Diagnostic tests for frailty

Transcription

1 Diagnosic ess for fraily P. Economou and C. Caroni Deparmen of Mahemaics School of Applied Mahemaics and Physical Sciences Naional Technical Universiy of Ahens 9 Iroon Polyechniou, Zografou Ahens, Greece ( polikon@mah.nua.gr, ccar@mah.nua.gr) Absrac. A common way of allowing heerogeneiy beween individuals in models for lifeime daa is o inroduce an unobservable individual random effec Z. In a proporional hazards framework, he individual s hazard becomes zh b () where h b () is he baseline hazard. The random variable Z is ofen assumed o follow he Gamma or Inverse Gaussian disribuion. We develop here diagnosic ess for hese assumpions. One simple graphical diagnosic is based on he form of he uncondiional survival funcion when h b () is assumed o be Weibull. Anoher plo uses a closure propery of a family of fraily disribuions, which implies ha he fraily among survivors a ime has he same form as he original disribuion of Z, wih he same shape parameer bu differen scale parameer. In his mehod, we esimae he shape parameer a differen imes and examine graphically wheher i is consan. We give simulaion resuls and examples o illusrae hese mehods. Keywords: Lifeime daa, fraily, proporional hazards, Burr disribuion, Generalized Inverse Gaussian disribuion. 1 Inroducion When modelling daa obained from ime-o-even sudies, i is ofen found ha here is heerogeneiy beween individuals, over and above wha can be accouned for by any available covariaes. One common way of allowing for his heerogeneiy is o inroduce an unobservable individual random effec Z, he so-called fraily. This is usually assumed o operae in a proporional hazards framework, so ha i acs muliplicaively on he baseline hazard funcion h b which is common o all individuals. Thus he hazard funcion for an individual wih fraily Z = z is given by h( z) = zh b () If here are also measured covariaes, he model is usually exended o h( z; X) = ze β x() h b () where x(.) is a q-dimensional vecor of possibly ime-dependen covariaes.

2 1202 Economou and Caroni Inroducing heerogeneiy in lifeime daa by means of an unobserved quaniy in his way was iniiaed by [Clayon, 1978], [Vaupel e al., 1979] and [Hougaard, 1984]. The disribuion of he random variable Z is ofen assumed o be Gamma [Vaupel e al., 1979] or Inverse Gaussian [Hougaard, 1984]. As wih any par of he process of saisical modelling, i is desirable o check ha he assumed disribuion is suppored by he daa. The purpose of he presen paper is o develop diagnosic plos for he fraily disribuion, wih he emphasis on hese wo common choices, he Gamma and Inverse Gaussian. We will be assuming ha he baseline hazard funcion h b has been specified correcly and ha he muliplicaive proporional hazards fraily model is he proper one o describe he daa. 2 Diagnosic plos for mixures From he proporional hazards assumpion, i follows ha he condiional survivor funcion for an individual wih fraily z is S( z) = [S b ()] z where S b is he baseline survivor funcion. In paricular, if he baseline model is aken o be Weibull (η, β), hen he survivor funcion condiional on fraily z is S( z) = exp( zs) where s = (/η) β. If Z has disribuion funcion G on (0, ), hen he uncondiional survivor funcion is given by 0 exp( zs)dg(z) If G is aken o be Gamma wih shape and scale parameers boh equal o ν (so ha he mean is one), hen S() = ( 1 + s ν ) ν (1) (he Burr disribuion), while aking G o be Inverse Gaussian wih scale 1 and shape λ yields he survivor funcion ( ( )) 2s S() = exp λ 1 λ + 1 (2) (In boh cases, a consrain has been applied o he parameers of G o make he model idenifiable. Oher choices of consrain are possible bu make no difference in principle.) We now look for appropriae diagnosic plos o enable us o check ha he assumpion of one of hese disribuions is in fac correc. The idea is o

3 Diagnosic ess for fraily 1203 plo some funcion of he non-parameric Kaplan-Meier esimae of survival, Ŝ(), agains some funcion of, o obain he characerisic shape associaed wih he disribuions (1) and (2). Taking logarihms of (1), and supposing ha s/ν is large enough so ha log(1 + s/ν) log(s/ν), we see ha logŝ() agains log should give a sraigh line. [Wolsenholme, 1999] suggess his plo for he Pareo disribuion. This is he special case of he Burr disribuion when he baseline hazard is exponenial, which is a special case of he Weibull (β = 1). However, we observe ha he above approximaion is poor for he early failures. These give he plo a characerisic horizonal secion, whose lengh depends on ν, disappearing as ν becomes small (high degree of heerogeneiy) (Figure 1). When he fraily Ln(S()) v=0.1 Ln(S()) v=0.5 Ln(S()) v= Fig.1. Diagnosic plo for Burr disribuion for various values of he parameer ν simulaions of samples of size 1000, baseline Weibull parameers 1000 (scale) and 2 (shape). Type I censoring a =3000. disribuion is he Inverse Gaussian, aking logarihms of (2) suggess he plo of log( logŝ()) agains log if λ is large. Under hese circumsances, here is only a small degree of heerogeneiy and he disribuion will no be very differen from he baseline Weibull disribuion. This is he sandard diagnosic plo used o check for he Weibull disribuion. I gives a sraigh line wih slope equal o he shape parameer β. On he oher hand, suppose ha λ is large. I is easy o show ha in his case he Weibull(η, β) - Inverse Gaussian(1,λ) mixure ends o he Weibull wih scale parameer η/(2λ) 1/β and shape β/2. Thus he same plo gives a sraigh line wih slope β/2. For inermediae values of λ, he plo should be curved wih slope falling from β o β/2 as ime increases. Examples are shown in Figure 2. 3 Closure propery of he fraily disribuions To develop anoher kind of diagnosic plo, we sar wih a closure propery of Gamma fraily in he proporional hazards model ([Vaupel e al., 1979]).

4 1204 Economou and Caroni Ln( Ln(S())) lamda=0.001 Ln( Ln(S())) lamda=1 Ln( Ln(S())) lamda= Fig. 2. Diagnosic plo for Weibull-Inverse Gaussian mixure for various values of he parameer λ. Deails of simulaions as in Figure 1. Given ha he fraily disribuion among all individuals is Gamma wih scale parameer κ and shape parameer λ, hen he fraily disribuion among he populaion of survivors a ime is again Gamma wih he same shape parameer λ bu differen scale parameer given by κ + H b (), where H b () is he cumulaive baseline hazard funcion. This propery can be generalized o a whole family of disribuions. 3.1 Generalizaion of he Gamma fraily propery The Gamma fraily propery given by [Vaupel e al., 1979] can be generalized firs o he case of available covariaes. More specifically, given ha he fraily disribuion is Gamma(κ, λ), hen he fraily disribuion among survivors a ime, condiional on he value of he covariaes, is Gamma (κ + Hb x (), λ), where Hb x () is given by H x b () = 0 e β x(u) h b (u)du. The closure propery is no a characerizaion of he Gamma disribuion only. I is quie easy o show ha a similar propery holds also for he Inverse Gaussian and he Generalized Inverse Gaussian (GIG). Furhermore, a similar propery holds for a whole class of disribuions ha belong o he exponenial family [Hougaard, 1984]. Le fraily Z be a random variable wih disribuion F(α) on (0, ), where α is he parameer vecor, wih p.d.f. of he form f z (z) = e [z,g(z)][η1(α),η2(α)] ξ(z) Φ(α) which is an exponenial family disribuion wih canonical saisics z and g(z) [Shao, 1998]. For his fraily disribuion he following heorem holds, which exends Hougaard s resul by including covariaes.

5 Diagnosic ess for fraily 1205 Theorem 1 Given he fraily disribuion F(α) wih p.d.f. as above, hen under he proporional hazards fraily model he fraily disribuion among survivors a ime is again F(.). The value of η 1 (α), he elemen of he parameer vecor corresponding o z, changes, bu he componens of η 2 (α) do no. More specifically, he p.d.f. of fraily among survivors a ime is given by f Z T > (z) = e [z,g(z)][η 1 (α),η2(α)] Φ ξ(z) (α) where η 1 (α) = η 1(α) + H x b () and Φ (α) = Φ(α)S T (). The GIG disribuion (and hence he Gamma and Inverse Gaussian disribuions which i includes) belongs o he above class of he exponenial family. Unforunaely, some oher disribuions, like he lognormal, which are also widely used as fraily disribuions, do no belong o his class because hey do no have z as a canonical saisic. This obsacle can be overcome by considering a generalized disribuion, adding one more parameer [Hougaard, 1986] which will be zero iniially. So, he Theorem can be applied o all disribuions F(α) wih p.d.f given by f z (z) = e T(z)[η(α)] Φ(α) ξ(z) where T(z) does no conain z as a componen, since he above disribuion can be seen as a special case of GF(α, β) wih p.d.f. given by f z (z) = e [z,t(z)][β,η(α)] Φ G (α, β) ξ(z) for β = 0. Φ G (α, β) is he inegral over he range of z, R(z), of he numeraor of he previous relaionship. Applying he Theorem o he disribuion GF(α, 0) shows ha he fraily disribuion among he survivors a ime will be again GF bu wih parameer vecor given by (α, H x b ()). 3.2 Plos Given ha he fraily disribuion has been chosen correcly, hen our above Theorem shows ha he vecor η 2 (α) of he iniial parameers does no change when we resric our aenion o he fraily disribuion among hose unis ha have survived unil ime. Le ˆη 2i (α) T> denoe componen i of he maximum likelihood esimae of his vecor among he survivors a ime. If our assumpion of he fraily disribuion is correc, hen ˆη 2i (α) T> for any ime is an asympoically unbiased esimaor of he same quaniy η 2i (α). Therefore, a plo of ˆη 2i (α) T> agains ime should give a sraigh line parallel o he ime axis.

6 1206 Economou and Caroni For he Gamma(κ, λ) and Inverse Gaussian(κ, λ) disribuions of fraily his plo reduces o ˆλ T> versus since η 2 (κ, λ) = λ. For he GIG(λ, δ, γ), he proposed plos are of ˆλ T> and ˆδ T> agains, since η 2 (λ, δ, γ) = (λ 1, δ 2 ) for his disribuion. In all cases, he maximum likelihood esimaes of he model s parameers ha are required for he plos are obained by maximising he logarihm of he usual likelihood funcion for lifeime daa ( i : i = 1, 2...n) L = n { h(i ) δi S( i ) } i=1 where δ i is he censoring indicaor which akes he value 1 if i is an observed lifeime and zero if i represens a righ censored observaion. The expressions for S() are given above in (1) and (2) for he Gamma and Inverse Gaussian fraily disribuions, respecively, and he hazard funcion h() can be obained as minus he derivaive of logs. We firs carry ou his esimaion using all he daa. Then we selec a sequence of convenien ime poins τ j (j = 1, 2...k) and repea he esimaion k imes, using in he ih esimaion only hose daa poins i saisfying i τ j. 3.3 Simulaions To illusrae he mehod, we simulaed a se of 1000 uncensored daa poins from he Burr disribuion (Weibull-Gamma mixure) and produced he above plo based on repeaed esimaes of ν. Then we fied he incorrec Weibull-Inverse Gaussian mixure o he same daa and produced he corresponding plo (Figure 3). Nex we repeaed he exercise wih he roles of he wo disribuions reversed. Thus we generaed a se of daa from he Weibull- Inverse Gaussian mixure and produced he plos for boh he correc model and for he incorrec Burr disribuion (Figure 4). In boh cases, he plos Gamma shape Inv Gaus shape T T Fig. 3. Plo defined in Secion 3.2, fiing (lef) correc Burr disribuion, (righ) incorrec Weibull-Inverse Gaussian mixure o daa generaed from Burr (ν = 1).

7 Diagnosic ess for fraily 1207 Gamma Shape T Inv Gaus Shape T Fig. 4. Plo defined in Secion 3.2, fiing (lef) incorrec Burr disribuion, (righ) correc Weibull-Inverse Gaussian mixure o daa generaed from Weibull-Inverse Gaussian mixure (λ = 0.5). discriminae exremely well beween he wo fraily disribuion; he plo for he correc disribuion is a horizonal sraigh line as prediced, bu he plo for he incorrec disribuion depars clearly from a horizonal line. 4 Example For a real-daa illusraion of our mehods, we used daa on he duraion of a readmill es underaken by 978 successive paiens a a cardiac clinic in Ahens. Figure 5 shows he simple diagnosic plos ha were developed in Secion 2. The plo for he Burr disribuion, on he righ, has he expeced shape of a sraigh line preceded by a horizonal secion. The plo for he Weibull-Inverse Gaussian mixure, on he lef, is curved as expeced, bu he curvaure is greaer han i should be if his is he correc model. Figure 6 shows he diagnosic plos ha were developed in Secion 3. These indicae Ln( Ln(S())) Ln Ln(S()) Ln Fig. 5. Plos defined in Secion 2 for daa on 978 cardiac paiens. Lef: Weibull- Inverse Gaussian mixure; righ: Burr.

8 1208 Economou and Caroni Inv Gaus Shape Gamma Shape Fig. 6. Plos defined in Secion 3 for daa on 978 cardiac paiens. Lef: Weibull- Inverse Gaussian mixure; righ: Burr. clearly ha he assumpion of a Gamma disribuion for fraily is accepable, because he esimaes of is shape parameer a differen imes are scaered abou a horizonal line, bu he Inverse Gaussian assumpion is no. 5 Furher research Alhough graphical diagnosics have proved very useful in saisical modelling, i can also be valuable o have formal saisical ess for he presence of any fraily, hus showing wheher or no i is necessary o use he models considered here. In hese models, one parameer of he disribuion conrols boh he presence and he degree of fraily. For example, when he fraily disribuion is Gamma(ν, ν), he parameer ν conrols he amoun of fraily since V (Z) = 1/ν 0 (ν ). If he basic disribuion is Weibull and he uncondiional disribuion is herefore Burr, he presence of any fraily can be examined by esing he null hypohesis ν = (or 1/ν = 0) by likelihood-based mehods applied o he Burr disribuion. The heory for one of hese mehods, he score es, was given by [Crowder and Kimber, 1997] for mulivariae lifeime daa and he deails for he univariae case which we are ineresed in by [Kimber, 1996]. (In fac, he es also holds for oher Weibull mixures, no jus for Weibull-Gamma = Burr.) Oher likelihood-based ess ha can be applied include a Wald es and a likelihood raio for his parameer. A difficuly ha arises is ha he null value of he parameer being esed falls on he boundary of he parameer space. In such cases, he disribuion of minus wice he log likelihood raio is no given by he usual chi-squared approximaion. Insead, a mixure of chi-squared disribuions usually applies. We inend o complee a sudy of he likelihood raio es for his model and hen carry ou a simulaion sudy o compare he properies of he differen ess in order o recommend he bes one for use in pracice.

9 Diagnosic ess for fraily 1209 References [Clayon, 1978]D.G. Clayon. A model for associaion in bivariae life ables and is applicaion in epidemiological sudies of familial endency in chronic disease incidence. Biomerika, 65: , [Crowder and Kimber, 1997]M. Crowder and A. Kimber. A score es for he mulivariae burr and oher weibull mixure disribuions. Scandinavian Journal of Saisics, 24: , [Hougaard, 1984]P. Hougaard. Life able mehods for heerogeneous populaions: Disribuions describing he heerogeneiy. Biomerika, 71:75 83, [Hougaard, 1986]P. Hougaard. Survival models for heerogeneous populaions derived from sable disribuions. Biomerika, 73: , [Kimber, 1996]A. Kimber. A weibull-based score es for heerogeneiy. Lifeime Daa Analysis, 2:63 71, [Shao, 1998]J. Shao. Mahemaical Saisics. Springer-Verlag, New York, [Vaupel e al., 1979]J.A. Vaupel, K.G. Manon, and E. Sallard. The impac of heerogeneiy in individual fraily on he dynamics of moraliy. Demography, 16: , [Wolsenholme, 1999]L.C. Wolsenholme. Reliabiliy Modelling: A Saisical Approach. Chapman and Hall/CRC, Boca Raon, 1999.