 melanoma and time to other causes based on VM (in gray) and on VMCF (in black) for a male individual with ulceration and for the mean values of the continuous covariates, together with the $95\%$ confidence intervals derived by fitting the former model.}\label{fig:CumI}
\end{center}
\end{figure}

We then applied the vertical modeling with a cured fraction (VMCF) to these data. This model serves three purposes: (1) to estimate the effect of covariates on time to death due to each of  melanoma cancer and other causes in the population of uncured patients, (2) to estimate the cause-specific cumulative incidences in the population of uncured patients and (3) to estimate the effect of covariates on the probability of being cured. A logistic regression model was postulated for the incidence part and for the latency part, a Cox proportional hazards model for the conditional (on $Y=1$) total hazard and a logistic model for the relative hazards where piece-wise constant time functions were used with cut-off points at the quartiles of the failure time distribution function. Note that this model for the relative hazards coincides with the one used in VM where the evidence of cure was ignored, because the conditional (on $T=t$) distribution of causes of failure depends only on the actual causes of failure observed. All baseline covariates were included in both parts (incidence and latency) of the model. No selection of covariates was performed to test whether some covariates could be removed. The estimated regression parameters are reported in Table~\ref{tab:reg_param}.

\begin{table}
\caption{\label{tab:reg_param}Regression parameters in VMCF.}
\centering
\begin{tabular}{c|c|cc}
    \hline\hline
                  &       \multicolumn{3}{c}{Regression parameters (SE)}   \\
    \hline\hline
    Covariate     &   Incidence model  &  \multicolumn{2}{|c}{Latency model}      \\
                  &                    &    Conditional Cox & Logistic      \\
    \hline
     Intercept    & -2.62 (0.88)  &                &               \\
     Thickness    &  0.07 (0.11)  &   0.10 (0.06)  &   0.01 (0.12) \\
     Ulcer        &  0.95 (0.64)  &   0.87 (0.50)  &   1.46 (0.86) \\
     Age          &  0.03 (0.15)  & -0.0006 (0.11)  &  -0.04 (0.24) \\
     Year(stand)  &  0.51 (0.89)  &  -1.49 (1.03)  &  -0.95 (1.70) \\
     Sex          &  0.57 (0.47)  &   0.42 (0.50)  &   0.30 (0.72) \\
  \hfill $\textbf{1}\{\textrm{t} \in (   0, 1.76]\}$  & &  & 1.93 (1.87) \\
  \hfill $\textbf{1}\{\textrm{t} \in (1.76, 2.90]\}$  & &  & 4.24 (2.08) \\
  \hfill $\textbf{1}\{\textrm{t} \in (2.90, 4.67]\}$  & &  & 3.84 (1.80) \\
  \hfill $\textbf{1}\{\textrm{t} \in (4.67,15.23]\}$  & &  & 2.61 (1.67) \\
  \hline\hline
\end{tabular}
\end{table}

The estimators of the survival curve $S_{\textrm{pop}}(t|\textbf{Z})$ and of the conditional (on $Y=1$) survival curve $S(t|Y=1, \textbf{Z})$, for a male individual with ulceration  and for average values of the continuous covariates,  are plotted in Figure~\ref{fig:KM} in dashed and dotted lines, respectively, and compared with the estimator of the survival curve $S(t|\textbf{Z})$ derived from VM (solid line).

\begin{figure}
\begin{center}
  \includegraphics[width=0.45\textwidth]{Survival_CoxPH_cure_model_CI95}
  \caption{The estimated survival curve of time to any event at the mean values of the continuous covariates for a male individual with ulceration, based on VM (solid line) and on VMCF for the combined population (dashed line) and for the susceptible sub-population (dotted line), together with the $95\%$ confidence intervals derived by fitting the former model.}\label{fig:KM}
\end{center}
\end{figure}

The estimated conditional (on $Y=1$) cumulative incidence function for a susceptible male patient with ulceration and for the mean values of the continuous covariates are plotted in Figure~\ref{fig:CumI_model_based}. The corresponding predicted probability of cure is $1-\widehat{p}=0.25$. We also plotted in Figure~\ref{fig:CumI} (in black) the estimated (unconditional) cumulative incidence functions of time to melanoma and time to other causes derived from VMCF, for a male individual with ulceration and average values of his other covariates. As expected in Figure~\ref{fig:KM}, the estimates for the population $S_{\textrm{pop}}(t|\textbf{Z})$ based on our VMCF model is similar to that  based on VM.

\begin{figure}
\begin{center}
  \includegraphics[width=0.5\textwidth]{CI_cure_model_male_ulcer}
  \caption{Model-based estimates of the conditional (on $Y=1$) cumulative incidences of time to melanoma and time to other causes for susceptible male patient with ulceration and for the mean values of the continuous covariates.}\label{fig:CumI_model_based}
\end{center}
\end{figure}

Finally, it is worth comparing Table~\ref{tab:reg_param_nocure} with Table~\ref{tab:reg_param}; it clearly shows that, especially for age, interpretation in terms of hazard ratios differs between $S(t|Y=1,\textbf{Z})$ and $S(t|\textbf{Z})$.



\section{Discussion}\label{sec:disc}

Little has been published about cure models in combination with competing risks. Several reviews have been recently published to highlight the advantages of cure models over the limitations of standard methods like Kaplan-Meier, Cox models or parametric models for survival data when statistical cure is a reasonable assumption~(\cite{Ji:13,Yu:13}). In this paper, we have illustrated a strategy for the statistical analysis of competing risks data with a cure fraction. In this way, we have argued that, when there is clinical evidence of a cured proportion in a cohort, special attention should be given to the heterogeneity present among individuals. 

An important advantage of the competing risks mixture cure model (VMCF) over the standard competing risks model (VM) is two-fold: (1) it allows inference of the susceptible sub-population, and therefore a better understanding and interpretation of the variability of the data and (2) it allows estimation and direct modeling of the cure indicator. Summary measures like these can be a useful tool to complement the existing statistical measures. Each covariate in VMCF can contribute with up to three sets of regression parameters, one parameter reflecting how the covariate affects the chance of cure, one parameter for the risk of failure, irrespective of the cause of failure, and one set of parameters for the relative position of each failure type among all failure types. 

An appealing technical feature of VMCF resides in its parametrisation, which is illustrated in equation~(\ref{eq:factorisation_full_lik}). VMCF naturally separates the observed likelihood into two factors: one where the cure indicator distribution is irrelevant and one where the cure indicator distribution is relevant. The former factor is pertinent to causes of failure when a failure occurs; in the absence of competing events, the observed likelihood~(\ref{eq:factorisation_full_lik}) reduces to the observed likelihood of the Cox proportional hazards mixture cure model of \cite{SyTay:00}. The latter factor uses information on the failure and censoring times and, therefore, it is sensitive to the joint distribution of $(T, Y)$. This feature makes our method straighforward to implement by means of \textrm{smcure} package available in~\cite{R:10}.

We believe that our approach can play a useful role, as it can accommodate complex competing risks data. It is worth mentioning that the approach can be naturally extended to deal with  missing causes of failure in competing risks. We refer to the work of~\cite{Nic:11} for a description of how this can be achieved. 

An important issue is that semi-parametric mixture cure models are by construction non-identifiable. We cannot precisely tell apart individuals who are cured or not among those who are censored. However