bigtcr-package {bigtcr}R Documentation

Bivariate Gap Time with Competing Risks

Description

This package implements the non-parametric estimator for the conditional cumulative incidence function and the non-parametric conditional bivariate cumulative incidence function for the bivariate gap times proposed in Huang et al. (2016).

Conditional Cumulative Incidence Functions

Denote by TT the time to a failure event of interest. Suppose the study participants can potentially experience any of several, say JJ, different types of failure events. Let ϵ=1,,J\epsilon=1, \ldots, J indicate the failure event type.

The cumulative incidence function (CIF) for the jjth competing event is defined as

Fj(t)=\mboxpr(Tt,ϵ=j),    j=1,,J. F_j(t)=\mbox{pr}(T\leq t, \epsilon =j), \;\; j=1,\ldots, J.

Huang et al. (2016) proposed a non-parametric estimator for the conditional cumulative incidence function (CCIF)

Gj(t)=\mboxpr(TtTη,ϵ=j),    t[0,η],    j=1,,J, G_j(t) = \mbox{pr}(T\le t \mid T\le \eta, \epsilon =j), \;\; t\in[0,\eta],\;\; j=1,\ldots, J,

where the constant η\eta is determined from the knowledge that survival times could potentially be observed up to time η\eta.

To compare the CCIF of different failure types jkj\neq k, we consider the following class of stochastic processes

Q(t)=K(t){G^j(t)G^k(t)}, Q (t) = K(t)\{\widehat G_j(t) - \widehat G_k(t)\},

where K(t)K(t) is a weight function. For a formal test, we propose to use the supremum test statistic

supt[0,η]Q(t), \sup_{t\in [0,\eta] } \mid Q(t) \mid,

an omnibus test that is consistent against any alternatives under which Gj(t)Gk(t)G_j(t) \neq G_k(t) for some t[0,η]t\in [0,\eta].

An approximate pp-value corresponding to the supremum test statistic is obtained by applying the technique of permutation test.

Bivariate Gap Time Distribution With Competing Risks

For bivariate gap times (e.g. time to disease recurrence and the residual lifetime after recurrence), let VV and WW denote the two gap times so that V+WV+W gives the total survival time TT. Note that, given the first gap time VV being uncensored, the observable region of the second gap time WW is restricted to CVC-V. Because the two gap times WW and VV are usually correlated, the second gap time WW is subject to induced informative censoring CVC-V. As a result, conventional statistical methods can not be applied directly to estimate the marginal distribution of WW.

Huang et al. (2016) proposed non-parametric estimators for the cumulative incidence function for the bivariate gap time (V,W)(V, W)

Fj(v,w)=\mboxpr(Vv,Ww,ϵ=j) F_j (v,w)=\mbox{pr}( V\le v, W\le w, \epsilon=j )

and the conditional bivariate cumulative incidence function

Hj(v,w)=\mboxpr(Vv,WwTη,ϵ=j). H_j(v, w)=\mbox{pr}(V\le v, W\le w \mid T \le \eta, \epsilon=j).

To compare the joint distribution functions Hj(v,w)H_j(v, w) and Hk(v,w)H_k(v, w) of different failure types jkj\neq k, we consider the supremum test supv+wηQ(v,w)\sup_{v+w\le\eta}\mid Q^*(v, w)\mid based on the following class of processes

Q(v,w)=K(v,w){H^j(v,w)H^k(v,w)}, Q^*(v, w) = K^*(v, w) \{\widehat H_j(v, w) - \widehat H_k(v, w)\},

where K(v,w)K^*(v, w) is a prespecified weight function.

The approximate pp-value can be obtained through simulation by applying the technique of permutation tests.

Nonparametric Association Measure for the Bivariate Gap Time With Competing Risks

To evaluate the association between the bivariate gap times, Huang et al. (2016) proposed a modified Kendall's tau measure that was estimable with observed data

τj=4×\mboxpr(V1>V2,W1>W2V1+W1η,V2+W2<η,ϵ1=j,ϵ2=j)1. \tau_j^*= 4\times \mbox{pr}(V_1>V_2, W_1>W_2\mid V_1+W_1\le\eta, V_2+W_2< \eta,\epsilon_1=j, \epsilon_2=j)-1.

References

Huang CY, Wang C, Wang MC (2016). Nonparametric analysis of bivariate gap time with competing risks. Biometrics. 72(3):780-90. doi: 10.1111/biom.12494


[Package bigtcr version 1.1 Index]