# Study of the bivariate survival data using frailty models based on Lévy processes

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## Abstract

Frailty models allow us to take into account the non-observable inhomogeneity of individual hazard functions. Although models with time-independent frailty have been intensively studied over the last decades and a wide range of applications in survival analysis have been found, the studies based on the models with time-dependent frailty are relatively rare. In this paper, we formulate and prove two propositions related to the identifiability of the bivariate survival models with frailty given by a nonnegative bivariate Lévy process. We discuss parametric and semiparametric procedures for estimating unknown parameters and baseline hazard functions. Numerical experiments with simulated and real data illustrate these procedures. The statements of the propositions can be easily extended to the multivariate case.

## Keywords

Frailty Lévy process Bivariate survival function Identifiability## 1 Introduction

*X*(

*t*) is a nonnegative Lévy process (subordinator) with Laplace exponent \( \Phi (c)\), \(R(t)=\int _0^tr(u)\mathrm{d}u\) defines the time transformation of subordinator for a nonnegative rate function

*r*(

*t*), and the nonnegative weight function

*a*(

*u*,

*s*) determines the extent to which the previous behavior of transformed subordinator influences the hazard function at time

*t*. The authors derived the expressions for the population survival and hazard functions in a general case:

The aim of this paper is to study the problem of identifiability for bivariate survival models with/without observed covariates and with time-dependent frailty when this frailty is given by a Lévy process (or Lévy processes). In addition, we demonstrate how these models can be used for longevity datasets based on simulated data.

In Sect. 2, we give the definitions of the univariate survival model under mixed proportional hazard specification and the bivariate correlated frailty model. We then discuss the definitions of the uni- and bivariate survival model with time-dependent frailty given by the nonnegative Lévy processes. At the end of this section, we discuss known findings related to the identifiability of survival models with time-independent frailty, formulate two new propositions about the identifiability of the bivariate survival models with time-dependent frailties, and give the EM algorithm for estimating unknown baseline hazard functions and parameters for the correlated bivariate model with time-dependent frailty. In Sect. 3, we discuss the results of estimations of the baseline hazard functions and unknown parameters (including the Cox-regression coefficients and parameters characterizing the frailty process) in experiments with simulated and real data for parametric and semiparametric approaches. The real-world example was based on the data from the Danish Twin Registry. Conclusions and outlook are presented in Sect. 4.

## 2 Survival analysis under a frailty setting

The assumption that the individual frailty is determined at birth and does not change with age seems to be too strong and unrealistic. To make the approach more flexible, we can weaken this assumption and suppose that the frailty is a random process.

*c*is the argument of Laplace transform. Note that

*d*and the jump measure \(\nu \) with support \((0,\infty )\) satisfying \(\int _{0}^{\infty }\min (1,x)\nu (\mathrm{d}x)<\infty \). The Laplace exponent is an increasing and concave function. The Laplace exponent and the jump measure for the gamma process are given by the formulas

*ht*and the scale parameter \(\gamma \) for the gamma-distributed random variable

*Z*(

*t*). This corresponds to the values of \(ht\gamma ^{-1}\) and \(ht\gamma ^{-2}\) for mean and variance of

*Z*(

*t*), respectively. To avoid non-identifiability of the model, we shall standardize the frailty distribution and put \(h=\gamma \). In this case, \(\mathbb {E}Z(t)=t\), \(\text {Var} Z(t)=t\gamma ^{-1}\) for \(t\geqslant 0\). In “Appendix 1,” we can find the formulas for calculating univariate population survivals in this case for a constant and an exponential (Gompertz) baseline hazard functions. We will denote the Laplace exponent for the univariate frailty processes \(Z_{j}(t)\), \(j=1,2\) by \(\Phi _{j}(.)\).

*A*, and the following integrability conditions for Lévy measure are satisfied:

### 2.1 Model identifiability

The identifiability of the univariate model with unspecified functional form of frailty distribution and baseline hazard has been studied by Elbers and Ridder (1982). This model is identifiable given information on *T* for finite \(\mathbb {E}Z\) and is not identifiable when frailty has an infinite mean. Identifiability of the correlated frailty models using data on the pair \((T_{1},T_{2})\) was proved by Honoré (1993) under the assumption of finite means of \(Z_{1}\) and \(Z_{2}\). Yashin and Iachine (1999) proved the identifiability of the correlated frailty model without observed covariates assuming that \(Z_{1}\) and \(Z_{2}\) are gamma distributed. Abbring and van den Berg (2003) studied the identifiability of the mixed proportional hazards competing risks model. We adopt this method to investigate the identifiability of the mixed bivariate survival model for time-dependent correlated frailties.

### Proposition 1

- Assumption 1.
Regressor functions \(\chi _{i}:U _{i}\rightarrow \mathbb {R_{+}}\) are continuous with supports \(U _{i}\), \(U_{i}\subset \mathbb {R}^{n}\), \(i=1,2\), and \(\chi _{1}({\mathbf {u}}_{\mathbf{1}}^{*})=\chi _{2}(\mathbf {u}_{\mathbf{2}}^{*})=1\) for some \(\mathbf {u}_{\mathbf{1}}^{*}\in U_{1}\), \(\mathbf {u}_{\mathbf{2}}^{*}\in U_{2}\). Set \(\Upsilon =\{(\chi _{1}({\mathbf {u}}_{1}),\chi _{2}({\mathbf {u}}_{2}))|\mathbf {u}_{\mathbf{1}}\in U_{1},\mathbf {u}_{\mathbf{2}}\in U_{2}\}\) contains a non-empty open set \(\Upsilon _{0}\subset \mathbb {R}^{2}_{+}\).

- Assumption 2.
Baseline hazard functions \(\lambda _{j}(.)\) are integrable on [0,

*t*] with \(\Lambda _{j}(t)=\int _{0}^{t}\lambda _{j}(\tau ) d \tau <\infty \) for all \(t\in \mathbb R_{+}\), and \(\Lambda _{1}(t^{*})=\Lambda _{2}(t^{*})=1\) for some \(t^{*}>0\), \(j=1,2\). - Assumption 3.Let \(\mu \) be a probability measure corresponding to the bivariate frailty variable \((Z_{1}(1),Z_2(1))\in \mathbb {R}_{+}^{2}\). Then,for some real number \(b>0\).$$\begin{aligned} \int _{0}^{\infty }\int _{0}^{\infty } e^{b(x_{1}+x_{2})}\mathrm{d}\mu <\infty \end{aligned}$$(8)

### Proof

*Identification of the regressor functions.*

*Identification of the hazard functions.*

For \(0\leqslant \tau \leqslant t\leqslant T<\infty \), it holds that \(\Lambda _{j}(\tau , t)\leqslant C< \infty \), \(j=1,2\). Therefore, there exists a real \(b(T)>0\) such that the bivariate function \(\Phi _\mathrm{biv}(\chi _{1}\Lambda _{1}(\tau ,t),\chi _{2}\Lambda _{2}(\tau ,t))\) and the univariate functions \(\Phi _{j}(\chi _{j}\Lambda _{j}(\tau ,t))\) are real analytic functions on the set \(\Upsilon _{T}=\{(\chi _{1},\chi _{2})|\chi _{1}>-b(T), \chi _{2}>-b(T)\}\) containing the point (0,0) for fixed \((\tau ,t)\), \(0\leqslant \tau \leqslant t\leqslant T<\infty \) and \(j=1,2\). Moreover, the univariate survival functions \(S_{j}(t_{j}|\chi _{j})\), their derivatives \(S_{j}^{\prime }(t_{j}|\chi _{j})\) in \(t_{j}\), \(j=1,2\), and the bivariate survival function \(S(t_{1},t_{2}|{\chi }_{1},\mathbf \chi _{2})\) are real analytic functions uniquely defined on \(\Upsilon _{T}\) for \(\{(t_{1},t_{2})|0\leqslant t_{1},t_{2}\leqslant T<\infty \}\).

*t*, dividing then by \(\chi _{j}\), and setting formally \(\chi _{j}\rightarrow 0\), we get the following equations:

*Identification of the Laplace exponent.*

### Proposition 2

- 1.
Assumption 1. Decomposition (11) holds for independent positive Lévy processes \(Y_{0}\), \(Y_{1}\), \(Y_{2}\) with zero drift and some \(\alpha >0\).

Assumption 2. Baseline hazard functions \(\lambda _{j}(.)\) are integrable on [0,

*t*] with \(\Lambda _{j}(t)=\int _{0}^{t}\lambda _{j}(\tau ) d \tau <\infty \) for all \(t\in \mathbb R_{+}\), \(\lim _{t\rightarrow \infty }\Lambda _{j}(t)=\infty \), and \(\Lambda _{j}(t^{*})=1\) for some \(t^{*}>0\), \(j=1,2\).Assumption 3. Jump measures \(\nu _{i}\) satisfy \(\int _{0}^{\infty }x\nu _{i}(\mathrm{d}x)<\infty \), \(i=0,1,2\).

Assumption 4. Let \(\mu _i\) is a probability measure corresponding to the frailty \(Y_i(1)\in \mathbb {R}_{+}^{1}\), \(i=0,1,2\). Then,for some real number \(b>0\) and \(i=0,1,2\).$$\begin{aligned} \int _{0}^{\infty }\int _{0}^{\infty } e^{bx}\mathrm{d}\mu _i<\infty \end{aligned}$$(12)

The proof of Proposition 2 is given in “Appendix 3.”

### 2.2 Model validation

*i*, \(i=1,\ldots ,n\). The estimate of the vector parameter \(\theta \) we can find by maximizing the log-likelihood function (13).

*j*in twin pair

*i*, \(i=1,\ldots ,n\), \(j=1,2\). Define random processes

- 1.
*Initialization.*Set \(l=0\). Put \(\hat{\zeta } _{0}=(0, \varrho _{0})\) for any \(0\leqslant \varrho _{0}\leqslant 1 \) in the case of time-dependent frailty and \(\hat{\zeta } _{0}=(0, \rho _{0})\) for any \(0\leqslant \rho _{0}\leqslant 1 \), otherwise. This corresponds to \(\hat{Z}_{ij}(t)=t\) and \(\hat{Z}_{ij}=1\), respectively. Calculate \(\hat{\beta } _{1}\) and \(\hat{\Lambda } _{1}(t)\) from (14) and (15), respectively. Given the estimates \(\hat{\beta } _{1}\) and \(\hat{\Lambda } _{1}(t)\), calculate \(\hat{\zeta } _{1}\) by maximizing (16). Set \(l=1\). - 2.
Given \(\hat{\Lambda } _{l}(t)\) and \(\hat{\zeta } _{l} \), calculate \(\hat{\beta } _{l+1}\) from (14).

- 3.
Given the estimates \(\hat{\zeta } _{l}\) and \(\hat{\beta } _{l+1} \), calculate \(\hat{\Lambda } _{l+1}(t)\) by using formula (15).

- 4.
Given the estimates \(\hat{\beta } _{l+1}\), \(\hat{\Lambda } _{l+1}(t)\), calculate \(\hat{\zeta } _{l+1}\) by maximizing (16).

- 5.
Stop if convergence is reached with respect to estimates of \(\beta \) and \(\zeta \). Otherwise, \(l\rightarrow l+1\) and repeat steps 2-5.

*n*is the number of pairs). The values of \(\Lambda _\mathrm{appr}^\mathrm{dyn}(t_i)\) can be calculated recurrently for \(i=1,2,\ldots ,2n\) using a simple bisectional procedure. Note that the function \(\Lambda _\mathrm{appr}^\mathrm{dyn}(t)\) converges pointwise to the solution of (19) as \(n\rightarrow \infty \) and the distance between neighboring moments \(t_i\) tends to zero.

## 3 Results

### 3.1 Experiments with simulated data

In this subsection, we will discuss the results of the consistency test for the correlated frailty models with time-dependent and time-independent frailties (11). It was assumed that \(\alpha =1\), \(\text{ Var }Z_1=\text{ Var }Z_2=\sigma ^2\), and \(\text{ Corr }(Z_1,Z_2)=\rho \) in the case of the time-independent frailty or \(\text{ Var }Z_1(1)=\text{ Var }Z_2(1)=\gamma ^{-1}\) and \(\text{ Corr }(Z_1(1),Z_2(1))=\varrho \) in the case of the time-dependent frailty, baseline hazard functions \(\lambda _j(t)\) followed Gompertz (exponential) form \(\lambda _j(t)=a\exp (bt)\), and an observed covariate \({\mathbf {u}}\) influenced longevity so that the conditional hazard function was defined by \(\mu _j(t_j|Z_j,{\mathbf {u}}_j)=Z_j\exp (\beta {\mathbf {u}}_j)\lambda _j(t)\), \(j=1,2\). The covariates were randomly generated from the uniform distribution on the interval [0,1] and were independent for the individuals. The (true) values for data generating are given in Tables 1 and 2 and have been chosen so that the mean and the standard deviation of the generated times-to-event were equal to approximately 75 and 12 years, respectively. The bivariate times-to-event have been generated using formula (4) with \(\sigma _1=\sigma _2=\sigma \) in the case of the time-independent frailty and using formula (25) given in “Appendix 2” in the case of time-dependent frailty. In both cases, it was assumed that \(\Lambda _1(t)=\Lambda _2(t)=(a/b)(\exp (bt)-1)\) and \(\chi _1({\mathbf {u}})=\chi _2({\mathbf {u}})=\exp (\beta {\mathbf {u}})\). We have not truncated or censored the generated data. We estimated unknown parameters and cumulative hazard functions using the classic maximum likelihood estimator (parametric method) and the EM algorithm (semiparametric method). In all cases, we simulated 100 datasets for 500 twin pairs.

Estimates of unknown parameters for the time-independent frailty model

True | Estimator | ||||
---|---|---|---|---|---|

Classic MLE | EM algorithm | ||||

Mean | SD | Mean | SD | ||

\(10^{5}\cdot a\) | 1 | 1.043 | 0.369 | – | – |

\(10\cdot b\) | 1 | 1.002 | 0.049 | – | – |

\(\beta \) | 3 | 3.012 | 0.231 | 2.889 | 0.244 |

\(\sigma ^{2}\) | 1 | 1.002 | 0.125 | 0.890 | 0.166 |

\(\rho \) | 0.5 | 0.497 | 0.086 | 0.543 | 0.100 |

Estimates of unknown parameters for the time-dependent frailty model

True | Estimator | ||||
---|---|---|---|---|---|

Classic MLE | EM algorithm | ||||

Mean | SD | Mean | SD | ||

\(10^{7}\cdot a\) | 1 | 1.059 | 0.464 | – | – |

\(10\cdot b\) | 1 | 1.005 | 0.052 | – | – |

\(\beta \) | 3 | 3.018 | 0.221 | 3.006 | 0.176 |

\(20\cdot \gamma \) | 1 | 1.088 | 0.579 | 1.094 | 0.195 |

\(\varrho \) | 0.5 | 0.465 | 0.240 | 0.535 | 0.088 |

### 3.2 Experiments with real data

For experiments with real data, we used the datasets from the Danish Twin Registry (DTR). This registry was created in the 1950s. It is one of the oldest population-based registries in the world and contains information about twins born in Denmark since 1870 and who survived to age 6. Multiple births were manually ascertained in birth registers from all 2200 parishes in Denmark. As soon as a twin was traced, a questionnaire was mailed to the twin, to her/his partner or to their closest relatives if neither of the twin partners were alive. The zygosity of twins was assessed on the basis of questions about phenotypic similarities. The reliability of the zygosity diagnosis was validated by comparing laboratory methods based on the blood, serum, and enzyme group determination. In general, the misclassification rates were less than 5\(\%\). Other information includes the data on sex, birth, cause of death, health, and lifestyle. An important feature of the Danish twin survival data is their right censoring and left truncation. In our study, we used the longevity data on the like-sex twins with known zygosity born between 1870 and 1900 and who survived until age 30. This non-censored data include 470 male monozygotic (MZ) twin pairs, 475 female MZ twin pairs, 780 male dizygotic (DZ) twin pairs, and 835 male DZ twin pairs. Further details on the Danish Twin Registry can be found in Hauge (1981).

Estimates of unknown parameters (standard errors) for the time-independent and \( ^{\dag }\)the time-dependent frailty models

Frailty | ||||
---|---|---|---|---|

Time-independent | Time-dependent | |||

Males | Females | Males | Females | |

\(10^{5}\cdot \tilde{a}\) | 7.004 | 6.735 | 7.005 | 6.710 |

(0.782) | (0.743) | (0.835) | (0.997) | |

\(10\cdot \tilde{b}\) | 0.916 | 0.887 | 0.916 | 0.888 |

(0.015) | (0.014) | (0.016) | (0.020) | |

\(s^2\) | 0 | 0 | 0 | 0 |

– | – | – | – | |

\(\sigma ^{2}\) or \(^{\dag }10^{2}\cdot \gamma \) | 1.751 | 1.166 | 0.626 | 1.718 |

(0.811) | (0.853) | (1.696) | (0.773) | |

\(\rho _{MZ}\) or \(^{\dag }\varrho _{MZ}\) | 0.507 | 0.538 | 0.477 | 0.671 |

(0.092) | (0.132) | (0.170) | (0.158) | |

\(\rho _{DZ}\) or \(^{\dag }\varrho _{DZ}\) | 0.191 | 0.318 | 0.176 | 0.391 |

(0.070) | (0.111) | (0.091) | (0.111) | |

| \(-\) 9873.705 | \(-\) 10,470.3 | \(-\) 9872.825 | \(-\) 10,470.14 |

| 19,757.41 | 20,950.6 | 19,755.65 | 20,950.28 |

## 4 Discussion

Frailty models are a powerful tool for studying non-observable inhomogeneity in a population related to time-to-failure (e.g., death or disease). Models with time-independent frailty have been intensively studied over the last decades and have found a wide range of applications in survival analysis and in searching for genes influencing longevity. However, the studies based on the models with time-dependent frailty are scarce. In this paper, we have attempted to improve the knowledge in this area and to study some properties of multivariate survival models with time-dependent frailty components.

Proposition 1 we have formulated and proved for the bivariate case. It is not difficult to generalize this result and to prove the identifiability of the frailty model with observed covariates for any number *J* of related individuals equal or greater than 1 if the time-dependent frailty is a multivariate Lévy process. Similarly, we can generalize Proposition 2 for the case of \(J\geqslant 2\). However, the number of frailty components in the multivariate analog of the decomposition (11) will be equal to \(2^J-1\). The shared frailty model where all individuals in a family or cluster share the same non-observable risk of failure does not meet this problem.

In experiments with simulated data, we tested for consistency and used parametric and semiparametric approaches. In the parametric approach, we assumed that the parametric form of the baseline hazard functions is known and follows the Gompertz form. All unknown parameters characterizing frailty distribution, baseline hazard function, and Cox-regression parameters were estimated directly by maximizing the likelihood function. In the semiparametric approach, we used the EM algorithm and estimated the Cox-regression parameters and the parameters of frailty distribution by maximizing the partial likelihood function. The cumulative baseline hazard function was estimated using the Breslow estimator regarding this function as infinite-dimensional parameters. The EM algorithm suffers from its slow convergence. Moreover, in the semiparametric approach, the number of calculations increases with the number of individuals much more rapidly than in the parametric one. It leads to the drastic slowing of the convergence of the EM algorithm and increases substantially the time of estimation. It makes implementing the EM algorithm in the case of the time-dependent frailty for analysis of the real data problematic.

Experiments with real data show that the proposed method and the method with time-independent frailty produce similar shapes of the estimated bivariate probability density functions. The baseline cumulative hazard functions have been chosen so that the estimated marginal survival functions guarantee the best fit to the empirical ones according to Eqs. (19)–(20). A large degree of similarity of the estimated bivariate density functions for the models with time-dependent and time-independent frailties in the range of ages 30–100 years guarantees the similar bivariate fit. The difference between the two approaches can involve the shape of the baseline hazard functions and the asymptotic behavior of the bivariate probability density functions. The models with time-dependent and time-independent frailties are not nested. Therefore, we cannot compare them using the likelihood ratio test. For this purpose, the AKAIKE Information Criterion can be used. In accordance with this criterion, the model with time-dependent frailty is slightly more informative compared to the one with time-independent frailty for the data we considered.

Gorfine and Hsu (2011) studied the robustness of the multivariate survival models with frailty components against the violations of the model assumptions. It was found that unnecessary modeling of the dependency between the frailty variates can lead to some efficiency loss of parameter estimates. Misspecification of the frailty distribution can introduce a bias in estimates. Misspecification of the baseline hazard functions can lead to severe bias of all estimates if we use the parametric maximum likelihood estimator, where the baseline hazard functions follow the parametric form. The nonparametric maximum likelihood estimator does not suffer from this drawback. Note that in experiments with real data, we have used a flexible parametrization of the baseline cumulative hazard functions given by formulas (19)–(20). This parameterization does not presume any knowledge about the form of the baseline hazard function. It is sufficient to have a good approximation of the marginal survival function.

An extension of the present study can include the investigation of identifiability of the survival models with competing risks and time-dependent frailty components. The piecewise-constant approximation of the cumulative hazard function has been used in experiments with real data [formulas (19)–(20)]. Other approximative functions such as piecewise linear or piecewise exponential can be used to improve the bivariate goodness-of-fit. Further, numerical experiments with real data are needed to understand whether the proposed method improves the goodness-of fit on the method with time-independent frailty.

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