Gamma and inverse Gaussian frailty models: A comparative study

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1 Iteratioal Joural of Mathematics ad Statistics Ivetio (IJMSI) E-ISSN: P-ISSN: Volume 4 Issue 4 April. 06 PP Gamma ad iverse Gaussia frailty models: A comparative study Samia A. Adham, Amai A. AlAhmadi (Departmet of Statistics/ Faculty of Sciece/ Kig Abdulaziz Uiversity/ Saudi Arabia) ABSTRACT:Frailty models have become very popular durig the last three decades ad their applicatios are umerous. The mai goal of this mauscript is to compare two frailty models (gamma frailty model ad iverse Gaussia frailty model) each of which has a log-logistic distributio to be its baselie hazard fuctio. A real data set is applied for the two cosidered frailty models i order to deal with models compariso. It has bee cocluded that the gamma frailty model is the best model fits this data set. The the iverse Gaussia frailty model, which provides a better fit of the cosidered data set tha the Cox s model. KEYWORDS Proportioal hazards; Heterogeeity;Shared Frailty Models; Survival Aalysis. I. INTRODUCTION I applicatios of survival aalysis, usually oly a few covariates such as age, sex, severity of disease or laboratory data are kow. It is kow that there are may other factors that cafluece survival, icludig health status, life style, smokig, occupatio ad geetic risk factors. I may applicatios, the populatio uder study caot be assumed to be homogeeous but must be cosidered as a heterogeeous. A popular regressio model for the aalysis of survival data is the Cox proportioal hazards regressio model. It allows testig for differeces i survival times of two or more groups. The frailty approach is a statistical modelig cocept which aims to accout for heterogeeity, caused by umeasured covariates. The frailty model is a radom effect model, where the radom effect (the frailty) has a multiplicative effect o the baselie hazard fuctio. This radom effect explais the depedece i the frailty models. The term frailty was first suggested by [] i the cotext of mortality studies. []suggested a radom effects model i order to accout for the uobserved heterogeeity due to uobserved covariates ad itroduced the model to the literature of ecoomics ad the model is called the mixed proportioal hazards model. [3][4] ad [5] cosidered distributios for the frailty model to fid the best model. [6]used frailty model to explai the deviat behavior of mortality rates at advaced ages. There are may applicatios of the gamma frailty model. [7]studied the expulsio of itrauterie cotraceptive devices. [8]studied recidivism amog crimials usig gamma-weibull model. [9]used the gamma frailty model to check the proportioal hazards assumptios i his study of maligat melaoma. A formal of the goodess-of-fit tests for the gamma frailties was costructed by [0]. They also costruct a ew class of frailty models that exted the gamma frailty model by usig certai polyomial expasios that are orthogoal with respect to the gamma desity. For that exteded family, they obtaied a explicit expressio for the margial likelihood of the data. The order selectio test is based o fidig the best fittig model i such a series of expaded models. A bootstrap was used to obtai p-values for the tests. Simulatios ad data examples illustrated the test s performace. []cosidered gamma distributio as frailty distributio ad the log-logistic distributio as baselie distributio for bivariate survival times. Because this distributio has the advatage of havig simple algebraic expressios for its survivor ad hazard fuctios ad a closed form for its distributio fuctio. []studied the case of severe acute malutritio (SAM) i developig coutries. The, they used expoetial, Weibull ad log-logistic as baselie hazard fuctios ad the gamma as well as iverse Gaussia for the frailty distributios ad the based o AIC criteria, all models were compared for their performace. I this mauscript the aalysis of the right cesored survival data are cosidered. A real data example is applied for illustratio. Sectio () cocers with maximum likelihood approach to the shared frailty models. Recostitutio data set: Recostitutio of blood milk barrier after mastitis is preseted i (3), i details. Fially, Sectio (4) discusses some importat coclusios. II. LIKELIHOOD APPROACH TO SHARED FRAILTY MODELS I this sectio, the shared frailty model, which we cosider i this mauscript, is explaied. The the likelihood fuctio accordig for the right cesored data is preseted. gamma frailty ad the iverse Gaussia frailty modelssuppose that we have a data set of dividuals from some populatio ad i =,,, C subgroups or clusters. Each subgroup cosists of idividuals. The idividuals i Page

2 Gamma ad iverse Gaussia frailty models: A comparative each subgroup have depedet evet times due to uobserved frailty u i. This frailty term may represet aggregate effect of commo gees or shared evirometal effect o survival of members of a give family, such as sibligs, husbad ad wives. The goal is to estimate the frailty variace, θ. The variace of the frailty distributios used to determie the degree of heterogeeity i the study populatio. The frailty model is give as ij t u, β, Z = ο t u i e βz ij, u > 0. (β, Z R) () where ο (t) is a commo baselie hazard fuctio, βis a vector of ukow regressio coefficiets ad, for i =,,, C ad j =,, Z ij is a vector of the observable covariates. The frailties u i are uobserved (radom) commo risk factor shared by all subjects i cluster i assumed to be idetically ad idepedetly distributed radom variables with a commo desity fuctio f(u, θ), where θis the parameter of the frailty distributio. The value of the frailty u i is commo to all idividuals i the cluster. I the literature, differet frailty distributios have bee proposed, such as gamma distributio, iverse Gaussia distributio, positive stable distributio, power variace fuctio distributio, compoud Poisso distributio ad logormal distributio. A more detailed presetatio of the shared frailty models ca be foud i [3]. The likelihood fuctio for right cesored survival data is give by L = G j t f j t δ j F j t g j t δ j,() whereδ j is the cesorig idicator, g ad G are the desity fuctio ad the cumulative distributio fuctio of the cesorig time, respectively; adf ad F are, respectively, the desity fuctio ad the cumulative distributio fuctio of the evet time. The distributio of cesorig times i the likelihood fuctio ca be igored because it does ot deped o the parameters of iterest related to the survival fuctio. Therefore, assumig right cesorig, the likelihood fuctio give by () ca be rewritte as L = (f j t ) δ j (S j (t)) δ j, (3) where S j t = F j t is the survival fuctio of the evet time. Cosiderig the shared frailty model preseted above, the likelihood fuctio for the j t subject i the i t subgroup is give by L i = f ij t δ ij i (Sij (t)) δ ij. (4) Sice ij t = f ij t S ij t, the the likelihood fuctio (4) reduces to L i = ij t δ ij i Sij t.(5) The coditioal likelihood fuctio for the i t subgroup is the give by L i ψ, β u i = ( ο t u i e βz ij ) δ ij e H ο t u ie β Zij, where, ψ is a vector of parameters of the baselie hazard fuctio. It follows that, the margial likelihood fuctio for the i th subgroup is L i ψ, θ, β = ( ο t u i e βz ij ) δ ij 0 e H ο t u ie β Z ij g k u i du where g k (u i ) is the probability desity fuctio of the frailty u i, k =,, i =,,, C. For k =,, the probability desity fuctios of frailties i the gamma frailty model ad the iverse Gaussia frailty model are, respectively, give by Page

3 g u i = u i θ exp ( u i Γ( θ )θ θ θ ) ad g u i = πθ Gamma ad iverse Gaussia frailty models: A comparative u 3 i e u i u iθ, (u i, θ > 0). Where u i > idicates that idividuals i groupi are frail, whereas u i < idicates that idividuals are strog ad have lower risk. The log-logistic hazard fuctio ad cumulative hazards fuctios with parameters ψ = α, κ are, respectively, give by exp (α)κt κ o t = ad H +exp (α)t κ o t = l exp (α)κt κ +exp (α)t κ. Hece, the margial log likelihood fuctio for the gamma frailty with the log-logistic baselie hazard fuctios l ψ, θ, β = G i= d i log θ log Γ + log Γ θ θ + d i + δ ij βz ij + log exp (α)κtκ + exp (α)t κ θ + d i log + θ l + exp (α)t κ e βz ij Ad the margial log likelihood fuctio for the iverse Gaussia frailty model with the log-logistic baselie hazard fuctios l ψ, θ, β = G i= δ ij log exp α κtκ + exp α t log l + exp α κ tκ θ l + exp α t κ + θ + θ + log θ + l + exp α t κ + θ By maximizig the log likelihood fuctio for each of the two frailty models proposed above, oe ca obtai the maximum likelihood estimates for the parametersψ, θad β, see [3]ad [4]. III. NUMERICAL ILLUSTRATION I this sectiowe applied a real data set to compare the Cox proportioal hazards model with its extesios, the gamma frailty model ad the iverse Gaussia frailty model, each of which has the loglogistic distributio as a baselie hazard fuctio. The comparisos betwee the Cox proportioal hazard model ad the cosidered gamma ad iverse Gaussia frailty models are based o the Akaike Iformatio Criteria (AIC) ad the Bayesia Iformatio Criteria (BIC). The effect of the cosidered frailty o the coefficiets of the treatmet effects is examied. The real right cesored data set, called Recostitutio data set: Recostitutio of blood milk barrier after mastitis, is applied for the required comparisos. (Source of the data set: I order to perform the required computatios, the statistical package R is used. This study assumes two covariates Drug ad heifer with coefficiets β ad β, respectively. However, the two treatmet times (active compoud ad placebo) for each cow is assumed idepedet.the, the frailty modelca be writte as: ij t = t u i exp β drug + β eifers. Hece,thegamma ad the iverse Gaussia frailty models with the log-logistic hazard fuctio are applied to this data seti order to compare the effects of these two frailty models. I additio, the Cox s model is also applied to this data set to compare it with the cosidered two frailty models. The maximum likelihood estimate of the parameters of the gamma ad the iverse Gaussia frailty models ad the Cox s modelare computed. Furthermore, the compariso of the gamma ad the ivers Gaussia frailty modelsare cosidered accordig to the AIC ad BIC. 3 Page

4 Gamma ad iverse Gaussia frailty models: A comparative Table : Parameters estimates parameter Gamma Iverse Gaussia Cox θ (SE) (0.5) 0.347(0.9) β (SE)(p-value) 0.453(0.7) (0.009) (0.73) (0.009) 0.3(0. 65) β SE (p-value) 0.378(0.3) (0.0) (0.0) (0.09) 0.340(0. 45) AIC BIC Table () provides the maximum likelihood estimates of the parameter θ ad the regressio parametersβ ad β of the gamma frailtyad iverse Gaussia frailty models the with log-logistic baselie hazard fuctio. I the case of ot icludig frailty (Cox s model), it is clear that the regressio coefficiets (β, β ) of the effect of the two covariate heifer ad Drug are biased dow. Whereas, for the gamma ad the ivers Gaussia frailty models, the regressio estimates ad their stadard errors (SE) icrease, which is predictable because the frailty variable u i is icluded i the model. The p-value of theregressio coefficiets (β, β )of drug ad heifer are ad 0.0, respectively; which idicates that β is sigificat while β is ot sigificat for the gamma frailty ad the same results are fouded for the ivers Gaussia frailty. It is clear that from Table () the parameter θ of the gamma frailty is less tha the parameter θofthe ivers Gaussia frailty, which idicates that icludig gamma frailty gives a better fit tha the ivers Gaussia frailty. The estimate of the variace of frailty term θ equal ad for the gamma ad the ivers Gaussia frailties, respectively. The p-value ofθ equal ad 0.004, respectively, for the gamma ad theivers Gaussia frailty models. That meas that the heterogeeity parameter θ is sigificat i the two cosidered frailty models. The AIC ad BIC value are computed for the gamma frailty ad the iverse Gaussia frailty models with the log-logistic baselie hazard fuctio ad for the Cox s model. The smallest AIC ad BIC values suggest the model that gives better fit for the data tha other models. Oe ca see from Table (), that the gamma frailty model gives the best fit to this data set the the ivers Gaussia frailty model is better tha Cox s model. Figure : The hazard fuctios for ivers Gaussia frailty ad gammafrailty with thelog-logistic baselie hazard fuctio. 4 Page

5 Gamma ad iverse Gaussia frailty models: A comparative The hazard fuctios for bothgamma ad iverse Gaussia frailty models are icreasig ad the decreasig ad this is expected. It is clear from Fig.() that the two curves of the hazard fuctios of the gamma frailty model ad the ivers Gaussia frailty model are compatible. The estimated values of the parameters of the log-logistic baselie hazard fuctios are give Table (). Table : The value of the parameter estimate of the log-logistic baselie hazard fuctio parameter gamma frailty ivers Gaussia frailty α.93.5 κ For the log-logistic baselie hazard fuctio, it is kow that,theegative sig of the parameterα idicates that the hazard fuctios icreasig ad the decreasig. IV. CONCLUSIONS This study compares the gamma ad the iverse Gaussia frailty models whe assumig the loglogistic distributio as their baselie hazard fuctio. The maximum likelihood estimatio method is cosidered to estimate the parameters of the cosidered models i order to compare them through estimatio ad testig the sigificace of the parameters of the models uder cosideratio. A real data set called Recostitutiodata set is applied to compare the two frailty models. The AIC ad BIC were computed to assess the cosidered frailty models which of them gives the best fit to this data set. It has bee foud that the gamma frailty model is the best model that fits this data set amog the other two models. The, the ivers Gaussia frailty model fits the databetter tha the Cox s model. Furthermore, it has bee foud that, the heterogeeity parameter θ is sigificat i both gamma ad iverse Gaussia frailty models. REFERENCES [] J.W. Vaupel,K.G. Mato, ad E. Stallard, The impact of heterogeeity idividual frailty o the dyamics of mortality,demography6,979, [] T. Lacaster, Aalysis of Trasitio Data (Uiversity of Cambridge, New work, 990). [3] P. Hougaard, Life Table Methods for Heterogeeous Populatios: Distributios Describig Heterogeeity.Biometrika,7, 984, [4] P. Hougaard, Survival models for heterogeeous populatios derived from stable distributios. Biometrika73, 986 a, [5] P. Hougaard, A class of multivariate failure time distributios. Biometrika73, 986 b, [6] J.W. Vaupel, ad A.I. Yashi, Heterogeeity s ruses: some surprisig effects of selectio o populatio dyamics. America Statisticia,39, 985, [7] O.O. Aale, Two examples of modellig heterogeeity i survival aalysis. ScadiaviaJouralofStatistics4, 987, 9-5. [8] R. Ellerma, S. Pasquale, ad J. M. Tie,A Alterative Approach to Modelig Recidivism Usig Quatile Residual Life Fuctios. OperatiosResearch40, 99, [9] P.K. Aderse, O. Borga, R.D. Gill, ad N. Keidig, Statistical models based o coutig processes (Spriger Verlag, New York, 993). [0] C. Geerdes,G. Claeskes, ad P. Jasse, Goodess-of-fit tests for the frailty distributio proportioal hazards models with shared frailty, Biostatistics, 0, -4. [] D.D. Haagal, ad R. Sharma, Modelig Heterogeeity for Bivariate Survival Data by Shared Gamma Frailty Regressio Model, Model Assisted Statistics ad Applicatios, 8, 03, [] Babeta.A, D. Seyoum, T. Belachew, B. Birlie, ad Y. Getachew, Modelig time-to-cure from severe acute malutritio applicatio of various parametric frailty models. Archives of Public Health, 73(6), 05. Doi: 0.86/ [3] LDuchateau, ad P. Jasse,The Frailty Model. (Spriger-Verlag, New York, 008). [4] U.A. Abdulkarimova, Frailty Models for Modellig Heterogeeity, McMaster Uiversity, MSc, Page