θ = θ Π Π Parametric counting process models θ θ θ Log-likelihood: Consider counting processes: Score functions:
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1 Paramerc coug process models Cosder coug processes: N,,..., ha cou he occurreces of a eve of eres for dvduals Iesy processes: Lelhood λ ( ;,,..., N { } λ < Log-lelhood: l( log L( Score fucos: U ( l( log λ ( ; dn λ ( ; d λ λ log ( ; dn ( ; d he MLE ˆ ( ˆ may be foud by maxmzg,..., ˆ q he log-lelhood or by solvg he lelhood equaos (umercally, f eeded U (,,..., q ˆ ( ˆ ˆ,..., q s approxmaely mulvarae ormally dsrbued aroud s rue value wh a covarace marx ha may be esmaed as Ι ( ˆ where Ι ( s he observed formao marx wh elemes Example.: Pecewse cosa hazard ad occurrece/exposure raes - Assume ha λ ( ; Y for < h ( U ( l( h h he lelhood rao, score ad Wald ess apply as usual Iroduce: O N ( N ( "occurrece" Y ( u du "exposure"
2 he lelhood may be wre N { } λ < N { Y } < Π ( ( exp Y O exp Log-lelhood: l( O log Score fucos U ( l( he MLEs become he occurrece/exposure raes: ˆ O he observed formao marx has elemes h ( U ( h O { h } Ι ( ( O for h for h 6 hus O O Ι ( dag,..., Now ˆ ( ˆ ˆ,..., q s approxmaely mulvarae ormally dsrbued aroud s rue value wh a covarace marx ha may be esmaed as ˆ O O Ι ( dag,..., ˆ ˆ se( ˆ ˆ O ˆ ˆ dag,..., O O I follows ha occurrece/exposure raes are approxmaely depede ad ormally dsrbued wh sadard errors ˆ 7 Posso regresso Assume proporoal hazards: α ( x α( ; exp( β x wh pecewse cosa basele hazard: α ( ; for < Iroduce: O N ( N ( Y ( u du 8
3 Lelhood: he lelhood s proporoal o he lelhood of "depede Posso varables" O wh "parameers" e N { } λ < β x N { } < β x Π ( e Y exp e Y β x O {( exp β x β x e ( e } O {( exp β x β x e ( e } 9 Esmao procedure: F model by GLM-sofware reag he O as "depede Posso varables" wh "parameers" wh { ψ β x log } β x e exp + + ψ log Use logarhmc l ad log as offses me erval s reaed as a caegorcal covarae A dvdual corbues oe record o he daa fle for each me erval whe he/she s a rs Posso regresso caegorcal covaraes Assume he covarae vecors x ca oly aa L dsc values: x (,, x (L ( l ( l O ( l β x ( ( β x l ( L( β, e exp e l Suffce sascs (sums are over wh x x (l l O O ad ( l ( l ( l ( l O ( l β x ( l ( ( β x l e exp( e Moraly hree Norwega coues Basele Grea compuaoal savgs for large daa ses (Number of records aggregaed daa se: 6**6
4 Shapes of hazard raes x x Moraly of paes wh myocardal farco. Dvorce raes. Moraly of cacer paes. Icdece of chldhood leuema. Geeral moraly. Icdece of mos cacers. How ca he dffere paers be erpreed? We eed o dsgush bewee he dvdual hazards ad he populao hazard Heerogeey survval aalyss Dffereces bewee dvduals are of wo ds: hose observed hrough covaraes hose ha are uobserved he laer oes are usually dsregarded survval aalyss hs may lead o dsoros as explaed he fraly heory here s a large leraure o fraly heory, bu we wll revew us a few basc ssues Populao hazard ad dvdual hazard We wll dsgush bewee he populao hazard rae ad he dvdual hazard raes (deals below he populao hazard rae s flueced by seleco: hose wh hghes rs experece he eve early he shape of he populao hazard may be erely dffere from ha of he dvdual hazards Hece he populao hazard ca o be erpreed as gvg formao o dvdual developme rs A smple example Assume a populao s composed of wo groups, a "low rs group" ad a "hgh rs group" A he ouse, he proporos he wo groups are p ad p - p Low rs group has hazard α { α u du} S exp ( Hgh rs group has hazard α { α } S exp ( u du ad survval fuco ad survval fuco 6
5 Illusrao: Populao survval fuco S p S + p S..... Hazard raes 6 8 me ed: low rs Blac: hgh rs Survval fucos 6 8 me 7 Populao hazard rae ( µ S S wh ps + ps p S + p S ps S ps S + ps + ps ( ( + ( ( S p S p S S { } w α + w α w ps p S + p S 8 Illusrao: Hazard me ed: low rs Blac: hgh rs Blue: populao 9
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