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).
Denote by \(T\) the time to a failure event of interest. Suppose the study participants can potentially experience any of several, say \(J\), different types of failure events. Let \(\epsilon=1, \ldots, J\) indicate the failure event type.
The cumulative incidence function (CIF) for the \(j\)th competing event is defined as $$ F_j(t)=\mbox{pr}(T\leq t, \epsilon =j), \;\; j=1,\ldots, J. $$
Huang et al. (2016) proposed a non-parametric estimator for the conditional cumulative incidence function (CCIF) $$ G_j(t) = \mbox{pr}(T\le t \mid T\le \eta, \epsilon =j), \;\; t\in[0,\eta],\;\; j=1,\ldots, J, $$ where the constant \(\eta\) is determined from the knowledge that survival times could potentially be observed up to time \(\eta\).
To compare the CCIF of different failure types \(j\neq k\), we consider the following class of stochastic processes $$ Q (t) = K(t)\{\widehat G_j(t) - \widehat G_k(t)\}, $$ where \(K(t)\) is a weight function. For a formal test, we propose to use the supremum test statistic $$ \sup_{t\in [0,\eta] } \mid Q(t) \mid, $$ an omnibus test that is consistent against any alternatives under which \(G_j(t) \neq G_k(t)\) for some \(t\in [0,\eta]\).
An approximate \(p\)-value corresponding to the supremum test statistic is obtained by applying the technique of permutation test.
For bivariate gap times (e.g. time to disease recurrence and the residual lifetime after recurrence), let \(V\) and \(W\) denote the two gap times so that \(V+W\) gives the total survival time \(T\). Note that, given the first gap time \(V\) being uncensored, the observable region of the second gap time \(W\) is restricted to \(C-V\). Because the two gap times \(W\) and \(V\) are usually correlated, the second gap time \(W\) is subject to induced informative censoring \(C-V\). As a result, conventional statistical methods can not be applied directly to estimate the marginal distribution of \(W\).
Huang et al. (2016) proposed non-parametric estimators for the cumulative incidence function for the bivariate gap time \((V, W)\) $$ F_j (v,w)=\mbox{pr}( V\le v, W\le w, \epsilon=j ) $$ and the conditional bivariate cumulative incidence function $$ H_j(v, w)=\mbox{pr}(V\le v, W\le w \mid T \le \eta, \epsilon=j). $$
To compare the joint distribution functions \(H_j(v, w)\) and \(H_k(v, w)\) of different failure types \(j\neq k\), we consider the supremum test \(\sup_{v+w\le\eta}\mid Q^*(v, w)\mid\) based on the following class of processes $$ Q^*(v, w) = K^*(v, w) \{\widehat H_j(v, w) - \widehat H_k(v, w)\}, $$ where \(K^*(v, w)\) is a prespecified weight function.
The approximate \(p\)-value can be obtained through simulation by applying the technique of permutation tests.
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 $$ \tau_j^*= 4\times \mbox{pr}(V_1>V_2, W_1>W_2\mid V_1+W_1\le\eta, V_2+W_2< \eta,\epsilon_1=j, \epsilon_2=j)-1. $$
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