Semiparametric Marginal Regression Analysis for Dependent Competing Risks under an Assumed Copula

J. R. Statist. Soc. B (2010) 72, Part 2, pp. 235–251

• Dependent censoring：dependent competing risks
• Examples : The end point and withdrawal as two dependent competing risks in AIDS Clinical Trials.

• Problem : semiparametric marginal regression analysis for dependent competing risks.

• Zheng & Klein(1995): Copula graphic estimator for marginal distributions, but without covariates.

• Huang & Zhang(2008):the competing risks are marginally modeled by PH model and are jointly modelled by a known copula. And they apply a 'redistribution - of - mass' algorithm for fitting with no theoretical justification.

• non-parametric maximum likelihood approach to estimation of the marginal semiparametric transformation models under any assumed copula. Generalization of Zheng and Klein(1995).

• Closed form expressions for score functions which facilitate efficient computation.

• more general and unifying framework of event time regression analysis

• \( S_k \): marginal survival function
• \( \tau \): the maximum fellow up time
• \( \Lambda_k \): cumulative intensity function
• \( G_k \): specified non-negative function
• \( R_k \): unspecified increasing function
• \( I_{*} \): the expected single-obs information matrix at \( \Theta_{*} \).
• \( \Theta_{*} \): true value of the parameter vectors

1. Copula \( C(S_1(t_1),S_2(t_2)) = P(T_1^{*}>t_1, T_2^{*}>t_2) \), \( S_k=exp(-\Lambda_k) \)

2. semiparametric transformation models: \[ \Lambda_k(t;\beta_k,R_k) = G_k[\int_0^t I(T_k^{*} ≥ s)exp\{\beta_k^T Z_k(s)\}dR_k(s)], k=1,2 \] We derive the MLE for \( (\{dR_k\}, \beta_k, k=1,2) \) where \( dR_k \) is the collection of the jump size of \( R_k(.) \) at the observed times of event k, because we view \( R_k(.) \) as a non-decreasing step function with jumps only at the time where event k is observed.

3. Writing likelihood based on 1 and 2.

the cause-specific intensity for the counting process Nki(t)(k = 1,2) is： \[ Y_i(t)exp\{β_k^T Z_{ki}(t)\}η_{ki}(t−;β,R)r_k(t) \]

This can be deduced by taking partial derivative of \( -log(C(S_1(t_1=t),S_2(t_2=t))) \) with respect to \( t_k \). Recall the relation between intensity and the survival function\( S= exp\{- \int \lambda\} \)

if the competing cause is independent censoring:
\[ log(f=\frac{\delta(1-\exp^{-\Lambda(t)})}{\delta t})= log(\lambda(t)) - \Lambda \] \[ log(f)= log(intensity) - cumulative~~intensity \]

data \( \{T_i,Z_{KI}(t); k =1,2, i=1...n\} \), para \( \{\beta , dR_k\} \):

• \( \hat \beta = (\hat \beta_1,\hat \beta_2)~~and~~\hat R = (\hat R_1,\hat R_2) \).

• The \( \hat \beta \) is strongly consistent.

• \( \hat R(.) \) converges to R(.) uniformly in the interval [0,\( \tau \)] with probability 1.

Zeng & Lin (2006)

• \( \Theta = (dR_1(t),dR_2(t),\beta_1,\beta_2) \),
• Taylor Expansion:\( n^{1/2}(\hat \Theta -\Theta_{*}) = n^{-1/2} I_{*}^{-1} U(\Theta_{*}) + O_p(1) \) where \( I_{*} \) is the expected single-obs information matrix at \( \Theta_{*} \).
• \( U(\Theta) \) are martingales, \( V_{n^{-1/2}U(\Theta)}=I_{*} \).
  

By martingale CLT, that the linear combination \( H^{T}n^{1/2}(\hat \Theta - \Theta) \):

\[ n^{1/2}\int_0^{\tau}w(t)^{T}\{\hat dR(t) - dR(t)\} + n^{1/2}b^{T}(\hat \beta-\beta) \rightarrow N(0,H^{T}I_{*}^{-1}H) \]

\( H^{T} = (W^T b^{T}) \)

By Cramér–Wold theorem, \[ n^{1/2}(\hat \Theta -\Theta_{*}) = n^{-1/2} I_{*}^{-1} U(\Theta_{*}) \rightarrow N(0,I_{*}^{-1}) \]

Quasi-Newton Methods(fminunc in MATLAB) which requires explicit expressions of the score functions and information matrix.

• simulation settings:
  The event T1 follows a proportional odds model
  The event T2 follows a proportional hazard model
  The joint distribution of (T1,T2) is given by a Frank, Clayton or Gumbel copula whose association parameter\( \alpha \) has been specified to give Kendall's \( \tau \) of 0.3 or 0.8. (see table1,2)

• copula function sensitivity:
  The bias of the estimates based on the Frank and Gumbel copulas when the data are indeed generated from a Clayton copula shows that the bias due to misspecification of the form of copula is usually moderate if the level of association between event times is correctly specified in the analysis. (see table3)

• see paper's section 6

• two dependent competing risks under an assumed copula model: end point and withdrawal

• marginal regression models for the end point and withdrawal are specified by Cox’s proportional hazards models.

see table 4

The identifiability difficulty in the competing risks problem.

Given assumed forms for the copula and marginal regression models, the variability of the resulting estimates can be quite large for both the regresion and association parameter.

Copulas are useful in portfolio/risk management and help us analyse the effects of downside regimes by allowing the modelling of the marginals and dependence structure of a multivariate probability model separately.

For example, the individualistic behaviour of each trader in a stock exchange can be described by modelling the marginals. However, each trader's actions have an interaction effect with other traders'. This interaction effect can be described by modelling the dependence structure. Therefore, copulas allow us to analyse the interaction effects which are of particular interest during downside regimes as investors tend to herd their trading behaviour and decisions.