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 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 jth 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.

Bivariate Gap Time Distribution With Competing Risks

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.

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

\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

Conditional Cumulative Incidence Function (CCIF) Estimation

Description

Estimate the conditional cumulative incidence function. See bigtcr-package.

Usage

get.ccif(obs.y, event, tau = Inf)

Arguments

Value

A matrix with class ccif that has J columns. Columns 1 to J correspond to G_1(t) to G_J(t). Each row represents a distinct observed time point t and the row name contains the value of t.

Examples

Gj <- get.ccif(obs.y = pancancer$obs.y, event = pancancer$min.type, tau   = 120);

Conditional Bivariate Cumulative Incidence Function Estimation

Description

Estimate the conditional bivariate cumulative incidence function. See bigtcr-package.

Usage

get.gap.ccif(obs.y, event, v, tau = Inf)

Arguments

Value

A matrix with class gap.ccif that has J+2 columns. Column 1 and 2 are (v,w). The rest columns correspond to H_1(v,w) to H_J(v,w). Each row represents a distinct observed time point and the row name contains the value of this time point.

Examples

Hj <- get.gap.ccif(obs.y=pancancer$obs.y, event=pancancer$min.type, v=pancancer$v, tau=120)

Cause-Specific Kendall's tau Estimation

Description

Estimate the modified cause-specific Kendall's tau for the evaluation of association for bivariate gap time with competing risks. See bigtcr-package.

Usage

get.gap.kt(obs.y, event, v, tau = Inf, nbs = 0)

Arguments

Value

A list of the estimation and variances of modified casue-specific Kendall's tau

Examples

Kt <- get.gap.kt(obs.y=pancancer$obs.y, event=pancancer$min.type,
               v=pancancer$v, tau=120, nbs=5)

Comparison of Bivariate CCIF

Description

Compare the bivariate CCIF of different failure typess by applying the technique of permutation test. See bigtcr-package.

Usage

get.gap.pval(obs.y, event, v, tau = Inf, comp.event = c(1, 2), np = 1000,
  Kt = function(x) {     1 })

Arguments

Value

P-value of the hypothesis test H_0: H_j = H_k = \ldots = H_l.

Examples

gap.pval <- get.gap.pval(pancancer$obs.y, pancancer$min.type, pancancer$v,
                         tau=120, comp.event=c(1,2), np=20);

Kendall's tau Estimation

Description

Estimate Kendall's tau association between two random variables

Usage

get.kendalltau(v, w)

Arguments

Examples

kt <- get.kendalltau(pancancer$v, pancancer$w);

Comparison of CCIF

Description

Compare the CCIF of different failure typess by applying the technique of permutation test. See bigtcr-package.

Usage

get.pval(obs.y, event, tau = Inf, comp.event = c(1, 2), np = 1000,
  Kt = function(x) {     1 })

Arguments

Value

P-value of the hypothesis test H_0: G_j = G_k = \ldots = G_l.

Examples

pval <- get.pval(pancancer$obs.y, pancancer$min.type,
                 tau=120, comp.event=c(1,2), np=20);

Example Pancreatic Cancer Dataset

Description

Simulated data used in bigtcr examples.

Format

A dataframe with 3 variables:

obs.y

Observed time to failure events or censoring in months

min.type

Type of failure events

0

Censored

1

death with metastasis limited to lung only

2

death with metastasis that involves sites other than lung (e.g. liver)

3

death without disease recurrence

v

Time to recurrence. NA if no recurrence observed

Details

Data simulated based on the patients who had surgical resection of pancreatic adenocarcinomas and had postoperative follow-up at the Johns Hopkins Hospital between 1998 and 2007.