5.3 Estimation of the conditional survival

Let \(S_1\) and \(S\) be the marginal survival functions of \(T_1\) and \(T\); that is, \(S_1(y)=P(T_1>y)\) and \(S(y)=P(T>y)\). Introduce also the conditional survival probabilities \(P(T>y|T_1>x)\) and \(P(T>y|T_1\leq x)\). without loss of generality, we only consider the estimation of \(S(y|x)=P(T>y|T_1>x)\).

The Kaplan-Meier estimator, also known as the product-limit estimator, is the most frequently used method to estimate survival for censored data. The most used representation of the Kaplan-Meier estimator of the total time is through a product of the following form

\[\begin{eqnarray*} \widehat S(y)=\prod_{\widetilde T_i\leq t}\left(1-\frac{\Delta_{2i}}{R(\widetilde T_i)}\right) \end{eqnarray*}\]

where \(R(t)=\sum_{i=1}^{n} I(\widetilde T_i \geq t)\) denote the number of individuals at risk just before time \(t\).

Below we introduce a weighted average representation of the Kaplan-Meier estimator which will be used later to introduce estimators for the conditional survival function

\[\begin{equation*} \widehat S(y)=1-\sum_{i=1}^{n}W_{i}I(\widetilde T_{(i)}\leq y),%\equiv 1-\widehat{F}_1(x), \end{equation*}\]

where \(\widetilde T_{\left( 1\right) }\leq ...\leq \widetilde T_{\left( n\right) }\) denotes the ordered \(\widetilde T\)-sample and

\[\begin{equation*} W_{i}=\frac{\Delta_{2\left[ i\right] }}{n-i+1}\prod_{j=1}^{i-1}\left[ 1-\frac{% \Delta _{2\left[ j\right] }}{n-j+1}\right] \end{equation*}\]

is the Kaplan-Meier weight attached to \(\widetilde T_{\left( i\right) }\). In the expression of \(W_{i}\) notation \(\Delta_{2\left[ i\right] }\) is used for the \(i\)-th concomitant value of the censoring indicator (that is, \(\Delta_{2\left[ i \right] }=\Delta _{2j}\) if \(\widetilde T_{\left( i\right) }=\widetilde T_{j}\)).

Well, we are interested in the estimation of the conditional survival function, \(S(y\mid x)=P(T>y\mid T_1>x)\). Below we provide estimators for this quantity, all based on the Kaplan-Meier estimator.

5.3.1 Kaplan-Meier Weighted Estimator (KMW)

Since \(S(y\mid x)\) can be expressed as \(S(y\mid x)=P(T > y|T_1 > x) = 1 - P(T\leq y\mid T_1 > x)= 1 - P(T_1 > x, T\leq y)/\left(1-P\left(T_1\leq x\right)\right),\) the conditional survival function may be estimated as

\[\begin{equation} \widehat S^{\texttt{KMW}}(y\mid x)=1-\frac{\sum_{i=1}^{n}{W_iI(\widetilde T_{1\left[i\right]} >x, \widetilde T_{\left(i\right)} \leq y)}}{\widehat S_1(x)}. \end{equation}\]

5.3.2 The Landmark approach (LDM)

The Landmark approach (Van Houwelingen 2007) states that, given the time point \(x\), to estimate \(S(y\mid x)=P(T> y\mid T_1>x)\) the analysis can be restricted to the individuals with an observed first event time greater than \(x\).

Let \(n_x\) be the cardinal of \(\left\{i:\widetilde T_{1i}>x\right\}\) and \(\left( \widetilde T_{\left( i\right) }^{x},\Delta_{\left[ i\right]}^{x}\right)\), \(i=1,...,n_{x}\), is the \(\left(\widetilde T,\Delta\right)\)-sample in \(\left\{i:\widetilde T_{1i}>x\right\}\) ordered with respect to \(\widetilde T\).

\[\begin{equation*} \widehat S^{\texttt{LDM}}(y\mid x)=1-\sum_{i=1}^{n_x}{W_i^{x}I(\widetilde T_{\left(i\right)}^x \leq y)}. \end{equation*}\]

where \(W_i^{x}\) denotes the Kaplan-Meier weight attached to the i-th ordered T-datum, computed from the subsample \(\left\{i:\widetilde T_{1i}>x\right\}\).

5.3.3 The Presmoothed Landmark approach (PLDM)

The standard error of the LDM approach may be large when the censoring is heavy, particularly with a small sample size. Interestingly, the variance of this estimator may be reduced by presmoothing (Dikta 1998). Here, the idea of presmoothing involves replacing the censoring indicators (in the expression of the Kaplan-Meier weights) by some smooth fit before the Kaplan-Meier formula is applied. This preliminary smoothing may be based on a certain parametric family such as the logistic (thus leading to a semiparametric estimator), or on a nonparametric estimator of the binary regression curve. The corresponding presmoothed landmark estimator is then given by

\[\begin{equation*} \widehat S^{\texttt{PDLM}}(y\mid x)=1-\sum_{i=1}^{n_x}{W_i^{x\star}I(\widetilde T_{\left(i\right)}^x \leq y)} \end{equation*}\]

where \(W_{i}^{x\star}\) is defined through

\[\begin{equation*} W_{i}^{x\star}=\frac {m(\widetilde T_{\left(i\right)}^{x})}{n_x-i+1}\prod_{j=1}^{i-1}\left[1-\frac {m(\widetilde T_{\left(j\right)}^{x})}{n_x-j+1}\right], \quad 1\leq i\leq n_{x}, \end{equation*}\]

where \(\left( \widetilde T_{\left( i\right) }^{x},\Delta_{\left[ i\right]}^{x}\right)\), \(i=1,...,n_{x}\), is the \(\left( \widetilde T,\Delta\right)\)-sample in \(\left\{i:\widetilde T_{1i}>x\right\}\) ordered with respect to \(\widetilde T\).

Here, \(m(t)= P(\Delta=1\mid \widetilde T^{x}=t)\). \(m(\widetilde T^{x})\) belongs to a parametric (smooth) family of binary regression curves, e.g., logistic.

According to the performance, it has been demonstrated that all of the estimators perform well, approaching their targets as the sample size increases. Besides, simulation results reveal that the landmark estimator (LDM) perform favorably when compared with the first method (KMW). Furthermore, the reported simulation results reveal relative benefits of presmoothing (PLDM) in the heavily censored scenarios or small sample sizes.

References

Van Houwelingen, H.C. 2007. “Dynamic Prediction by Landmarking in Event History Analysis.” Scandinavian Journal of Statistics 34: 70–85.

Dikta, G. 1998. “On Semiparametric Random Censorship Models.” Journal of Statistical Planning and Inference 66: 253–79.