Analyzing multivariate survival data using composite likelihood and flexible parametric modeling of the hazard functions

Research Article

Received 25 March 2009, Accepted 4 March 2010 Published online 7 May 2010 in Wiley Interscience

(www.interscience.wiley.com) DOI: 10.1002/sim.3934

Analyzing multivariate survival data usingcomposite likelihood and flexible parametricmodeling of the hazard functionsJan Nielsena∗† and Erik T. Parnerb

In this paper, we model multivariate time-to-event data by composite likelihood of pairwise frailty likelihoods and marginalhazards using natural cubic splines. Both right- and interval-censored data are considered. The suggested approach is appliedon two types of family studies using the gamma- and stable frailty distribution: The first study is on adoption data where theassociation between survival in families of adopted children and their adoptive and biological parents is studied. The secondstudy is a cross-sectional study of the occurrence of back and neck pain in twins, illustrating the methodology in the context ofgenetic epidemiology. Copyright © 2010 John Wiley & Sons, Ltd.

Keywords: survival analysis; multivariate time-to-event data; composite likelihood; shared frailty; copulas; cubic splines;parametric models

1. Introduction

Clustered multivariate time-to-event data appear increasingly in clinical and epidemiological research, e.g. several typesof events in an individual, recurrent events or family studies. Two approaches are commonly used when modelingmultivariate time-to-event data. The marginal approach is used when the dependencies between the event times are notof interest, but rather a regression analysis of the marginal event rates [1]. Regression parameters are estimated in asurvival model as if the event times were independent, but the dependence between event times is taken into accountwhen estimating the variance of the parameter estimates [2, 3]. A random effect model [4] is often used when thedependence between the event times is of interest. Here the dependency of the related event times is introduced by ashared, unobserved risk factor. The most commonly used random effect model for clustered time-to-event data is theshared frailty model, where a single risk factor (frailty) introduces a symmetric dependence structure, i.e. the samedependencies between every pair of waiting times. The random effect is typically modeled assuming proportional ratesfor individuals with different frailties. A symmetric dependence structure may in many cases be unrealistic, especiallywhen event times for individuals in a family are studied.

Attempts have been made to generalize the shared frailty to more than two event times to allow for more complicateddependence structures, so that each bivariate marginal model would be a shared frailty model. Although such modelsdo exist [5], they are not easily formulated as explicit statistical models. Parner [6] suggested making inference fora multivariate survival model with bivariate shared frailty margins by composite likelihood. The composite likelihoodis obtained by multiplying pairwise likelihood functions, thereby avoiding formulating a complete multivariate model.Non-parametric modeling of hazard functions is not feasible using the full composite likelihood function, especially forlarge data sets. The composite likelihood approach was applied on a real data example, of right-censored data, assuminggamma frailty and Weibull marginal rates.

aSouthern Center for National Clinical Databases, Odense University Hospital, Sdr. Boulevard 29, Entrance 101, 3rd floor, DK-5000 OdenseC, Denmark

bDepartment of Biostatistics, University of Aarhus, Vennelyst Boulevard 6, DK-8000 Aarhus C, Denmark∗Correspondence to: Jan Nielsen, Southern Center for National Clinical Databases, Odense University Hospital, Sdr. Boulevard 29, Entrance

101, 3rd floor, DK-5000 Odense C, Denmark.†E-mail: [email protected]

2126

Copyright © 2010 John Wiley & Sons, Ltd. Statist. Med. 2010, 29 2126--2136

J. NIELSEN AND E. T. PARNER

In this paper, we present a parametric approach to composite likelihood for frailty models, with a flexibility similar tothe non-parametric modeling of hazard functions. The hazards are modeled as flexible parametric functions using cubicsplines, as proposed in [7] for censored univariate survival data. This parametric modeling of the hazards has the usualadvantages of a parametric model, while retaining much of the flexibility of a non-parametric method. Each bivariatesurvival model is formulated as a shared frailty model. The time-to-event data may be both right and interval censored.

The suggested analysis of multivariate time-to-event data is illustrated in two examples: an adoption study, where thesurvival of the adopted child and the adoptive and biological parents are studied to quantify genetic and environmentalfactors on longevity, and a study of back pain in twins to study the influence of genetic and environmental factors onback and neck pain. These data sets are introduced in Section 2. In Section 3, we introduce the proposed statisticalmethods; first the composite likelihood approach to multivariate data, and subsequently the two frailty distributions:gamma frailty and stable frailty. Finally we review the cubic spline techniques from [7]. Detailed analyses of the datasets are presented in Section 4, and Section 5 contains a discussion of the methods.

2. Example data sets

The methods are applied on data from a cohort study of the survival of adopted children and their biological and adoptiveparents (right-censored data) and a cross-sectional study of back pain in twins (interval-censored data).

2.1. Adoption study

The purpose of the adoption study was to analyze the impact of environmental and genetic factors of longevity [8].The basic idea of studying adopted children is that the similarity, with respect to the survival, between the adoptivechild and its biological parents can be ascribed to genetic factors, whereas the similarity between the adoptive child andits adoptive parents can be due to environmental factors. The data have been used in several methodological papers toillustrate approaches for analyzing multivariate data; Nielsen et al. [9], who proposed a shared gamma frailty model foreach pair of survival times between the adoptive child and one of the parents. However, this approach did not allowfor a statistical comparison for the dependence parameters; Parner [6], who proposed a composite likelihood methodfor the estimation of frailty and regression parameters and their standard deviations, giving the possibility for statisticalcomparison of the dependence parameters; and Tibaldi et al. [10], who proposed a Plackett-Dale model for describingthe dependence between event times with Weibull margins.

Here we consider children born during the period 1924–1926 who early in life were adopted by couples unrelated tothem. A total of 772 families are included.

2.2. Back and neck pain in twins

Back and neck pain has become an epidemic in modern society, as in the general Danish population. There is evidencethat once you have had back or neck pain, there is profound increased risk of recurrent pain [11, 12], hence it wouldbe interesting to study the time to the first onset of back or neck pain. The underlying causes of back and neck painsare poorly understood, like the influences of genetic and environmental factors. The use of twin data is often used ingenetic epidemiology for answering these questions on genetic and environmental influences (for example by variancecomponent models [13]). A difference in the dependence of back or neck pain between mono- and dizygotic twins in apair would indicate that genetic factors exist which would make one predisposed to getting back pains. Furthermore, itis also of interest to quantify the dependence between pain in different regions of the spine, e.g. neck versus low back,and to examine if this dependence can be credited to genetic or environmental factors.

In 2002, a cross-sectional study was performed on 34 902 Danish twins aged 20–71 recruited from The Danish TwinRegistry. The study consisted of a postal survey in which there were several questions regarding spinal pain, includingquestions on whether or not the person ever had neck or (low) back pain, respectively [14].

3. Methods

Suppose we have a cluster of q related event times T1, . . . ,Tq . The event times may be right-censored or interval-censored.For right-censored event times, we observe (T ∗

j ,� j ), where T ∗j is equal to the event time Tj when the indicator � j is

one. When the indicator � j is zero, the event time Tj is only known to be larger than T ∗j . In many applications, one

never observes the exact event time, but it is only possible to construct lower and upper bounds for the event times.So here, the observations consist of intervals [L j , R j ], where 0�L j<R j�∞ and L j<Tj�R j . The event time is then


2127


termed interval-censored. If the individual is recruited to the study on condition of having no event up to a time Vj , thetime-to-event is said to be left-truncated.

A special case of interval-censored data is current status data; there is only one monitoring time C j , per individual,and it is recorded if the event has occurred or not. So all intervals are on the form [Vj ,C j ] or [C j ,∞]. Current statusdata type of data are often encountered in cross-sectional studies.

The observations D1, . . . , Dq therefore consist of D j = (T ∗j ,� j ,Vj ) in the case of right-censored event times and

D j = ([Li , Ri ],Vj ) in the case of interval-censored event times. Let each cluster be denoted by D∗ = (D1, . . . , Dq ).

3.1. Composite likelihood

In many applications, not all pairwise dependencies of event times that are of interest. Let M be the set of pairs of eventtimes of interest. The composite likelihood is defined by multiplying the marginal likelihood of all pairs of survivaltimes of interest

CL(�)= ∏( j,k)∈M

Ljk(�),

where Ljk(�) is the marginal likelihood function for individuals j and k and � is the parameter vector we intend toestimate.

Consider the sample D∗1 , . . . , D∗

n of i.i.d. replicates of D∗, and let CL(i)(�) denote the composite likelihood of D∗i .

Then the composite likelihood of this sample is

CLn(�)=n∏

i=1CL(i)(�).

Composite likelihood is a pseudo-likelihood, but typically shares the same properties as the ordinary likelihood, i.e. itproduces asymptotically normal estimates under the same regularity condition as the ordinary likelihood, although theparameters should be identifiable from marginal bivariate distributions [15, 16]. There may be some loss in efficiencyby applying composite likelihood as compared with specifying a full model for the data and using the maximumlikelihood estimation. Composite likelihood can be considered to be more general than the full likelihood, as there areonly assumptions on the bivariate distributions, and not the full q-dimensional distribution; in the composite likelihoodapproach one only needs to specify the bivariate distributions.

Let CSn(�) denote the composite score function

CSn(�)= ��

logCLn(�)=n∑

i=1

��

logCL(i)(�)=n∑

i=1CS(i)(�)

and cin(�) the composite Fisher information matrix

cin(�)=E

{− �2

��T logCLn(�)

}.

Further, let �̂n denote the parameter value that maximizes the composite likelihood of D∗1 , D∗

2 , . . . , D∗n , and �0 the

true parameter value. It has been proven that√

n(�̂n −�0) converges in distribution to a normal distribution with meanzero and covariance matrix

cin(�0)−1E{CS(1)(�0)CS(1)(�0)T}cin(�0)−1.

The covariance matrix can consistently be estimated by

cjn(�̂n)−1 Bn(�̂n)cjn(�̂n)−1,

where

cjn(�)=n−1n∑

i=1− �2

��T logCL(i)(�)

is the normed observed composite information matrix, and

Bn(�)=n−1n∑

i=1CS(i)(�)CS(i)(�)T

is the matrix of products of score functions [16, 17].

2128



3.2. Frailty models for bivariate survival times

A frailty is a latent variable that generates dependence between the waiting times. In the shared frailty model, conditionallyon the frailty Z , the hazard function for individual j is assumed to be of the form h j (t |Z )= Z� j (t). The conditionalbivariate survival function is then

S(t1, t2|Z )=exp(−Z (M1(t1)+ M2(t2))),

where

M j (t)=∫ t

0�(u)du

is the integrated conditional hazard. Integrating Z out gives

S(t1, t2)=E{exp(−Z (M1(t1)+ M2(t2)))}= L(M1(t1)+ M2(t2)),

where L(s) is the Laplace transform of Z . Similarly, the marginal univariate survival function is S j (t)= L(M j (t)),hence M j (t)= L−1(S j (t)). The bivariate survival function can thus also be written in terms of the two marginal survivalfunctions

S(t1, t2)= L(L−1(S1(t1))+L−1(S2(t2))).

The above model is a copula model, since it models the marginal distribution and the correlation. The copula formulationhas the advantage that we can model the same observed marginal hazards for different groups, e.g. mono- and dizygotictwin pairs, with different dependences in each group. A further advantage of the copula formulation is that it allows foradjusting for covariates on the observed marginal survival functions. We will use the copula formulation in the remainderof the paper.

Frailty models, such as gamma frailty and stable frailty [4, 18], are often used. The gamma frailty model assumesZ is gamma distributed with mean 1 and variance �. The Laplace transform of the gamma distribution is

L(s)= (�s+1)−1/�.

The bivariate survival function for the gamma frailty model is then

S(t1, t2;�)= (S1(t1)−�+S2(t2)−�−1)−�−1, �>0.

We can show that the above function is a bivariate survival function, even for negative � [9], as long as S1(t1)−�+S2(t2)−�−1>0. Negative values correspond to a negative correlation between T1 and T2. It is important to note thatwhen � is negative there is no frailty interpretation of the model, and � has no interpretation as a variance. However,the model keeps the interpretation of a copula model [19], in which case the model is called the Clayton copula.

An interesting alternative to the gamma frailty model is the stable frailty model, which assumes that Z follows astrictly positive stable distribution. This family of distribution is parametrized by �� 1

2 , but to achieve a distribution onthe positive numbers we need to have �>1. The Laplace transform of the stable distribution is

L(s)=exp(−s1/�).

Hence the bivariate survival function for the stable copula (stable frailty model) is

S(t1, t2;�)=exp(−[(− log S1(t1))�+(− log S2(t1))�]�−1

), ��1.

The gamma and stable frailty models assume different types of dependency between the event times. The gammafrailty models primarily assume late dependence, hence the dependence estimates are unchanged by a large number ofleft-truncated event times. On the other hand, stable frailty models primarily assume early dependence, which often doesnot work well in practice with heavily truncated data [4].

The frailty parameters can be difficult to interpret. A simple summary measure of the association of the two waitingtimes is Kendall’s tau. If (T (1)

1 ,T (1)2 ) and (T (2)

1 ,T (2)2 ) are two independent realizations of (T1,T2), then the two pairs

are termed concordant, if (T (1)1 −T (2)

1 )(T (1)2 −T (2)

2 )>0, and discordant, if (T (1)1 −T (2)

1 )(T (1)2 −T (2)

2 )<0. Kendall’s tau isdefined as the difference between the probabilities of the pairs being either concordant or discordant. When T1 and T2are continuous variables then Kendall’s tau can be written as

� = Pr{(T (1)1 −T (2)

1 )(T (1)2 −T (2)

2 )>0}−Pr{(T (1)1 −T (2)

1 )(T (1)2 −T (2)

2 )<0}

= 2Pr{(T (1)1 −T (2)

1 )(T (1)2 −T (2)

2 )>0}−1.


2129


Table I. Relationship between frailty parameter and Kendall’s tau.

Frailty Gamma Stable

Kendall’s � �2+� 1− 1

�

Table II. Recommended placement of the internal knots.

#knots Centiles

1 502 33 673 25 50 754 20 40 60 805 17 33 50 67 83

It is easy to see that −1��1, and if T1 and T2 are independent, then �=0. The value of � can in the shared frailtymodel be found by means of the formula from [4]

�=4∫ ∞

0sL(s)L ′′(s)ds−1.

The formulas for � in the gamma and stable frailty are given in Table I.In cases where no frailty distribution a priori has a lower preference to another, we suggest fitting more than one

frailty model, and calculating Kendall’s tau for all frailty models. If the estimates of Kendall’s tau are comparablebetween frailty models, then one may conclude that the results are robust with respect to the choice of frailty.

3.3. Spline-based parametric survival models

Royston and Parmer [7] proposed to model the log-integrated hazard function by natural cubic splines as a function oflog-time. If T is Weibull distributed, say, with scale parameter � and shape parameter p, i.e. S(t)=exp(−(t/�)p), thenthe logarithm of the integrated hazard function H (t)=− log S(t) is

log H (t)= log(t/�)p = p log t − p log�=�0 +�1 log t.

Hence, for the Weibull distribution the log-integrated hazard function is a linear function of log-time. In more generalcases, log H (t) will be related to �= log(t) by a non-linear function s(�;�), which we will model by cubic splinefunctions.

In general, cubic spline functions are piecewise cubic polynomials. If we let k1< · · ·<km denote time points, knots,combining the polynomials, a cubic spline function can be written as

s(�;�)=�0 +�01�+�02�2 +�03�

3 +�1(�−k1)3++·· ·+�m(�−km)3

+,

where (�−a)+ =max(0,�−a). Cubic splines are known to behave poorly in the tails. Therefore, one restricts the splineto be linear outside boundary knots kmin<k1 and kmin>km [20, 21]. A restricted cubic spline is then given as

s(�;�)=�0 +�1�+�2�1(�)+·· ·+�m+2�m(�),

where the j th basis function is defined as

� j (�)= (�−k j )3+− j (�−kmin)3

+−(1− j )(�−kmax)3+

and

j = kmax −k j

kmax −kmin.

The above spline is termed as a natural cubic spline. The survival function is then given by

S(t)=exp(−exp(s(log t;�))).

The curve complexity is governed by the number of internal knots; zero knots meaning s(�;�)=�0 +�1�, which isthe Weibull distribution. The placement of the internal knots is an issue as discussed in [7, 22]. We have chosen tofollow the recommendations in [7], the boundary knots are placed at the extreme uncensored times, and the internal isplaced at the centile-based positions of the uncensored times given in Table II. For interval-censored data, the knots aredetermined from the finite right limits of the intervals.

2130



Figure 1. Adopted child (AC), adoptive father (AF), adoptive mother (AM), biological father (BF), biological mother (BM), eachline corresponds to an association, �i is the parameter from the copula model.

Adjusting for covariates can be done using the proportional hazards model. Let x be the covariate vector, then theproportional hazard model for survival data is defined through the hazard function as h(t;x)=h0(t)exp(bTx), whereh0(t)=h(t;0) is the baseline hazard and b the regression parameters. For the log-integrated hazard this gives

log H (t;x)= log H0(t)+bTx=s(log t;�)+bTx,

so the spline model for the marginal hazards can easily be extended to include covariates.

4. Applications

The methods described above were implemented as a procedure in Stata [23], using the built-in maximum likelihoodfacilities, ml, with the robust option for variance estimates of the parameters. The spline basis functions wereorthogonalized by the function orthog to obtain more stable numerical properties.

4.1. Adoption study

We fitted a composite likelihood consisting of a bivariate copula model for each pair of adoptive children and theirparents, and also for the biological parents and the adoptive parents respectively (Figure 1). The four main dependenciesof interest are between the adopted child and each of the four parents (�1, . . . ,�4). A separate hazard was fitted for eachof the five family members. The number of knots for each marginal was determined graphically, starting with zero knots(Weibull distribution), and increasing until the estimated parametric survival function appeared to be satisfactorily closeto the Kaplan–Meier estimate. The resulting model will, in the following, be called the ‘Flexible parametric model’,and the model, where all marginal hazards are assumed to be Weibull distributed, will be called the ‘Weibull model’.Figure 2 shows the estimated survival curves for both models, as well as the Kaplan–Meier estimates. Owing to thehigh number of left-truncated data (each parent is truncated at the birth of the child), fitting the stable copula model tothe data would be unfeasible. Hence only the results of Clayton copula composite analysis’ are shown (Table III). Sincethere are negative estimates of dependence parameters, the model cannot be interpreted as a frailty model.

The results are different from those obtained in Parner [6], due to a longer follow-up and a smaller number of familiesin the present data set.

Comparing the Weibull model to the Flexible parametric model, we see that the estimates of the dependence parametersare roughly the same, but the standard errors are smaller in the Flexible parametric model.

There does not seem to be any difference in genetic and environmental effects on mortality; all estimates of dependenceparameters are close to zero. Testing the hypothesis that the environmental effect between the adoptive father and theadopted child and the genetic effect between the biological father and the adopted child are equal, �1 =�3, gives ap-value of 0.79. Testing the hypothesis that the environmental effect between the adoptive mother and the adoptedchild and the genetic effect between the biological mother and the adopted child are equal, �2 =�4, gives a p-value of0.49. Testing the hypothesis that the mortality of the adopted child is independent of biological and adoptive parents,�1 =�2 =�3 =�4 =0, gives a p-value of 0.52. All p-values where obtained using a Wald test.

4.2. Back and neck pain in twins

Several dependencies are of interest in the twin data (Figure 3). In the analysis of dependencies, we wish to take fulladvantage of the data structure. A basic assumption in twin analyses is that the dependence within a person does notdepend on whether or not this person is a mono- or dizygotic twin, but the cross-twin dependencies depend on thezygosity, as one would assume that the dependence is larger for monozygotic pairs if there is a genetic effect. Tosummarize the frailty parameters: �1 is the individual association between neck N and low back L , �2 is the cross-twincross-trait association, �3 is the cross-twin association of the low back trait, and �4 is the cross-twin association of theneck trait.


2131


0.00

0.25

0.50

0.75

1.00

0.00

0.25

0.50

0.75

1.00

0 20 40 60 80 100

0 20 40 60 80 100 0 20 40 60 80 100

Adopted child Adoptive father Adoptive mother

Biological father Biological mother

Flexible parametric model

Weibull model

Sur

viva

l

Age

Figure 2. Graphs of the Kaplan–Meier, fitted Weibull survival function and fitted flexible parametric survival function. In thecases where the data are truncated (the parents), the survival function is conditional upon survival until first person becomes at

risk.

Table III. Estimates of the frailty parameters, for both the Weibull model and the flexible parametric model, andKendall’s tau with 95 per cent confidence intervals.

Weibull model Flexible parametric model

Frailty estimate Frailty estimate Kendall’s tau

Est. s.e. 95 per cent CI Est. s.e. 95 per cent CI Est. 95 per cent CI

AC-AF (1) 0.04 0.059 (−0.07,0.16) 0.04 0.054 (−0.07,0.14) 0.02 (−0.03,0.07)AC-AM (2) −0.03 0.059 (−0.15,0.08) −0.02 0.051 (−0.12,0.08) −0.01 (−0.06,0.04)AC-BF (3) 0.08 0.070 (−0.06,0.22) 0.06 0.058 (−0.05,0.17) 0.03 (−0.02,0.08)AC-BM (4) −0.09 0.066 (−0.22,0.04) −0.06 0.049 (−0.16,0.03) −0.03 (−0.08,0.02)AF-AM (5) 0.09 0.048 (−0.01,0.18) 0.08 0.044 (−0.01,0.17) 0.04 (−0.00,0.08)BF-BM (6) −0.03 0.061 (−0.15,0.09) −0.03 0.045 (−0.12,0.06) −0.02 (−0.06,0.03)

Figure 3. Dependence structure of the event times in a twin pair. L is the age-at-onset of low back pain and N isthe age-at-onset of neck pain. Abbreviation: zyg, zygosity.

Current status data are usually analyzed under the assumption of no calendar time effect, i.e. the probability of anevent may depend on the age of the individual, but not on calendar time. In these twin data there is, however, a clearcalendar time effect; persons born later in calendar time have a higher probability of answering yes to low back or neckpains.

2132



0.00

0.25

0.50

0.75

1.00

20 30 40 50 60 70 20 30 40 50 60 70

Low back pain Neck pain

Pre

vale

nce

Age in 2002

WomenMen

Women, Flexible parametric modelMen, Flexible parametric model

Figure 4. Graphs of the observed and fitted prevalences in 2002.

Table IV. Estimates of the frailty parameters for both gamma and stable frailty, with 95 per centconfidence intervals.

Gamma frailty Stable frailty

MZ DZ MZ DZ

�1 0.94 1.47(0.88,0.99) (1.44,1.50)

�2 0.66 0.19 1.40 1.11(0.55,0.77) (0.14,0.23) (1.33,1.47) (1.08,1.15)

�3 0.44 0.13 1.21 1.06(0.36,0.52) (0.09,0.17) (1.17,1.25) (1.04,1.08)

�4 0.75 0.21 1.28 1.08(0.61,0.89) (0.15,0.28) (1.23,1.34) (1.05,1.10)

To take this calender time effect into account, we need to extend the notation: as previously we let t denote the ageof the individual and now let b denote the birth year of the individual. The analysis of the dependence parameters issimilar to the situation without calendar time effect if we make the assumption that the frailty model holds across allbirth cohorts, h j (t,b|Z )= Z� j (t,b), i.e. the frailty effect is assumed the same for all cohorts. The conditional bivariatesurvival function for cohort b is

S(t1, t2,b|Z )=exp(−Z (M1(t1,b)+ M2(t2,b))),

where

M j (t,b)=∫ t

0� j (u,b)du

is the integrated conditional hazard. The observed bivariate survival function for cohort b, S(t1, t2,b), can similarly bewritten as in terms of the marginal survival probabilities

S(t1, t2,b)= L(L−1(S1(t1,b))+L−1(S2(t2,b))),

where S j (t,b) is the marginal univariate survival function. For a cross-sectional study at time b0, then t1 = t2 = t(b)=b0 −b. So, we have the same model structure as without calendar time effect, and can therefore estimate frailty parameters,but the function S j (t(b),b) is no longer a survival function.

We fitted four marginal hazard functions: for neck and for low back, separately for men and women. Each marginalwas fitted using three internal knots. The fitted and observed prevalences are shown in Figure 4; the observed prevalenceis simply the percentage of yes-answers for each of the ages 20–71, where the age of an individual was defined as theage at the end of 2002. There is a rather good fit of the prevalences 1−S j (t(b),b).

The model was fitted using both gamma and stable frailty (Table IV). To examine the robustness of the results to thespecification of the frailty distribution, we computed Kendall’s tau under the two frailty models (Table V). The pairwise


2133


Table V. Estimates of Kendall’s tau with 95 per cent confidence intervals.

Gamma frailty Stable frailty

MZ DZ MZ DZ

�1 0.32 0.32(0.31,0.33) (0.31,0.33)

�2 0.25 0.08 0.29 0.10(0.22,0.28) (0.06,0.11) (0.25,0.32) (0.08,0.13)

�3 0.18 0.06 0.17 0.06(0.15,0.21) (0.04,0.08) (0.15,0.20) (0.04,0.08)

�4 0.27 0.10 0.22 0.07(0.23,0.31) (0.07,0.12) (0.19,0.25) (0.05,0.09)

estimates look very similar for gamma and stable frailty, hence the results seem robust for the specification of the frailtydistribution.

The cross-twin dependencies are allowed to depend on the zygosity of the pair. Under the assumption that theenvironment is similar for each twin in a pair, any difference in the dependence between mono- and dizygotic twin pairscould be explained by a genetic effect on the predisposition to low back and neck pain. If we look at the cross-twindependence of low back pain, then a difference in dependence between mono- and dizygotic twins could be explainedby a genetic influence on low back pain. Testing the hypothesis of no difference in dependence between mono- anddizygotic twins, �3(mz)=�3(dz), gives a p-value lower than 0.001. Similarly for neck pain, testing the hypothesis of nodifference in dependence between mono- and dizygotic twins, �4(mz)=�4(dz), gives a p-value lower than 0.001. So, itis likely that genetic factors have an effect on the predisposition of neck and low back pain.

The cross-twin cross-trait dependencies are a little more complicated to interpret. First of all, the cross-twin cross-traitdependence between twins should be less than the cross-trait dependence within a person, so the dependence betweenneck and low back pain should be less cross-twin than within a person. If we look at mono- versus dizygotic pairs,then a difference in dependence between mono- and dizygotic twins could be interpreted as a genetic effect on thedependence between neck and low back pain, since we assume that the environmental effect on spinal pain is the sameregardless of zygosity. One often interprets a difference in the cross-twin cross-trait dependencies between mono- anddizygotic twins as: it is likely that there are genetic factors that promote predisposition to both neck and low back pain.Testing this hypothesis �2(mz)=�2(dz) gives a p-value <0.001, so it is likely there are genetic factors that makes youpredisposed to both neck and low back pain.

To sum up, if genetic influences are present some, but not necessarily all, of the following inequalities would hold.

�1 > �2(mz)>�2(dz),

�3(mz) > �3(dz),

�4(mz) > �4(dz).

In this example all of the above expected relations hold for both gamma and stable frailty. Thus it is likely that there aregenetic effects on the predispositions of developing low back and neck pain, and that it is likely that there are geneticfactors that influence the predisposition of both low back and neck pain.

5. Discussion

In the paper, we present a computationally simple and flexible approach for analyzing multivariate time-to-event dataunder right-censoring, interval-censoring and left-truncation. The hazard functions are modeled using cubic splines,which apply very mild assumptions on the shape of the hazard functions. The approach allows advanced dependencestructures between the event time, as shown for example in the twin back pain example, by using composite likelihood.

The composite likelihood approach to multivariate time-to-event analysis was suggested in Parner [6]. The approachwas, however, only investigated in a special case of Weibull marginal hazard and a gamma frailty distribution. TheWeibull hazard was here extended to flexible modeling of the hazard using cubic splines and the frailty distributionwas supplemented with the stable frailty. Several choices of the frailty distribution make it possible to investigate therobustness of the results of the various frailty distributions. We suggest transforming the frailty parameter to the Kendall’s

2134



tau correlation measure to compare the robustness of the results to the assumption of the frailty distribution. The use ofcubic splines to model hazard function was proposed for univariate time-to-event data in [7]. Nelson [24] applied thecubic spline approach to model relative survival.

Tibaldi et al. [10] also used composite likelihood to analyze multivariate time-to-event data, similarly to Parner [6].They modeled the marginal hazards by the Weibull distribution, but used the Plankett-Dale model for dependence insteadof the frailty model. In addition, they used all possible pairs of event times, rather than reducing the number of nuisanceparameters by considering the subset of pairs of interest. Andersen [25] proposed combining the composite likelihoodmethod with a two-stage estimation procedure; in the first stage, the marginal survival functions are estimated usingonly the univariate time-to-event data, and in the second stage, the frailty parameter is estimated using a compositelikelihood of bivariate frailty models. It is then possible to use semi-parametric modeling with non-parametric estimationof the marginal hazards and parametric estimation of the dependences, although the estimation of standard errors of thedependence parameters becomes much more complicated.

Statistical tests were for simplicity performed using Wald tests. Likelihood ratio tests for composite likelihood arepossible [16], but the asymptotic distribution of the likelihood ratio test statistic is a linear combination of chi-squaredistributions. Furthermore, model selection can be performed using the composite likelihood information criterion,developed in [26].

In twin studies, authors often estimate heritabilities by means of variance component models. We did, however,not estimate heritabilities in the back pain example, since we are investigating censored time-to-event data. The mostcommon method to model dependences in time-to-event data is through frailty models, and heritability, in the classicalsense, presumes a different structure of the variance component.

The combination of composite likelihood and flexible modeling of hazards using cubic spline is a versatile tool foranalyzing many types of multivariate time-to-event data, which we illustrate in our two data examples.

A copy of the Stata program (ado file) can be obtained by contacting the corresponding author.

Acknowledgements

Thorkild Sørensen kindly made the adoption study available. The twin study was kindly made available by The Danish TwinRegistry. The authors thank Philip Hougaard, Torben Martinussen and Jacob Hjelmborg for their comments.

References1. Wei LJ, Lin DY, Weissfeld L. Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. Journal

of the American Statistical Association 1989; 84(408):1065--1073.2. Huber PJ. The behavior of maximum likelihood estimates under nonstandard conditions. Proceedings of the Fifth Berkeley Symposium on

Mathematical Statistics and Probability 1967; 1:221--233.3. White H. A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica 1980; 48(4):

817--838. DOI: 10.1016/0304-4076(85)90158-7.4. Hougaard P. Analysis of Multivariate Survival Data. Springer: New York, 2000.5. Hoffmann-Jørgensen J. The marginal problem. In Functional Analysis II, Kurepa S, Kraljevic H, Butkovic D (eds). Springer: Berlin, 1987.6. Parner E. A composite likelihood approach to multivariate survival data. Scandinavian Journal of Statistics 2001; 28:295--302. DOI:

10.1111/1467-9469.00238.7. Royston P, Parmer MKB. Flexible parametric proportional-hazards and proportional-odds models for censored survival data, with application

to prognostic modelling and estimation of treatment effect. Statistics in Medicine 2002; 21:2175--2197. DOI: 10.1002/sim.1203.8. Sørensen TIA, Nielsen GG, Andersen PK, Teasdale TW. Genetic and environmental influences on premature death in adult adoptees. New

England Journal of Medicine 1988; 318(12):727--732.9. Nielsen GG, Gill RD, Andersen PK, Sørensen TIA. A counting process approach to maximum likelihood estimation in frailty models.

Scandinavian Journal of Statistics 1992; 19:25--44.10. Tibaldi F, Molenberghs G, Burzykowski T, Geys H. Pseudo-likelihood estimation for a marginal multivariate survival model. Statistics in

Medicine 2004; 23:947--963. DOI: 10.1002/sim.1664.11. Hestbaek L, Leboeuf-Yde C, Manniche C. Low back pain: what is the long-term course? a review of studies of general patient populations.

European Spine Journal 2003; 12:149--165. DOI: 10.1007/s00586-002-0508-5.12. Hestbaek L, Leboeuf-Yde C, Engberg M, Lauritzen T, Bruun NH, Manniche C. The course of low back pain in a general population results

from a 5-year prospective study. Journal of Manipulative and Physiological Therapeutics 2003; 26(4):213--219. DOI: 10.1016/S0161-4754(03)00006-X).

13. Neale MC, Cardon LR. Methodology for Genetic Studies of Twins and Families. Kluwer Academic Publishers: Dordrecht, 1992.14. Leboeuf-Yde C, Nielsen J, Kyvik KO, Fejer R, Hartvigsen J. Pain in the lumbar, thoracic or cervical regions: do age and gender matter?

a population-based study of 34,902 danish twins 20-71 years of age. BMC Musculoskeletal Disorders 2009; 10:39. DOI: 10.1186/1471-2474-10-39.

15. Lindsay BG. Composite likelihood methods. Contemporary Mathematics 1988; 80:221--239.16. Varin C. On composite marginal likelihoods. Advances in Statistical Analysis 2008; 92:1--28. DOI: 10.1007/s10182-008-0060-7.17. Cox DR, Reid N. Miscellenea, a note on pseudolikelihood constructed from marginal densities. Biometrika 2004; 91(3):729--737. DOI:

10.1093/biomet/91.3.729.


2135


18. Shemyakin AE, Youn H. Copula models of joint last survivor analysis. Applied Stochatic Models in Business and Industry 2006;22(2):211--224. DOI: 10.1002/asmb.629.

19. Goethals K, Janssen P, Duchateau L. Frailty models and copulas: similarities and differences. Journal of Applied Statistics 2008; 35(9):1071--1079. DOI: 10.1080/02664760802271389.

20. Stone CJ, Koo CY. Additive splines in statistics. Proceedings of the Statistical Computing Section ASA, Washington DC, 1985; 45--48.21. Harrell Jr FE. Regression Modelling Strategies. Springer: New York, 2001.22. Durrleman S, Simon R. Flexible regression models with cubic splines. Statistics in Medicine 1989; 8:551--561. DOI:

10.1002/sim.4780080504.23. StataCorp, Reference Manual, Release 10.0.24. Nelson CP, Lambert PC, Squire IB, Jones DR. Flexible parametric models for relative survival, with application in coronary heart disease.

Statistics in Medicine 2007; 26:5486--5498. DOI: 10.1002/sim.3064.25. Andersen EW. Composite likelihood and two-stage estimation in family studies. Biostatistics 2004; 5(1):15--30. DOI:

10.1093/biostatistics/5.1.15.26. Varin C, Vidoni P. A note on composite likelihood inference and model selection. Biometrika 2005; 92:519--528. DOI:

10.1093/biomet/92.3.519.

2136


Documents

Analyzing multivariate survival data using composite likelihood and flexible parametric modeling of the hazard functions