# Generate a sample of time to event dataset with dependent right censoring under an Archimedean copula Generate a sample of time to event dataset with, dependent right censoring based on one of the Archimedean copulas the given Kendall's tau, sample size n and covariates matrix Z. ```r surv_data_dc(n, a, Z, lambda, betas, phis, cons7, cons9, tau, copula, distr.ev, distr.ce) ``` ## Arguments - `n`: the sample size, or the number of the subjects in a sample. - `a`: the shape parameter of baseline hazard for the event time $T$. - `Z`: the covariate matrix with dimension of $n$ by $p$, where $p$ is the number of covariates. - `lambda`: the scale parameter of baseline hazard for event time $T$. - `betas`: the regression coefficient vector of proportional hazard model for the event time $T$ with dimenion of $p$ by $1$. - `phis`: the regression coefficient vector of proportional hazard model for dependent censoring time $C$ with dimenion of $p$ by $1$. - `cons7`: the parameter of baseline hazard for the dependent censoring time $C$ if assuming an exponential distribution. - `cons9`: the upper limit parameter of uniform distribution for the independent censoring time $A$, i.e. $A$~U(0, cons9). - `tau`: the Kendall's correlation coefficient between $T$ and $C$. - `copula`: the Archemedean copula that captures the dependence between $T$ and $C$, a characteristc value, i.e. 'independent', 'clayton', 'gumbel' or 'frank'. - `distr.ev`: the distribution of the event time, a characteristc value, i.e. 'weibull' or 'log logit'. - `distr.ce`: the distribution of the dependent censoring time, a characteristc value, i.e. 'exponential' or 'weibull'. ## Returns A sample of time to event dataset under dependent right censoring, which includes observed time $X$, event indicator $\delta$ and dependent censoring indicator $\eta$. ## Details surv_data_dc allows to generate a survival dataset under dependent right censoring, at sample size `n`, based on one of the Archimedean `copula`, Kendall's `tau`, and covariates matrix `Z` with dimension of $n$ by $p$. For example, at `p=2`, we have `Z=cbind(Z1, Z2)`, where `Z1` is treatment generated by distribution of bernoulli(0.5), i.e. 1 represents treatment group and 0 represents control group; `Z2` is the age generated by distribution of U(-10, 10). The generated dataset includes three varaibles, which are $X_i$, $\delta_i$ and $\eta_i$, i.e. $X_i=min(T_i, C_i, A_i)$, $\delta_i=I(X_i=T_i)$ and $\eta_i=I(X_i=C_i)$, for $i=1,\ldots,n$. 'T' represents the event time, whose hazard function is $$ h_T(x)=h_{0T}(x)exp(Z^{\top}\beta) $$ , where the baseline hazard can take weibull form, i.e. $h_{0T}(x) = ax^{a-1} / \lambda^a$, or log logistic form, i.e. $$ h_{0T}(x) = \frac{ \frac{ 1 }{ a exp( \lambda ) } ( \frac{ x }{ exp( \lambda ) } )^{1/a -1 } }{ 1 + ( \frac{ x }{ exp( \lambda ) } )^{1/a} } $$ . 'C' represents the dependent censoring time, whose hazard function is $ h_{C}(x) = h_{0C}(x)exp( Z^{\top}\phi) $, where the baseline hazard can take exponential form, i.e. $h_{0C}(x)=cons7$, or weibull form, i.e. $h_{0C}(x) = ax^{a-1} / \lambda^a$.'A' represents the administrative or independent censoring time, where A~U(0, cons9). ## Examples ```r ##-- Copula types copula3 <- 'frank' ##-- Marginal distribution for T, C, and A a <- 2 lambda <- 2 cons7 <- 0.2 cons9 <- 10 tau <- 0.8 betas <- c(-0.5, 0.1) phis <- c(0.3, 0.2) distr.ev <- 'weibull' distr.ce <- 'exponential' ##-- Sample size n <- 200 ##-- One sample Monte Carlo dataset cova <- cbind(rbinom(n, 1, 0.5), runif(n, min=-10, max=10)) surv <- surv_data_dc(n, a, cova, lambda, betas, phis, cons7, cons9, tau, copula3, distr.ev, distr.ce) n <- nrow(cova) p <- ncol(cova) ##-- event and dependent censoring proportions colSums(surv)[c(2,3)]/n X <- surv[,1] # Observed time del<-surv[,2] # failure status eta<-surv[,3] # dependent censoring status ``` ## References Xu J, Ma J, Connors MH, Brodaty H. (2018). **"Proportional hazard model estimation under dependent censoring using copulas and penalized likelihood"**. Statistics in Medicine 37, 2238–2251. ## See Also `coxph_mpl_dc` ## Author(s) Jing Xu, Jun Ma, Thomas Fung

surv_data_dc function