t ✶✼ ♦✉t ♦❢ ✶✵✵ ♣❛t✐❡♥ts ✇❤♦ ❛❧❧ r❡❝❡✐✈❡❞ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ✶✼✪✳ ❆ st❛t✐st✐❝❛❧ ♠♦❞❡❧ t❤❛t ♣r❡❞✐❝ts t❤❡ ❛❜s♦❧✉t❡ r✐s❦ ♦❢ ❛♥ ❡✈❡♥t s❤♦✉❧❞ ❜❡ ❝❛❧✐❜r❛t❡❞ ✐♥ t❤❡ s❡♥s❡ t❤❛t ✐t ♣r♦✈✐❞❡s r❡❧✐❛❜❧❡ ♣r❡❞✐❝t✐♦♥s ❢♦r ❛❧❧ s✉❜❥❡❝ts✳ ❆ ❝❛❧✐❜r❛t✐♦♥ ♣❧♦t ❞✐s♣❧❛②s ❤♦✇ ✇❡❧❧ ♦❜s❡r✈❡❞ ❛♥❞ ♣r❡❞✐❝t❡❞ ❡✈❡♥t st❛t✉s ❝♦♥♥❡❝t ♦♥ t❤❡ ❛❜s♦❧✉t❡ ♣r♦❜❛❜✐❧✐t② s❝❛❧❡✳

✷ ✴ ✷✽

❈❛❧✐❜r❛t✐♦♥ ♣❧♦t

Predicted event probability 0 % 25 % 50 % 75 % 100 % Observed event status 0 % 25 % 50 % 75 % 100 %

Cause−specific Cox regression Fine−Gray regression

✸ ✴ ✷✽

Pr❡❞✐❝t✐♥❣ ❛❜s♦❧✉t❡ r✐s❦s ✐♥ t✐♠❡✲t♦✲❡✈❡♥t ❛♥❛❧②s✐s

❋✐rst ♣✐❝❦ ❛ t✐♠❡ ♦r✐❣✐♥ ❛t ✇❤✐❝❤ ✐t ✐s ♦❢ ✐♥t❡r❡st t♦ ♣r❡❞✐❝t t❤❡ ❢✉t✉r❡ st❛t✉s ♦❢ ❛ ♣❛t✐❡♥t✳ ❯♥t✐❧ t✐♠❡ t ❛❢t❡r t❤❡ t✐♠❡ ♦r✐❣✐♥ t❤r❡❡ t❤✐♥❣s ❝❛♥ ❤❛♣♣❡♥✿ ✶✳ t❤❡ ❡✈❡♥t ❤❛s ♦❝❝✉rr❡❞ ✷✳ ❛ ❝♦♠♣❡t✐♥❣ ❡✈❡♥t ❤❛s ♦❝❝✉rr❡❞ ✸✳ t❤❡ ♣❛t✐❡♥t ✐s ❛❧✐✈❡ ❛♥❞ ❡✈❡♥t✲❢r❡❡✳ ❚❤❡ ♣❛t✐❡♥t ♥❡❡❞s t♦ ❦♥♦✇ t❤❡ ❛❜s♦❧✉t❡ r✐s❦s ♦❢ ❛❧❧ ❡✈❡♥ts ✭❞❡❛t❤✱ ❞✐s❡❛s❡✱ r❡❝✉rr❡♥❝❡✱ ❡t❝✳✮✳

✹ ✴ ✷✽

❏♦❤♥ ❑❧❡✐♥✬s ❞❛t❛ ❢r♦♠ ❜♦♥❡ ♠❛rr♦✇ tr❛♥s♣❧❛♥t ♣❛t✐❡♥ts

❆ ❞❛t❛ ❢r❛♠❡ ✇✐t❤ ✶✼✶✺ ♦❜s❡r✈❛t✐♦♥s✶

Transplant Relapse Death n= 557 n= 311

❚❤❡ r❡♠❛✐♥✐♥❣ n = ✽✹✼ ♣❛t✐❡♥ts ✇❡r❡ ✐♥ r❡♠✐ss✐♦♥ ❜② t❤❡ ❡♥❞ ♦❢ t❤❡ ❢♦❧❧♦✇✲✉♣ ♣❡r✐♦❞✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♣r❡❞✐❝t✐♥❣ t❤❡ ❝✉♠✉❧❛t✐✈❡ ✐♥❝✐❞❡♥❝❡s ♦❢ r❡❧❛♣s❡ ❛♥❞ ❞❡❛t❤✳

✶❙③②❞❧♦✱ ●♦❧❞♠❛♥✱ ❑❧❡✐♥ ❡t ❛❧✳ ❏♦✉r♥❛❧ ♦❢ ❈❧✐♥✐❝❛❧ ❖♥❝♦❧♦❣②✱ ✶✾✾✼✳ ✺ ✴ ✷✽

❖❜s❡r✈❡❞ ♦✉t❝♦♠❡

Months since transplantation Cumulative incidence 12 36 60 84 0 % 25 % 50 % 75 % 100 %

Aalen−Johansen estimate

Event Relapse Death without relapse Months since transplantation Cumulative incidence 12 36 60 84 0 % 25 % 50 % 75 % 100 %

Kaplan−Meier estimate

• f censoring probability

❲✐t❤♦✉t ❝♦✈❛r✐❛t❡s t❤❡ ♠❛r❣✐♥❛❧ ❆❛❧❡♥✲❏♦❤❛♥s❡♥ ❡st✐♠❛t❡ ✐s t❤❡ ❜❡st ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧✳

✻ ✴ ✷✽

❋♦r♠✉❧❛ ■

▲❡t ❳ ❜❡ ❛ ✈❡❝t♦r ♦❢ ❝♦✈❛r✐❛t❡s✿ F✶(t|X) = ❈✉♠✉❧❛t✐✈❡ ✐♥❝✐❞❡♥❝❡ ♦❢ ❡✈❡♥t ✶ t

✵

❡①♣

• −

s

✵

{λ✶(u|X) + λ✷(u|X)}❞u

• ◆♦ ❡✈❡♥t ♦❢ ❛♥② ❝❛✉s❡ ✉♥t✐❧ s

λ✶(s|X)

❊✈❡♥t t②♣❡ ✶ ❛t s

❞s. ❘❡q✉✐r❡s ❛ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❢♦r t❤❡ ❤❛③❛r❞ ♦❢ t❤❡ ❝♦♠♣❡t✐♥❣ r✐s❦s ♦r ❛ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❢♦r t❤❡ ❡✈❡♥t✲❢r❡❡ s✉r✈✐✈❛❧ ♣r♦❜❛❜✐❧✐t②✳

✼ ✴ ✷✽

❋♦r♠✉❧❛ ■■

❚r❛♥s❢♦r♠❛t✐♦♥ ♠♦❞❡❧ h(F✶(t|X)) = β✵✶(t) + β✶X✶ + · · · + βKXK

◮ ❤✭♣✮ ❂ ❧♦❣✭✲❧♦❣✭♣✮✮

✭❋✐♥❡✲●r❛② ♠♦❞❡❧✮

◮ ❤✭♣✮ ❂ ❧♦❣✭♣✴✭✶✲♣✮✮

✭▲♦❣✐st✐❝ ♠♦❞❡❧✮

◮ ❤✭♣✮ ❂ ❧♦❣✭♣✮

✭▲♦❣✲❜✐♥♦♠✐❛❧ ♠♦❞❡❧✮ ❘❡q✉✐r❡s ❛ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❢♦r t❤❡ ❝✉♠✉❧❛t✐✈❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❜❡✐♥❣ ✉♥❝❡♥s♦r❡❞✿ ●✭t⑤❳✮ ❂ P✭❚❃t⑤❳✮ ✐♥ ✇❤❛t ❢♦❧❧♦✇s✿ ●✭t⑤❳✮❂●✵✭t✮✳

✽ ✴ ✷✽

■♥t❡r♣r❡t❛t✐♦♥ ❝r✐s✐s ✐♥ ❝♦♠♣❡t✐♥❣ r✐s❦s

Pr♦❜❧❡♠s✿

◮ ❚❤❡ ❤❛③❛r❞ r❛t✐♦s ♦❜t❛✐♥❡❞ ❜② ❝❛✉s❡✲s♣❡❝✐✜❝ ❈♦① r❡❣r❡ss✐♦♥

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

◮ ❚❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡

❋✐♥❡✲●r❛② ♠♦❞❡❧ ❤❛✈❡ ♥♦ ❞✐r❡❝t ✐♥t❡r♣r❡t❛t✐♦♥✳ Pr♦♣♦s❛❧✿ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s ❢♦r t❤❡ ❛❜s♦❧✉t❡ r✐s❦ ♦❢ r❡❧❛♣s❡ ✐♥ ✇❤✐❝❤ t❤❡ r❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r♣r❡t❛t✐♦♥✿ ❚❤❡ ✺✲②❡❛r r✐s❦ ♦❢ r❡❧❛♣s❡ ❝❤❛♥❣❡s ✇✐t❤ ❛ ❢❛❝t♦r ❡①♣

✶ ❢♦r ❛ ♦♥❡

✉♥✐t ❝❤❛♥❣❡ ♦❢

✶ ❛♥❞ ❣✐✈❡♥ ✈❛❧✉❡s ❢♦r t❤❡ ♦t❤❡r ♣r❡❞✐❝t♦r ✈❛r✐❛❜❧❡s ✷

✳

✾ ✴ ✷✽

■♥t❡r♣r❡t❛t✐♦♥ ❝r✐s✐s ✐♥ ❝♦♠♣❡t✐♥❣ r✐s❦s

Pr♦❜❧❡♠s✿

◮ ❚❤❡ ❤❛③❛r❞ r❛t✐♦s ♦❜t❛✐♥❡❞ ❜② ❝❛✉s❡✲s♣❡❝✐✜❝ ❈♦① r❡❣r❡ss✐♦♥

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

◮ ❚❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts ✐♥ t❤❡

❋✐♥❡✲●r❛② ♠♦❞❡❧ ❤❛✈❡ ♥♦ ❞✐r❡❝t ✐♥t❡r♣r❡t❛t✐♦♥✳ Pr♦♣♦s❛❧✿ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s ❢♦r t❤❡ ❛❜s♦❧✉t❡ r✐s❦ ♦❢ r❡❧❛♣s❡ ✐♥ ✇❤✐❝❤ t❤❡ r❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥t❡r♣r❡t❛t✐♦♥✿ ❚❤❡ ✺✲②❡❛r r✐s❦ ♦❢ r❡❧❛♣s❡ ❝❤❛♥❣❡s ✇✐t❤ ❛ ❢❛❝t♦r ❡①♣(β✶) ❢♦r ❛ ♦♥❡ ✉♥✐t ❝❤❛♥❣❡ ♦❢ X✶ ❛♥❞ ❣✐✈❡♥ ✈❛❧✉❡s ❢♦r t❤❡ ♦t❤❡r ♣r❡❞✐❝t♦r ✈❛r✐❛❜❧❡s (X✷, ..., XK)✳

✾ ✴ ✷✽

❆❜s♦❧✉t❡ r✐s❦ r❡❣r❡ss✐♦♥

❚❤❡ r❡❣r❡ss✐♦♥ ♣❛r❛♠❡t❡rs ✐♥ t❤❡ ❧♦❣✲❜✐♥♦♠✐❛❧ ♠♦❞❡❧ ❤❛✈❡ t❤❡ ❞❡s✐r❡❞ ✐♥t❡r♣r❡t❛t✐♦♥✿ F✶(t|X) = exp(β✵✶(t)) ❡①♣(β✶X✶ + · · · + βKXK) ❆ ♦♥❡ ✉♥✐t ❝❤❛♥❣❡ ♦❢ t❤❡ ❦t❤ ❝♦✈❛r✐❛t❡✿ F✶(t|X✶, . . . , Xk = xk, . . . , XK) F✶(t|X✶, . . . , Xk = (xk + ✶), . . . , XK) = ❡①♣{βk(xk − xk + ✶)} = ❡①♣(βk).

✶✵ ✴ ✷✽

❇♦♥❡ ♠❛rr♦✇ tr❛♥s♣❧❛♥t ❞❛t❛✿ ❛❜s♦❧✉t❡ r✐s❦ ♦❢ r❡❧❛♣s❡

❋❛❝t♦r ❡①♣✭β✮ ❈■✳✾✺ P✲✈❛❧✉❡ ❞✐s❡❛s❡✿❆▲▲ ✕ ✕ ✕ ❞✐s❡❛s❡✿❆▼▲ ✵✳✽✻ ❬✵✳✻✽❀✶✳✵✽❪ ✵✳✶✾✽✷✷✾✷ ❞✐s❡❛s❡✿❈▼▲ ✵✳✺✽ ❬✵✳✹✹❀✵✳✼✻❪ ✵✳✵✵✵✶✵✶✼ ❦❛r♥♦❢s❦② ✶✳✸ ❬✶✳✵✸❀✶✳✻✽❪ ✵✳✵✷✺✸✾✼✺ ❞♦♥♦r✿s✐❜❧✐♥❣ ✕ ✕ ✕ ❞♦♥♦r✿♠❛t❝❤❡❞ ✵✳✼✷ ❬✵✳✺✺❀✵✳✾✺❪ ✵✳✵✷✷✷✻✻✸ ❞♦♥♦r✿♠✐s♠❛t❝❤❡❞ ✵✳✷✼ ❬✵✳✶✸❀✵✳✺✼❪ ✵✳✵✵✵✻✷✾✹ st❛❣❡✿❡❛r❧② ✕ ✕ ✕ st❛❣❡✿✐♥t❡r♠❡❞✐❛t❡ ✶✳✽ ❬✶✳✸✼❀✷✳✹✻❪ ❁ ✵✳✵✵✵✶ st❛❣❡✿❛❞✈❛♥❝❡❞ ✸✳✶ ❬✷✳✹✼❀✹✳✵✷❪ ❁ ✵✳✵✵✵✶ t✐♠❡❞①t① ✵✳✾✾ ❬✵✳✾✽❀✶❪ ✵✳✵✷✶✾✾✸✽ ❊✳❣✳✱ ❚❤❡ r✐s❦ ♦❢ r❡❧❛♣s❡ ✇❛s ❡st✐♠❛t❡❞ ❛s ✶✳✽ t✐♠❡s ❤✐❣❤❡r ❢♦r ❞✐s❡❛s❡ st❛❣❡ ✐♥t❡r♠❡❞✐❛t❡ ❝♦♠♣❛r❡❞ t♦ ❞✐s❡❛s❡ st❛❣❡ ❡❛r❧②✳

✶✶ ✴ ✷✽

❉♦❡s t❤✐s ♠♦❞❡❧ ✜t❄

❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❝♦♠♠♦♥ ❛❧t❡r♥❛t✐✈❡s✿

◮ ❈♦♠❜✐♥❛t✐♦♥ ♦❢ ❝❛✉s❡✲s♣❡❝✐✜❝ ❈♦① r❡❣r❡ss✐♦♥s ✭❋♦r♠✉❧❛ ■✮ ◮ ❋✐♥❡✲●r❛② r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ✭❋♦r♠✉❧❛ ■■✿ ❞✐✛❡r❡♥t ❧✐♥❦ ❢✉♥❝t✐♦♥✮ ◮ ❋❧❡①✐❜❧❡ ❛❜s♦❧✉t❡ r✐s❦ r❡❣r❡ss✐♦♥✿ ❛❧❧♦✇ t✐♠❡✲❞❡♣❡♥❞❡♥t

❝♦✈❛r✐❛t❡ ❡✛❡❝ts βk(t) ❋♦❝✉s✿ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♠♦❞❡❧ ❢♦r ♣r❡❞✐❝t✐♦♥ P❡rs♦♥❛❧✐③❡❞✿ r❡✲❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♣r❡❞✐❝t❡❞ ♣r♦❜❛❜✐❧✐t✐❡s ❈❛❧✐❜r❛t✐♦♥ ♣❧♦t✿ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣r❡❞✐❝t❡❞ ❡①♣❡❝t❡❞ ♣r♦❜❛❜✐❧✐t✐❡s ❇r✐❡r s❝♦r❡✿ ♠❡❛♥ sq✉❛r❡❞ ❡rr♦r ❢♦r ♣r❡❞✐❝t❡❞ ♣r♦❜❛❜✐❧✐t✐❡s

✶✷ ✴ ✷✽

❉♦❡s t❤✐s ♠♦❞❡❧ ✜t❄

❈♦♠♣❛r✐s♦♥ ✇✐t❤ ❝♦♠♠♦♥ ❛❧t❡r♥❛t✐✈❡s✿

◮ ❈♦♠❜✐♥❛t✐♦♥ ♦❢ ❝❛✉s❡✲s♣❡❝✐✜❝ ❈♦① r❡❣r❡ss✐♦♥s ✭❋♦r♠✉❧❛ ■✮ ◮ ❋✐♥❡✲●r❛② r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ✭❋♦r♠✉❧❛ ■■✿ ❞✐✛❡r❡♥t ❧✐♥❦ ❢✉♥❝t✐♦♥✮ ◮ ❋❧❡①✐❜❧❡ ❛❜s♦❧✉t❡ r✐s❦ r❡❣r❡ss✐♦♥✿ ❛❧❧♦✇ t✐♠❡✲❞❡♣❡♥❞❡♥t

❝♦✈❛r✐❛t❡ ❡✛❡❝ts βk(t) ❋♦❝✉s✿ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ♠♦❞❡❧ ❢♦r ♣r❡❞✐❝t✐♦♥

◮ P❡rs♦♥❛❧✐③❡❞✿ r❡✲❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♣r❡❞✐❝t❡❞ ♣r♦❜❛❜✐❧✐t✐❡s ◮ ❈❛❧✐❜r❛t✐♦♥ ♣❧♦t✿ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ♣r❡❞✐❝t❡❞ ❡①♣❡❝t❡❞

♣r♦❜❛❜✐❧✐t✐❡s

◮ ❇r✐❡r s❝♦r❡✿ ♠❡❛♥ sq✉❛r❡❞ ❡rr♦r ❢♦r ♣r❡❞✐❝t❡❞ ♣r♦❜❛❜✐❧✐t✐❡s

✶✷ ✴ ✷✽

❈♦♠♣❛r✐s♦♥ ♦❢ ♣r❡❞✐❝t❡❞ ♣r♦❜❛❜✐❧✐t✐❡s

Pr❡❞✐❝t❡❞ r✐s❦ ♦❢ r❡❧❛♣s❡ ✇✐t❤✐♥ ✸ ②❡❛r ❛❢t❡r tr❛♥s♣❧❛♥t❛t✐♦♥

Absolute risk regression Gray−Fine regression

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• Absolute risk regression

Cause−specific Cox regression

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• Absolute risk regression

Time−dependent effects

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• ❘✐s❦ r❡✲❝❧❛ss✐✜❝❛t✐♦♥ ♣❧♦ts

✶✸ ✴ ✷✽

❈❛❧✐❜r❛t✐♦♥ ❝✉r✈❡

■♥❣r❡❞✐❡♥ts✿

◮ ❚❤❡ ❡✈❡♥t st❛t✉s ✐♥❞✐❝❛t♦r ✈❛r✐❛❜❧❡✿

N(t) = ✶{T ≤ t, D = ✶}

◮ ❚❤❡ r✐s❦ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧✿

r(t|X) ∈ [✵, ✶]

◮ ❚❤❡ r✐s❦ ❣r♦✉♣ ❛t p ∈ [✵, ✶]

Gr(t; p) = {x ∈ Rd : r(t|x) = p} ❚❤❡ ❝❛❧✐❜r❛t✐♦♥ ❝✉r✈❡ ❛t t✐♠❡ t✿ ❊ ❊

✶✹ ✴ ✷✽

❈❛❧✐❜r❛t✐♦♥ ❝✉r✈❡

■♥❣r❡❞✐❡♥ts✿

◮ ❚❤❡ ❡✈❡♥t st❛t✉s ✐♥❞✐❝❛t♦r ✈❛r✐❛❜❧❡✿

N(t) = ✶{T ≤ t, D = ✶}

◮ ❚❤❡ r✐s❦ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧✿

r(t|X) ∈ [✵, ✶]

◮ ❚❤❡ r✐s❦ ❣r♦✉♣ ❛t p ∈ [✵, ✶]

Gr(t; p) = {x ∈ Rd : r(t|x) = p} ❚❤❡ ❝❛❧✐❜r❛t✐♦♥ ❝✉r✈❡ ❛t t✐♠❡ t✿ p → C(p, t, r) = ❊{N(t) | r(t|X) = p}. = ❊{N(t) | X ∈ Gr(t; p)}

✶✹ ✴ ✷✽

❊st✐♠❛t✐♦♥

❚♦ ♦❜t❛✐♥ t❤❡ ❣r❛♣❤ ✇❡ ♥❡❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❡①♣❡❝t❛t✐♦♥ ❊{N(t) | X ∈ Gr(t; p)} ❚❤r❡❡ ♦❢t❡♥ ❡♥❝♦✉♥t❡r❡❞ ♣r❛❝t✐❝❛❧ ♣r♦❜❧❡♠s ❛r✐s❡✿ ❘✐❣❤t ❝❡♥s♦r✐♥❣✿ ✐❢ ♣❛t✐❡♥t ✐s ♥♦t ❢♦❧❧♦✇❡❞ ✉♥t✐❧ t✐♠❡ ✱ t❤❡ st❛t✉s ✐s ✉♥❦♥♦✇♥✳ ❈♦♥t✐♥✉✐t②✿ t❤❡ s✐③❡ ♦❢ t❤❡ s❡ts ♠❛② ❜❡ s♠❛❧❧ ❛♥❞ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t ❛ s❡t ✐♥❝❧✉❞❡s ♦♥❧② ❛ s✐♥❣❧❡ ♣❛t✐❡♥t✳

• ❡♥❡r❛❧✐③❛❜✐❧✐t②✿ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❦♥♦✇ ✐❢ t❤❡ ♠♦❞❡❧ ✇✐❧❧ ❜❡

r❡❧✐❛❜❧❡ ❢♦r ♥❡✇ ♣❛t✐❡♥ts✱ ♥♦t t❤♦s❡ ✐♥ t❤❡ ❞❛t❛ s❡t ✇❤✐❝❤ ✇❛s ✉s❡❞ t♦ s♣❡❝✐❢② ❛♥❞ ❡st✐♠❛t❡ t❤❡ ♠♦❞❡❧s✳

✶✺ ✴ ✷✽

❊st✐♠❛t✐♦♥

❚♦ ♦❜t❛✐♥ t❤❡ ❣r❛♣❤ ✇❡ ♥❡❡❞ t♦ ❡st✐♠❛t❡ t❤❡ ❡①♣❡❝t❛t✐♦♥ ❊{N(t) | X ∈ Gr(t; p)} ❚❤r❡❡ ♦❢t❡♥ ❡♥❝♦✉♥t❡r❡❞ ♣r❛❝t✐❝❛❧ ♣r♦❜❧❡♠s ❛r✐s❡✿

◮ ❘✐❣❤t ❝❡♥s♦r✐♥❣✿ ✐❢ ♣❛t✐❡♥t i ✐s ♥♦t ❢♦❧❧♦✇❡❞ ✉♥t✐❧ t✐♠❡ t✱ t❤❡

st❛t✉s Ni(t) ✐s ✉♥❦♥♦✇♥✳

◮ ❈♦♥t✐♥✉✐t②✿ t❤❡ s✐③❡ ♦❢ t❤❡ s❡ts Gr(t; p) ♠❛② ❜❡ s♠❛❧❧ ❛♥❞ ✐t

♠❛② ❤❛♣♣❡♥ t❤❛t ❛ s❡t ✐♥❝❧✉❞❡s ♦♥❧② ❛ s✐♥❣❧❡ ♣❛t✐❡♥t✳

◮ ●❡♥❡r❛❧✐③❛❜✐❧✐t②✿ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❦♥♦✇ ✐❢ t❤❡ ♠♦❞❡❧ ✇✐❧❧ ❜❡

r❡❧✐❛❜❧❡ ❢♦r ♥❡✇ ♣❛t✐❡♥ts✱ ♥♦t t❤♦s❡ ✐♥ t❤❡ ❞❛t❛ s❡t ✇❤✐❝❤ ✇❛s ✉s❡❞ t♦ s♣❡❝✐❢② ❛♥❞ ❡st✐♠❛t❡ t❤❡ ♠♦❞❡❧s✳

✶✺ ✴ ✷✽

❊st✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤

◮ ❘✐❣❤t ❝❡♥s♦r✐♥❣✿ ✐❢ ♣❛t✐❡♥t i ✐s ♥♦t ❢♦❧❧♦✇❡❞ ✉♥t✐❧ t✐♠❡ t✱ t❤❡

st❛t✉s Ni(t) ✐s ✉♥❦♥♦✇♥✿ ❏❆❈❑◆■❋❊ P❙❊❯❉❖✲❱❆▲❯❊❙ ❈♦♥t✐♥✉✐t②✿ t❤❡ s✐③❡ ♦❢ t❤❡ s❡ts ♠❛② ❜❡ s♠❛❧❧ ❛♥❞ ✐t ♠❛② ❤❛♣♣❡♥ t❤❛t ❛ s❡t ✐♥❝❧✉❞❡s ♦♥❧② ❛ s✐♥❣❧❡ ♣❛t✐❡♥t✿ ◆❊❆❘❊❙❚ ◆❊■●❍❇❖❘❍❖❖❉ ❙▼❖❖❚❍■◆●

• ❡♥❡r❛❧✐③❛❜✐❧✐t②✿ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❦♥♦✇ ✐❢ t❤❡ ♠♦❞❡❧ ✇✐❧❧ ❜❡

r❡❧✐❛❜❧❡ ❢♦r ♥❡✇ ♣❛t✐❡♥ts✱ ♥♦t t❤♦s❡ ✐♥ t❤❡ ❞❛t❛ s❡t ✇❤✐❝❤ ✇❛s ✉s❡❞ t♦ s♣❡❝✐❢② ❛♥❞ ❡st✐♠❛t❡ t❤❡ ♠♦❞❡❧✿ ❇❖❖❚❙❚❘❆P✲❈❘❖❙❙❱❆▲■❉❆❚■❖◆

✶✻ ✴ ✷✽

❊st✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤

◮ ❘✐❣❤t ❝❡♥s♦r✐♥❣✿ ✐❢ ♣❛t✐❡♥t i ✐s ♥♦t ❢♦❧❧♦✇❡❞ ✉♥t✐❧ t✐♠❡ t✱ t❤❡

st❛t✉s Ni(t) ✐s ✉♥❦♥♦✇♥✿ ❏❆❈❑◆■❋❊ P❙❊❯❉❖✲❱❆▲❯❊❙

◮ ❈♦♥t✐♥✉✐t②✿ t❤❡ s✐③❡ ♦❢ t❤❡ s❡ts Gr(t; p) ♠❛② ❜❡ s♠❛❧❧ ❛♥❞ ✐t

♠❛② ❤❛♣♣❡♥ t❤❛t ❛ s❡t ✐♥❝❧✉❞❡s ♦♥❧② ❛ s✐♥❣❧❡ ♣❛t✐❡♥t✿ ◆❊❆❘❊❙❚ ◆❊■●❍❇❖❘❍❖❖❉ ❙▼❖❖❚❍■◆●

• ❡♥❡r❛❧✐③❛❜✐❧✐t②✿ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❦♥♦✇ ✐❢ t❤❡ ♠♦❞❡❧ ✇✐❧❧ ❜❡

r❡❧✐❛❜❧❡ ❢♦r ♥❡✇ ♣❛t✐❡♥ts✱ ♥♦t t❤♦s❡ ✐♥ t❤❡ ❞❛t❛ s❡t ✇❤✐❝❤ ✇❛s ✉s❡❞ t♦ s♣❡❝✐❢② ❛♥❞ ❡st✐♠❛t❡ t❤❡ ♠♦❞❡❧✿ ❇❖❖❚❙❚❘❆P✲❈❘❖❙❙❱❆▲■❉❆❚■❖◆

✶✻ ✴ ✷✽

❊st✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤

◮ ❘✐❣❤t ❝❡♥s♦r✐♥❣✿ ✐❢ ♣❛t✐❡♥t i ✐s ♥♦t ❢♦❧❧♦✇❡❞ ✉♥t✐❧ t✐♠❡ t✱ t❤❡

st❛t✉s Ni(t) ✐s ✉♥❦♥♦✇♥✿ ❏❆❈❑◆■❋❊ P❙❊❯❉❖✲❱❆▲❯❊❙

◮ ❈♦♥t✐♥✉✐t②✿ t❤❡ s✐③❡ ♦❢ t❤❡ s❡ts Gr(t; p) ♠❛② ❜❡ s♠❛❧❧ ❛♥❞ ✐t

♠❛② ❤❛♣♣❡♥ t❤❛t ❛ s❡t ✐♥❝❧✉❞❡s ♦♥❧② ❛ s✐♥❣❧❡ ♣❛t✐❡♥t✿ ◆❊❆❘❊❙❚ ◆❊■●❍❇❖❘❍❖❖❉ ❙▼❖❖❚❍■◆●

◮ ●❡♥❡r❛❧✐③❛❜✐❧✐t②✿ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❦♥♦✇ ✐❢ t❤❡ ♠♦❞❡❧ ✇✐❧❧ ❜❡

r❡❧✐❛❜❧❡ ❢♦r ♥❡✇ ♣❛t✐❡♥ts✱ ♥♦t t❤♦s❡ ✐♥ t❤❡ ❞❛t❛ s❡t ✇❤✐❝❤ ✇❛s ✉s❡❞ t♦ s♣❡❝✐❢② ❛♥❞ ❡st✐♠❛t❡ t❤❡ ♠♦❞❡❧✿ ❇❖❖❚❙❚❘❆P✲❈❘❖❙❙❱❆▲■❉❆❚■❖◆

✶✻ ✴ ✷✽

❊st✐♠❛t❡❞ ❝❛❧✐❜r❛t✐♦♥ ❝✉r✈❡

ˆ Can,B(p, t, r) = ✶ n

n

• i=✶

✶ mi

• b:i∈Vb

˜ Ni(t)Kan(p, ˆ rb(t|Xi)) .

✶✼ ✴ ✷✽

❊st✐♠❛t❡❞ ❝❛❧✐❜r❛t✐♦♥ ❝✉r✈❡

ˆ Can,B(p, t, r) = ✶ n

n

• i=✶

✶ mi

• b:i∈Vb

˜ Ni(t)Kan(p, ˆ rb(t|Xi)) .

◮

˜ Ni(t) ❂ ❥❛❝❦♥✐❢❡ ♣s❡✉❞♦ ✈❛❧✉❡ ❢♦r ❡✈❡♥t st❛t✉s ❛t t✐♠❡ t ❜❛s❡❞ ♦♥ ❆❛❧❡♥✲❏♦❤❛♥s❡♥ ❡st✐♠❛t❡ ♦❢ E(N(t))

◮ ❑❛♥✭♣✱q✮❂ s♠♦♦t❤✐♥❣ ❦❡r♥❡❧ ◮ ❛♥ ❂ ❜❛♥❞✇✐❞t❤ ◮ ❇ ❂ ♥✉♠❜❡r ♦❢ ❜♦♦tstr❛♣ s♣❧✐ts✿ ❉❛t❛ ❂ ▲❜ ✰ ❱❜ ◮ ˆ

rb ❂ ♠♦❞❡❧ ✜tt❡❞ ✐♥ ❧❡❛r♥✐♥❣ s❛♠♣❧❡ ▲❜

◮ mi ❂ t❤❡ ♥✉♠❜❡r ♦❢ s♣❧✐ts ✇❤❡r❡ ♣❛t✐❡♥t ✐ ✐s ✐♥ ❱❜ ◮ ˆ

rb(t, Xi) ❂ ♣r❡❞✐❝t✐♦♥ ❢♦r ♣❛t✐❡♥t ✐♥ ✈❛❧✐❞❛t✐♦♥ s❛♠♣❧❡ ❱❜✳

✶✽ ✴ ✷✽

❊st✐♠❛t❡❞ ❝❛❧✐❜r❛t✐♦♥ ❝✉r✈❡

ˆ Can,B(p, t, r) = ✶ n

n

• i=✶

✶ mi

• b:i∈Vb

˜ Ni(t)Kan(p, ˆ rb(t|Xi)) .

◮

˜ Ni(t) ❂ ❥❛❝❦♥✐❢❡ ♣s❡✉❞♦ ✈❛❧✉❡ ❢♦r ❡✈❡♥t st❛t✉s ❛t t✐♠❡ t ❜❛s❡❞ ♦♥ ❆❛❧❡♥✲❏♦❤❛♥s❡♥ ❡st✐♠❛t❡ ♦❢ E(N(t))

◮ ❑❛♥✭♣✱q✮ ❂ s♠♦♦t❤✐♥❣ ❦❡r♥❡❧ ◮ ❛♥ ❂ ❜❛♥❞✇✐❞t❤ ◮ ❇ ❂ ♥✉♠❜❡r ♦❢ ❜♦♦tstr❛♣ s♣❧✐ts✿ ❉❛t❛ ❂ ▲❜ ✰ ❱❜ ◮ ˆ

rb ❂ ♠♦❞❡❧ ✜tt❡❞ ✐♥ ❧❡❛r♥✐♥❣ s❛♠♣❧❡ ▲❜

◮ mi ❂ t❤❡ ♥✉♠❜❡r ♦❢ s♣❧✐ts ✇❤❡r❡ ♣❛t✐❡♥t ✐ ✐s ✐♥ ❱❜ ◮ ˆ

rb(t, Xi) ❂ ♣r❡❞✐❝t✐♦♥ ❢♦r ♣❛t✐❡♥t ✐♥ ✈❛❧✐❞❛t✐♦♥ s❛♠♣❧❡ ❱❜✳

✶✾ ✴ ✷✽

❊st✐♠❛t❡❞ ❝❛❧✐❜r❛t✐♦♥ ❝✉r✈❡

ˆ Can,B(p, t, r) = ✶ n

n

• i=✶

✶ mi

• b:i∈Vb

˜ Ni(t)Kan(p, ˆ rb(t|Xi)) .

◮

˜ Ni(t) ❂ ❥❛❝❦♥✐❢❡ ♣s❡✉❞♦ ✈❛❧✉❡ ❢♦r ❡✈❡♥t st❛t✉s ❛t t✐♠❡ t ❜❛s❡❞ ♦♥ ❆❛❧❡♥✲❏♦❤❛♥s❡♥ ❡st✐♠❛t❡ ♦❢ E(N(t))

◮ ❑❛♥✭♣✱q✮❂ s♠♦♦t❤✐♥❣ ❦❡r♥❡❧ ◮ ❛♥ ❂ ❜❛♥❞✇✐❞t❤ ◮ ❇ ❂ ♥✉♠❜❡r ♦❢ ❜♦♦tstr❛♣ s♣❧✐ts✿ ❉❛t❛ ❂ ▲❜ ✰ ❱❜ ◮ ˆ

rb ❂ ♠♦❞❡❧ ✜tt❡❞ ✐♥ ❧❡❛r♥✐♥❣ s❛♠♣❧❡ ▲❜

◮ mi ❂ t❤❡ ♥✉♠❜❡r ♦❢ s♣❧✐ts ✇❤❡r❡ ♣❛t✐❡♥t ✐ ✐s ✐♥ ❱❜ ◮ ˆ

rb(t, Xi) ❂ ♣r❡❞✐❝t✐♦♥ ❢♦r ♣❛t✐❡♥t ✐♥ ✈❛❧✐❞❛t✐♦♥ s❛♠♣❧❡ ❱❜✳

✷✵ ✴ ✷✽

❊✛❡❝t ♦❢ ❝❡♥s♦r✐♥❣✿ ✸ ♠♦♥t❤s ❛❢t❡r tr❛♥s♣❧❛♥t❛t✐♦♥

Relapse

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

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• Death without relapse

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

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❋✐❣✉r❡✿ ❘✐s❦s ♣r❡❞✐❝t❡❞ ❜② t✇♦ ✐♥❞❡♣❡♥❞❡♥t ❛❜s♦❧✉t❡ r✐s❦ r❡❣r❡ss✐♦♥ ♠♦❞❡❧s✱ ♦♥❡ ❢♦r r❡❧❛♣s❡ ❛♥❞ ♦♥❡ ❢♦r ❞❡❛t❤ ✇✐t❤♦✉t r❡❧❛♣s❡✳

✷✶ ✴ ✷✽

❊✛❡❝t ♦❢ ❝❡♥s♦r✐♥❣✿ ✶ ②❡❛r ❛❢t❡r tr❛♥s♣❧❛♥t❛t✐♦♥

Relapse

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

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• Death without relapse

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

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• ❋✐❣✉r❡✿ ❘✐s❦s ♣r❡❞✐❝t❡❞ ❜② t✇♦ ✐♥❞❡♣❡♥❞❡♥t ❛❜s♦❧✉t❡ r✐s❦ r❡❣r❡ss✐♦♥

♠♦❞❡❧s✱ ♦♥❡ ❢♦r r❡❧❛♣s❡ ❛♥❞ ♦♥❡ ❢♦r ❞❡❛t❤ ✇✐t❤♦✉t r❡❧❛♣s❡✳

✷✷ ✴ ✷✽

❊✛❡❝t ♦❢ ❝❡♥s♦r✐♥❣✿ ✸ ②❡❛rs ❛❢t❡r tr❛♥s♣❧❛♥t❛t✐♦♥

Relapse

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

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• Death without relapse

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

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• ❋✐❣✉r❡✿ ❘✐s❦s ♣r❡❞✐❝t❡❞ ❜② t✇♦ ✐♥❞❡♣❡♥❞❡♥t ❛❜s♦❧✉t❡ r✐s❦ r❡❣r❡ss✐♦♥

♠♦❞❡❧s✱ ♦♥❡ ❢♦r r❡❧❛♣s❡ ❛♥❞ ♦♥❡ ❢♦r ❞❡❛t❤ ✇✐t❤♦✉t r❡❧❛♣s❡✳

✷✸ ✴ ✷✽

❊✛❡❝t ♦❢ ❜❛♥❞✇✐❞t❤✿ ❡✈❡♥t❂ r❡❧❛♣s❡✱ t❂✸✻ ♠♦♥t❤s

Calibration in the large bandwidth=1

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

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• Localized calibration

bandwidth=0

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

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automatically selected bandwidth=0.044

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

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• Kernel smoother

bandwidth=0.1

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

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• ✷✹ ✴ ✷✽

❊✛❡❝t ♦❢ ❝r♦ss✲✈❛❧✐❞❛t✐♦♥

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

1000 bootstrap cross−validation steps Same data used twice

✷✺ ✴ ✷✽

❈♦♠♣❛r✐s♦♥ ♦❢ ♠♦❞❡❧s

Relapse (t=36 months) Same data twice

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

Absolute risk regression Cause−specific Cox Fine−Gray

Bootstrap cross−validation B=1000

Predicted event probability 0 % 25 % 50 % 75 % 100 % Pseudo−observed event status 0 % 25 % 50 % 75 % 100 %

Absolute risk regression Cause−specific Cox Fine−Gray ✷✻ ✴ ✷✽

❙✉♠♠❛r② ♦❢ ❝❛❧✐❜r❛t✐♦♥✿ ❇r✐❡r s❝♦r❡

BS(t, r) = ❊{N(t) − r(t|X)}✷ ❆♣♣❛r❡♥t ♣❡r❢♦r♠❛♥❝❡ ✭s❛♠❡ ❞❛t❛ t✇✐❝❡✮ t✐♠❡ ❘❡❢❡r❡♥❝❡ r✐s❦❘❡❣r❡ss✐♦♥ ❈❛✉s❡❙♣❡❝✐✜❝❈♦① ❋●❘ t✐♠❡✈❛r ✸ ✺✳✷ ✹✳✾ ✹✳✽ ✹✳✾ ✹✳✼ ✶✷ ✶✷✳✶ ✶✵✳✺ ✶✵✳✹ ✶✵✳✸ ✶✵✳✸ ✸✻ ✶✺✳✷ ✶✸✳✷ ✶✸✳✷ ✶✸✳✷ ✶✸✳✶ ❈r♦ss✈❛❧✐❞❛t✐♦♥ ♣❡r❢♦r♠❛♥❝❡ ✭❇❂✶✵✵✵✮ t✐♠❡ ❘❡❢❡r❡♥❝❡ r✐s❦❘❡❣r❡ss✐♦♥ ❈❛✉s❡❙♣❡❝✐✜❝❈♦① ❋●❘ t✐♠❡✈❛r ✸ ✺✳✷ ✺ ✹✳✾ ✹✳✾ ✺ ✶✷ ✶✷✳✶ ✶✵✳✼ ✶✵✳✻ ✶✵✳✻ ✶✵✳✼ ✸✻ ✶✺✳✸ ✶✸✳✺ ✶✸✳✺ ✶✸✳✺ ✶✸✳✺

◮ ❚❤❡ ❧♦✇❡r t❤❡ ❜❡tt❡r ◮ ❚❤❡ ♥✉❧❧ ♠♦❞❡❧ ✐❣♥♦r❡s t❤❡ ❝♦✈❛r✐❛t❡s ◮ ❈♦♥❝❧✉s✐♦♥✿ ❆❧❧ ♠♦❞❡❧s ❛r❡ ❜❡tt❡r t❤❛♥ r❡❢❡r❡♥❝❡✱ ❜✉t ♦t❤❡r✇✐s❡ ❝♦♠♣❛r❛❜❧❡

✷✼ ✴ ✷✽

❙✉♠♠❛r② ❛♥❞ ❞✐s❝✉ss✐♦♥

◮ ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♠♦❞❡❧ ✇✐t❤ ❧♦❣✲❧✐♥❦ ②✐❡❧❞s ❛❜s♦❧✉t❡ r✐s❦

r❛t✐♦s ❛❞❥✉st❡❞ ❢♦r ❝♦♥❢♦✉♥❞❡rs✳

◮ ❆ ❝❛❧✐❜r❛t✐♦♥ ♣❧♦t ✐s ❛ ❣r❛♣❤✐❝❛❧ t♦♦❧ t♦ ✐♥✈❡st✐❣❛t❡ t❤❡

r❡❧✐❛❜✐❧✐t② ♦❢ ❛ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧✳

◮ ■t ❝❛♥ ❜❡ ❡st✐♠❛t❡❞ ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝♦♠♣❡t✐♥❣ r✐s❦s ❛♥❞

r✐❣❤t ❝❡♥s♦r❡❞ ❞❛t❛ ❜❛s❡❞ ♦♥

◮ ❡①t❡r♥❛❧ ✈❛❧✐❞❛t✐♦♥ ❞❛t❛ ◮ ❝r♦ss✲✈❛❧✐❞❛t✐♦♥

◮ ❚❤❡ s❝❛tt❡r♣❧♦t ♦❢ ♣s❡✉❞♦✲✈❛❧✉❡s ✐♥❞✐❝❛t❡s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢

t❤❡ ♣r❡❞✐❝t❡❞ r✐s❦s ❛♥❞ t❤❡ ❧❡✈❡❧ ♦❢ ❝❡♥s♦r✐♥❣✳

◮ ❊st✐♠❛t✐♥❣ ❛ ❝❛❧✐❜r❛t✐♦♥ ♣❧♦t ✐s ❛s ❤❛r❞ ❛s ❡st✐♠❛t✐♥❣ ❛ ❞❡♥s✐t②

❛♥❞ t❤❡ ❝❤♦✐❝❡ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❜❛♥❞✇✐❞t❤ ❛❧❧♦✇s t❤❡ ✉s❡r t♦ ♠❛♥✐♣✉❧❛t❡ t❤❡ ❝❛❧✐❜r❛t✐♦♥ ♣❧♦t✳

✷✽ ✴ ✷✽