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8/18/2019 B. Hall Intro to Frailty Models
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STA635 Project by Benjamin Hall
8/18/2019 B. Hall Intro to Frailty Models
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Cox proportional hazards
modelModel hyi !t" # h$!t"e%&%'(((' )&) is the hazard
*+nction o* the ith indi,id+al
Ass+mption The hazard *+nction *or eachindi,id+al is proportional to the basine hazard-h$!t"( This ass+mption implies that the hazard*+nction is *+lly determined by the co,ariate
,ector(Problem There may be +nobser,ed
co,ariates that ca+se this ass+mption to be,iolated(
8/18/2019 B. Hall Intro to Frailty Models
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Siml+ation .x /nobser,ed
co,ariateConsider a sit+ation 0ith the *ollo0in1 pop+lation
2b,io+sly r+1 A is e4ecti,e *or the entirepop+lation(
B+t 0hat happens in the Cox Model i* the 1ro+p is+nobser,able
Group ProportionofPopulation
HazardRate withplacebo
HazardRate withDrug A
% $7 % (5
8 $7 8 %
3 8$7 %$ 9
8/18/2019 B. Hall Intro to Frailty Models
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Sim+lation .xample-contin+ed:et;s sim+late this example and apply the Cox
model
Ha,e < sim+late %$$ people accordin1 to thepre,io+s table;s probabilities and randomlyassi1n them to treatment or placebo
=or each person- ha,e < sim+late > o*
incidents 0ithin period o* len1th %( At the endo* period o* len1th % ri1ht?censorin1 occ+rs(
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Sim+lation .xample-contin+edHere is some o* the data 1enerated by < !see
@nal slide *or code"
id group treat time status
%-
% 3 $ (39 %
8-
% 3 $ ($ %
3-
% 3 $ (83 %
-
% 3 $ (%5 %
5-
% 3 $ (%% %
6-
% 3 $ (9D $
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Sim+lation .xample-contin+edEo0 0e r+n coxph on o+r data
F my@t%Call
coxph!*orm+la # S+r,!time- stat+s" G treat"
coe* exp!coe*" se!coe*" z p
treat ?$(%89 $(99 $(%% ?%(% $(8
:i)elihood ratio test#%(3 on % d*- p#$(88 n# 5
Eotice that the :
8/18/2019 B. Hall Intro to Frailty Models
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Sim+lation .xample-contin+ed The problem is that 0e ha,e hetero1eneity in
the data d+e to the +nobser,able 1ro+ps(
Since 0e cannot incl+de 1ro+p in o+r model-the ass+mption o* proportional hazards is,iolated(
hat can 0e do to sol,e this problem /se a
*railty model(
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=railty Model=railty models can help explain the +nacco+nted
*or hetero1eneity(
=railty Model hyi !t" # z h$!t"e%&%'(((' )&) is the
hazard *+nction o* the ith indi,id+al
The distrib+tion o* z is speci@ed to be- say-Iamma( !Eote z m+st be non?ne1ati,e since
the hazard is non?ne1ati,e("Jn this sit+ation- the shared *railty model is
appropriate- that is m+ltiple obser,ations o* thesame indi,id+al al0ays has the same ,al+e o* z(
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=railty Model in <:et;s apply the *railty model to o+r sim+lated
dataF my@t8
Callcoxph!*orm+la # S+r,!time- stat+s" G treat ' *railty!id""
coe* se!coe*" se8 ChisK = p
treat ?$(% $(%6$ $(%%% $(95 %($ 3(6e?$%
*railty!id" D3(9D 3(5 %(e?$5
Jterations 5 o+ter- % Ee0ton?
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=railty Model in <Eo0 let;s try implementin1 the *railty model
to a real data set- the )idnet data set(
Here are the res+lts *or the re1+lar Cox ModelF )@t%Call
coxph!*orm+la # S+r,!time- stat+s" G a1e ' sex- data # )idney"
coe* exp!coe*" se!coe*" z p
a1e $($$8$3 %($$8 $($$D85 $(88$ $(93$$
sex ?$(98D3% $(36 $(8D9D5 ?8( $($$55
:i)elihood ratio test#(%8 on 8 d*- p#$($895 n# 6
Here the :
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=railty Model in <Ho0e,er- a *railty model seems applicable in
this sit+ation since their are m+ltipleoberser,ations !i(e( 8 )idneys" per person( Belo0considers *railty
F )@t8Call
coxph!*orm+la # S+r,!time- stat+s" G a1e ' sex ' *railty!id"- data # )idney"
coe* se!coe*" se8 ChisK = p
a1e $($$585 $($%%D $($$99 $(8 % $(66$$$
sex ?%(59D $(6$6 $(358$ %%(D % $($$$5
*railty!id" 83(% %3 $($$$$Jterations o+ter- D Ee0ton?
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< Codelibrary!s+r,i,al"
>I.ETJM.
1entime Q? *+nction!1ro+p- treat" R
i* !1ro+p ## %" R
ret+rn !ro+nd!rexp!%- %?!(5treat""-8""
i* !1ro+p ## 8" R
ret+rn !ro+nd!rexp!%- 8?treat"-8""
i* !1ro+p ## 3" R
ret+rn !ro+nd!rexp!%- %$?8treat"-8""
> P.