Weibull Extension of Bivariate Exponential Regression

Stat Papers (2009) 50:29–49DOI 10.1007/s00362-007-0057-4

R E G U L A R A RT I C L E

Weibull extension of bivariate exponential regressionmodel with different frailty distributions

David D. Hanagal

Received: 2 August 2006 / Revised: 15 February 2007 / Published online: 7 March 2007© Springer-Verlag 2007

Abstract We propose bivariate Weibull regression model with frailty whichis generated by a gamma or positive stable or power variance function distri-bution. We assume that the bivariate survival data follows bivariate Weibull ofHanagal (Econ Qual Control 19:83–90, 2004; Econ Qual Control 20:143–150,2005a; Stat Pap 47:137–148, 2006a; Stat Methods, 2006b). There are some inter-esting situations like survival times in genetic epidemiology, dental implantsof patients and twin births (both monozygotic and dizygotic) where geneticbehavior (which is unknown and random) of patients follows known frailtydistribution. These are the situations which motivate to study this particularmodel.

Keywords Bivariate Weibull · Frailty · Gamma · Positive stable ·Power variance function · Parametric regression · Survival times

1 Introduction

The shared gamma frailty models was suggested by Clayton (1978) for theanalysis of the correlation between clustered survival times in genetic epidemi-ology. An advantage is that without covariates its mathematical properties areconvenient for estimation [See (Oakes 1982, 1986)]. However, when adjusting

David D. Hanagal is on leave from Department of Statistics, University of Pune, Pune 411007,India.

D. D. Hanagal (B)Department of Statistics and Probability, Michigan State University,East Lansing, MI 48824, USAe-mail: [email protected]; david−[email protected]

30 D. D. Hanagal

for environment risk factors the analysis of the clustering is more difficult [See(Parner 1998)]. Until recently, a lack of theory and reliable software had pre-vented widespread use of the model.

In a frailty model, it is absolutely necessary to be able to include explana-tory variables. The reason is that the frailty describes the influence of commonunknown factors. If some common covariates are included in the model, thevariation owing to unknown covariates should be reduced. Common covariatesare common for all members of the group.

For monozygotic twins, examples are sex and any other genetically basedcovariate. Both monozygotic and dizygotic twins share date of birth and com-mon pre-birth environment. By measuring some potentially important cova-riates, we can examine the influence of the covariates, and we can examine,whether they explain the dependence, that is, whether the frailty has no effect(or more correctly, no variation), when the covariate is included in the model.

It is not possible in practice to include all relevant covariates. For example,we might know that some given factor is important, but if we do not knowthe value of the factor for each individual, we can not include the variable inthe analysis. For example, it is known that excretion of small amounts of albu-min in the urine is a diagnostic marker for increased mortality, not only fordiabetic patients, but also for general population. However, we are unable toinclude this variable, unless we actually obtain urine and analyze samples foreach individual under study. It is furthermore possible that we are not awarethat there exist variables that we ought to include. For example, this couldbe a genetic factor, as we do not know all possible genes having influence onsurvival. This consideration is true for all regression models, not only survivalmodels. If it is known that some factor is important, it makes sense to try toobtain the individual values, but if it is not possible, the standard is to ignorethe presence of such variables. In general terms, we let the heterogeneity gointo the error term. This will, of course, lead to an increase in the variabil-ity of the response compared to the case, when the variables are included. Inthe survival data case, however, the increased variability implies a change inthe form of the hazard function, as will be illustrated by some more detailedcalculations.

There are some situations where T1 and T2 are dependent, for example,paired organs like kidneys, lungs, eyes, ears, dental implants, etc. are depen-dent on each other. So, we are interested mainly on paired data with commonshared frailty. Hence we have two types dependencies. The first one is dueto the dependence of life times of the components in such a way that failureof one component increases the load on the other component and hence in-crease in the failure rate of survived component or the dependence may bedue to a common shock which results simultaneous failure of components.The second type of dependence is due to frailty, the unobserved covariate, forexample, the influence of genes on the survival. There may be several type ofdependencies in the components of a system. One has to take into accountall these type of dependencies in order to get correct model without losinginformation.

Weibull extension of bivariate exponential regression model 31

The regression model is derived conditionally on the shared frailty (Y).Conditionally on Y, the hazard function of lifetimes (T1, T2) is assumed to beof the form

Y · M(t1, t2)

where M(t1, t2) is the bivariate cumulative or integrated hazard function of the(T1, T2) and that the value of Y is common to two components in a group. Whenthere is no variability in the distribution of Y, that is, when Y has a degener-ate distribution. When the distribution is not degenerate, the dependence ispositive. The value of Y can be considered as generated from unknown valuesof some explanatory variables. Conditional on Y = y, the bivariate survivalfunction is

S(t1, t2 | y) = e−y·M(t1,t2) (1)

When T1 and T2 are independent, M(t1, t2) = M1(t1) + M2(t2), where Mi(ti),i = 1, 2 are the integrated hazards of T1 and T2, respectively. From this, weimmediately derive the bivariate survival function by integrating Y out

S(t1, t2) = Ee−Y·M(t1,t2)

= L(M(t1, t2)

)(2)

where L(·) is the Laplace transform of the distribution of Y. Thus, the bivariatesurvivor function is easily expressed by means of the Laplace transform of thefrailty distribution, evaluated at the total integrated conditional hazard.

The natural parametric distribution to consider is the Weibull, because itallows for both the proportional hazard model and the accelerated failure timemodel. There is no unique natural extension of Weibull distribution in thebivariate or multivariate situation. So, we have different versions of bivariateor multivariate Weibull distributions, each has its own merits and demerits.These distributions have been derived from the exponential distribution bytaking power transformation.

Lu (1989) extended Weibull extensions of Freund and Marshall–Olkin bivari-ate exponential models. Later, Hanagal (1996) extended to multivariate Weibull(MVW) distribution which is the extension of bivariate Weibull of Lu (1989)and also the extension of the multivariate exponential of Marshall and Olkin(1967). Hanagal (2004, 2005a, 2006a,b) proposed bivariate Weibull (BVW)regression models for the survival data without using frailty distribution. TheseBVW models are based on the extension of bivariate exponential of Marshalland Olkin (1967) and Freund (1961). Recently Hanagal (2005b, 2006c) pro-posed BVW regression model (Weibull extension of bivariate exponential ofMarshall–Olkin) with positive stable and gamma frailties respectively and alsoobtained maximum likelihood estimators (MLEs) of the parameters in themodel.

32 D. D. Hanagal

Lee (1979) extended BVE of Gumbel (1960) model to bivariate Weibull(BVW) distribution by introducing shape parameter. Hanagal (2004, 2005a,2006a) proposed bivariate regression analysis based on Weibull model withidentical covariates for both components. There are some situations wherewe find some non-identical covariates in addition to identical covariates for apaired components in a system. For example, failure times of a pair of den-tal implants in a jaw of a patient. Here the identical covariates are age andsex of a patient and non-identical covariates may be (1) different materials(ceramic, metal) of dental implant, (2) different shapes (screw, anchor, pillar,hollow cylinder) of dental implant, (3) dental implants in different locations(front, premolar, molar) of a jaw, (4) dental implants in different jaws (lower,upper).

Dental implant study in medical science is one branch called Implantology.Implantology studies the scientific technique of installation of dental implantthrough surgical procedure after loss of natural tooth. After installation ofdental implant, one usually waits three to six months for the placement of anartificial tooth on this implant. With some procedures, placement of an artifi-cial tooth after the installation of the implant is immediate. Implants within apatient are dependent and their life times are correlated. See (Haas et al. 1996;Ivanoff et al. 1999).

We propose BVW regression model based on identical and non-identicalcovariates on both components. In this paper, we consider bivariate WeibullBVW of Hanagal (2004, 2005a, 2006a,b) with covariates with three differentfrailty distributions namely gamma, positive stable and power variance func-tion distributions. We introduce the BVW regression model with gamma frailty,positive stable frailty and power variance function (PVF) frailty distributionsin Sects. 2, 3 and 4, respectively.

2 Gamma frailty

Gamma distributions have been used for many years to generate mixtures inexponential and Poisson models. From a computational point of view, gammamodels fit very well to survival models, because it is easy to derive the for-mulas for any number of events. This is due to simplicity of the derivatives ofthe Laplace transform. This is also the reason why this distribution has beenapplied in most of the applications published until now. The probability densityfunction (PDF) of gamma distribution as

f (y) = θαyα−1e−θy/�(α) (3)

For many calculations, it makes sense to restrict the scale parameter, andthe standard restriction is θ = α, as this implies a mean of 1 for Y, whenα −→ ∞, the distribution becomes degenerate. The PDF and Laplace trans-form of gamma distribution (or unconditional bivariate survivor function) are,


respectively, as follows:

f (y) = ααyα−1e−αy/�(α) (4)

Sα(t1, t2) = L(M(t1, t2)

)

= αα

[α + M(t1, t2)

]α

={

1 + M(t1, t2)α

}−α

. (5)

The various families of distributions have some theoretical advantages. If thefrailty distribution is a natural exponential family, selection, that is truncation(updating when no events have happened), implies that the conditional distri-bution of the frailty is still within the same family. The cross-ratio function isdefined as [See (Clayton 1978; Oakes 1989)]

θ(t1, t2) = λ(t1 | T2 = t2

)

λ(t1 | T2 > t2

)

=(

∂2Sα(t1,t2)∂t1∂t2

) (Sα(t1, t2)

)

(∂Sα(t1,t2)

∂t1

) (∂Sα(t1,t2)

∂t2

)

= [1 + α−1] + µ12(t1, t2)

[α + M(t1, t2)

]

αµ1(t1, t2)µ2(t1, t2)(6)

where

µi(t1, t2) = ∂M(t1, t2)∂ti

, i = 1, 2

µ12(t1, t2) = ∂2M(t1, t2)∂t1∂t2

The above expression is true for any underlying bivariate distribution of (T1, T2)

and Y has gamma with PDF given in Eq. (4). When T1 and T2 are independent,then M(t1, t2) = M1(t1) + M2(t2), µ(t1, t2) = 0 and

θ(t1, t2) = 1 + α−1.

If µ(t1, t2) = 0, then T1 and T2 may not necessarily be independent. An exam-ple for this is Weibull extension of bivariate exponential of Marshall and Olkin(1967) discussed in Sect. 2.2 in which µ(t1, t2) = 0.

The expression (6) can be interpreted as the relative risk for an individ-ual if the other one has experienced the event rather than being event free ata given time [See (Liang 1991)]. Therefore, it is an association function suchthat θ(t1, t2) > 1 represents positive association, θ(t1, t2) < 1 indicates negativeassociation and θ(t1, t2) = 1 implies no association.

34 D. D. Hanagal

2.1 Weibull extension of BVE of Gumbel

The Weibull extension of BVE-type I of Gumbel (1960) with survival functionis given by

S(t1, t2) = P[T1 > t1, T2 > t2

]

= e−λ1tc11 −λ2t

c22 −δλ1λ2t

c11 t

c22 , t1, t2 > 0 (7)

where λ1, λ2, c1, c2 > 0, 0 ≤ δ ≤ 1Here the marginal distribution of T1 and T2 are distributed as Weibull with

scale parameters λ1 and λ2 and shape parameters c1 and c2, respectively. Theparameter δ corresponds to the dependence parameter in BVW model.

Now the conditional survival function of BVW given the frailty (Y = y) isgiven by

S(t1, t2 | y) = e−y(λ1t

c11 +λ2t

c22 +δλ1λ2t

c11 t

c22

)(8)

where Y follows gamma distribution given in (4).Integrating over Y, we get unconditional survival function and is given by

Sα(t1, t2) =[

1 + λ1tc11 + λ2tc2

2 + δλ1λ2tc11 tc2

2

α

]−α

(9)

This is the bivariate Burr distribution, generalizing the Pareto power distri-bution. The marginal distributions are Burr distribution with survival functionsgiven by

Sα(ti) =[

1 + λitcii

α

]−α

, i = 1, 2 (10)

The unconditional bivariate survival function in (9) has two types of depen-dencies, one parameter (δ) is due to dependence parameter and the otherparameter (α) is due to frailty. These two parameters are identifiable.

As we have seen in the univariate Weibull regression, the scale parameterof the univariate Weibull distribution can be expressed in terms of regression

coefficients. If λ is the scale parameter of the Weibull distribution, then λ = e−β′z

or λ = eβ′z where β is the vector of regression parameters and z is the vector

of regressors or covariates. In the similar manner, the scale parameters λ1 andλ2 can be expressed in terms of regression parameters in the following way.

λ1 = e−(β′0z0+β

′1z1)

λ2 = e−(β′0z0+β

′2z2)

(11)


where

β′0 = (

β01, . . . , β0p), −∞ < β0 < ∞

β′1 = (

β11, . . . , β1q), −∞ < β1 < ∞

β′2 = (

β21, . . . , β2q), −∞ < β2 < ∞

z′0 = (

z01, . . . , z0p)

z′1 = (

z11, . . . , z1q)

z′2 = (

z21, . . . , z2q)

The exponent terms in the above expressions, we can take either positive ornegative but in either case λ1, λ2 > 0. β

′0z0 corresponds to the term containing

identical covariates for both components. β′1z1 corresponds to the term contain-

ing covariates for first component and β′2z2 corresponds to the term containing

covariates for second component.Now the survival function of BVW in terms of covariates is given by

Sα(t1, t2) ={

1 + tc11 e−

(β

′0z0+β

′1z1

)+ tc2

2 e−(β

′0z0+β

′2z2

)

α

+ δtc11 tc2

2 e−(

2β′0z0+β

′1z1+β

′2z2

)

α

}−α

. (12)

The another Weibull extension of BVE-type II of Gumbel (1960) with sur-vival function is given by

S(t1, t2) = S1(t1)S2(t2)[1 + δ

(1 − S1(t1)

)(1 − S2(t2)

) ], t1, t2 > 0 (13)

where −1 ≤ δ ≤ +1,The above survival function is derived from the given two independent mar-

ginal distributions. This relation is true for any bivariate distribution. SupposeTi, i = 1, 2 follows Weibull (λi, ci) distribution then BVW of (T1, T2) withsurvival function is given by

S(t1, t2) = (1 + δ)e−λ1tc11 −λ2t

c22 − δe−2λ1t

c11 −λ2t

c22

−δe−λ1tc11 −2λ2t

c22 + δe−2λ1t

c11 −2λ2t

c22 , (14)

where t1, t2 > 0, λ1, λ2, c1, c2 > 0, −1 ≤ δ ≤ +1.The above survival function is the weighted combination of four bivariate

Weibull distributions with weights (1 + δ), −δ, −δ, δ. Here the marginal distri-bution of T1 and T2 are distributed as Weibull with scale parameters λ1 andλ2 and shape parameters c1 and c2, respectively. The parameter δ corresponds

36 D. D. Hanagal

to the dependence parameter in BVW model. When δ = 0, T1 and T2 areindependent.

Now the conditional survival distribution of the above BVW given the frailty(Y = y) is given by the weighted combination of the four conditional survivalfunctions given the frailty, that is,

S(t1, t2 | y) = (1 + δ)e−y(λ1t

c11 +λ2t

c22

)− δe−y

(2λ1t

c11 +λ2t

c22

)

−δe−y(λ1t

c11 +2λ2t

c22

)+ δe−y

(2λ1t

c11 +2λ2t

c22

), (15)


Sα(t1, t2) = (1 + δ)

[

1 + λ1tc11 + λ2tc2

2

α

]−α

− δ

[

1 + 2λ1tc11 + λ2tc2

2

α

]−α

−δ

[

1 + λ1tc11 + 2λ2tc2

2

α

]−α

+ δ

[

1 + 2λ1tc11 + 2λ2tc2

2

α

]−α

(16)

The above bivariate survival function is the weighted combination of thefour bivariate Burr distributions with weights (1 + δ), −δ, −δ, δ. The marginaldistributions are Burr distribution with survival functions given by

Sα(ti) =[

1 + λitcii

α

]−α

, i = 1, 2 (17)

The parameters λ1 and λ2 here also expressed in terms of the regression param-eters and covariates as we did in the Eq. (11).

The another Weibull extension of BVE-type III of Gumbel (1960) with sur-vival function is given by

S(t1, t2) = e−(λ1t

c1/δ

1 +λ2tc2/δ

2

)δ

, t1, t2 > 0 (18)

where λ1, λ2, c1, c2 > 0, 0 ≤ δ ≤ 1Here the marginal distribution of T1 and T2 are distributed as Weibull with

scale parameters λ1 and λ2 and shape parameters c1 and c2, respectively. Theparameter δ corresponds to the dependence parameter in BVW model.

Now the conditional survival distribution of the above BVW given the frailty(Y = y) is given by

S(t1, t2 | y) = e−y(λ1t

c1/δ

1 +λ2tc2/δ

2

)δ

, t1, t2 > 0 (19)

where Y follows gamma distribution given in (4).


Integrating over Y, we get unconditional survival function and is given by

Sα(t1, t2) =[

1 +(λ1tc1/δ

1 + λ2tc2/δ2

)δ

α

]−α

(20)

The marginal distributions are Burr distribution with survival functions givenby

Sα(ti) =[

1 + λitcii

α

]−α

, i = 1, 2 (21)

The parameters λ1 and λ2 here also expressed in terms of the regression param-eters as we did in the Eq. (11). The survival function in terms of covariates isgiven by

Sα(t1, t2) =

⎡

⎢⎢⎢⎣

1 +

(tc1/δ

1 e−(β

′0z0+β

′1z1

)+ tc2/δ

2 e−(β

′0z0+β

′2z2

))δ

α

⎤

⎥⎥⎥⎦

−α

(22)

2.2 Weibull extension of BVE of Marshall–Olkin

Hanagal (1996) proposed k + 2 parameter family of k-variate Weibull distribu-tion. Specifying k = 2, we get BVW with survival function given by

S(t1, t2) = e−λ1tc1−λ2tc2−λ3tc(2) (23)

where t(2) = max(t1, t2), λ1, λ2, λ3, c > 0.The BVW model has important properties. Both marginal distribution of

BVW are univariate Weibull. More specifically speaking, we can state that

(T1, T2) ∼ BVW(λ1, λ2, λ3, c)�⇒T1 ∼Weibull((λ1 + λ3), c

)

T2 ∼Weibull((λ2 + λ3), c

)

T(1) ≡min(T1, T2)∼Weibull((λ1+ λ2+ λ3), c

)

where (λ1, +λ3), (λ2 + λ3) and (λ1 + λ2 + λ3) are the scale parameters of T1,T2 and T(1), respectively and c is the common shape parameter. This BVWmodel is not absolutely continuous with respect to Lebesque measure in R2. Ithas singularity on the diagonal T1 = T2. Here the parameter λ3 correspondsto the dependence between the two variables (T1, T2) and λ3 = 0 impliesT1 and T2 and independent. The probability of simultaneous failures, that is,P[T1 = T2] is λ3/(λ1 +λ2 +λ3) which is the correlation between T1 and T2. The

38 D. D. Hanagal

probability in the remaining two regions are P[T1 < T2] = λ1/(λ1 + λ2 + λ3)

and P[T1 > T2] = λ2/(λ1 + λ2 + λ3).Now the conditional survival function of BVW given the frailty (Y = y) is

given by

S(t1, t2 | y) = e−y(λ1tc1+λ2tc2+λ3tc

(2)

)(24)


Sα(t1, t2) =[

1 + λ1tc1 + λ2tc2 + λ3tc(2)

α

]−α

(25)

This is the bivariate Burr distribution, generalizing the Pareto power distri-bution. The marginal distributions and T(1) = min(T1, T2) are Burr distributionwith survival functions given by

Sα(ti) =[

1 + (λi + λ3)tciα

]−α

, i = 1, 2

Sα(t(1)) =[

1 + (λ1 + λ2 + λ3)tc(1)

α

]−α

.

(26)

The unconditional bivariate survival function in (25) has two types of depen-dencies, one is due to simultaneous failures and the other is due to frailty.Now we develop a regression model for the two component system. The scaleparameters λ1 and λ2 can be expressed in terms of regression parameters andcovariates as in Eq. (11) and the parameter λ3 expressed as follows.

λ3 = e−(β

′0z0+β

′1z1+β

′2z2

)(27)

Note: From here onwards, λ1, λ2 and λ3 are expressed in terms of covariatesfrom Eqs. (11) and (27) when one wants to express survival function in theform of regression model.

2.3 Weibull extension of BVE of Block–Basu

Block and Basu (1974) proposed an absolutely continuous BVE as an alterna-tive model to Marshall and Olkin (1967) which has a singular component onthe diagonal arising out of possible simultaneous failures. Marshall–Olkin is notabsolutely continuous with respect to Lebesque measure in R2, there are somesituations when this model is not appropriate. The survival function of BVE of


Block and Basu (1974) is given by

S(x1, x2) = λ

λ1 + λ2e−λ1x1−λ2x2−λ3x(2) − λ3

λ1 + λ2e−λx(2) (28)

where x(2) = max(x1, x2), λ1, λ2, λ3 > 0, λ = λ1 + λ2 + λ3.Taking the transformation T1 = Xc

1 and T2 = Xc2, c > 0, we get BVW and its

survival function is given by

S(t1, t2) = P[T1 > t1, T2 > t2

]

= λ

λ1 + λ2e−λ1tc1−λ2tc2−λ3tc

(2) − λ3

λ1 + λ2e−λtc

(2) (29)

where t(2) = max(t1, t2),Here the marginal distribution of T1 and T2 are now not Weibull but a

weighted combination of two Weibull with scales (λi + λ3) and λ, i=1,2 withweights 1 + λ3/(λ1 + λ2) and −λ3/(λ1 + λ2).

The above model turns out to be the absolutely continuous part of BVW ofHanagal (1996, 2004) and also a special case of BVW of Hanagal (2006a). Theindependence of two components corresponds to λ3 = 0.

Now the conditional survival function of BVW given the frailty (Y = y) isgiven by the weighted combination of the conditional survival functions giventhe frailty, that is,

S(t1, t2 | y) = λλ1+λ2

e−y(λ1tc1+λ2tc2+λ3tc

(2)

)− λ3

λ1+λ2e−y

(λtc

(2)

)(30)


Sα(t1, t2) = λλ1+λ2

[1 + λ1tc1+λ2tc2+λ3tc

(2)

α

]−α

− λ3λ1+λ2

[1 + λtc

(2)

α

]−α

(31)

The above bivariate survival function is weighted combination of bivari-ate Burr distribution and univariate Burr distribution with weights [1 + λ3/

(λ1 + λ2)] and −λ3/(λ1 + λ2), respectively. The marginal distributions areweighted combinations of two Burr distributions with weights [1+λ3/(λ1 +λ2)]and −λ3/(λ1 + λ2). The distribution of T(1) = min(T1, T2) is Burr distributionwith survival function given by

Sα(t(1)) =[

1 + (λ1 + λ2 + λ3)tc(1)

α

]−α

(32)

The unconditional bivariate survival function in (31) has two types of depen-dencies, one is due to the dependence parameter λ3 and the other is due tofrailty.

40 D. D. Hanagal

2.4 Weibull extension of BVE of Freund and Proschan–Sullo

Freund (1961) proposed BVE as a model for failure time distribution of a sys-tem with life-times (X1, X2) operating in the following manner. Initially X1 andX2 are independent exponential with failure rates λ1 and λ2, respectively, λ1,λ2, > 0. The interdependence of the components is such that failure of a com-ponent changes the failure rate of other component from λ1 to λ11 ( λ2 to λ22 ).The BVE of Freund (1961) with its joint PDF is given by

f (x1, x2) ={

λ1λ22e−λ22x2−(λ1+λ2−λ22)x1 , 0 < x1 < x2 < ∞λ2λ11e−λ11x1−(λ1+λ2−λ11)x2 , 0 < x2 < x1 < ∞ (33)

where λ1, λ2, λ11, λ22 > 0.Proschan and Sullo (1974) proposed BVE which is the combination of both

Marshall–Olkin and Freund models and the two component system operate inthe following manner. Initially X1 and X2 follow BVE of Marshall and Olkin(1967). When a component fails the failure of a component changes the failurerate of other component from λ1 + λ3 to λ11 + λ3 ( λ2 + λ3 to λ22 + λ3). TheBVE of Proschan and Sullo (1974) with its PDF is given by

f (x1, x2) =

⎧⎪⎨

⎪⎩

λ1(λ22 + λ3)e−(λ22+λ3)x2−(λ1+λ2−λ22)x1 , 0 < x1 < x2 < ∞λ2(λ11 + λ3)e−(λ11+λ3)x1−(λ1+λ2−λ11)x2 , 0 < x2 < x1 < ∞λ3e−(λ1+λ2+λ3)x, 0 < x1 = x2 = x < ∞

(34)

where λ1, λ2, λ3, λ11, λ22 > 0.Taking transformation T1 = Xc

1 and T2 = Xc2, c > 0 we get bivariate Weibull

model (BVW) which was introduced by Hanagal (2005a,b) with PDF given by

f (t1, t2) =

⎧⎪⎨

⎪⎩

λ1(λ22 + λ3)c2(t1t2)c−1e−(λ22+λ3)tc2−(λ1+λ2−λ22)tc1 , 0 < t1 < t2 < ∞λ2(λ11 + λ3)c2(t1t2)c−1e−(λ11+λ3)tc1−(λ1+λ2−λ11)t

c2 , 0 < t2 < t1 < ∞

λ3e−(λ1+λ2+λ3)tc , 0 < t1 = t2 = t < ∞(35)

Re-parameterize λ11 = φ1λ1, λ22 = φ2λ2 and rewrite the above PDF as

f (t1, t2) =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

λ1(λ2φ2 + λ3)c2(t1t2)c−1e−(λ2φ2+λ3)tc2−(λ1+λ2−λ2φ2)tc1 ,0 < t1 < t2 < ∞

λ2(λ1φ1 + λ3)c2(t1t2)c−1e−(λ1φ1+λ3)tc1−(λ1+λ2−λ1φ1)tc2 ,

0 < t2 < t1 < ∞λ3e−(λ1+λ2+λ3)tc , 0 < t1 = t2 = t < ∞

(36)

As we know in BVE of Proschan and Sullo (1974), the marginals are weightedcombinations of two exponential distributions. Here in the BVW also, the marg-inals are weighted combinations of two Weibull distributions with same weights.


The min(T1, T2) is Weibull with scale parameter (λ1+λ2+λ3) and shape param-eter c. When λ3 = 0, the BVW in Eq. 36 reduces to BVW of Hanagal (2004)and when φ1 = φ2 = 1, it reduces to BVW of Hanagal (2006a). When φi = 1,i = 1, 2 and λ3 = 0 then T1 and T2 are independent. The probabilities in thethree regions are given by

P[T1 < T2

] = λ1/(λ1 + λ2 + λ3),

P[T1 > T2

] = λ2/(λ1 + λ2 + λ3) and P[T1 = T2

] = λ3/(λ1 + λ2 + λ3).

The survival function of this BVW is given by

S(t1, t2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

λ2(1−φ2)e−(λ1+λ2+λ3)tc2

λ1+λ2(1−φ2)

+λ1e−(λ1+λ2(1−φ2))tc1−(λ2φ2+λ3)tc2

λ1+λ2(1−φ2), 0 < t1 ≤ t2

λ1(1−φ1)e−(λ1+λ2+λ3)tc1

λ2+λ1(1−φ1)

+λ2e−(λ2+λ1(1−φ1))tc2−(λ1φ1+λ3)tc1

λ2+λ1(1−φ1), 0 < t2 ≤ t1

(37)

Now the conditional survival function of BVW given the frailty (Y = y) isgiven by the weighted combination of the conditional survival functions giventhe frailty, that is,

S(t1, t2 | y) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

λ2(1−φ2)e−y(λ1+λ2+λ3)tc2

λ1+λ2(1−φ2)

+λ1e−y[(λ1+λ2(1−φ2))tc1+(λ2φ2+λ3)tc2

]

λ1+λ2(1−φ2), 0 < t1 ≤ t2

λ1(1−φ1)e−y(λ1+λ2+λ3)tc1

λ2+λ1(1−φ1)

+λ2e−y[(λ2+λ1(1−φ1))tc2+(λ1φ1+λ3)tc1

]

λ2+λ1(1−φ1), 0 < t2 ≤ t1

(38)


Sα(t1, t2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

λ2(1−φ2)[

1+ (λ1+λ2+λ3)tc2α

]−α

λ1+λ2(1−φ2)

+λ1

[1+ (λ1+λ2(1−φ2))tc1+(λ2φ2+λ3)tc2

α

]−α

λ1+λ2(1−φ2), 0 < t1 ≤ t2

λ1(1−φ1)

[1+ (λ1+λ2+λ3)tc1

α

]−α

λ2+λ1(1−φ1)

+λ2

[1+ (λ2+λ1(1−φ1))tc2+(λ1φ1+λ3)tc1

α

]−α

λ2+λ1(1−φ1), 0 < t2 ≤ t1

(39)

42 D. D. Hanagal

The above bivariate survival function is the weighted combination of thebivariate Burr distribution and univariate Burr distribution with same weightsas in BVE of Proschan and Sullo (1974). The unconditional bivariate survivalfunction in (39) has three types of dependencies, one is due to simultaneousfailures and second is due to load on one component due to the failure ofanother component and the third is due to frailty.

Substituting λ3 = 0 in all the above expressions we get the correspondingexpressions for the Weibull extension of BVE of Freund (1961).

3 Positive stable frailty

In practice, the gamma frailty specification may not fit well (Shih (1998); Glidden(1999); Fan et al. (2000). The positive stable model (Hougaard 2000) is a usefulalternative, in part because it has the attractive feature that predictive haz-ard ratio decrease to one over time (Oakes 1989). The property is observed infamilial associations of the ages of onset of diseases with etiologic heterogeneity,where genetic cases occur early and long-term survivors are weakly correlated.The gamma model has predictive hazard ratios which are time invariant andmay not be suitable for these patterns of failures (Fine et al. 2003). The PDF ofpositive stable distribution with two parameters α and δ is given by

f (y) = − 1πy

∞∑

k=1

�(kα + 1)

k! (−y−αδ/α)k sin(αkπ), y > 0, 0 < α < 1 (40)

with Laplace transform

E{e−sY} = e−δsα/α . (41)

[See Hougaard (2000, p. 503)]The unconditional bivariate survival function with positive stable frailty is

given by

Sα,δ(t1, t2) = e−δ[M(t1,t2)]α/α (42)

The PDF of positive stable distribution with (α = δ) is given by

f (y) = 1π

∞∑

k=1

�(kα + 1)

k!(

−1y

)αk+1

sin(αkπ), y > 0, 0 < α < 1 (43)

with Laplace transform

E{e−sY} = e−sα . (44)


The unconditional bivariate survival function with positive stable frailty isgiven by

Sα(t1, t2) = e−[M(t1,t2)]α . (45)

When α = 1, the frailty distribution is degenerate at Y = 1.The stable Weibull model The Weibull distribution is particularly well suited

to the positive stable frailty model. The bivariate Weibull model is obtained byassuming Mi(t) = λitci , i = 1, 2. This means that conditionally on Y = y, thedistribution of Ti is Weibull (λi, y, ci), i = 1, 2. When T1 and T2 are independent,the bivariate survival function is

Sα(t1, t2) = e−[λ1t

c11 +λ2t

c22

]α

. (46)

The advantage of this model is that the marginal distributions are also ofWeibull form. Also, the time to the first event, T(1) = min(T1, T2) is of Weibullform when c1 = c2.

The positive stable model has the advantage that it fits proportional hazardswhich means that if the conditional model has proportional hazards, so doesthe marginal distribution. This is an advantage, when considering the model asa random effects model.

3.1 Weibull extension of BVE of Gumbel

Now the conditional survival distribution of Weibull extension of the BVE typeI of Gumbel (1960) given the frailty (Y = y) is given by

S(t1, t2 | y) = e−y(λ1t

c11 +λ2t

c22 +δλ1λ2t

c11 t

c22

)(47)

where λ1, λ2, c1, c2 > 0, 0 ≤ δ ≤ 1where Y follows positive stable distribution given in (43).


Sα(t1, t2) = e−(λ1t

c11 +λ2t

c22 +δλ1λ2t

c11 t

c22

)α

(48)

The above bivariate survival function has two types of dependencies, oneis due to dependence parameter δ and other is due to frailty. The marginaldistributions are also Weibull and are given by

Sα(ti) = e−(λit

cii

)α

, i = 1, 2 (49)

The parameters λ1 and λ2 here also expressed in terms of the regression param-eters as we did in the Eq. (11).

44 D. D. Hanagal

Now the conditional survival distribution of Weibull extension of the BVEtype II of Gumbel (1960) given the frailty (Y = y) is given by

S(t1, t2 | y) = (1 + δ)e−y(λ1t

c11 +λ2t

c22

)− δe−y

(2λ1t

c11 +λ2t

c22

)

− δe−y(λ1t

c11 +2λ2t

c22

)+ δe−y

(2λ1t

c11 +2λ2t

c22

), (50)

where Y follows positive stable distribution given in (43).Integrating over Y, we get unconditional survival function and is given by

Sα(t1, t2) = (1 + δ)e−(λ1t

c11 +λ2t

c22

)α

− δe−(

2λ1tc11 +λ2t

c22

)α

− δe−(λ1t

c11 +2λ2t

c22

)α

+ δe−(

2λ1tc11 +2λ2t

c22

)α

, (51)

The marginal distributions are also of Weibull form with survival functionsgiven by

Sα(ti) = e−(λit

cii

)α

, i = 1, 2 (52)

Now the conditional survival distribution of Weibull extension of BVE type IIIof Gumbel (1960) given the frailty (Y = y) is given by

S(t1, t2 | y) = e−y(λ1t

c1/δ

1 +λ2tc2/δ

2

)δ

, t1, t2 > 0 (53)

where λ1, λ2, c1, c2 > 0, 0 ≤ δ ≤ 1where Y follows positive stable distribution given in (43).


Sα(t1, t2) = e−(λ1tc1/δ

1 +λ2tc2/δ

2 )δα , (54)

The marginal distributions are also of Weibull form with survival functionsgiven by

Sα(ti) = e−(λitcii )α , i = 1, 2 (55)

3.2 Weibull extension of BVE of Marshall–Olkin

Suppose Y has a positive stable distribution which we assume as a frailty distri-bution. Now the conditional survival function of Weibull extension of BVE of


Marshall and Olkin (1967) given the frailty (Y = y) is given by

S(t1, t2 | y) = e−y(λ1tc1+λ2tc2+λ3tc

(2)

)(56)


Sα(t1, t2) = e−(λ1tc1+λ2tc2+λ3tc

(2)

)α

(57)

The above bivariate survival function has two types of dependencies, one isdue to simultaneous failures and other is due to frailty. When α = 1, the frailtydistribution is degenerate at Y = 1.

The main advantage of this model is that the marginal distributions are alsoof Weibull and T(1) = min(T1, T2) is also of Weibull form and correspondingsurvival functions are given by

Sα(ti) = e−(λi+λ3)α tcαi , i = 1, 2

Sα(t(1)) = e−(λ1+λ2+λ3)α tcα

(1) (58)

3.3 Weibull extension of BVE of Block–Basu

Now the conditional survival function of Weibull extension of BVE of Marshalland Olkin (1967) given the frailty (Y = y) is given by

S(t1, t2 | y) = λ

λ1 + λ2e−y(λ1tc1+λ2tc2+λ3tc

(2)) − λ3

λ1 + λ2e−yλtc

(2) (59)


Sα(t1, t2) = λ

λ1 + λ2e−

(λ1tc1+λ2tc2+λ3tc

(2)

)α

− λ3

λ1 + λ2e−(λtc

(2))α (60)

The above bivariate survival function has two types of dependencies, one isdue to dependence parameter λ3 and other is due to frailty. When α = 1, thefrailty distribution is degenerate at Y = 1.

The marginal distributions are of weighted combinations of two Weibull andT(1) = min(T1, T2) is of Weibull form and corresponding survival function ofT(1) is given by

Sα(t(1)) = e−(λ1+λ2+λ3)α tcα

(1) (61)

46 D. D. Hanagal

3.4 Weibull extension of BVE of Freund and Proschan–Sullo

Now the conditional survival function of Weibull extension of BVE of Proschanand Sullo (1974) given the frailty (Y = y) is given by

S(t1, t2 | y) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

λ2(1−φ2)e−y

(λ1+λ2+λ3)tc2

λ1+λ2(1−φ2

)

+λ1e−y[(λ1+λ2(1−φ2))tc1+(λ2φ2+λ3)tc2]λ1+λ2(1−φ2)

, 0 < t1 ≤ t2

λ1(1−φ1)e−y(λ1+λ2+λ3)tc1

λ2+λ1(1−φ1)

+λ2e−y[(λ2+λ1(1−φ1))tc2+(λ1φ1+λ3)tc1]λ2+λ1(1−φ1)

, 0 < t2 ≤ t1

(62)


Sα(t1, t2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

λ2(1−φ2)e−((λ1+λ2+λ3)tc2)α

λ1+λ2(1−φ2)

+λ1e−[(λ1+λ2(1−φ2))tc1+(λ2φ2+λ3)tc2]α

λ1+λ2(1−φ2), 0 < t1 ≤ t2

λ1(1−φ1)e−((λ1+λ2+λ3)tc1)α

λ2+λ1(1−φ1)

+λ2e−[(λ2+λ1(1−φ1))tc2+(λ1φ1+λ3)tc1]α

λ2+λ1(1−φ1), 0 < t2 ≤ t1

(63)

Substituting λ3 = 0 in all the above expressions we get the correspondingexpressions for the Weibull extension of BVE of Freund (1961).

4 Power variance function frailty

This distribution is a three-parameter family uniting gamma and positive stabledistributions. The distribution is denoted PVF(α, δ

′, θ). For α = 0, the gamma

distributions are obtained, with same parametrization. Some formulas are valid,but many are others are different in this case. For θ = 0, the positive stable dis-tributions are obtained. For α = 1/2, the inverse Gaussian distributions areobtained. For α = −1, the non-central gamma distribution of shape parameterzero is obtained. For α = 1, a degenerate distribution is obtained.

The parameter set is (α ≤ 1, δ′> 0), with (θ ≥ 0 for α > 0), and (θ > 0 for

α ≤ 0). The distribution is concentrated on the positive numbers for α ≥ 0, andis positive or zero for α < 0. In the case α > 0, the PDF of PVF is given by [SeeHougaard (2000, p. 504)]

f (y) = e−θy+δθα/α 1π

∞∑

k=1

�(kα + 1)

k! (−1y)αk+1 sin(αkπ), y > 0 (64)


In the case α < 0, the �−term in the density is not necessarily defined,and therefore we can use the alternative expression for p.d.f. of PVF as [SeeHougaard (2000, p. 504)]

f (y) = e−θy+δθα/α 1y

∞∑

k=1

(−δyα/α)k

k!�(−kα), y > 0 (65)

This expression is valid for all α values, except 0 and 1, with the conventionthat when the �-function in the denominator is undefined (which happens whenkα is a positive integer), the whole term in the sum is zero. For α < 0, thereis probability exp(δ

′θα/α) of the random variable being zero. For α ≥ 0, the

distribution is unimodal.If Y1 and Y2 are independent, and Yi follows PVF(α, δ

′i , θ), i = 1, 2 the distri-

bution of Y1 +Y2 is PVF(α, δ′1 +δ

′2, θ). So, PVF distribution is infinitely divisible.

When θ > 0, all (positive) moments exist, and the mean is δ′θα−1. The variance

is δ′(1 − α)θα−2.

The Laplace transform of PVF distribution is

L(s) = e−δ′ {(θ+s)α−θα}/α . (66)

The unconditional bivariate survival function with PVF frailty is given by

Sα,δ′ ,θ (t1, t2) = e−δ

′ {θ+M(t1,t2)}α/α+δ′θα/α . (67)

4.1 Weibull extension of BVE Models

The unconditional survival function of BVW with PVF frailty distribution cor-responding to BVE type I of Gumbel (1960) is given by

Sα,δ′ ,θ (t1, t2) = e−δ

′(θ+λ1t

c11 +λ2t

c22 +δλ1λ2t

c11 t

c22

)α/α+δ

′θα/α (68)

The unconditional survival function of BVW with PVF frailty distributioncorresponding to BVE type II of Gumbel (1960) is given by

Sα,δ′ ,θ (t1, t2) = (1 + δ)e−δ

′(θ+λ1t

c11 +λ2t

c22

)α/α+δ

′θα/α

− δe−δ′(

θ+2λ1tc11 +λ2t

c22

)α/α+δ

′θα/α

− δe−δ′(

θ+λ1tc11 +2λ2t

c22

)α/α+δ

′θα/α

+ δe−δ′(

θ+2λ1tc11 +2λ2t

c22

)α/α+δ

′θα/α , (69)

48 D. D. Hanagal

The unconditional survival function of BVW with PVF frailty distributioncorresponding to BVE type III of Gumbel (1960) is given by

Sα,δ′ ,θ (t1, t2) = e−δ

′ {θ+(λ1tc1δ

1 +λ2tc2δ

2 )δ}α/α+δ′θα/α , (70)

The unconditional survival function of BVW with PVF frailty distributioncorresponding to BVE of Marshall and Olkin (1967) is given by

Sα,δ′ ,θ (t1, t2) = e−δ

′(θ+λ1tc1+λ2tc2+λ3tc

(2)

)α/α+δ

′θα/α (71)

The unconditional survival function of BVW with PVF frailty distributioncorresponding to BVE of Block and Basu (1974) is given by

Sα,δ′ ,θ (t1, t2) = λ

λ1 + λ2e−δ

′(θ+λ1tc1+λ2tc2+λ3tc

(2)

)α/α+δ

′θα/α

− λ3

λ1 + λ2e−δ

′(θ+λtc

(2)

)α/α+δ

′θα/α (72)

The unconditional survival function of BVW with PVF frailty distributioncorresponding to BVE of Proschan and Sullo (1974) is given by

Sα(t1, t2) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

λ2(1−φ2)e−δ

′(θ+λtc2)α/α+δ

′θα/α

λ1+λ2(1−φ2)

+λ1e−δ

′ [θ+

(λ1+λ2(1−φ2)

)tc1+

(λ2φ2+λ3

)tc2

]α/α+δ

′θα

/α

λ1+λ2(1−φ2), t1 ≤ t2

λ1(1−φ1)e−δ

′(θ+λtc1)

α/α+δ

′θα/α

λ2+λ1(1−φ1)

+λ2e−δ

′ [θ+

(λ2+λ1(1−φ1))tc2+(λ1φ1+λ3

)tc1

]α/α+δ

′θα/α

λ2+λ1(1−φ1), t2 ≤ t1

(73)

where λ = λ1 + λ2 + λ3.Substituting λ3 = 0 in all the above expressions we get the corresponding

expressions for the Weibull extension of BVE of Freund (1961).

Acknowledgements I thank both the referees for the suggestions and comments.

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Weibull Extension of Bivariate Exponential Regression

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