B. Hall Intro to Frailty Models

8/18/2019 B. Hall Intro to Frailty Models

1/13

STA635 Project by Benjamin Hall


2/13

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(


3/13

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


4/13

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(


5/13

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 $


6/13

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 :


7/13

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(


8/13

=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(


9/13

=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?


10/13

=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 :


11/13

=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?


12/13


13/13

< 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.

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B. Hall Intro to Frailty Models